Maintenance for Industrial Systems (Springer Series in Reliability Engineering)

Citation preview

Springer Series in Reliability Engineering

Series Editor Professor Hoang Pham Department of Industrial and Systems Engineering Rutgers, The State University of New Jersey 96 Frelinghuysen Road Piscataway, NJ 08854-8018 USA

Other titles in this series The Universal Generating Function in Reliability Analysis and Optimization Gregory Levitin

Human Reliability and Error in Transportation Systems B.S. Dhillon

Warranty Management and Product Manufacture D.N.P. Murthy and Wallace R. Blischke

Complex System Maintenance Handbook D.N.P. Murthy and Khairy A.H. Kobbacy

Maintenance Theory of Reliability Toshio Nakagawa

Recent Advances in Reliability and Quality in Design Hoang Pham

System Software Reliability Hoang Pham Reliability and Optimal Maintenance Hongzhou Wang and Hoang Pham Applied Reliability and Quality B.S. Dhillon Shock and Damage Models in Reliability Theory Toshio Nakagawa Risk Management Terje Aven and Jan Erik Vinnem Satisfying Safety Goals by Probabilistic Risk Assessment Hiromitsu Kumamoto

Product Reliability D.N.P. Murthy, Marvin Rausand and Trond Østerås Mining Equipment Reliability, Maintainability, and Safety B.S. Dhillon Advanced Reliability Models and Maintenance Policies Toshio Nakagawa Justifying the Dependability of Computerbased Systems Pierre-Jacques Courtois

Offshore Risk Assessment (2nd Edition) Jan Erik Vinnem

Reliability and Risk Issues in Large Scale Safety-critical Digital Control Systems Poong Hyun Seong

The Maintenance Management Framework Adolfo Crespo Márquez

Risks in Technological Systems Torbjörn Thedéen and Göran Grimvall

Riccardo Manzini · Alberto Regattieri Hoang Pham · Emilio Ferrari

Maintenance for Industrial Systems With 504 figures and 174 tables

123

Prof. Riccardo Manzini Università di Bologna Dipartimento Ingegneria delle Costruzioni Meccaniche, Nucleari, Aeronautiche e di Metallurgia (DIEM) Viale Risorgimento, 2 40136 Bologna Italy [email protected]

Prof. Hoang Pham Rutgers University Department of Industrial and Systems Engineering 96 Frelinghuysen Road Piscataway NJ 08854-8018 USA [email protected]

Prof. Alberto Regattieri Università di Bologna Dipartimento Ingegneria delle Costruzioni Meccaniche, Nucleari, Aeronautiche e di Metallurgia (DIEM) Viale Risorgimento, 2 40136 Bologna Italy [email protected]

Prof. Emilio Ferrari Università di Bologna Dipartimento Ingegneria delle Costruzioni Meccaniche, Nucleari, Aeronautiche e di Metallurgia (DIEM) Viale Risorgimento, 2 40136 Bologna Italy [email protected]

ISSN 1614-7839 ISBN 978-1-84882-574-1 e-ISBN 978-1-84882-575-8 DOI 10.1007/978-1-84882-575-8 Springer Dordrecht Heidelberg London New York British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Control Number: 2009937576 © Springer-Verlag London Limited 2010 MaintiMizer™ is a trademark of Ashcom Technologies, Inc., 3917 Research Park Drive, Suite B4, Ann Arbor, MI 48108, USA, http://www.ashcomtech.com MATLAB® and Simulink® are registered trademarks of The MathWorks, Inc., 3 Apple Hill Drive, Natick, MA 01760-2098, USA, http://www.mathworks.com Portions of the input and output contained in this publication/book are printed with permission of Minitab Inc. All material remains the exclusive property and copyright of Minitab Inc. All rights reserved. MINITAB® and all other trademarks and logos for the Company’s products and services are the exclusive property of Minitab Inc. All other marks referenced remain the property of their respective owners. See minitab.com for more information. Relex® is a registered trademark of Relex Software Corporation, 540 Pellis Road, Greensburg, PA 15601, USA, http://www.relex.com ReliaSoft® is a registered trademark of ReliaSoft Corporation, 1450 S. Eastside Loop, Tucson, Arizona 85710-6703, USA, http://www.reliasoft.com Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be repro-duced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licenses issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. The use of registered names, trademarks, etc., in this publication does not imply, even in the ab-sence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the informa-tion contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. Cover design: deblik, Berlin, Germany Typesetting and production: le-tex publishing services GmbH, Leipzig, Germany Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

to Sara and Marta

Preface

Billions of dollars are currently spent producing high-technology products and services in a variety of production systems operating in different manufacturing and service sectors (e. g., aviation, automotive industry, software development, banks and financial companies, health care). Most of these products are very complex and sophisticated owing to the number of functions and components. As a result, the production process that realizes these products can be very complicated. A significant example is the largest passenger airliner in the world, the Airbus A380, also known as the “Superjumbo,” with an operating range of approximately 15,200 km, sufficient to fly directly from New York City to Hong Kong. The failure and repair behaviors of the generic part of this system can be directly or indirectly associated with thousands of different safety implications and/or quality expectations and performance measurements, which simultaneously deal with passengers, buildings, the environment, safety, and communities of people. What is the role of maintenance in the design and management of such a product, process, or system? Proper maintenance definitely helps to minimize problems, reduce risk, increase productivity, improve quality, and minimize production costs. This is true both for industrial and for infrastructure assets, from private to government industries producing and supplying products as well as services. We do not need to think about complex production systems, e. g., nuclear power plants, aerospace applications, aircraft, and hospital monitoring control systems, to understand the strategic role of maintenance for the continuous functioning of production systems and equipment. Concepts such as safety, risk, and reliability are universally widespread and maybe abused, because daily we make our choices on the basis of them, willingly or not. That is why we prefer a safer or a more reliable car, or why we travel with a safer airline instead of saving money with an ill-famed company. The acquisition of a safer, or high-quality, article is a great comfort to us even if we pay more. The strategic role of maintenance grows in importance as society grows in complexity, global competition increases, and technological research finds new applications. Consequently the necessity for maintenance actions will continue to increase in the future as will the necessity to further reduce production costs, i. e., increase efficiency, and improve the safety and quality of products and processes. In particular, during the last few decades the so-called reliability and maintenance engineering

vii

viii

discipline has grown considerably in both universities and industry as well as in government. The activities of planning, design, management, control, and optimization of maintenance issues are very critical topics of reliability and maintenance engineering. These are the focus of this book, whose aim is to introduce practitioners and researchers to the main problems and issues in reliability engineering and maintenance planning and optimization. Several supporting decision models and methods are introduced and applied: the book is full of numerical examples, case studies, figures, and tables in order to quickly introduce the reader to very complicated engineering problems. Basic theory and fundamentals are continuously combined with practical experience and exercises useful to practitioners but also to students of undergraduate and graduate schools of engineering, science, and management. The most important keywords used in this book are as follows: product, process, production system, productivity, reliability, availability, maintainability, risk, safety, failure modes and criticality analyses (failure modes and effects analysis and failure mode, effects, and criticality analysis), prediction and evaluation, assessment, preventive maintenance, inspection maintenance, optimization, cost minimization, spare parts fulfillment and management, computerized maintenance management system, total productive maintenance, overall equipment effectiveness, fault tree analysis, Markov chains, Monte Carlo simulation, numerical example, and case study. The book consists of 12 chapters organized as introduced briefly below. Chapter 1 identifies and illustrates the most critical issues concerning the planning activity, the design, the management, and the control of modern production systems, both producing goods (manufacturing systems in industrial sectors) and/or supplying services (e. g., hospital, university, bank). This chapter identifies the role of maintenance in a production system and the capability of guaranteeing a high level of safety, quality, and productivity in a proper way. Chapter 2 introduces quality assessment, presents statistical quality control models and methods, and finally Six Sigma theory and applications. A brief illustration and discussion of European standards and specifications for quality assessment is also presented. Chapter 3 introduces the reader to the actual methodology for the implementation of a risk evaluation capable of reducing risk exposure and guaranteeing the desired level of safety. Chapter 4 examines the fundamental definitions concerning maintenance, and discusses the maintenance question in product manufacturing companies and service suppliers. The most important maintenance engineering frameworks, e. g., reliability-centered maintenance and total productive maintenance, are presented. Chapter 5 introduces the reader to the definition, measurement, management, and control of the main reliability parameters that form the basis for modeling and evaluating activities in complex production systems. In particular, the basic maintenance terminology and nomenclature related to a generic item as a part, component, device, subsystem, functional unit, piece of equipment, or system that can be considered individually are introduced. Chapter 6 deals with reliability evaluation and prediction. It also discusses the elementary reliability configurations of a system in order to introduce the reader to the basic tools used to evaluate complex production systems.

Preface

Preface

ix

Chapter 7 discusses about the strategic role of the maintenance information system and computerized maintenance management systems in reliability engineering. Failure rate prediction models are also illustrated and applied. Chapter 8 introduces models and methods supporting the production system designer and the safety and/or maintenance manager to identify how subsystems and components could fail and what the corresponding effects on the whole system are, and to quantify the reliability parameters for complex systems. In particular models, methods, and tools (failure modes and effects analysis and failure mode, effects, and criticality analysis, fault tree analysis, Markov chains, Monte Carlo dynamic simulation) for the evaluation of reliability in complex production systems are illustrated and applied to numerical examples and case studies. Chapter 9 presents basic and effective models and methods to plan and conduct maintenance actions in accordance with corrective, preventive, and inspection strategies and rules. Several numerical examples and applications are illustrated. Chapter 10 discusses advanced models and methods, including the block replacements, age replacements, and inspection policies for maintenance management. Chapter 11 presents and applies models and tools for supporting the activities of fulfillment and management of spare parts. Chapter 12 presents two significant case studies on reliability and maintenance engineering. In particular, several models and methods introduced and exemplified in previous chapters are applied and compared. We would like to thank our colleagues and students, particularly those who deal with reliability engineering and maintenance every day, and all professionals from industry and service companies who supported our research and activities, Springer for its professional help and cooperation, and finally our families, who encouraged us to write this book. Bologna (Italy) and Piscataway (NJ, USA) Autumn 2008

Riccardo Manzini Alberto Regattieri Hoang Pham Emilio Ferrari

Contents

1

A New Framework for Productivity in Production Systems . . . . . . . .

1

1.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

A Multiobjective Scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.2.1

Product Variety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.2.2

Product Quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.3

Production System Design Framework . . . . . . . . . . . . . . . . . . . . . .

4

1.4

Models, Methods, and Technologies for Industrial Management

5

1.4.1

The Product and Its Main Features . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.4.2

Reduction of Unremunerated Complexity: The Case of Southwest Airlines . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

1.4.3

The Production Process and Its Main Features . . . . . . . . . . . . . . .

7

1.4.4

The Choice of Production Plant . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

1.5

Design, Management, and Control of Production Systems . . . . .

10

1.5.1

Demand Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

1.5.2

Product Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

1.5.3

Process and System Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

1.5.4

Role of Maintenance in the Design of a Production System . . . .

11

1.5.5

Material Handling Device Design . . . . . . . . . . . . . . . . . . . . . . . . . .

11

1.5.6

System Validation and Profit Evaluation . . . . . . . . . . . . . . . . . . . .

11

1.5.7

Project Planning and Scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

1.5.8

New Versus Existing Production Systems . . . . . . . . . . . . . . . . . . .

11

1.6

Production System Management Processes for Productivity . . . .

13

1.6.1

Inventory and Purchasing Management . . . . . . . . . . . . . . . . . . . . .

14

1.6.2

Production Planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

1.6.3

Distribution Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

1.7

Research into Productivity and Maintenance Systems . . . . . . . . .

14 xi

xii

2

3

Contents

Quality Management Systems and Statistical Quality Control . . . . . . 2.1 Introduction to Quality Management Systems . . . . . . . . . . . . . . . . 2.2 International Standards and Specifications . . . . . . . . . . . . . . . . . . . 2.3 ISO Standards for Quality Management and Assessment . . . . . . . 2.3.1 Quality Audit, Conformity, and Certification . . . . . . . . . . . . . . . . . 2.3.2 Environmental Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Introduction to Statistical Methods for Quality Control . . . . . . . . 2.4.1 The Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Terms and Definition in Statistical Quality Control . . . . . . . . . . . 2.5 Histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Control Charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Control Charts for Means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.1 The R-Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.2 Numerical Example, R-Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.3 The x-Chart N ............................................ 2.7.4 Numerical Example, x-Chart N .............................. 2.7.5 The s-Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.6 Numerical Example, s-Chart and x-Chart N ................... 2.8 Control Charts for Attribute Data . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.1 The p-Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.2 Numerical Example, p-Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.3 The np-Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.4 Numerical Example, np-Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.5 The c-Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.6 Numerical Example, c-Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.7 The u-Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.8 Numerical Example, u-Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Capability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9.1 Numerical Example, Capability Analysis and Normal Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9.2 Numerical Examples, Capability Analysis and Nonnormal Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10 Six Sigma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10.1 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10.2 Six Sigma in the Service Sector. Thermal Water Treatments for Health and Fitness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17 17 19 19 19 21 23 23 24 25 25 26 26 29 29 30 30 33 33 35 36 37 37 37 39 40 40 40

Safety and Risk Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction to Safety Management . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Terms and Definitions. Hazard Versus Risk . . . . . . . . . . . . . . . . . . 3.3 Risk Assessment and Risk Reduction . . . . . . . . . . . . . . . . . . . . . . . 3.4 Classification of Risks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Protective and Preventive Actions . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Risk Assessment, Risk Reduction, and Maintenance . . . . . . . . . . 3.7 Standards and Specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53 53 54 57 58 60 63 63

42 46 48 51 51

Contents

xiii

4

Introduction to Maintenance in Production Systems . . . . . . . . . . . . . . 4.1 Maintenance and Maintenance Management . . . . . . . . . . . . . . . . . 4.2 The Production Process and the Maintenance Process . . . . . . . . . 4.3 Maintenance and Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Maintenance Workflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Maintenance Engineering Frameworks . . . . . . . . . . . . . . . . . . . . . . 4.6 Reliability-Centered Maintenance . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Total Productive Maintenance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.1 Introduction to TPM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.2 The Concept of TPM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.3 TPM Operating Instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.4 From Tradition to TPM: A Difficult Transition . . . . . . . . . . . . . . . 4.8 Maintenance Status Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9 Maintenance Outsourcing and Contracts . . . . . . . . . . . . . . . . . . . .

65 65 66 69 70 70 72 73 73 74 75 76 80 83

5

Basic Statistics and Introduction to Reliability . . . . . . . . . . . . . . . . . . . . 5.1 Introduction to Reliability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Components and Systems in Reliability . . . . . . . . . . . . . . . . . . . . . 5.3 Basic Statistics in Reliability Engineering . . . . . . . . . . . . . . . . . . . 5.4 Time to Failure and Time to Repair . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Probability Distribution Function . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Repairable and Nonrepairable Systems . . . . . . . . . . . . . . . . . . . . . 5.7 The Reliability Function – R(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Hazard Rate Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.1 Hazard Rate Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.2 Mean Time to Failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9 Stochastic Repair Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10 Parametric Probability Density Functions . . . . . . . . . . . . . . . . . . . 5.10.1 Constant Failure Rate Model: The Exponential Distribution . . . . 5.10.2 Exponential Distribution. Numerical example . . . . . . . . . . . . . . . . 5.10.3 The Normal and Lognormal Distributions . . . . . . . . . . . . . . . . . . . 5.10.4 Normal and Lognormal Distributions. Numerical example . . . . . 5.10.5 The Weibull Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10.6 Weibull Distribution. Numerical Example . . . . . . . . . . . . . . . . . . . 5.11 Repairable Components/Systems: The Renewal Process and Availability A(t) . . . . . . . . . . . . . . . . . . 5.12 Applications and Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.12.1 Application 1 – Nonrepairable Components . . . . . . . . . . . . . . . . . 5.12.2 Application 2 – Repairable System . . . . . . . . . . . . . . . . . . . . . . . . .

87 88 88 89 90 90 91 91 92 94 95 95 97 97 99 103 106 110 112

Reliability Evaluation and Reliability Prediction Models . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Data Collection and Evaluation of Reliability Parameters . . . . . . 6.2.1 Empirical Functions Direct to Data . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Theoretical Distribution Research . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Introduction to Reliability Block Diagrams . . . . . . . . . . . . . . . . . . 6.4 Serial Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Numerical Example – Serial Configuration . . . . . . . . . . . . . . . . . . 6.5 Parallel Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

133 133 134 135 145 152 153 154 161

6

113 117 117 122

xiv

Contents

6.5.1 6.6 6.7 6.8 6.8.1 6.9 6.9.1

Numerical Example – Parallel Configuration . . . . . . . . . . . . . . . . . Combined Series–Parallel Systems . . . . . . . . . . . . . . . . . . . . . . . . . Combined Parallel–Series Systems . . . . . . . . . . . . . . . . . . . . . . . . . k-out-of-n Redundancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Examples, k-out-of-n Redundancy . . . . . . . . . . . . . . . . Simple Standby System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Example – Time-Dependent Analysis: Standby System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.10 Production System Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.10.1 Water Supplier System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.10.2 Continuous Dryer System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

163 168 170 170 171 174

7

Maintenance Information System and Failure Rate Prediction . . . . . 7.1 The Role of a Maintenance Information System . . . . . . . . . . . . . . 7.2 Maintenance Information System Framework . . . . . . . . . . . . . . . . 7.2.1 Data Collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Maintenance Engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Interventions and Workload Analysis . . . . . . . . . . . . . . . . . . . . . . . 7.2.4 Spare Parts and Equipment Management . . . . . . . . . . . . . . . . . . . . 7.3 Computer Maintenance Management Software . . . . . . . . . . . . . . . 7.4 CMMS Implementation: Procedure and Experimental Evidence 7.4.1 System Configuration and Integration . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Training and Data Entry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.3 Go Live . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.4 Postimplementation Phase and Closing . . . . . . . . . . . . . . . . . . . . . 7.4.5 Experimental Evidence Concerning CMMS Implementation . . . 7.5 Failure Rate Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 Accelerated Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.2 Failure Data Prediction Using a Database . . . . . . . . . . . . . . . . . . . 7.6 Remote Maintenance/Telemaintenance . . . . . . . . . . . . . . . . . . . . . 7.6.1 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

189 189 190 190 192 194 195 196 199 199 200 200 200 200 204 204 206 214 216

8

Effects Analysis and Reliability Modeling of Complex Production Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction to Failure Modes Analysis and Reliability Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Failure Modes and Effects Analysis . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Product Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Failure Mode, Effects, and Causes Analysis . . . . . . . . . . . . . . . . . 8.2.3 Risk Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.4 Corrective Action Planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.5 FMEA Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Failure Mode, Effects, and Criticality Analysis . . . . . . . . . . . . . . . 8.3.1 Qualitative FMECA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Quantitative FMECA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Introduction to Fault Tree Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Qualitative FTA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.1 Fault Tree Construction Guidelines . . . . . . . . . . . . . . . . . . . . . . . . .

180 183 185 187

219 220 220 221 222 222 225 229 229 231 231 232 236 239 239

Contents

xv

8.5.2 8.5.3 8.5.4 8.6 8.6.1 8.6.2 8.6.3

Numerical Example 1. Fault Tree Construction . . . . . . . . . . . . . . . Boolean Algebra and Application to FTA . . . . . . . . . . . . . . . . . . . Qualitative FTA: A Numerical Example . . . . . . . . . . . . . . . . . . . . . Quantitative FTA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantitative FTA, Numerical Example 1 . . . . . . . . . . . . . . . . . . . . Quantitative FTA, Numerical Example 2 . . . . . . . . . . . . . . . . . . . . Numerical Example. Quantitative Analysis in the Presence of a Mix of Statistical Distributions . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Application 1 – FTA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7.1 Fault Tree Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7.2 Qualitative FTA and Standards-Based Reliability Prediction . . . . 8.7.3 Quantitative FTA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8 Application 2 – FTA in a Waste to Energy System . . . . . . . . . . . . 8.8.1 Introduction to Waste Treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8.2 Case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8.3 Emissions and Externalities: Literature Review . . . . . . . . . . . . . . 8.8.4 SNCR Plant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8.5 SNCR Plant. Reliability Prediction and Evaluation Model . . . . . 8.8.6 Qualitative FTA Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8.7 NOx Emissions: Quantitative FTA Evaluation . . . . . . . . . . . . . . . . 8.8.8 Criticality Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8.9 Spare Parts Availability, What-If Analysis . . . . . . . . . . . . . . . . . . . 8.8.10 System Modifications for ENF Reduction and Effects Analysis . 8.9 Markov Analysis and Time-Dependent Components/Systems . . . 8.9.1 Redundant Parallel Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.9.2 Parallel System with Repairable Components . . . . . . . . . . . . . . . . 8.9.3 Standby Parallel Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.10 Common Mode Failures and Common Causes . . . . . . . . . . . . . . . 8.10.1 Unavailability of a System Subject to Common Causes . . . . . . . . 8.10.2 Numerical Example, Dependent Event . . . . . . . . . . . . . . . . . . . . . . 9

Basic Models and Methods for Maintenance of Production Systems . 9.1 Introduction to Analytical Models for Maintenance of Production Systems . . . . . . . . . . . . . . . . . . . . . 9.1.1 Inspection Versus Monitoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Maintenance Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Introduction to Preventive Maintenance Models . . . . . . . . . . . . . . 9.4 Component Replacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1 Time-Related Terms and Life Cycle Management . . . . . . . . . . . . 9.4.2 Numerical Example. Preventive Replacement and Cost Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Time-Based Preventive Replacement – Type I Replacement Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.1 Numerical Example. Type I Replacement Model . . . . . . . . . . . . . 9.5.2 Numerical Example. Type I Model and Exponential Distribution of ttf . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.3 Type I Replacement Model for Weibull distribution of ttf . . . . . . 9.5.4 The Golden Section Search Method . . . . . . . . . . . . . . . . . . . . . . . .

240 241 242 244 248 252 254 263 264 266 269 277 277 278 279 280 281 283 287 292 295 300 301 302 304 306 309 310 311 313 314 315 315 318 319 319 320 323 324 325 326 326

xvi

Contents

9.5.5 9.6 9.6.1 9.6.2 9.6.3 9.6.4 9.7 9.7.1 9.7.2 9.7.3 9.7.4 9.7.5 9.7.6 9.7.7 9.8 9.8.1 9.9 9.9.1 9.9.2 9.10 9.11 9.11.1 9.11.2 9.12 9.12.1 9.12.2 9.13 9.14 9.15 9.15.1 9.15.2 9.16 9.17 9.18 9.18.1 9.18.2 9.18.3 9.19

Numerical Example. Type I Model and the Golden Section Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time-Based Preventive Replacement Including Duration of Replacements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Example 1: Type I Replacement Model Including Durations Tp and Tf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Type I Model with Duration of Replacement for Weibull Distribution of ttf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Example 2: Type I Model with Durations Tp and Tf . . Practical Shortcut to tp Determination . . . . . . . . . . . . . . . . . . . . . . Block Replacement Strategy – Type II . . . . . . . . . . . . . . . . . . . . . . Renewal Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Laplace Transformation: W(t) and w(t) . . . . . . . . . . . . . . . . . . . . . . Renewal Process and W(t) Determination, Numerical Example . Numerical Example, Type II Model . . . . . . . . . . . . . . . . . . . . . . . . Discrete Approach to W(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Practical Shortcut to W(t) and tp Determination . . . . . . . . . . . . . . Maintenance Performance Measurement in Preventive Maintenance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Minimum Total Downtime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Type I – Minimum Downtime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Type II – Downtime Minimization . . . . . . . . . . . . . . . . . . . . . . . . . Group Replacement: The Lamp Replacement Problem . . . . . . . . Preventive Maintenance Policies for Repairable Systems . . . . . . . Type I Policy for Repairable Systems . . . . . . . . . . . . . . . . . . . . . . . Type II Policy for Repairable Systems . . . . . . . . . . . . . . . . . . . . . . Replacement of Capital Equipment . . . . . . . . . . . . . . . . . . . . . . . . . Minimization of Total Cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Literature Discussion on Preventive Maintenance Strategies . . . . Inspection Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Single Machine Inspection Model Based on a Constant Value of Conditional Probability Failure . . . . . . . . Numerical Example 1, Elementary Inspection Model . . . . . . . . . . Numerical Example 2, Elementary Inspection Model . . . . . . . . . . Inspection Frequency Determination and Profit per Unit Time Maximization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inspection Frequency Determination and Downtime Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inspection Cycle Determination and Profit per Unit Time Maximization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exponential Distribution of ttf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Weibull Distribution of ttf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Single Machine Inspection Model Based on Total Cost per Unit Time Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

328 333 333 335 335 335 339 340 341 341 343 348 349 352 353 354 355 355 357 358 359 360 370 372 372 372 372 373 375 376 377 378 380 381 381 382 382 383

Contents

xvii

9.20

Single Machine Inspection Model Based on Minimal Repair and Cost Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.21 Inspection Model Based on Expected Availability per Unit Time Maximization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.22 Group of Machines Inspection Model . . . . . . . . . . . . . . . . . . . . . . . 9.23 A Note on Inspection Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.24 Imperfect Maintenance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.24.1 Imperfect Preventive Maintenance p – q . . . . . . . . . . . . . . . . . . . . . 9.25 Maintenance-Free Operating Period . . . . . . . . . . . . . . . . . . . . . . . . 9.25.1 Numerical Example (Kumar et al. 1999) . . . . . . . . . . . . . . . . . . . . 9.25.2 MFOPS and Weibull Distribution of ttf . . . . . . . . . . . . . . . . . . . . . 9.26 Opportunistic Maintenance Strategy . . . . . . . . . . . . . . . . . . . . . . . .

384 385 386 387 388 388 390 391 392 393

10 Advanced Maintenance Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Maintenance Policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Age Replacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.2 Block Replacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Modeling of Nonrepairable Degraded Systems . . . . . . . . . . . . . . . 10.4 Modeling of Inspection-Maintenance Repairable Degraded Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.1 Calculate EŒNI  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.2 Calculate Pp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.3 Expected Cycle Length Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.4 Optimization of Maintenance Cost Rate Policy . . . . . . . . . . . . . . . 10.4.5 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Warranty Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

397 397 398 398 399 399 402 403 404 405 405 406 406 408

11 Spare Parts Forecasting and Management . . . . . . . . . . . . . . . . . . . . . . . 11.1 Spare Parts Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Spare Parts Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Forecasting Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Croston Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 Poisson Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6 Binomial Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6.1 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.7 Spare Parts Forecasting Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . 11.8 Spare Parts Forecasting Methods: Application and Case Studies 11.8.1 Case Study 1: Spare Parts Forecasting for an Aircraft . . . . . . . . . 11.8.2 Case Study 2: Spare Parts Forecasting in a Steel Company . . . . . 11.9 Methods of Spare Parts Management . . . . . . . . . . . . . . . . . . . . . . . 11.9.1 Spare Parts Management: Qualitative Methods . . . . . . . . . . . . . . . 11.9.2 Spare Parts Management: Quantitative Methods . . . . . . . . . . . . . .

409 409 410 411 412 413 414 415 416 417 417 418 422 423 426

12 Applications and Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Preventive Maintenance Strategy Applied to a Waste to Energy Plant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1.1 Motor System Reliability Evaluation . . . . . . . . . . . . . . . . . . . . . . .

433 433 434

xviii

Contents

12.1.2 12.1.3 12.1.4 12.1.5 12.1.6 12.1.7 12.1.8 12.1.9 12.2

Bucket Reliability Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Motor System. Determination of Maintenance Costs . . . . . . . . . . Time-Based Preventive Replacement for the Motor System . . . . Time-Based Preventive Replacement for the Bucket Component Time-Based Preventive Replacement with Durations Tp and Tf . Downtime Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Monte Carlo Dynamic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . Monte Carlo Analysis of the System . . . . . . . . . . . . . . . . . . . . . . . . Reliability, Availability, and Maintainability Analysis in a Plastic Closures Production System for Beverages . . . . . . . . RBD construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rotating Hydraulic Machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Data Collection and Reliability Evaluation of Components . . . . . Reliability Evaluation, Nonrepairable Components/Systems . . . . Data on Repairs and Maintenance Strategies . . . . . . . . . . . . . . . . . Monte Carlo Analysis of the Repairable System . . . . . . . . . . . . . . Alternative Scenarios and System Optimization . . . . . . . . . . . . . . Conclusions and Call for New Contributions . . . . . . . . . . . . . . . . .

436 437 439 439 441 442 442 446

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1 Standardized Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Control Chart Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3 Critical Values of Student’s Distribution with  Degree of Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

463 463 464

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

467

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

475

12.2.1 12.2.2 12.2.3 12.2.4 12.2.5 12.2.6 12.2.7 12.3 A

446 448 449 449 454 456 456 460 462

465

1

A New Framework for Productivity in Production Systems

Contents 1.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

A Multiobjective Scenario . . . . . . . . . . . . . . . . . . . . . 1.2.1 Product Variety . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Product Quality . . . . . . . . . . . . . . . . . . . . . . . . .

2 3 3

1.3

Production System Design Framework . . . . . . . . .

4

1.4

Models, Methods, and Technologies for Industrial Management . . . . . . . . . . . . . . . . . . . . 1.4.1 The Product and Its Main Features . . . . . . . . . 1.4.2 Reduction of Unremunerated Complexity: The Case of Southwest Airlines . . . . . . . . . . . 1.4.3 The Production Process and Its Main Features 1.4.4 The Choice of Production Plant . . . . . . . . . . .

1.5

1.6

1.7

Design, Management, and Control of Production Systems . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Demand Analysis . . . . . . . . . . . . . . . . . . . . . . . 1.5.2 Product Design . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.3 Process and System Design . . . . . . . . . . . . . . . 1.5.4 Role of Maintenance in the Design of a Production System . . . . . . . . . . . . . . . . . . 1.5.5 Material Handling Device Design . . . . . . . . . 1.5.6 System Validation and Profit Evaluation . . . . 1.5.7 Project Planning and Scheduling . . . . . . . . . . 1.5.8 New Versus Existing Production Systems . . . Production System Management Processes for Productivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.1 Inventory and Purchasing Management . . . . . 1.6.2 Production Planning . . . . . . . . . . . . . . . . . . . . . 1.6.3 Distribution Management . . . . . . . . . . . . . . . .

5 5 6 7 7 10 10 10 10 11 11 11 11 11 13 14 14 14

Research into Productivity and Maintenance Systems . . . . . . . . . . . . . . . . . . . . . 14

The pressure of the global market ... we all face increased competition for share. The fundamental key is the productivity of the system. All players in the indus-

try are in the same race to become low cost producers, including manufacturers, our suppliers, and their suppliers, too. And each of us must do it while improving quality, because consumers require it (Alain Batty, CEO, Ford Motor Company of Canada, 2004). High levels of product personalization and quality standardization are essential requirements in current market conditions, in which prices are falling, and in which a new production paradigm for a production system has come into existence. The planning, management, and control of a production system are crucial activities requiring an integrated approach examining the internal features of available production resources and guiding their rational exploitation. Maintenance techniques play a major role in supporting research into productivity, and these very effective tools must be adopted by modern companies.

1.1 Introduction In this book explicitly devoted to maintenance, the first chapter aims to identify and to illustrate the most critical issues concerning the planning activity, the design, the management, and the control of modern production systems, both producing goods (manufacturing systems in industrial sectors) and/or supplying services (e. g., hospital, university, bank). By this discussion it is possible to identify the role of maintenance in a production system and the capability of guaranteeing a high level of safety, quality, and productivity in a proper way. In particular, the expression

R. Manzini, A. Regattieri, H. Pham, E. Ferrari, Maintenance for Industrial Systems © Springer 2010

1

2

“research for productivity” frequently animates the sections of this chapter. The following section introduces the uncertain operating scenario that modern companies have to face to compete in a globalized market. Section 1.3 illustrates a meta-framework for the design of a production system with an enterprise perspective. The aim is to underline the most important tasks and decisional steps affecting the performance of the system with particular attention being given to the business and corporate strategies of the enterprise and its related companies. Section 1.4 briefly discusses the models, methods, and technologies currently available to support the decision-making process dealing with production systems. Section 1.5 presents a conceptual framework, proposed by the authors, for the integration of the design, management, and control of a production system.

1.2 A Multiobjective Scenario Vaughn et al. (2002) identified the most critical factors affecting the performance of a production system as part of an enterprise system. The enterprise does not have complete control over these factors: • Market uncertainty. This is defined as the demand fluctuations for the product, including both shortterm random variability and long-term step/cyclical variability. The uncertainty of demand can create overcapacity or undercapacity, generating customer dissatisfaction. • Production volume, i. e., the number of products to be manufactured over a time period. Market uncertainty and production volume are tightly coupled. Production volume determines the production system capacity and most of the factory physical design, e. g., floor space needed, machine selection, layout, and number of workers. • Product mix. This is the number of different products to be manufactured. The production system has to be capable of producing various versions of a product, or different products simultaneously in the same plant in order to fulfill the market need with the best exploitation of the resources. Product mix and product volume are closely related (Manzini et al. 2004).

1 A New Framework for Productivity in Production Systems

• Frequency of changes. This is the number of engineering changes per time period. The changes can be either structural or upgrades to existing systems. It is not possible to foresee all the changes that might be introduced into a product in the future. For example, the frequency of changes is a very critical issue for the electronic control systems of packaging machines. A packaging system can be used by a generic customer for a few decades: the electronic technologies change very quickly and the customer could need to replace failed parts with new, different spare parts. • Complexity. There are several ways to measure product, process, or system complexity. A few examples are the number of parts, the number of process steps, and the number of subsystems. Complexity deals with the level of difficulty to design, manufacture, assemble, move, etc. a part, and it is affected by the available process capability (see Chap. 2). • Process capability, as the ability to make something repeatedly with minimal interventions. This factor deals with the quality of the process, product, and production systems, as properly illustrated in Chap. 2. • Type of organization and in particular the innovation of the workforce participating in product, process, and system improvements. • Worker skill level, i. e., the availability of high-level employee skills. This factor is strongly linked to the necessary and/or available level of automation. • Investment, as the amount of financial resources required. This is one of the most critical constraints in the production system design, management, and control. • Time to first part. This is another very critical constraint and represents the time from the initial system design to the full rate of production. • Available/existing resources (financial, technological, human, etc.). Current markets have changed a great deal from those of a few years ago. Mass production (large quantities of a limited range of products) has declined in several production systems and been replaced by customeroriented production. Sales and quantities have essentially remained constant, but the related product mix is growing ever larger. Companies are attempting to spread risk over a wider range of base products and

1.2 A Multiobjective Scenario

3

meet (or anticipate) customer needs and desires. This trend is intensified by global competition: different players throughout the world are supplying “similar” products to the same markets. This situation has produced significant changes in production systems (which either produce products or supply services): production batches are very small, production lead times are kept very short, product life cycle is also brief, and consequently product time to market is very compressed. In conclusion, production systems must possess two important features: flexibility and elasticity. Flexibility deals with the ability of the production system to evolve continuously and manufacture wide ranges of products. On the other hand, elasticity allows great variation in production volumes without a significant change in the production system configuration (i. e., without needing time-consuming and expensive work). The literature also names these concepts “capability flexibility” and “capacity flexibility.”

1.2.1 Product Variety The great increase in product variety is easily verified in several case studies. It is sufficient to investigate a single product in order to see how many different versions are now offered in comparison with 10 years ago. Some significant results from the research conducted by Thonemann and Bradley (2002) on product variety analysis are reported below. Table 1.1 shows the increase of product mix in different industrial sectors in the decade 1990–2000. The smallest increase of a little over 50% occurred in commodities. Table 1.1 Product variety increase in various industrial sectors Sector

Percentage variety increase (1990–2000)

Commodities Telecommunications Information technology Automotive Defense

52 57 77 80 81

Table 1.2 Increase in variety of different products Product

1970

2001

Car models Newspapers TV sizes Breakfast cereals Types of milk Running shoes Brands of sparkling waters Pantyhose

140 339 5 160 4 5 16 5

260 790 15 340 19 285 50 90

The change in several product mix ratios is relevant and, as Table 1.2 illustrates, these have more than doubled in some cases.

1.2.2 Product Quality In addition to the range of the product mix, another feature has also greatly increased in significance and is a growing trend: product quality. Consumers have developed great sensitivity and their perception of the quality of products and services is increasing. Consequently, companies must not only produce but also supply products and services to very high quality standards, meaning stand-alone quality is no longer a marginal success factor. In addition to these observations of “new market trends,” industrial and service companies also need their industrial investments to be remunerated. This field is also significantly affected by global competition: with prices falling, companies are forced to reduce production costs. Therefore, modern companies must expand their product mix, increase the quality of the product and the process, and reduce costs: a very stimulating challenge! Moreover, companies are striving to improve the productivity and quality of their production systems, with the most relevant targets in this multiscenario decision-making process including: • a great degree of flexibility and elasticity in the production system; • short lead times; • high-quality products and production processes; • short time to market; • control of production costs.

4

1 A New Framework for Productivity in Production Systems

1.3 Production System Design Framework

INFRASTRUCTURE

This section presents a conceptual framework for supporting the design of a production system with an enterprise perspective. It takes inspiration from the study by Fernandes (2001) in the aerospace industry and lean production. The illustration of this framework is very useful for identification of the operating context of modern production systems and for justification of the introduction of an integrated quality-, safety-, and reliability-based approach to support the design, management, and control of a complex system. In particular, maintenance models and methods reveal themselves as very effective tools to conduct this process. Figure 1.1 presents the meta-framework which also contain other tools, methods, and processes applicable to the design process of production systems operating in different industrial and service sectors, such as auto-

motive, food, health care, pharmaceutical, education, and public administration. The proposed framework is made of three main and distinct elements: 1. Infrastructure, as a result of the enterprise strategy formulation which defines important and critical attributes of the system as operating policy, organizational structure, location, and environment (see the top portion of Fig. 1.1). This strategy is the result of long-term objectives and programs, and is focused on creating operating capabilities. The corporate-level strategy balances the conflicting needs of the numerous stakeholders (e. g., customers, employees, and owners) facing the overall enterprise the production system belongs to, by the formulation of a corporate strategy which is transferred to the business units throughout the corporation. 2. Structure (see the bottom portion Fig. 1.1). It is the physical manifestation of the detailed produc-

Corporate level

Strategy formulation

Stakeholders

Corporate level

Corporate business strategy

Business unit

Suppliers Make/buy

Product Production Marketing design system DFA, DFM, conCustomer needs current engineering technical feasibility

Product strategy

Manufacturing

Physical implications

STRUCTURE

Requirements/considerations/constraints

Manufacturing system design/selection Analytical & simulation tools Implement Fine tune

Trial & error Evaluate/ validate

Finalized product design

RATE PRODUCTION

Fig. 1.1 Production system design framework. DFA design for assembly, DFM manufacturing. (Fernandes 2001)

System design

Marketing

PRODUCT STRATEGY

Suppliers

Product design

1.4 Models, Methods, and Technologies for Industrial Management

tion system design and is the result of the factory layout, number and configuration of machines, and production methods and processes. 3. Product strategy. congruence between the corporate-level business strategy and the functional strategies. It involves functional elements such as marketing, product design, supplier, and manufacturing (see the concurrent engineering overlapping of functions in Fig. 1.1) This meta-framework gives the concurrent engineering approach a great and strategic importance and provides enlightenment on the validation analysis, and the continuous improvement (see the so-called modification loop in Fig. 1.1).

1.4 Models, Methods, and Technologies for Industrial Management Which resources are capable of supporting companies in meeting the challenge introduced in the previous section? First of all, it is important to state that only resources relating to products (or services) and to production processes (i. e., manufacturing and assembly activities in industrial companies) are considered in this chapter. It is not the authors’ purpose to take into account some other factors associated with advertising, marketing, or administrative areas. In brief, research supports productivity via three fundamental and interrelated drivers: the product, the process, and the production system.

1.4.1 The Product and Its Main Features Products are usually designed with reference to their performance (i. e., the ability to satisfy customer needs) and to the aesthetic appearance required by the market. Requirements derived from the production system are sometimes neglected, thus having a negative effect on final production costs. As a consequence, during the last few decades several strategies or techniques for product design, such as design for manufacturing (DFM) and design for assembly (DFA), which, respectively, consider manufacturing and assembly requirements during the design process, have

5

been proposed in the literature and applied in modern production systems. They provide a valid support to the effective management of total production costs. In recent years, the matter of reuse and/or recycling of products has become extremely pressing worldwide, and many countries are facing problems relating to waste evaluation and disposal. The significance of these topics is demonstrated by the wide diffusion of product life cycle management, as the process of managing the entire life cycle of a product from its conception, through design and manufacture, to service and disposal. Figure 1.2 presents a conceptual model of the product life cycle, including the design, production, support, and ultimate disposal activities. Maintenance of production facilities and recovery of products explicitly play a strategic role in product life cycle management. As a consequence, a product design process that also considers product disassembly problems at the end of the product life cycle has become a success factor in modern production systems. This approach to the design process is known as “design for disassembly” (DFD). In several supply chains (e. g., tires and batteries) the manufacturer is burdened with the reuse or final disposal of the product, and DFD is a particularly effective tool for the reduction of production costs. Section 1.2 discusses the advantages and disadvantages associated with the production of a wide variety of products: wide ranges of product mix are an effective strategy in meeting customer expectations, but companies must reach this goal with the minimum number of components and parts. In particular, any part or function not directly perceived by the customer implies both an unnecessary and a harmful addition of complexity because it is not remunerated. Research and trials examining this special kind of complexity lead to the formulation of the following production strategy: what is visible over the skin of the product is based on a very high degree of modularity under the skin. The so-called product platforms are a good solution to support product variability, and so have been adopted in modern production systems. Several families of similar products are developed on the same platform using identical basic production guidelines for all “derivative” products. A well-known example is the “spheroid platform” developed by Piaggio (the Italian manufacturer of the famous Vespa scooter): the products named Zip, Storm, Typhoon, Energy, Skip-

6

1 A New Framework for Productivity in Production Systems

Fig. 1.2 Product life cycle model

per, Quartz, and Free are all derived from the same underlying fundamental design of the scooter called “Sphere” (hence the spheroid platform). Another significant example is the standardization of car speed indicators in the automotive sector: the manufacturers tend to use the same component in every product mix regardless of the speed attainable by each individual car model. As a result of this strategy, the range of the product mix is reduced and the management of parts is simplified without affecting product performance. Every remark or comment about the techniques and strategies cited is also effective both in production systems and in supply services such as hospitals, banks, and consultants.

1.4.2 Reduction of Unremunerated Complexity: The Case of Southwest Airlines Southwest Airlines has developed several interesting ideas for reducing complexity in the service sector.

Figure 1.3 shows the cost per passenger for each mile traveled with the main US airlines. Two fundamental facts can be observed in Fig. 1.3: since 2004 the cost per passenger for each mile traveled (extrapolated from available seat miles) for Southwest Airlines has been lower than for its competitors, clearly competing in the same market and over the same routes. Moreover, the available seat mile costs of Southwest Airlines have continued to decrease since 11 September 2001, in contrast to those of its competitors. Moreover, these costs have significantly increased owing to the increase in the cost of petroleum and owing to the introduction of new safety and security standards. How can this be explained? The answer lies in the efforts of Southwest Airlines, since 1996, to reduce the variety and complexity of services offered to its customers but not directly perceived by them. A significant analysis of the fleet configurations of major American airlines is reported in Table 1.3. The average number of different models of airplane used by the major USA airlines is 14, but Southwest Airlines employs only Boeing 737 airplanes. In fact,

1.4 Models, Methods, and Technologies for Industrial Management

7

Table 1.3 Number of different models of airplane used by USA airlines (June 2008)

No. of different models of airplane in fleet 

United Airlines

Delta Airlines

American Airlines

Average for USA airlines

Southwest Airlines

13

9

6

7

1

Boeing 737

very effectively. Among a great many original approaches proposed during the last two decades for the reduction of complexity in a production system, the well-known Variety Reduction Program (VRP) developed by Koudate and Suzue (1990) is worthy of mention.

1.4.3 The Production Process and Its Main Features

Fig. 1.3 Cost per passenger for each mile traveled. ASM available seat miles. (United States Securities and Exchange Commission 2000)

in June 2008, Southwest Airlines owned 535 airplanes of this particular type but using various internal configurations, ranging from 122 to 137 seats. Variety based on the type of airplane is completely irrelevant to customers. Furthermore, when a passenger buys a ticket, the airline companies do not communicate the model of airplane for that flight. However, reducing the number of different models of airplane in the fleet directly results in a significant saving for the airline company: only one simulator for pilot training is required, only one training course for technicians and maintenance staff, spare parts management and control activities are optimized, “on ground” equipment such as systems for towing and refueling are standard, etc. In spite of critical safety problems and high fuel costs, Southwest Airlines has been able to compete

Production processes in several industrial sectors have recently been forced to undergo significant transformations in order to ensure both cost reductions and high quality. A good example from the wood sector is the nonstop pressing process used to simplify the assembly process by using small flaps, gluing, and other techniques instead of screw junctions. Every process innovation capable of consuming too many production resources such as energy, manpower, and raw materials is a very useful motivating factor driving research into productivity. Consequently, when a new production investment is being made in a manufacturing or service sector, a benchmark investigation is required in order to check the state of the art of the production processes. In addition to this, from an economic or technical point of view, scouting for alternative processes that could be more effective is also recommended.

1.4.4 The Choice of Production Plant An effective production process is a basic condition in making an entire production system effective. Thorough analysis of the specific characteristics of production factors, e. g., resources and equipment required by the available processes, is one of the most important aspects of research into productivity.

8

It is possible to have two different production plants carrying out the same process with their own specifications and production lead times to get the same results, but at different costs. A great deal of effort in innovation of the plant equipment has taken place in recent years, but innovation in the production process is a very difficult problem to solve, often involving contributions from various industrial disciplines (e. g., electronics, robotics, industrial instrumentation, mechanical technology). One of the most significant trends in equipment innovation developments is represented by flexible automation, which provides the impetus for a production system to achieve high levels of productivity. Presently, industrial equipment and resources are highly automated. However, flexible automation is required so that a wide mix of different products and services is achieved without long and expensive setups. One of the best expressions of this concept, i. e., the simultaneous need for automation and flexibility, is the so-called flexible manufacturing system (FMS). A flexible manufacturing system is

1 A New Framework for Productivity in Production Systems

a melting pot where several automatic and flexible machines (e. g., computer numerical control (CNC) lathes or milling machines) are grouped and linked together using an automatic and flexible material handling system. The system can operate all job sequences, distinguish between different raw materials by their codes, download the correct part program from the logic controller, and send each part to the corresponding machine. This basic example of the integration of different parts shows how successful productivity in a modern production system can be. The potential offered by flexible automation can only be exploited effectively if every element of the integrated system is capable of sharing information quickly and easily. The information technology in flexible systems provides the connectivity between machines, tool storage systems, material feeding systems, and each part of the integrated system in general. Figure 1.4 presents a brief classification, proposed by Black and Hunter (2003), of the main manufac-

Fig. 1.4 Different kinds of manufacturing systems (Black and Hunter 2003)

1.4 Models, Methods, and Technologies for Industrial Management

turing systems in an industrial production context by comparing different methodologies based on production rates and flexibility, i. e., the number of different parts the generic system can handle. In conclusion, the required system integration means developing data exchange and sharing of information, and the development of production systems in the future will be based on this critical challenge. The current advanced information technology solutions (such as local area networks, the Internet, wire-

9

less connectivity, and radio-frequency identification (RFID)) represent a valid support in the effective integration of production activities. Figure 1.5 is extracted from a previous study by the authors and briefly summarizes the productivity paradigm discussed in this chapter. This figure was proposed for the first time by Rampersad (1995). Research into productivity also requires technical, human, and economic resources. Consequently, before a generic production initiative is embarked upon, it is

EXTERNAL DEVELOPMENTS RESOURCES MARKET DEVELOPMENTS PRODUCT DEVELOPMENTS

SELLING MARKET • • • • • • •

•

New design strategies (DFM, DFA, DFD,..) • New materials

International competition Shorter product life cycle Increasing product diversity Decreasing product quantity Shorter delivery times Higher delivery reliability Higher quality requirements

PROCESS DEVELOPMENTS • • •

LABOR MARKET • •

Increasing labour costs Lack of well-motivated and qualified personnel

Innovative processes New process strategies New joining methods

SYSTEM DEVELOPMENTS • • •

Flexible automation Integration Information technology

COMPANY COMPANY

POLICY

COMPANY OBJECTIVES

ACTIVITIES

•

High flexibility

•

Effective system design

•

Constant and high product quality

•

Effective system management

•

Short throughput times

•

Low production costs

Fig. 1.5 The new productivity paradigm for a production system. DFM design for manufacturing, DFA design for assembly, DFD design for disassembly. (Rampersad 1995)

10

essential to carry out a feasibility study and an appraisal of the economic impact. At the design stage of a product or service, a multidecision approach is often required before the production start-up is initiated. Moreover, as it involves a broad spectrum of enterprise roles and functions, an integrated management approach is achieved because brilliant design solutions can be compromised by bad management. The following section deals with the design, management, and control of a production system in accordance with a new productivity paradigm proposed by the authors.

1.5 Design, Management, and Control of Production Systems A systematic and integrated approach to the management and control of a production system is essential for rational and effective use of the above-mentioned resources and equipment. In other words, productivity must be designed and managed correctly, otherwise the enterprise will risk not being appropriately remunerated for its investment. In both the manufacturing and the service sectors, every new industrial initiative at its start-up needs a complete design process taking the following critical aspects into consideration: market demand analysis, design activity, validation of design, and sequencing and scheduling of project activities. Once the production system has been designed and installed, modern management and optimization techniques and tools need to be applied. Because of this complex scenario, the productivity goal for a complex production system can be effectively achieved by using the integrated and systematic approach shown in Fig. 1.6 (Manzini et al. 2006a). This approach summarizes the complete design procedure for a generic production system according to the current state of the art supporting decision-making techniques and methods

1.5.1 Demand Analysis The starting point of the proposed method is the product (or service) market analysis, based on up-to-date statistical forecasting methods (e. g., time series, exponential smoothing, moving average) for the extrapolation of the future demand from the current one.

1 A New Framework for Productivity in Production Systems

The logical sequence of events is therefore the design phase, and only after its approval is it possible to move on to process design, and lastly the production plant can be designed. Once system optimization has been carried out, the product can be launched on the market.

1.5.2 Product Design The product design phase involves the very important strategies and methodologies of DFM, DFA, and DFD which support management decision making in manufacturing and service companies. These two strategies take manufacturing and assembly problems, respectively, into consideration during the product design activity. The results bring about a drastic reduction in the number of redesign cases, a significant improvement in production system performance, and a noteworthy compression of product time to market. Another supporting decision-making technique is the previously mentioned VRP, which focuses on reduction of complexity. All these supporting design strategies are implemented by using several computerized system solutions: the well-known design automation tools, particularly computer-aided design and computer-aided manufacturing. The design of a new product (or service) is generally based on an interactive loop that verifies and modifies the project by the execution of several fine-tuning iterations.

1.5.3 Process and System Design The product specification forms the input data used in the production process design, which is therefore strictly dependent on the product or service to be supplied. A benchmarking analysis is fundamental to effective process design because it analyses the state of the art in process technologies. The detailed definition of the production process immediately outlines the system structure (i. e., plant, production resources, and equipment), thus choosing the right number and type of machines, tools, operators, etc., and defining the corresponding facility layout design. The plant layout problem can be solved

1.5 Design, Management, and Control of Production Systems

11

using a dedicated software platform (Ferrari et al. 2003; Gamberi et al. 2009).

impossible to experiment on a reliable prototype, performance analysis and system validation are usually conducted by using simulation (e. g., visual interactive simulation, Monte Carlo simulation, what-if analysis). This ex ante evaluation checks the formal congruity of the whole design process, supporting the final choice of system configuration and the fine-tuning of the solution adopted. The technical analysis of the configuration examined is not a guarantee of a rapid return on the industrial investment: the economic evaluation, in terms of total amount of money over time, is the most important deciding factor. For an investment analysis methods such as the well-known net present value, payback analysis, and discounted cash flow rate of return are very frequently used. The best solution results from this double-check, both technical and economic, and forms the foundation for the following phase related to execution of the activities, i. e., project planning and activity scheduling.

1.5.4 Role of Maintenance in the Design of a Production System The maintenance function is a strategic resource during the preliminary design process of a production system. The analysis and forecasting of the reliability performance of a piece of equipment significantly improve the effectiveness of the design of the whole production system. It is very important to foresee future maintenance operations and costs both in the resources/facilities and in the plant layout design so as to avoid lengthy downtimes due to, e. g., the incorrect location of machines, or to a bad assignment and scheduling of manufacturing tasks to resources and workload. The role of maintenance has been increasing in importance, thus leading to a new conceptual framework: the so-called design for maintenance directly embodies maintenance principles in the design process.

1.5.5 Material Handling Device Design In order to complete the illustration of the design process of a production system, the material handling device design has to be considered. Several decisionmaking models and methods have been developed to support this critical issue (Gamberi et al. 2009), in particular in logistics and in operations research, e. g., vehicle routing algorithms and traveling scheduling procedures.

1.5.6 System Validation and Profit Evaluation Each design activity, for product, process, material handling device, etc., is very complex. As a whole they form a set of interlaced tasks whose global solution is not the sum of individual optimizations. An integrated approach generates a set of suitable solutions to be investigated in depth from an economic and technical point of view. In conclusion, the final design must be fully validated. As the production system does not exist during the design process, and it is often almost

1.5.7 Project Planning and Scheduling The effective planning and control of each task in a generic project is crucial in avoiding any delay. To respect the project deadline means to save money, especially when several activities must be performed simultaneously or according to several precedence constraints. A great many project scheduling models and methods are presented in the literature, such as the well-known program evaluation and review technique (PERT), the critical path method (CPM), and Gantt analysis. Figure 1.6 presents a nonexhaustive list of supporting techniques and tools for the execution of the design tasks previously illustrated in general. Most of them have already been mentioned and briefly described or are discussed in the following sections.

1.5.8 New Versus Existing Production Systems Some previous considerations concern research into productivity from the design process of a new production system. But what are the requirements for a production system that has already been set up and is working?

12

1 A New Framework for Productivity in Production Systems

Obviously the challenge of productivity also involves existing production systems. The techniques previously discussed are illustrated in Fig. 1.6 and also represent a useful benchmark in the process of rationalization and optimization of existing production systems.

PROCESSES

SUPPORTING TECHNIQUES AND TOOLS

DEMAND ANALYSIS PRODUCT VARIANTS ANALYSIS

An existing production system must follow a continuous improvement process based on the multitarget scenario, as described in Sect. 1.2. First of all, the company must analyze the structure of the product mix in the production system, seeking to rationalize it, e. g., by applying some effective supporting

NEW PRODUCTS DATA MINING & DATA WAREHOUSING HISTORICAL & TREND ANALYSIS MARKET INVESTIGATION

DEMAND & FORECASTS

PRODUCTION QUANTITIES-VARIANCE ANALYSIS

PRODUCT-PROCESS-MHD INTEGRATED DESIGN

PROCESS DESIGN

PRODUCT DESIGN

MHD DESIGN

ALTERNATIVE SOLUTIONS PRODUCT DEFINITION MANUFACTURING PROCESS DEFINITION MANUFACTURING SYSTEM DEFINITION PLANT LAYOUT DEFINITION MATERIAL HANDLING SYSTEM DEFINITION

DESIGN FOR MANUFACTURING - DFM DESIGN FOR ASSEMBLY - DFA DESIGN FOR DISASSEMBLY - DFD MODULARITY AND STANDARDIZATION VARIETY REDUCTION PROGRAM - VRP DESIGN AUTOMATION TOOL (CAD/CAE, CAPP...) RESOURCES DETERMINATION LAYOUT DESIGN MATERIAL HANDLING DEVICE DESIGN VEHICLE ROUTING OPTIMIZATION RELIABILITY & MAINTAINABILITY ANALYSIS

VISUAL INTERACTIVE SIMULATION - VIS MONTE CARLO SIMULATION WHAT-IF ANALYSIS

SYSTEM VALIDATION

PROFIT ANALYSIS SYSTEM SELECTION PROJECT PLANNING & SCHEDULING

NET PRESENT VALUE - NPV ECONOMIC VALUE ADDED - EVA DISCOUNTED CASH FLOW RATE OF RETURN PAY BACK ANALYSIS DECISION TREE ANALYSIS MONTE CARLO SIMULATION PROJECT SCHEDULING ALGORITHMS PROGRAM EVALUATION & REVIEW TECHNIQUE PERT SCHEDULING CRITICAL PATH METHOD - CPM GANTT ANALYSIS

PROJECT EXECUTION

Fig. 1.6 Production system: a complete design framework. MHD material handling device. (Manzini et al. 2006a)

1.6 Production System Management Processes for Productivity

decision-making techniques such as DFM, DFA, and VRP. Modern companies must put continuous monitoring and evaluation of the degree of innovation of their processes into operation. Consequently, process innovation is an important key factor in company success. In recent years, flexible automation has become a valid reference point in process innovation. Any production plant needs some revision during its life cycle, including partial or total substitution of resources, upgrades, and plant layout reengineering. Consequently, planning and execution of prior decisions are also important for a company already on-thejob. In conclusion, the general framework in Fig. 1.6 is also valid for existing production systems. The most important question remains how to choose the most convenient strategy and effective supporting decision methods from the very large collection of solutions available in the literature. The generic case study has its specific peculiarities making it different from all the others. That is why, at a first

13

glance, it is not easy to detect a suitable tool from the wide set of models and methods that can be used to support management decision making.

1.6 Production System Management Processes for Productivity This book discusses a set of effective management procedures, models, methods, and techniques, directly affecting the productivity performance of a production system. Even though they mainly deal with maintenance, safety, and quality assessments, we now illustrate a conceptual framework which classifies the most important management activities into three macro classes: materials and inventory management, production planning, and product/service distribution management (Fig. 1.7). All these activities have to be managed and optimized by whoever in a business unit, in a production system, or in an enterprise is concerned with research for productivity.

Distribution management Production planning Inventory and purchasing management Production system management

Inventory and purchasing management Economic order quantity Safety stock Just in time Comakership with suppliers

Production planning

Aggregate programming Material requirement planning Manufacturing resource planning

Distribution management Distribution resource planning Location allocation problem Transporation

Scheduling Vehicle routing

Fig. 1.7 Production system management activities

14

This book can effectively support the managers, analysts, and practitioners in a generic production system in making the best decisions regarding products, processes, and production plants, in accordance with customer’s expectations of quality and minimizing production costs with particular attention to the reduction of the production system downtimes and to the reliability/availability of products, processes, and equipment. The focus of this work is coherent with the definition of maintenance as “the combination of all technical, administrative and managerial actions during the life cycle of an item intended to retain it in, or restore it to, a state in which it can perform the required function” (European standard EN 13306:2001 – Maintenance terminology), and with the definition of quality management as the system which assists in enhancing customer satisfaction (European standard EN ISO 9000:2006 Quality management systems – Fundamentals and vocabulary). Consequently, the main keywords of this book are as follows: productivity, quality and safety by reliability engineering, maintenance, quality, and safety assessment.

1.6.1 Inventory and Purchasing Management A generic production system needs a fulfillment system for the continuous supply of raw materials and therefore has to cope with material management. In modern companies the traditional economic order quantity (EOQ) and safety stock methods are combined with a great many effective techniques based on pull logics, such as just-in-time strategy. Other eligible methodologies, such as consignment stock, electronic data interchange, comakership, business to business, and e-marketplaces, provide for very close cooperation between customers (service clients) and manufacturers (service providers).

1.6.2 Production Planning Production planning is a second management macroarea with significant impact on productivity. The aim of a preliminary definition of production planning is to provide a fundamental prerequisite for resource requirement planning. These programs are scheduled

1 A New Framework for Productivity in Production Systems

with reference to different time fences, or planning periods, with an increasing degree of detail: from a wide and outermost time fence, related to aggregate programming, to a narrow and very close time fence, related to detailed programming. After the aggregated programming phase, material and resource requirements need to be quantified. Techniques such as the well-known material requirement planning and manufacturing resource planning are usually suitable for this purpose, but the literature also contains several models and methods for so-called advanced planning: advanced planning systems (APS). Lastly, the final step requires the direct “load” of machines and assignment of workload. This is shortterm scheduling. The goal is to define the priorities of different jobs on different items of equipment and machines.

1.6.3 Distribution Management The third important management problem relates to the final distribution of products and services. The main problems are the following: the planning of shipments, generally issued as distribution resource planning; the location–allocation problem along the distributive network, i. e., the simultaneous location of equipment and logistic resources such as distribution centers and warehousing systems; the allocation of customer demand to the available set of resources; the optimal selection of transportation systems; the vehicle routing; and, finally, the execution of distribution activities.

1.7 Research into Productivity and Maintenance Systems The frameworks for the design and management of a production system, illustrated in Figs. 1.1, 1.5, and 1.6, underline how important the contributions of reliability, availability, and quality of resources (equipment, employees, and production plants) are to the production of products or services. In particular, there is a very strong positive link between maintenance and productivity. For example, the availability of a production plant is an absolute necessity for the scheduling of work orders, and spare parts forecasting

1.7 Research into Productivity and Maintenance Systems

is a fundamental part of the planning and design processes (see Chap. 11). A very important factor in purchasing is the quality control of raw materials, and the new design techniques, such as DFM and DFA, must guarantee quality levels set as targets. Modern companies must consider maintenance strategies, rules, procedures, and actions to be some of the most important issues and factors in their success. In other words, the effective design and management of a production system requires the effective design and management of the correlated maintenance process and system. A maintenance system requires strategic planning, dedicated budgets, relevant investments in terms of money and human resources, equipment, and spare parts too. In particular, the availability and commitment of personnel at all levels of an organization also includes the application of the maintenance process. An effective maintenance system provides supporting decision-making techniques, models, and methodologies, and enables maintenance personnel to apply them in order to set the global production costs at a minimum and to ensure high levels of customer service. To achieve this purpose in a production system, those elements such as the ability, skill, and knowledge required by the whole organization and in particular by product designers, production managers, and people who directly operate in the production plants, are crucial. In conclusion, as illustrated in Fig. 1.8, maintenance techniques, including also quality and safety assessment tools and procedures, represent very effective instruments for research into productivity, safety, and quality as modern companies are now forced to pursue them relentlessly. This issue will be demonstrated and supported in detail in the following chapters. The following chapters are organized as follows: • Chapter 2 introduces quality assessment and presents statistical quality control models and methods and Six Sigma theory and applications. A brief illustration and discussion of European standards and specifications for quality assessment is also presented. • Chapter 3 deals with safety assessment and risk assessment with particular attention being given to

15

Fig. 1.8 Maintenance engineering, safety assessment, and quality assessment

•

•

•

•

•

risk analysis and risk reduction procedures. Some exemplifying standards and specifications are illustrated. Chapter 4 introduces maintenance and maintenance management in production systems. An illustration of total productive maintenance production philosophy is also presented. Chapter 5 introduces the main reliability and maintenance terminology and nomenclature. It presents and applies basic statistics and reliability models for the evaluation of failure (and repair) activities in repairable (and nonrepairable) elementary components. Chapter 6 illustrates some effective statistics-based models and methods for the evaluation and prediction of reliability. This chapter also discusses the elementary reliability configurations of a production system, the so-called reliability block diagrams. Chapter 7 discusses the maintenance information systems and their strategic role in maintenance management. A discussion on computer maintenance management software (CMMS) is also presented. Finally, failure rate prediction models are illustrated and applied. Chapter 8 presents and applies models for the analysis and evaluation of failure mode, effects, and criticality in modern production systems. Then models, methods, and tools (failure modes and effects analysis and failure mode, effects, and criti-

16

cality analysis, fault tree analysis, Markov chains, Monte Carlo dynamic simulation) for the evaluation of reliability in complex production systems are illustrated and applied to numerical examples and case studies. • Chapter 9 presents several models and methods to plan and conduct maintenance actions in accordance with corrective, preventive, and inspection

1 A New Framework for Productivity in Production Systems

strategies and rules. Several numerical examples and applications are illustrated. • Chapter 10 illustrates advanced models and methods for maintenance management. • Chapter 11 discusses spare parts management and fulfillment models and tools. • Chapter 12 presents and discusses significant case studies on reliability and maintenance engineering.

2

Quality Management Systems and Statistical Quality Control

Contents 2.1

Introduction to Quality Management Systems . . . 17

2.2

International Standards and Specifications . . . . . . 19

2.3

ISO Standards for Quality Management and Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.3.1 Quality Audit, Conformity, and Certification 19 2.3.2 Environmental Standards . . . . . . . . . . . . . . . . . 21

2.4

Introduction to Statistical Methods for Quality Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.4.1 The Central Limit Theorem . . . . . . . . . . . . . . . 23 2.4.2 Terms and Definition in Statistical Quality Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.5

Histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.6

Control Charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.7

Control Charts for Means . . . . . . . . . . . . . . . . . . . . . 2.7.1 The R-Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.2 Numerical Example, R-Chart . . . . . . . . . . . . . 2.7.3 The x-Chart N ........................... 2.7.4 Numerical Example, x-Chart N ............. 2.7.5 The s-Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.6 Numerical Example, s-Chart and x-Chart N ...

26 26 29 29 30 30 33

2.8

Control Charts for Attribute Data . . . . . . . . . . . . . . 2.8.1 The p-Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.2 Numerical Example, p-Chart . . . . . . . . . . . . . . 2.8.3 The np-Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.4 Numerical Example, np-Chart . . . . . . . . . . . . 2.8.5 The c-Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.6 Numerical Example, c-Chart . . . . . . . . . . . . . . 2.8.7 The u-Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.8 Numerical Example, u-Chart . . . . . . . . . . . . . .

33 35 36 37 37 37 39 40 40

2.9

Capability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.9.1 Numerical Example, Capability Analysis and Normal Probability . . . . . . . . . . . . . . . . . . 42 2.9.2 Numerical Examples, Capability Analysis and Nonnormal Probability . . . . . . . . . . . . . . . 46

2.10 Six Sigma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.10.1 Numerical Examples . . . . . . . . . . . . . . . . . . . . 51 2.10.2 Six Sigma in the Service Sector. Thermal Water Treatments for Health and Fitness . . . . 51

Organizations depend on their customers and therefore should understand current and future customer needs, should meet customer requirements and strive to exceed customer expectations... Identifying, understanding and managing interrelated processes as a system contributes to the organization’s effectiveness and efficiency in achieving its objectives (EN ISO 9000:2006 Quality management systems – fundamentals and vocabulary). Nowadays, user and consumer assume their own choices regarding very important competitive factors such as quality of product, production process, and production system. Users and consumers start making their choices when they feel they are able to value and compare firms with high quality standards by themselves. This chapter introduces the reader to the main problems concerning management and control of a quality system and also the main supporting decision measures and tools for so-called statistical quality control (SQC) and Six Sigma.

2.1 Introduction to Quality Management Systems The standard EN ISO 8402:1995, replaced by EN ISO 9000:2005, defines “quality” as “the totality of characteristics of an entity that bear on its ability to satisfy stated and implied needs,” and “product” as

R. Manzini, A. Regattieri, H. Pham, E. Ferrari, Maintenance for Industrial Systems © Springer 2010

17

18

2 Quality Management Systems and Statistical Quality Control

“the result of activities or processes and can be tangible or intangible, or a combination thereof.” Consequently, these definitions refer to production systems both in industrial sectors, such as insurance, banking, and transport, and service sectors, in accordance with the conceptual framework introduced in Chap. 1. Another synthetic definition of quality is the “degree to which a set of inherent characteristics fulfills requirements” (ISO 9000:2005). A requirement is an expectation; it is generally related to the organization, customers, or other interested, or involved, parties. We choose to name all these entities, i. e., the stakeholders of the organization, as customers and, consequently, the basic keyword in quality management is customer satisfaction. Another basic term is capability as the ability of the organization, system, or process to realize a product fulfilling the requirements. A quality management system is a particular management system driving the organization with regard to quality. In other words, it assists companies and organizations in enhancing customer satisfaction. This is the result of products capable of satisfying the everchanging customer needs and expectations that consequently require the continuous improvement of products, processes, and production systems. Quality management is a responsibility at all levels of management and involves all members of an organi-

zation. For this reason, in the 1980s total quality management (TQM) as a business management strategy aimed at embedding awareness of quality in all organizational processes found very great success. According to the International Organization for Standardization (ISO) standards (ISO 9000:2006), the basic steps for developing and implementing a quality management system are: • determination of needs and expectations of customers and other involved parties; • definition of the organization’s quality policy and quality objectives; • determination of processes and responsibilities for quality assessment; • identification and choice of production resources necessary to attain the quality objectives; • determination and application of methods to measure the effectiveness and efficiency of each process within the production system; • prevention of nonconformities and deletion of the related causes; • definition and application of a process for continuous improvement of the quality management system. Figure 2.1 presents the model of a process-based quality management system, as proposed by the ISO standards. Continual improvement of the quality management system

Management responsability

Interested parties

Requirements

Fig. 2.1 Process-based quality management system (ISO 9000:2005)

Mesaurement analysis & improvement

Resource management realization

Interested parties

IN

Value adding activities Information flow

Product realization

Product

Satisfaction

OUT

2.3 ISO Standards for Quality Management and Assessment

2.2 International Standards and Specifications According to European Directive 98/34/EC of 22 June 1998, a “standard” is a technical specification for repeated or continuous application approved, without a compulsory compliance, by one of the following recognized standardization bodies: • ISO; • European standard (EN); • national standard (e. g., in Italy UNI). Standards are therefore documents defining the characteristics (dimensional, performance, environmental, safety, organizational, etc.) of a product, process, or service, in accordance with the state of the art, and they are the result of input received from thousands of experts working in the European Union and elsewhere in the world. Standards have the following distinctive characteristics: • Consensuality: They must be approved with the consensus of the participants in the works of preparation and confirmed by the result of a public enquiry. • Democracy: All the interested economic/social parties can participate in the works and, above all, have the opportunity to make observations during the procedure prior to final and public approval. • Transparency: UNI specifies the basic milestones of the approval procedure for a draft standard, placing the draft documents at the disposal of the interested parties for consultation. • Voluntary nature: Standards are a source of reference that the interested parties agree to apply freely on a noncompulsory basis. In particular CEN, the European Committee for Standardization founded in 1961 by the national standards bodies in the European Economic Community and EFTA countries, is contributing to the objectives of the European Union and European Economic Area with voluntary technical standards promoting free trade, safety of workers and consumers, interoperability of networks, environmental protection, exploitation of research and development programs, and public procurement. CEN works closely with the European Committee for Electrotechnical Standardization (CENELEC), the European Telecommunications Standards Institute

19

(ETSI), and the ISO. CEN is a multisectorial organization serving several sectors in different ways, as illustrated in the next sections and chapters dealing with safety assessment.

2.3 ISO Standards for Quality Management and Assessment The main issues developed by the technical committee for the area of quality are: 1. CEN/CLC/TC 1 – criteria for conformity assessment bodies; 2. CEN/SS F20 – quality assurance. Table 2.1 reports the list of standards belonging to the first technical committee since 2008. Similarly, Table 2.2 reports the list of standards belonging to the technical committee CEN/SS F20 since 2008, while Table 2.3 shows the list of standards currently under development. Quality issues are discussed in several standards that belong to other technical groups. For example, there is a list of standards of the aerospace series dealing with quality, as reported in Table 2.4. Table 2.5 presents a list of standards for quality management systems in health care services. Similarly, there are other sets of standards for specific sectors, businesses, or products.

2.3.1 Quality Audit, Conformity, and Certification Quality audit is the systematic examination of a quality system carried out by an internal or external quality auditor, or an audit team. It is an independent and documented process to obtain audit evidence and to allow its objective evaluation, in order to verify the extent of the fulfillment of the audit criteria. In particular, third-party audits are conducted by external organizations providing certification/registration of conformity to a standard or a set of standards, e. g., ISO 9001 or ISO 14001. The audit process is the basis for the declaration of conformity. The audit process is conducted by an auditor, or an audit team, i. e., a person or a team, with competence

20

2 Quality Management Systems and Statistical Quality Control

Table 2.1 CEN/CLC/TC 1 criteria for conformity assessment bodies, standards published since 2008 Standard

Title

EN 45011:1998

General requirements for bodies operating product certification systems (ISO/IEC Guide 65:1996) Attestation Standard for the assessment of contract award procedures of entities operating in the water, energy, transport and telecommunications sectors Conformity assessment – Vocabulary and general principles (ISO/IEC 17000:2004) Conformity assessment – General requirements for accreditation bodies accrediting conformity assessment bodies (ISO/IEC 17011:2004) General criteria for the operation of various types of bodies performing inspection (ISO/IEC 17020:1998) Conformity assessment – Requirements for bodies providing audit and certification of management systems (ISO/IEC 17021:2006) Conformity assessment – General requirements for bodies operating certification of persons (ISO/IEC 17024:2003) General requirements for the competence of testing and calibration laboratories (ISO/IEC 17025:2005) General requirements for the competence of testing and calibration laboratories (ISO/IEC 17025:2005/Cor.1:2006) Conformity assessment – General requirements for peer assessment of conformity assessment bodies and accreditation bodies (ISO/IEC 17040:2005) Conformity assessment – Supplier’s declaration of conformity – Part 1: General requirements (ISO/IEC 17050-1:2004) Conformity assessment – Supplier’s declaration of conformity – Part 2: Supporting documentation (ISO/IEC 17050-2:2004)

EN 45503:1996 EN ISO/IEC 17000:2004 EN ISO/IEC 17011:2004 EN ISO/IEC 17020:2004 EN ISO/IEC 17021:2006 EN ISO/IEC 17024:2003 EN ISO/IEC 17025:2005 EN ISO/IEC 17025:2005/AC:2006 EN ISO/IEC 17040:2005 EN ISO/IEC 17050-1:2004 EN ISO/IEC 17050-2:2004

Table 2.2 CEN/SS F20 quality assurance, standards published since 2008 Standard

Title

EN 45020:2006 EN ISO 10012:2003

Standardization and related activities – General vocabulary (ISO/IEC Guide 2:2004) Measurement management systems – Requirements for measurement processes and measuring equipment (ISO 10012:2003) Primary packaging materials for medicinal products – Particular requirements for the application of ISO 9001:2000, with reference to good manufacturing practice (GMP) (ISO 15378:2006) Guidelines for quality and/or environmental management systems auditing (ISO 19011:2002) Quality management systems – Fundamentals and vocabulary (ISO 9000:2005) Quality management systems – Requirements (ISO 9001:2000) Quality management systems – Guidelines for performance improvements (ISO 9004:2000)

EN ISO 15378:2007

EN ISO 19011:2002 EN ISO 9000:2005 EN ISO 9001:2000 EN ISO 9004:2000

Table 2.3 CEN/SS F20 quality assurance, standards under development as of October 2008 Standard

Title

ISO 15161:2001

Guidelines on the application of ISO 9001:2000 for the food and drink industry (ISO 15161:2001) Quality management systems – Requirements (ISO/FDIS 9001:2008) Guidelines for auditing management systems Managing for the sustained success of an organization – A quality management approach (ISO/DIS 9004:2008)

prEN ISO 9001 prEN ISO 19011 rev prEN ISO 9004

2.3 ISO Standards for Quality Management and Assessment

21

Table 2.4 Aerospace series, quality standards Standard

Title

EN 9102:2006 EN 9103:2005

Aerospace series – Quality systems – First article inspection Aerospace series – Quality management systems – Variation management of key characteristics Aerospace series – Quality systems – Model for quality assurance applicable to maintenance organizations Aerospace series – Quality management systems –Requirements for stockist distributors (based on ISO 9001:2000) Aerospace series – Quality management systems –Requirements for Aerospace Quality Management System Certification/Registrations Programs Aerospace series – Quality management systems – Assessment applicable to maintenance organizations (based on ISO 9001:2000) Aerospace series – Quality management systems – Assessment applicable to stockist distributors (based on ISO 9001:2000) Aerospace series – Quality management systems – Data Matrix Quality Requirements for Parts Marking Aerospace series – Qualification and approval of personnel for nondestructive testing Aerospace series – Recommended practices for standardizing company standards Aerospace series – Quality management systems – Assessment (based on ISO 9001:2000) Aerospace series – Quality management systems – Part 002: Requirements for Oversight of Aerospace Quality Management System Certification/Registrations Programs

EN 9110:2005 EN 9120:2005 EN 9104:2006 EN 9111:2005 EN 9121:2005 EN 9132:2006 EN 4179:2005 EN 4617:2006 EN 9101:2008 EN 9104-002:2008

Table 2.5 CEN/TC 362, health care services, quality management systems Standard

Title

CEN/TR 15592:200

Health services – Quality management systems – Guide for the use of EN ISO 9004:2000 in health services for performance improvement Health services – Quality management systems – Guide for the use of EN ISO 9001:2000

CEN/TS 15224:2005

to conduct an audit, in accordance with an audit program consisting of a set of one or more audits planned for a specific time frame. Audit findings are used to assess the effectiveness of the quality management system and to identify opportunities for improvement. Guidance on auditing is provided by ISO 19011:2002 (Guidelines for quality and/or environmental management systems auditing). The main advantages arising from certification are: • • • •

improvement of the company image; increase of productivity and company profit; rise of contractual power; quality guarantee of the product for the client.

In the process of auditing and certification, the documentation plays a very important role, enabling communication of intent and consistency of action. Several types of documents are generated in quality management systems.

2.3.2 Environmental Standards Every standard, even if related to product, service, or process, has an environmental impact. For a product this can vary according to the different stages of the product life cycle, such as production, distribution, use, and end-of-life. To this purpose, CEN has recently been playing a major role in reducing environmental impacts by influencing the choices that are made in connection with the design of products and processes. CEN has in place an organizational structure to respond to the challenges posed by the developments within the various sectors, as well as by the evolution of the legislation within the European Community. The main bodies within CEN are: 1. The Strategic Advisory Body on the Environment (SABE) – an advisory body for the CEN Technical Board on issues related to environment. Stakeholders identify environmental issues of importance

22

2 Quality Management Systems and Statistical Quality Control

to the standardization system and suggest corresponding solutions. 2. The CEN Environmental Helpdesk provides support and services to CEN Technical Bodies on how to address environmental aspects in standards. 3. Sectors – some sectors established a dedicated body to address environmental matters associated with their specific needs, such as the Construction Sector Network Project for the Environment (CSNPE). 4. Associates – two CEN associate members provide a particular focus on the environment within standardization: • European Environmental Citizens Organization for Standardization (ECOS); • European Association for the Coordination of Consumer Representation in Standardization (ANEC).

Table 2.6 reports the list of technical committees on the environment. There are several standards on environmental management. To exemplify this, Table 2.7 reports the list of standards grouped in accordance with the committee CEN/SS S26 – environmental management. ISO 14000 is a family of standards supporting the organizations on the containment of the polluting effects on air, water, or land derived by their operations, in compliance with applicable laws and regulations. In particular, ISO 14001 is the international specification for an environmental management system (EMS). It specifies requirements for establishing an environmental policy, determining environmental aspects and impacts of products/activities/services, planning environmental objectives and measurable targets, implementation and operation of programs to meet objectives and targets, checking and corrective action, and management review.

Table 2.6 Technical committees on the environment Technical commitee

Title

CEN/TC 223 CEN/TC 230 CEN/TC 264 CEN/TC 292 CEN/TC 308 CEN/TC 345 CEN/TC 351

Soil improvers and growing media Water analysis Air quality Characterization of waste Characterization of sludges Characterization of soils Construction Products – Assessment of release of dangerous substances

Table 2.7 Committee CEN/SS S26 – environmental management Standard

Title

EN ISO 14031:1999

Environmental management – Environmental performance evaluation – Guidelines (ISO 14031:1999) Environmental management systems – Requirements with guidance for use (ISO 14001:2004) Environmental labels and declarations – Type I environmental labeling – Principles and procedures (ISO 14024:1999) Environmental labels and declarations – Self-declared environmental claims (Type II environmental labelling) (ISO 14021:1999) Environmental labels and declarations – General principles (ISO 14020:2000) Environmental management – Life cycle assessment – Principles and framework (ISO 14040:2006) Environmental management – Life cycle assessment – Requirements and guidelines (ISO 14044:2006) Environmental management systems – Guidelines for a staged implementation of an environmental management system, including the use of environmental performance evaluation

EN ISO 14001:2004 EN ISO 14024:2000 EN ISO 14021:2001 EN ISO 14020:2001 EN ISO 14040:2006 EN ISO 14044:2006 prEN ISO 14005

2.4 Introduction to Statistical Methods for Quality Control

uniform

μ

23

n=2

x

n = 10

x

μ

x

μ

exponential

n = 10 n=2

μ

x

x

μ

μ

Gauss

n = 10 n=2

x

x μ

μ

μ

Fig. 2.2 Central limit theorem, examples

2.4 Introduction to Statistical Methods for Quality Control The aim of the remainder of this chapter is the introduction and exemplification of effective models and methods for statistical quality control. These tools are very diffuse and can be used to guarantee also the reliability,1 productivity and safety of a generic production system in accordance with the purpose of this book, as illustrated in Chap. 1.

2.4.1 The Central Limit Theorem This section briefly summarizes the basic result obtained by this famous theorem. Given a population or process, a random variable x, with mean  and standard deviation , let xN be the mean of a random sample made of n elements x1 ; x2 ; : : : ; xn extracted from this population: when the sample size n is sufficiently large, the sampling distribution of the random vari1

Reliability, properly defined in Chap. 5, can be also defined as “quality in use.”

able xN can be approximated by a normal distribution. The larger the value of n, the better the approximation. This theorem holds irrespective of the shape of the population, i. e., of the density function of the variable x. The analytic translation of the theorem is given by the following equations: M.x/ N D xN D ; O O .x/ N Dp ; n

(2.1) (2.2)

where O is the estimation of  and O is the estimation of . Figure 2.2 graphically and qualitatively demonstrates these results representing the basis for the development and discussion of the methods illustrated and applied below. In the presence of a normal distribution of population, the variable xN is normal too for each value of size n. Figure 2.3 quantitatively demonstrates the central limit theorem starting from a set of random values distributed in accordance with a uniform distribution Œ0; 10: the variable xN is a normally distributed variable when the number of items used for the calculus of mean xN i is sufficiently large. In detail, in Fig. 2.3 the size n is assumed be 2, 5, and 20.

24

2 Quality Management Systems and Statistical Quality Control

Histogram of DATA (n=1) and means (n>1) DATA (n=1)

Mean (n=2) 0.20

0.16 0.15

0.12

0.10

Density

0.08

0.05

0.04 0.00

0.00 0.0

1.5

3.0

4.5

6.0

7.5

9.0

Mean (n=5)

1.5

3.0

4.5

6.0

7.5

9.0

Mean (n=20)

0.8

0.3 0.6 0.2

0.4

0.1

0.2

0.0

0.0 1

2

3

4

5

6

7

8

3.6

4.2

4.8

5.4

6.0

6.6

Fig. 2.3 Central limit theorem, histogram of xN for n D f1; 2; 5; 20g. Uniform distribution of variable x

2.4.2 Terms and Definition in Statistical Quality Control Quality control is a part of quality management (ISO 9000:2005) focused on the fulfillment of quality requirements. It is a systematic process to monitor and improve the quality of a product, e. g., a manufactured article, or service by achieving the quality of the production process and the production plant. A list of basic terms and definitions in accordance with the ISO standards follows: • Process, set of interrelated activities turning input into output. It is a sequence of steps that results in an outcome. • Product, result of a process. • Defect, not fulfillment of a requirement related to an intended or specified use. • Measurement process, set of operations to determine the value of a quantity. • Key characteristic, a quality characteristic the product or service should have to fulfill customer requirements and expectations. • Value of a key characteristic. For several products a single value is the desired quality level for a characteristic. • Nominal or target value. It is the expected value for the key characteristic. It is almost impossible to make each unit of product or service identical to the

next; consequently it is nonsense to ask for identical items having a key characteristic value exactly equal to the target value. This need for flexibility is supported by the introduction of limits and tolerances. • Specification limit, or tolerance, conformance boundary, range, specified for a characteristic. The lower specification limit (LSL) is the lower conformance boundary, the upper specification limit (USL) is the upper conformance boundary. The following equation summarizes the relationship among these terms: Specification limits D (nominal value) ˙ tolerance: (2.3) • One-sided tolerance. It relates to characteristics with only one specification limit. • Two-sided tolerance. It refers to characteristics with both USLs and LSLs. • Nonconformity. It is a nonfulfillment of a requirement. It is generally associated with a unit: a nonconformity unit, i. e., a unit that does not meet the specifications. • Nonconforming product or service. A product or service with one or more nonconformities. A nonconforming product is not necessary defective, i. e., no longer fit for use.

2.6 Control Charts

25

2.5 Histograms Histograms are effective and simple graphic tools for the comprehension and analysis of a process behavior with regards to the target value and the specification limits. The histograms illustrate the frequency distribution of variable data: the values assumed by the variable are reported on the abscissa, while the vertical axis reports the absolute or relative frequency values. The specification limits are generally included in the graph and give warnings of possible process problems. Figure 2.4 exemplifies a few histogram shapes. The control charts illustrated in the next section represent a more effective tool for the analysis of a production process.

2.6 Control Charts Control charts, introduced by W.A. Shewhart in 1924, are effective tools for the analysis of the variation of repetitive processes. They are able to identify possible sources of process variation in order to control and eventually eliminate them. In a generic process, two different kinds of variations can be distinguished:

configurations

1. Common causes variations. They are the noise of a production system and are uncontrollable variations. 2. Assignable (or special) causes variations. They can be properly identified and controlled. Some examples are turnover in workman load, breakdowns, machine or tool wear out, and tool change. Control charts are a family of tools for detecting the existence of special causes variations in order to avoid them, i. e., eliminate all anomalous controllable patterns, and bring the process into a state called “of statistical control,” or simply “in control,” whose random behavior is justified by the existence of common causes variations. The “in control” state is necessary to obtain conforming products, as properly discussed in the following sections on capability analysis and Six Sigma. Control charts can be constructed by extracting successive samples from the variable output of the process. These samples, also called “subgroups,” all have size n and have to be taken at regular intervals of time. For each group a summary statistic is calculated and plotted as illustrated in Fig. 2.5. Typical statistical measures calculated for each subgroup are reported in Table 2.8, where the related statistical distribution is cited together with the values of

reasons

configurations

reasons

"Special (assignable) causes" of variation, i.e. errors of measurement or in the activity of data collection LSL

process variation too large for the specification limits LSL

USL

USL

process shifted to the "right" or measurements are out of calibration LSL

"truncated" data LSL

USL

USL

process shifted to the "left" or measurements are out of calibration LSL

USL

granularity, i.e. "granular process" LSL

USL

mix of two different processes, e.g. data from two operators, two machines, or collected at different points in time LSL

USL

stable process within specifications LSL

USL

Fig. 2.4 Exemplifying histograms shapes. LSL lower specification limit, USL upper specification limit

2 Quality Management Systems and Statistical Quality Control

Subgroup statistic (e.g. R, p, u)

26

UCL centerline LCL

1

2

3

4

5 …

Subgroup number

Fig. 2.5 Control chart

the centerline and control limits, as properly defined in the next subsections. A control chart is made of three basic lines as illustrated in Fig. 2.5: 1. Centerline. It is the mean of the statistic quantified for each subgroup, the so-called subgroup statistic (e. g., mean, range, standard deviation). 2. Control limits. These limits on a control chart delimit that region where a data point falls outside, thus alerting one to special causes of variation. This region is normally extended three standard deviations on either side of the mean. The control limits are: • upper control limit (UCL), above the mean; • lower control limit (LCL), below the mean. The generic point of the chart in Fig. 2.5 may represent a subgroup, a sample, or a statistic. k different samples are associated with k different points whose temporal sequence is reported on the chart. Control limits are conventionally set at a distance of three standards errors, i. e., three deviations of the subgroup statistic, from the centerline, because the distribution of samples closely approximates a normal statistical distribution by the central limit theorem. Consequently, the analyst expects that about 99.73% of samples lie within three standard deviations of the mean. This corresponds to a probability of 0.27% that a control chart point falls outside one of the previously defined control limits when no assignable causes are present. In some countries, such as in the UK, the adopted convention of ˙ three standard deviations is different. Figure 2.6 presents eight different anomalous patterns of statistic subgroups tested by Minitab® Statis-

tical Software to find reliable conditions for the in, or out, control state of the process. A process is said to be “in control” when all subgroups on a control chart lie within the control limits and no anomalous patterns are in the sequence of points representing the subgroups. Otherwise, the process is said to be “out of control,” i. e., it is not random because there are special causes variations affecting the output obtained. What happens in the presence of special causes? It is necessary to identify and eliminate them. Consequently, if a chart shows the possible existence of special causes by one of the anomalous behaviors illustrated in Fig. 2.6, the analyst and the person responsible for the process have to repeat the analysis by eliminating the anomalous subgroups. Now, if all the tests are not verified, the process has been conducted to the state of statistical control.

2.7 Control Charts for Means These charts refer to continuous measurement data, also called “variable data” (see Table 2.8), because there are an infinite number of data between two generic ones.

2.7.1 The R-Chart This is a chart for subgroup ranges. The range is the difference between the maximum and the minimum values within a sample of size n: Ri D max fxij g  min fxij g; j D1;:::;n

j D1;:::;n

(2.4)

where i is a generic sample and xij is the j th value in the sample i . Consequently, the centerline is R D RN D

k 1X Ri : k

(2.5)

i D1

This value is a good estimation of the mean value of variable Ri , called “R .” We also define the statistic measure of variability of the variable Ri , the standard deviation R . By the central limit theorem, the distribution of values Ri is normal. As a consequence, the

Variable

Variable

Variable

Attribute

Attribute

Attribute

Attribute

R-chart

s-chart

x-chart N

p-chart

np-chart

c-chart

u-chart

Discrete – Poisson distribution

Discrete – Poisson distribution

Discrete – binomial distribution

Discrete – binomial distribution

Continuous – normal distribution

Continuous – normal distribution

Continuous – normal distribution

Statistical distribution

Nonconformities per unit u

Number of nonconformities

Number of nonconformities np

Nonconforming proportion p

Mean xN

Standard deviation s

Range R

Statistic measure

iD1

iD1

k 1 X ui k

uN D

iD1

k 1 X ci k

cN D

npN

x1 C x2 C    C xk1 C xk n1 C n2 C    C nk1 C nk

pN D

iD1

k 1 X pi k

pN D

iD1

k 1 X N Xi  O D XNN D k

 O S D sN D

k 1 X si k

sN =c4 O UCL D  O C 3 p  XNN C 3 p D XNN C A3 sN n n s N =c O 4 N LCL D  O  3 p  XN  3 p D XNN  A3 sN n n s p.1 N  p/ N UCL D pN C 3 n s p.1 N  p/ N LCL D pN  3 n s p.1 N  p/ N UCLi D pN C 3 ni s p.1 N  p/ N LCLi D pN  3 ni p N  p/ N UCL D npN C 3 np.1 p LCL D npN  3 np.1 N  p/ N p UCL D cN C 3 cN p LCL D cN  3 cN s uN UCLi D uN C 3 ni s uN LCLi D uN  3 ni

N 2 R=d O UCL D  O C 3 p  XNN C 3 p D XNN C A2 RN n n N R=d O 2 LCL D  O  3 p  XNN  3 p D XNN  A2 RN n n

UCL D B4 sN LCL D B3 sN

UCL D R C 3R Š D4 RN LCL D R  3R Š D3 RN

k 1 X RN D Ri k iD1

Control limits

Centerline

UCL upper control limit, LCL lower control limit, UCLi upper control limit for sample i , LCLi lower control limit for sample i

Type of data

Name

Table 2.8 Statistical measures and control chart classification

O D

O D

sN c4

RN d2

Standard deviation

2.7 Control Charts for Means 27

28

2 Quality Management Systems and Statistical Quality Control

Test 1 – 1 point beyond 3 std.dev. (zone A)

Test 2 – 9 points in a row on same side of the center line

Test 3 – 6 points in a row all increasing (or decreasing)

Test 4 – 14 points in a row alternang up and down

Test 5 – 2/3 points in a row more than 2 std.dev.

Test 6 – 4 out of 5 points more than 1 std.dev.

Test 7 – 15 points in a row within 1 std.dev. (either side)

Test 8 – 8 points in a row more than 1 std.dev. (either side)

Fig. 2.6 Eight tests for special causes investigation, Minitab® Statistical Software

control limits are defined as N UCLR D R C 3R Š D4 R; N LCLR D R  3R Š D3 R;

(2.6)

where R is the standard deviation of the variable R and D4 is a constant value depending on the size of

the generic subgroup. The values are reported in Appendix A.2. The following equation represents an estimation of the standard deviation of the variable and continuous data xij : RN (2.7) O D d2

2.7 Control Charts for Means

29

2.7.2 Numerical Example, R-Chart

2.7.3 The x-Chart N

This application refers to the assembly process of an automotive engine. The process variable is a distance, D, measured in tenths of millimeters, between two characteristic axes in the drive shafts and heads. Table 2.9 presents the data collected over 25 days of observation and grouped in samples of size n D 5. By the application of Eqs. 2.5 and 2.6, we have

This is a chart for subgroup means. In the x-chart, N also called “x-chart,” the problem is the estimation of the standard deviation of the population of values. In Sect. 2.7.1, Eq. 2.7 is an effective estimation. Consequently, this chart is generally constructed after the creation of the R-chart and reveals the process to be in the state of statistical control. The centerline of the statistic variable xN is the average of the subgroup means: n k k X X 1X O D . O x/ N D xN D xN i D xij : (2.8) k

k 1X 1 .R1 C    C R25 / Š 6:50; RN D Ri D k 25 i D1

UCL D R C 3R Š D4 RN LCL D R  3R Š D3 RN

D

D4 .nD5/D2:114

i D1

13:74;

The control limits are D

D4 .nD5/D0

0:

The R-chart obtained is reported in Fig. 2.7. Previously introduced tests for anomalous behaviors are not verified. As a consequence, the process seems to be random and “coherent with itself” and its characteristic noise and variance. There are no special causes of variation.

N 2 O R=d N UCLxN D O C 3 p  xN C 3 p D xN C A2 R; n n N 2 O R=d N LCLxN D O  3 p  xNN  3 p D xNN  A2 R; n n (2.9) where d2 and A2 are constant values as reported in Appendix A.2.

Table 2.9 Data – 25 subgroups, numerical example Sample

Month

Day

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8

25 26 27 28 29 30 31 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

i D1 j D1

D (mm=10) 0:387 4.251 2:727 6.980 3.978 3.424 4:285 1:756 4.184 3:577 2:467 1.199 4.312 3.282 2.000 3.268 3.356 0:240 4:524 0.837 1:016 4.547 0.159 0.842 4.435

5.192 3.333 2:806 3.280 3.479 1.758 2:369 1:434 1.005 1:684 2:752 0.817 1.127 3.319 3:364 1.519 3:335 3:811 0:091 4:536 2.023 0.262 3.786 3:550 1.730

1.839 4.398 4.655 3.372 7.034 0.009 2:666 1.887 0.825 1.800 4:029 0:213 2.534 3:564 1:996 2.704 3:358 1:615 1.945 4.249 4.539 4:108 1:951 1:805 0:185

0.088 6.082 0.494 1:914 4.388 0:216 2.639 1:678 6:427 4.339 2:798 0:737 1.618 3.430 1:830 0.138 4:302 3:510 4.515 0.114 0.075 1:881 6.315 2:731 0.242

1.774 1.706 2:807 2.478 1:790 1.832 3.081 7.060 4:598 0.027 2:152 1:757 0:665 1.556 0.015 0:050 2:856 4:377 1:667 0:087 2:724 0:004 5.161 1:610 4:689

30

2 Quality Management Systems and Statistical Quality Control

R Chart of D 14

UCL=13.74

Sample Range

12 10 8 _ R=6.50

6 4 2

LCL=0

0 Sample 1 Month 7 Day 25

3 7 27

5 7 29

7 7 31

9 8 2

11 8 4

13 8 6

15 8 8

17 8 10

19 8 12

21 8 14

23 8 16

25 8 18

Fig. 2.7 R-chart, numerical example. Minitab® Statistical Software. UCL upper control limit, LCL lower control limit

2.7.4 Numerical Example, x-Chart N Consider the application introduced in Sect. 2.7.2. By Eqs. 2.8 and 2.9, O D xN D

k 1X 1 .xN 1 C    C xN 25 / D 0:389; xN i D k 25 i D1

N 2 O R=d UCL D O C 3 p  xNN C 3 p D xNN C A2 RN n n D 0:389 C 0:577  6:5 Š 4:139; A2 D0:577

N 2 O R=d LCL D O  3 p  xNN  3 p D xNN  A2 RN n n D 0:389  0:577  6:5 Š 3:361: A2 D0:577

The chart obtained is reported in Fig. 2.8. Test 6 for anomalous behaviors is verified in sample 5, month 7, and day 29, i. e., there are four of five points in zone B or beyond. As a consequence, the process seems to be “out of control.” There is in fact a very scarce probability of having a sample in those points when the process is “in control.” We assume we are able to properly identify this special cause of variation and to eliminate it. Figure 2.9 presents the charts obtained from the pool of samples without the anomalous subgroup 5. The chart shows another potential anomalous behav-

ior regarding subgroup 4. In this way, assuming we identify and eliminate new special causes, we obtain Figs. 2.10 and 2.11. In particular, Fig. 2.11 presents a process in the state of statistical control: subgroups 2, 4, and 5 have been eliminated.

2.7.5 The s-Chart This chart for subgroup standard deviation can be used to support the construction of the x-chart by the estimation of the standard deviation of the continuous variable xij . In particular, the control limits of the xchart use the centerline of the s-chart. The average of standard deviation of subgroups, sO , is the centerline of the s-chart: O S D .s O i / D sN D

k 1X si ; k

(2.10)

i D1

where .s O i / is the estimation of the mean of the variable si , the standard deviation of a subgroup. The control limits are .s O i/ O i / C 3 p D B4 sN ; UCLs D .s n .s O i/ LCLs D .s O i /  3 p D B3 sN ; n

(2.11)

2.7 Control Charts for Means

31

Xbar-R Chart of D Sample Mean

4

UCL=4.136

6

2 _ _ X=0.389

0 -2

LCL=-3.359 -4 Sample Month Day

1 7 25

3 7 27

5 7 29

7 7 31

9 8 2

11 8 4

13 8 6

15 8 8

17 8 10

19 8 12

21 8 14

23 8 16

25 8 18

15

Sample Range

UCL=13.74 10 _ R=6.50 5

LCL=0

0 Sample Month Day

1 7 25

3 7 27

5 7 29

7 7 31

9 8 2

11 8 4

13 8 6

15 8 8

17 8 10

19 8 12

21 8 14

23 8 16

25 8 18

Fig. 2.8 x-chart from R. Numerical example (25 samples). Minitab® Statistical Software

Xbar-R Chart [rif.no sub.5] 4

UCL=3.984

Sample Mean

5

2 _ _ X=0.262

0 -2

-4 Sample [24] Month [24] Day [24]

LCL=-3.459 1 7 25

3 7 27

6 7 30

8 8 1

10 8 3

12 8 5

14 8 7

16 8 9

18 8 11

20 8 13

22 8 15

24 8 17

15

Sample Range

UCL=13.64 10 _ R=6.45 5

0 Sample [24] Month [24] Day [24]

LCL=0 1 7 25

3 7 27

6 7 30

8 8 1

10 8 3

12 8 5

14 8 7

16 8 9

18 8 11

Fig. 2.9 x-chart from R. Numerical example (24 samples). Minitab® Statistical Software

20 8 13

22 8 15

24 8 17

32

2 Quality Management Systems and Statistical Quality Control

Xbar-R Chart [no sub 4 and 5] 1

Sample Mean

4

UCL=3.847

2 _ _ X=0.150

0 -2

-4 Sample [23] Month [23] Day [23]

LCL=-3.546 1 7 25

3 7 27

7 7 31

9 8 2

11 8 4

13 8 6

15 8 8

17 8 10

19 8 12

21 8 14

23 8 16

25 8 18

15

Sample Range

UCL=13.55 10 _ R=6.41 5

LCL=0

0 Sample [23] Month [23] Day [23]

1 7 25

3 7 27

7 7 31

9 8 2

11 8 4

13 8 6

15 8 8

17 8 10

19 8 12

21 8 14

23 8 16

25 8 18

Fig. 2.10 x-chart from R. Numerical example (23 samples). Minitab® Statistical Software

Xbar-R Chart [no sub 2, 4 and 5] Sample Mean

4

UCL=3.729

2 _ _ X=-0.022

0 -2

-4 Sample [22] Month [22] Day [22]

LCL=-3.774 1 7 25

6 7 30

8 8 1

10 8 3

12 8 5

14 8 7

16 8 9

18 8 11

20 8 13

22 8 15

24 8 17

15

Sample Range

UCL=13.75 10 _ R=6.50 5

0 Sample [22] Month [22] Day [22]

LCL=0 1 7 25

6 7 30

8 8 1

10 8 3

12 8 5

14 8 7

16 8 9

18 8 11

Fig. 2.11 x-chart from R. Numerical example (22 samples). Minitab® Statistical Software

20 8 13

22 8 15

24 8 17

2.8 Control Charts for Attribute Data

where .s O i / is the estimation of the standard deviation of the variable si , the standard deviation of a subgroup, and B3 and B4 are constant values reported in Appendix A.2. The standard deviation of the process measurement is sN O D . O xN i / D : (2.12) c4

33

The x-chart is now created assuming the centerline of the R-chart and in accordance with Eqs. 2.8 and 2.9: UCLx from R Š xNN C A2 RN D 0:009237 C 0:577  0:004155 LCLx from R

D 0:009237  0:577  0:004155

As a consequence, the control limits of the x-chart are sN =c4 O UCLxN D O C 3 p  xN C 3 p D xN C A3 sN; n n O s N =c 4 LCLxN D O  3 p  xNN  3 p D xNN  A3 sN; n n (2.13) where A3 is a constant value reported in Appendix A.2.

D 0:0116; Š xN  A2 RN D 0:0068:

The upper section of Fig. 2.12 presents the x-chart where some subgroups verify a few tests, as illustrated also in Fig. 2.13. Consequently, the process is not in a state of control. Similarly, by the application of Eqs. 2.10, 2.11, and 2.13, O S D sN D

k 1X si Š 0:00170: k i D1

2.7.6 Numerical Example, s-Chart and x-Chart N Table 2.10 reports a set of measurement data made for 20 samples of size n D 5. They are the output of a manufacturing process in the automotive industry. The last three columns report some statistics useful for the construction of the control charts and for verification of the status of the control of the process. With use of the values of the constant parameters in Appendix A.2, the following control limits and centerlines have been obtained. Firstly, we propose the results related to the Rchart. By Eq. 2.5 the centerline is k 1X N Ri Š 0:004155: R D R D k

UCLs Š B4 sN D 2:089  0:00170 D 0:00355; LCLs Š B3 sN D 0  0:00170 D 0; UCLx from s  xNN C A3 sN D 0:009237 C 1:427  0:00170 LCLx from s

D 0:01166;  xNN  A3 sN D 0:009237  1:427  0:00170 D 0:00681:

All these values are also reported in Fig. 2.14, showing that the process is not in the state of statistical control. Consequently, a survey for the identification and deletion of special causes of variations, and the subsequent repetition of the control analysis, is required.

i D1

By Eq. 2.6 UCLR  D4 RN D 2:114  0:004155 Š 0:008784: LCLR  D3 RN D 0  0:004155 D 0: These results are very close to those proposed by the R-chart, as constructed by the tool Minitab® Statistical Software (Fig. 2.12). From the R-chart the process seems to be in the state of statistical control.

2.8 Control Charts for Attribute Data These charts refer to counted data, also called “attribute data.” They support the activities of monitoring and analysis of production processes whose products possess, or do not possess, a specified characteristic or attribute. Attributes measurement is frequently obtained as the result of human judgements.

34

2 Quality Management Systems and Statistical Quality Control

Table 2.10 Measurement data and subgroup statistics. Numerical example ID sample – i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Measure – X 0.0073 0.0106 0.0096 0.0080 0.0104 0.0071 0.0078 0.0087 0.0074 0.0081 0.0078 0.0089 0.0087 0.0084 0.0074 0.0069 0.0077 0.0076 0.0069 0.0063

0.0101 0.0083 0.0080 0.0076 0.0084 0.0052 0.0089 0.0094 0.0081 0.0065 0.0098 0.0090 0.0075 0.0083 0.0091 0.0093 0.0089 0.0069 0.0077 0.0071

0.0091 0.0076 0.0132 0.0090 0.0123 0.0101 0.0122 0.0120 0.0120 0.0105 0.0113 0.0111 0.0125 0.0101 0.0116 0.0090 0.0091 0.0062 0.0073 0.0078

0.0091 0.0074 0.0105 0.0099 0.0132 0.0123 0.0091 0.0102 0.0116 0.0125 0.0087 0.0122 0.0106 0.0140 0.0109 0.0084 0.0068 0.0077 0.0074 0.0063

M.xi /

Ri

si

0.0053 0.0059 0.0098 0.0123 0.0120 0.0073 0.0095 0.0099 0.0122 0.0136 0.0118 0.0126 0.0113 0.0097 0.0108 0.0090 0.0094 0.0067 0.0074 0.0088

0.0082 0.0080 0.0102 0.0094 0.0113 0.0084 0.0095 0.0101 0.0103 0.0102 0.0099 0.0107 0.0101 0.0101 0.0100 0.0085 0.0084 0.0070 0.0073 0.0073

0.0048 0.0047 0.0052 0.0047 0.0048 0.0071 0.0044 0.0033 0.0048 0.0071 0.0040 0.0037 0.0050 0.0057 0.0042 0.0024 0.0026 0.0015 0.0008 0.0025

0.0019 0.0017 0.0019 0.0019 0.0019 0.0028 0.0016 0.0012 0.0023 0.0029 0.0017 0.0017 0.0020 0.0023 0.0017 0.0010 0.0011 0.0006 0.0003 0.0011

Mean

0.009237

0.004155

0.0016832

Xbar-R Chart

Sample Mean

UCL=0.011666 0.011

6 6

0.010

6

2

_ _ X=0.009237

0.009 0.008 0.007

3

1

3

5

7

9

11 Sample

13

15

17

5

5

LCL=0.006808

19

UCL=0.008905

Sample Range

0.008 0.006 _ R=0.004211

0.004 0.002

LCL=0

0.000 1

3

5

7

9

11 Sample

13

15

Fig. 2.12 R-chart and x-chart from R. Numerical example. Minitab® Statistical Software

17

19

2.8 Control Charts for Attribute Data

35

Test Results for Xbar Chart TEST 2. 9 points in a row on same side of center line. Test Failed at points: 15 TEST 3. 6 points in a row all increasing or all decreasing. Test Failed at points: 18 TEST 5. 2 out of 3 points more than 2 standard deviations from center line (on one side of CL). Test Failed at points: 19; 20 TEST 6. 4 out of 5 points more than 1 standard deviation from center line (on one side of CL). Test Failed at points: 12; 13; 14; 20

Fig. 2.13 x-chart from R, test results. Numerical example. Minitab® Statistical Software

Xbar-S Chart

Sample Mean

UCL=0.011666 0.011

6 6

0.010

6

2

_ _ X=0.009237

0.009 0.008 5

0.007

3

1

3

5

7

9

11 Sample

13

15

17

5

LCL=0.006808

19

0.004

Sample StDev

UCL=0.003555 0.003 0.002

_ S=0.001702

0.001 0.000

LCL=0 1

3

5

7

9

11 Sample

13

15

17

19

Fig. 2.14 s-chart and x-chart from s. Numerical example. Minitab® Statistical Software

2.8.1 The p-Chart The p-chart is a control chart for monitoring the proportion of nonconforming items in successive subgroups of size n. An item of a generic subgroup is said to be nonconforming if it possesses a specified characteristic. Given p1 ; p2 ; : : : ; pk , the subgroups’ proportions of nonconforming items, the sampling random

variable pi for the generic sample i has a mean and a standard deviation: p D ; r p D

.1  / ; n

(2.14)

where  is the true proportion of nonconforming items of the process, i. e., the population of items.

36

2 Quality Management Systems and Statistical Quality Control

The equations in Eq. 2.14 result from the binomial discrete distribution of the variable number of nonconformities x. This distribution function is defined as ! n x (2.15) p.x/ D  .1  /nx ; x where x is the number of nonconformities and  is the probability the generic item has the attribute. The mean value of the standard deviation of this discrete random variable is X xp.x/ D n; D x

s D

X

.x  /2 p.x/ D n.1  /:

(2.16)

Table 2.11 Rejects versus tested items. Numerical example Day

Rejects

Tested

32 25 21 23 13 14 15 17 19 21 15 16 21 9 25

286 304 304 324 289 299 322 316 293 287 307 328 304 296 317

21=10 22=10 23=10 24=10 25=10 26=10 27=10 28=10 29=10 30=10 31=10 1=11 2=11 3=11 4=11

Day 5=11 6=11 7=11 8=11 9=11 10=11 11=11 12=11 13=11 14=11 15=11 16=11 17=11 18=11 19=11

Rejects

Tested

21 14 13 21 23 13 23 15 14 15 19 22 23 24 27

281 310 313 293 305 317 323 304 304 324 289 299 318 313 302

x

By the central limit theorem, the centerline, as the estimated value of , and the control limits of the p-chart are O i / D pN D O p D .p r UCLp D pN C 3 r LCLp D pN  3

k 1X pi ; k

(2.17)

i D1

p.1 N  p/ N ; n p.1 N  p/ N : n

(2.18)

If the number of items for a subgroup is not constant, the centerline and the control limits are quantified by the following equations: pN D

x1 C x2 C    C xk1 C xk ; n1 C n2 C    C nk1 C nk

UCLp;i D pN C 3 s LCLp;i D pN  3

p.1 N  p/ N ; ni

Table 2.11 reports the data related to the number of electric parts rejected by a control process considering 30 samples of different size. By the application of Eqs. 2.19 and 2.20,

(2.20)

p.1 N  p/ N ; ni

where UCLi is the UCL for sample i and LCLi is the LCL for sample i .

573 x1 C x2 C    C xk1 C xk D n1 C n2 C    C nk1 C nk 9171 Š 0:0625; s p.1 N  p/ N D pN C 3 ni s 0:0625.1  0:0625/ Š 0:0625 C 3 ; ni s p.1 N  p/ N D pN  3 ni s 0:0625.1  0:0625/ Š 0:0625  3 : ni

pN D

UCLp;i

(2.19)

where xi is the number of nonconforming items in sample i and ni is the number of items within the subgroup i , and s

2.8.2 Numerical Example, p-Chart

LCLp;i

Figure 2.15 presents the p-chart generated by Minitab® Statistical Software and shows that test 1 (one point beyond three standard deviations) occurs for the first sample. This chart also presents the noncontinuous trend of the control limits in accordance with the equations in Eq. 2.20.

2.8 Control Charts for Attribute Data

37

P Chart of Rejects 1

0.11 UCL=0.1043

0.10

Proportion

0.09 0.08 0.07

_ P=0.0625

0.06 0.05 0.04 0.03

LCL=0.0207

0.02 21/10 24/10 27/10 30/10

2/11

5/11

8/11

11/11 14/11 17/11

Day Tests performed with unequal sample sizes

Fig. 2.15 p-chart with unequal sample sizes. Numerical example. Minitab® Statistical Software

Table 2.12 Rejected items. Numerical example

2.8.3 The np-Chart

Day

This is a control chart for monitoring the number of nonconforming items in subgroups having the same size. The centerline and control limits are N O np D np; UCLnp LCLnp

p D npN C 3 np.1 N  p/; N p D npN  3 np.1 N  p/: N

(2.21) (2.22)

2.8.4 Numerical Example, np-Chart The data reported in Table 2.12 relate to a production process similar to that illustrated in a previous application, see Sect. 2.8.2. The size of the subgroups is now constant and equal to 280 items. Figure 2.16 presents the np-chart generated by Minitab® Statistical Software: test 1 is verified by two consecutive samples (collected on 12 and 13 November). The analyst has to find the special causes, then he/she must eliminate them and regenerate the chart, as in Fig. 2.17. This second chart presents another anomalous subgroup: 11=11. Similarly, it is necessary to eliminate this sample and regenerate the chart.

21=10 22=10 23=10 24=10 25=10 26=10 27=10 28=10 29=10 30=10 31=10 1=11 2=11 3=11 4=11

Rejects 19 24 21 23 13 32 15 17 19 21 15 16 21 12 25

Day 5=11 6=11 7=11 8=11 9=11 10=11 11=11 12=11 13=11 14=11 15=11 16=11 17=11 18=11 19=11

Rejects 21 14 13 21 23 13 34 35 36 15 19 22 23 24 27

2.8.5 The c-Chart The c-chart is a control chart used to track the number of nonconformities in special subgroups, called “inspection units.” In general, an item can have any number of nonconformities. This is an inspection unit, as a unit of output sampled and monitored for determination of nonconformities. The classic example is a single printed circuit board. An inspection unit can be a batch, a collection, of items. The monitoring activity of the inspection unit is useful in a continuous pro-

38

2 Quality Management Systems and Statistical Quality Control

NP Chart of Rejects 1

1

35

UCL=34.35

Sample Count

30 25 __ NP=21.1

20 15 10

LCL=7.85 21/10 24/10 27/10 30/10 2/11

5/11 Day

8/11 11/11 14/11 17/11

Fig. 2.16 np-chart, equal sample sizes. Numerical example. Minitab® Statistical Software

NP Chart of Rejects (no 12 & 13 /11) 35

1

UCL=33.02

Sample Count

30

25 __ NP=20.07

20

15

10 LCL=7.12 21/10 24/10 27/10 30/10 2/11 5/11 8/11 Day (no 12 & 13 /11)

11/11 16/11 19/11

Fig. 2.17 np-chart, equal sample sizes. Numerical example. Minitab® Statistical Software

duction process. The number of nonconformities per inspection unit is called c. The centerline of the c-chart has the following average value: k 1X O i / D cN D ci : O c D .c k

(2.23)

i D1

The control limits are p UCLc D cN C 3 c; N p LCLc D cN  3 c: N

The mean and the variance of the Poisson distribution, defined for the random variable number of nonconformities units counted in an inspection unit, are N  .ci / D .ci / D c:

The density function of this very important discrete probability distribution is e x ; xŠ where x is the random variable. f .x/ D

(2.24)

(2.25)

(2.26)

2.8 Control Charts for Attribute Data

39 Table 2.13 Errors in inspection unit of 6,000 digits. Numerical example

2.8.6 Numerical Example, c-Chart Table 2.13 reports the number of coding errors made by a typist in a page of 6,000 digits. Figure 2.18 shows the c-chart obtained by the sequence of subgroups and the following reference measures: k 1X ci D 6:8; cN D k i D1

p p UCLc D cN C 3 cN D 6:8 C 3 6:8 Š 14:62; p p LCLc D cN  3 cN D maxf6:8  3 6:8; 0g Š 0; where ci is the number of nonconformities in an inspection unit. From Fig. 2.18 there are no anomalous behaviors suggesting the existence of special causes of variations in the process, thus resulting in a state of statistical control. A significant remark can be made: why does this numerical example adopt the c-chart and not the pchart? If a generic digit can be, or cannot be, an object of an error, it is in fact possible to consider a binomial process where the probability of finding a digit with an

Day 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Errors 10 11 6 9 12 12 14 9 5 0 1 2 1 11 9

Day

Errors

16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

error is pi D

8 7 1 2 3 5 1 11 9 14 1 9 1 8 12

ci ci D ; n 6;000

where n is the number of digits identifying the inspection unit. The corresponding p-chart, generated by Minitab® Statistical Software and shown in Fig. 2.19, is very similar to the c-chart in Fig. 2.18.

C Chart of errors 16 UCL=14.62

14

Sample Count

12 10 8

_ C=6.8

6 4 2 LCL=0

0 1

4

7

10

13

16 Day

19

22

25

28

Fig. 2.18 c-chart. Inspection unit equal to 6,000 digits. Numerical example. Minitab® Statistical Software

40

2 Quality Management Systems and Statistical Quality Control

P Chart of errors 0.0025

UCL=0.002436

Proportion

0.0020

0.0015 _ P=0.001133 0.0010

0.0005

0.0000

LCL=0 1

4

7

10

13

16 Day

19

22

25

28

Fig. 2.19 p-chart. Inspection unit equal to 6,000 digits. Numerical example. Minitab® Statistical Software

2.8.7 The u-Chart

ure 2.21 shows the chart obtained by the elimination of those samples. A new sample, i D 30, is “irregular.”

If the subgroup does not coincide with the inspection unit and subgroups are made of different numbers of inspection units, the number of nonconformities per unit, ui , is ci (2.27) ui D : n The centerline and the control limits of the so-called u-chart are O u D .u O i / D uN D s UCLu;i D uN C 3 s LCLu;i D uN  3

k 1X ui ; k i D1

uN ; ni

(2.28)

uN : ni

2.8.8 Numerical Example, u-Chart Table 2.14 reports the number of nonconformities as defects on ceramic tiles of different sizes, expressed in feet squared. Figure 2.20 presents the u-chart obtained; five different subgroups reveal themselves as anomalous. Fig-

2.9 Capability Analysis A production process is said to be capable when it is in state of statistical control and products meet the specification limits, i. e., the customers’ requirements. In other words, the process is capable when it produces “good” products. This is the first time the lower and upper specifications are explicitly considered in the analysis of the process variations. Nonconformity rates are the proportions of process measurements above, or below, the USL, or LSL. This proportion can be quantified in parts per million (PPM), as   USL  O ; PPM > USL D P .x > USL/  P z > O (2.29)   LSL  O ; PPM < LSL D P .x < LSL/  P z < O (2.30) where x is a normal random variable and z is a standard normal variable (see Appendix A.1).

2.9 Capability Analysis

41

Table 2.14 Errors/defects in ceramic tiles. Numerical example Sample i

ci [nonconform. number]

Size [ft2 ]

ui

Sample i

ci [nonconform. number]

Size [ft2 ]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

14 47 21 6 16 27 21 22 43 17 32 14 9 16 19

7.1 3.3 5.9 5.2 5.6 8 8.9 5.6 6.1 4.2 8.4 6.8 4.4 5.2 7.8

1.972 14.242 3.559 1.154 2.857 3.375 2.360 3.929 7.049 4.048 3.810 2.059 2.045 3.077 2.436

16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

25 32 41 13 0 14 16 17 18 26 14 23 35 42 31

9.8 8.8 7.1 3.3 6.8 4.4 5.6 8 8.9 5.3 3.1 6.2 4.8 13.5 5.9

ui 2.551 3.636 5.775 3.939 0.000 3.182 2.857 2.125 2.022 4.906 4.516 3.710 7.292 3.111 5.254

U Chart of number of nonconformities 16

Sample Count Per Unit

1

14 12 10 8

1

1 1

6

UCL=5.76 _ U=3.46

4 2

LCL=1.16 0

1

1

4

7

10

13

16 Sample

19

22

25

28

Tests performed with unequal sample sizes

Fig. 2.20 u-chart, tile industry numerical example – chart 1. Minitab® Statistical Software

Consequently, by the application of the central limit theorem, Eqs. 2.29 and 2.30 can be applied to the mean value of the random variable x, x, N assuming the generic statistical probability density function when the size n of the generic sample is over a threshold and critical value. From Eqs. 2.29 and 2.30 it is necessary to estimate  and , i. e., quantify O and O . In particular, in the presence of a normal distribution of values x, in order to quantify O it can be useful to use Eq. 2.7 or 2.12. In general, for a generic statistical distribution of the random variable, i. e., the process characteristic x,

there are two different kinds of standard deviations, called “within” and “overall”: the first relates to the within-subgroup variation, while the second relates to the between-subgroup variation. In particular, the “overall” standard deviation is a standard deviation of all the measurements and it is an estimate of the overall process variation, while the “within” standard deviation is a measure of the variations of the items within the same group. In a “in control” process these standard deviation measures are very close to each other. In the following, an in-depth illustration of the statistical models

42

2 Quality Management Systems and Statistical Quality Control

U Chart of nonconformities

Sample Count Per Unit

6 1

UCL=5.199

5 4

_ U=3.044

3 2 1

LCL=0.889

0 1

4

6

8

11

13 15 17 sample ID

21

23

25

27

30

Tests performed with unequal sample sizes

Fig. 2.21 u-chart, tile industry numerical example – chart 2. Minitab® Statistical Software

related to capability analysis is substituted by a few significant numerical examples created with the support of a statistical tool such as Minitab® Statistical Software. For this purpose, it is necessary to introduce the following process capability indexes, specifically designed for normally distributed data, i. e., measurements: USL  LSL ; 6O USL  O ; D 3O O  LSL D 3O  USL  O O  LSL I : D min 3O 3O

Cp D CP U CPL Cpk

(2.31) (2.32) (2.33) (2.34)

When Cp < 1 the process is said to be “noncapable,” otherwise it is “capable” because the quality control variability, represented by 6, can be included by the specification limits LSL and USL, i. e., the production process can meet the customer requirements. The 6 variation is also called “process spread,” while USLLSL is called “specification spread.” A capable process is able to produce products or services that meet specifications. Nevertheless, this index measures the capability only from a potential point of view, because Cp does not tell us if the range of values ˙3 above and below the mean value, called “centerline” in the

control charts, is really included in the specification range, i. e., in other words it does not tell the analyst if the process is centered on the target value. For this purpose, the index Cpk is preferable to Cp because, if we assume values greater than 1, it guarantees the process is centered on the target value, thus telling the analyst what capability the process could achieve if centered, while Cp does not consider the location of the process mean. Finally, the CP U and CPL indexes relate the process spread, the 3 variation, to a single-sided specification spread: -LSL O or USL-, O respectively. A conventionally accepted minimum value for these indexes is 1.33, corresponding to the so-called four sigma production process, as defined in Sect. 2.9. The performance of an in-control process is predictable. Therefore, the capability analysis following the “in-control analysis” can assess the ability of the production process to produce units that are “in spec” and predict the number of parts “out-of-spec.”

2.9.1 Numerical Example, Capability Analysis and Normal Probability Table 2.15 reports the measurements, in millimeters, obtained on 100 products produced by a manufacturing process of cutting metal bars when the expected

2.9 Capability Analysis

43

Table 2.15 Measurement data – process 1, numerical example Sample 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Data – process 1 600.3333 600.2929 599.8586 599.2491 600.4454 599.4055 600.1634 600.3021 600.1666 600.9336 600.3714 599.7379 599.797 600.2411 599.4932 600.6162 599.1419 600.5005 600.7689 599.9661

600.8494 598.789 599.706 599.537 599.9179 599.5074 599.5934 600.3307 599.8434 600.5842 601.2756 601.112 599.9101 599.643 599.6578 599.3922 599.8016 599.3184 599.1993 598.7038

600.693 599.8655 599.8773 599.848 599.5341 599.5099 599.9918 600.6115 600.612 599.7249 599.7404 600.5713 599.1727 599.6155 599.9164 600.6494 600.4682 599.424 599.8779 600.4608

values of the target and specification limits are 600, 601, and 599 mm. Consequently, the tolerances are ˙1 mm. First of all, it is useful to conduct the variability analysis by generating the control chart: Figure 2.22 reports the x-chart based on the s-chart. There are no anomalous behaviors of the sequence of subgroups. It is now possible to quantify the capability indexes and the nonconformity rates by adopting both the overall and the within standard deviations. Figure 2.23 is a report generated by Minitab® Statistical Software for the analysis of the capability of the production process. The Cp value obtained is 0.55, i. e., the process is not potentially capable, both considering the within capability analysis and the overall capability analysis. Figure 2.23 quantifies also the PPM over and under the specifications by Eqs. 2.29 and 2.30, distinguishing: • “Observed performance.” They are related to the observed frequency distribution of data (see the histogram in Fig. 2.23). • “Expected within performance.”2 They relate to the parametric distribution, and in particular to the nor2

Minitab® Statistical Software calls the performance indices Pp and Ppk in the “overall capability” analysis to distinguish them from Cp and Cpk defined by Eqs. 2.31–2.34 for the “within analysis” (see Fig. 2.23).

599.2493 599.3179 600.8859 600.0593 600.3004 599.9597 600.2792 599.0412 600.7174 599.5842 601.0146 600.287 600.8716 600.2896 600.6215 599.6583 599.3786 600.7875 600.7521 599.3556

Mean value

Range

600.6724 599.4127 600.3385 599.2632 598.8681 599.2939 599.41 599.4191 599.9917 599.8445 600.3568 599.922 600.1579 598.6065 599.3805 599.216 600.4624 600.2031 599.9077 601.4034

600.35948 599.5356 600.13326 599.59132 599.81318 599.53528 599.88756 599.94092 600.26622 600.13428 600.55176 600.32604 599.98186 599.67914 599.81388 599.90642 599.85054 600.0467 600.10118 599.97794

1.6001 1.5039 1.1799 0.8102 1.5773 0.6658 0.8692 1.5703 0.874 1.3494 1.5352 1.3741 1.6989 1.6831 1.241 1.4334 1.3263 1.4691 1.5696 2.6996

Average

599.971628

1.40152

mal distribution, obtained by a best-fitting statistical evaluation conducted with the within standard deviation. • “Expected overall performance.” They relate to the parametric distribution obtained by a best-fitting evaluation conducted with the overall standard deviation. In particular, the maximum expected value of PPM is about 96,620. The so-called six-pack capability analysis, illustrated in Fig. 2.24, summarizes the main results presented in Figs. 2.22 and 2.23 and concerning the variability of the process analyzed. The normal probability plot verifies that data are distributed as a normal density function: for this purpose the Anderson–Darling index and the P value are properly quantified. Similarly to the s-chart reported in Fig. 2.22, the R-chart is proposed to support the generation of the x-chart. The standard deviations and capability indexes are hence quantified both in “overall” and “within” hypotheses. Finally, the so-called capability plot illustrates and compares the previously defined process spread and specification spread. The analyst decides to improve the performance of the production process in order to meet the customer specifications and to minimize the process variations.

44

2 Quality Management Systems and Statistical Quality Control

Xbar-S Chart of manufacturing measurements 601.0

Sample Mean

UCL=600.778 600.5 _ _ X=599.972

600.0 599.5

LCL=599.165 599.0 1

3

5

7

9

11 Sample

13

15

17

19

Sample StDev

1.2

UCL=1.181

0.9 _ S=0.565

0.6 0.3

LCL=0

0.0 1

3

5

7

9

11 Sample

13

15

17

19

Fig. 2.22 x-chart and s-chart – process 1, numerical example. Minitab® Statistical Software

LSL

Target

USL

Process Data LSL 599 Target 600 USL 601 Sample Mean 599.972 Sample N 100 StDev(Within) 0.60121 StDev(Overall) 0.603415

Within Overall Potential (Within) Capability Cp 0.55 CPL 0.54 CPU 0.57 Cpk 0.54 Overall Capability Pp PPL PPU Ppk Cpm

0.55 0.54 0.57 0.54 0.55

598.5 599.0 599.5 600.0 600.5 601.0 601.5 Observed Performance PPM < LSL 40000.00 PPM > USL 40000.00 PPM Total 80000.00

Exp. Within Performance PPM < LSL 53034.23 PPM > USL 43586.49 PPM Total 96620.72

Exp. Overall Performance PPM < LSL 53675.43 PPM > USL 44166.87 PPM Total 97842.30

Fig. 2.23 Capability analysis – process 1, numerical example. Minitab® Statistical Software

Table 2.16 reports the process data as a result of the process improvement made for a new set of k D 20 samples with n D 5 measurements each. Figure 2.25 presents the report generated by the six-pack analysis.

It demonstrates that the process is still in statistical control, centered on the target value, 600 mm, and with a Cpk value equal to 3.31. Consequently, the negligible expected number of PPM outside the specification

2.9 Capability Analysis

45

Process Capability Sixpack of DATA Sample Mean

Xbar Chart

Capability Histogram

601

LSL

600

Specifications LSL 599 Target 600 USL 601

LCL=599.165 1

3

5

7

9

11

13

15

17

19

598.5 599.0 599.5 600.0 600.5 601.0 601.5

R Chart Sample Range

USL

_ _ X=599.972

599

Normal Prob Plot

3.0

UCL=2.957

1.5

_ R=1.398

0.0

AD: 0.481, P: 0.228

LCL=0 1

3

5

7

9

11

13

15

17

19

598

600

602

Capability Plot

Last 20 Subgroups Values

Target

UCL=600.778

Within StDev 0.60121 Cp 0.55 Cpk 0.54

601 600

Within

Overall

599

Overall StDev 0.603415 Pp 0.55 Ppk 0.54 Cpm 0.55

Specs 5

10 Sample

15

20

Fig. 2.24 Six-pack analysis – process 1, numerical example. Minitab® Statistical Software

Table 2.16 Measurement data – process 2, numerical example Sample 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18 2.19 2.20

Data – process 2 600.041 599.8219 600.0089 600.1896 600.1819 599.675 600.0521 600.0002 600.02 600.1571 600.0934 599.8668 599.9859 599.9456 600.0487 599.8959 600.1891 600.0002 599.9228 599.7843

600.0938 599.9173 600.075 600.1723 600.0538 599.9778 600.1707 600.0831 599.9963 600.0212 599.9554 599.8757 599.9269 600.0405 600.0569 599.979 600.1168 600.1121 600.092 599.9597

600.1039 600.0308 600.0148 599.8368 599.9957 599.9633 599.9446 599.9298 599.9278 599.9061 599.7975 600.0414 599.8124 600.0576 599.9321 600.1418 600.1106 599.93 599.9225 600.011

600.0911 600.07 599.9714 600.0947 600.0995 599.9895 599.8487 599.9329 599.9793 599.9786 600.0221 599.7939 600.0288 599.7819 599.9164 600.1157 599.9148 599.9924 600.1062 600.0409

Mean value

Range

600.1096 600.0732 600.0271 599.9781 599.9639 599.8853 600.012 599.9142 600.0456 600.0626 599.8821 600.1153 600.0261 600.0603 599.9984 599.9525 600.0013 600.0458 600.1794 600.0436

600.08788 599.98264 600.01944 600.0543 600.05896 599.89818 600.00562 599.97204 599.9938 600.02512 599.9501 599.93862 599.95602 599.97718 599.9905 600.01698 600.06652 600.0161 600.04458 599.9679

0.0686 0.2513 0.1036 0.3528 0.218 0.3145 0.322 0.1689 0.1178 0.251 0.2959 0.3214 0.2164 0.2784 0.1405 0.2459 0.2743 0.1821 0.2569 0.2593

Average

600.001124

0.23198

46

2 Quality Management Systems and Statistical Quality Control

Process Capability Sixpack of DATA2 Sample Mean

Xbar Chart

Capability Histogram UCL=600.1362

LSL

Target

USL

600.1

Specifications LSL 599 Target 600 USL 601

_ _ X=600.0011

600.0 599.9

LCL=599.8660 1

3

5

7

9

11

13

15

17

19

599.1 599.4

599.7

Sample Range

R Chart

600.6

600.9

Normal Prob Plot AD: 0.408, P: 0.340

UCL=0.4952 0.4 _ R=0.2342

0.2

0.0

LCL=0 1

3

5

7

9

11

13

15

17

19

599.6

600.0

Last 20 Subgroups

600.4

Capability Plot

600.2

Values

600.0 600.3

Within StDev 0.100682 Cp 3.31 Cpk 3.31

600.0 599.8

Within

Overall

Overall StDev 0.101659 Pp 3.28 Ppk 3.28 Cpm 3.28

Specs 5

10 Sample

15

20

Fig. 2.25 Six-pack analysis – process 2, numerical example. Minitab® Statistical Software

limits is quantified as ˇ   ˇ USL  O Total PPM D ˇˇP z > O ˇ  LSL  O ˇˇ CP z < ˇ O

2.9.2 Numerical Examples, Capability Analysis and Nonnormal Probability O D0:101659 NN D O xD600:0011

Š 0:

These numerical examples refer to data nondistributed in accordance with a normal density function. Consequently, different parametric statistical functions have to be adopted.

Table 2.17 Measurement data (mm=10), nonnormal distribution. Numerical example Sample 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Measurement data 1.246057 0.432057 3.289106 4.740917 1.03499 4.864409 3.045406 0.936205 4.55721 5.635049 4.693689 1.063906 2.902382 4.24421 1.667182

0.493869 1.573958 4.26632 1.38156 6.639968 1.546174 3.160609 0.940518 1.902965 1.851431 1.903728 0.821599 2.769513 4.099892 0.717635

2.662834 2.361707 3.597959 1.618083 6.071461 3.875799 2.901201 3.15243 4.462141 5.076608 6.866619 1.658612 4.439952 0.813895 1.420329

5.917727 0.178515 1.511217 5.597763 1.552255 1.098431 6.760744 4.550744 3.509317 1.630322 3.064651 5.847757 0.912794 4.460482 2.365193

3.020594 1.945173 3.783617 3.05798 0.151038 5.50208 6.04942 1.732531 1.995514 2.673297 0.565978 4.024718 3.192323 3.007995 2.011729

3.233249 3.891315 0.323979 2.404994 1.659891 1.281942 1.39276 5.629206 4.803485 0.777941 2.093118 3.41589 0.774273 3.84575 4.629

0.890597 2.222251 5.367135 1.409824 3.580737 0.921708 3.495365 0.397718 1.95335 7.998625 5.058873 2.196106 3.936241 3.755018 1.934723

1.107955 3.295799 0.429597 1.266203 6.482635 4.884044 2.494509 6.539783 2.53267 0.864797 4.96973 2.153251 2.605119 3.018857 1.844031

1.732582 2.521666 2.179387 3.864219 2.282011 3.054542 3.865445 4.46137 4.884973 5.338903 4.40998 1.59855 6.360237 2.535924 6.976545

2.963924 2.398454 1.945532 0.735855 3.062937 3.225921 1.390489 2.886115 0.882012 6.03149 1.459153 3.074742 5.220038 3.867536 1.01383

2.9 Capability Analysis

47

Process Capability of data Calculations Based on Weibull Distribution Model Target

USL

Process Data LSL * Target 0 USL 10 Sample Mean 3.09791 Sample N 150 Shape 1.70519 Scale 3.47777

Overall Capability Pp * PPL * PPU 0.93 Ppk 0.93 Exp. Overall Performance PPM < LSL * PPM > USL 2344.33 PPM Total 2344.33

Observed Performance PPM < LSL * PPM > USL 6666.67 PPM Total 6666.67

0.0

1.5

3.0

4.5

6.0

7.5

9.0

Fig. 2.26 Capability analysis – Weibull distribution, numerical example. Minitab® Statistical Software

Process Capability Sixpack - Weibull distribution Capability Histogram

Sample Mean

Xbar Chart 5

UCL=4.926

3

_ _ X=3.098

Specifications USL 10

LCL=1.270

1 1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

0.0

1.5

3.0

4.5

6.0

7.5

9.0

Weibull Prob Plot

S Chart Sample StDev

USL

AD: 0.248, P: > 0.250

UCL=3.217

3

_ S=1.874

2 1

LCL=0.532 1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

0.01

Last 15 Subgroups

1.00

10.00

Capability Plot

10

Values

0.10

Overall Shape 1.70519 Scale 3.47777 Pp * Ppk 0.93

5

Overall

Specs

0 5

10

15

Sample

Fig. 2.27 Six-pack analysis – Weibull distribution, numerical example. Minitab® Statistical Software

2.9.2.1 Weibull Distribution Table 2.17 reports data regarding the output of manufacturing process of tile production in the ceramics industry. This measurement refers to the planarity of the tile surface as the maximum vertical distance of cou-

ples of two generic points on the surface, assuming as the USL a maximum admissible value of 1 mm. Figures 2.26 and 2.27 present the report generated by Minitab® Statistical Software for the capability analysis. The production process generates products, i. e., output, that are “well fitted” by a Weibull statisti-

48

2 Quality Management Systems and Statistical Quality Control Table 2.18 Number of calls and “no answer”, numerical example

cal distribution, shape parameter ˇ D 1:71 and scale parameter  D 3:48. The process is therefore “in statistical control” but it does not meet customer requirements in terms of an admissible USL. In other words, the process is “predictable” but “not capable.” In particular, the number of expected items over the USL is about 6,667 PPM.

2.9.2.2 Binomial Distribution This application deals with a call center. Table 2.18 reports the number of calls received in 1 h, between 3 and 4 p.m., and the number of calls that were not answered by the operators. The measurement data can be modeled by assuming a binomial distribution of values. Figure 2.28 presents the results of the capability analysis conducted on this set of values, called “data set 1.” The process is not in statistical control because sample 15 is over the UCL. As a consequence, it is not correct to quantify the production process capability. This figure nevertheless shows that the process is difficultly capable, also in the absence of sample 15. In order to meet the demand of customers properly it is useful to increase the number of operators in the call center.

Sample No answer Calls

Sample No answer Calls

day 1 day 2 day 3 day 4 day 5 day 6 day 7 day 8 day 9 day 10

day 11 day 12 day 13 day 14 day 15 day 16 day 17 day 18 day 19 day 20

421 392 456 436 446 429 470 455 427 424

1935 1945 1934 1888 1894 1941 1868 1894 1938 1854

“Six Sigma” stands for six standard deviations and can be defined as a business management strategy, originally developed by Motorola, that enjoys widespread application in many sectors of industry and services. Six Sigma was originally developed as a set of practices designed to improve manufacturing processes and eliminate defects. This chapter presents a synthetic recall of the basic purpose of Six Sigma, assuming that a large number of the models and methods illustrated here and in the following can properly

Rate of Defectives

1

26

UCL=0.25684

0.24

% Defective

Proportion

P Chart

_ P=0.22769

0.22 0.20

LCL=0.19853 1

3

5

7

9 11 13 Sample

15

17

24 22 20 1840

19

1920 Sample Size

2000

Tests performed with unequal sample sizes Cumulative % Defective

Histogram Tar 8

Summary Stats

% Defective

22.5 22.0 21.5 21.0 5

10 Sample

15

20

%Defective: Lower CI: Upper CI: Target: PPM Def: Lower CI:

22.77 22.35 23.19 0.20 227686 223487

Upper CI: Process Z: Lower CI: Upper CI:

231926 0.7465 0.7325 0.7605

Frequency

(95,0% confidence)

23.0

1937 1838 2025 1888 1894 1941 1868 1894 1933 1862

2.10 Six Sigma

Binomial Process Capability Analysis of no answer 0.26

410 386 436 424 497 459 433 424 425 441

6 4 2 0

0

4

8 12 16 20 % Defective

Fig. 2.28 Binomial process capability, numerical example. Minitab® Statistical Software

24

2.10 Six Sigma

49

support it. Nevertheless, there are a lot of ad hoc tools and models specifically designed by the theorists and practitioners of this decisional and systematic approach, as properly illustrated in the survey by Black and Hunter (2003). Six Sigma is a standard and represents a measure of variability and repeatability in a production process. In particular, the 6 specifications, also known as Six Sigma capabilities, ask a process variability to be capable of producing a very high proportion of output

Lower specification limit

within specification. The “process spread” has to be included twice in the “specification spread” and centered on the target value. Figure 2.29 presents the results generated by a process capability conducted on an “in control” process in accordance with the Six Sigma philosophy. Configuration c identifies a capable process, as previously defined, whose variability meets the Six Sigma requirements. In other words, in a Six Sigma process there is a number of defects lower than two parts per billion,

Upper specification limit

The process is very capable with 6σ’ 0:75, p 2 Œ0:5; 0:75, p 2 Œ0:25; 0:5, etc. The following categories of risk can be conventionally adopted:

Risk estimation

Risk evaluation

Has the risk been adequately reduced?

specific/ minor risks

Yes

End

No

Risk reduction

Fig. 3.2 Iterative process for reducing risk (ISO 14121-1:2007)

gonomics aspects), availability and suitability of protective measures, information for use, etc.

3.4 Classification of Risks MIL-STD-882 identified four main categories of hazard severity: catastrophic (e. g., generation of death and loss of production), critical (generation of severe injury and major damage to the system), marginal

• Specific risks. This category has small values of magnitude M , assumed as a measure of the outcomes, and high likelihood of occurrence P , as typically for a continuative exposure. These risks are referred to in laws and technical regulations concerning health and safety at work, risk of noise, vibrations, thermal discomfort, etc. • Conventional risks. In comparison with the previous category there are slightly greater values of M and lower values of P . • Great risks, or potentially relevant accidents. In this case we have a very high level of magnitude M regardless of the value of P , e. g., in the case of risk of fire or explosion in a production plant. In Fig. 3.3 all these occurrences are placed on the M –P diagram. Depending on the position in the M –P diagram, the quantification of the risk expressed by the parameter R is carried out in three different ways: 1. Qualitative approach. Both M and P are ranked according to explanatory denominations quite similar to verbal expressions (e. g., high, low; strong,

3.4 Classification of Risks

59 Start

Time Series analysis on-field historical data collection

Check Lists

Hazard Operability HazOp

Hazard identification

Failure Modes & Effects Analysis FMEA

Failure modes and events identification

Failure Modes & Criticality Analysis FMECA

Reliability evaluation models and methods (see Chapter 5, 6, 7 and 8)

Probability of occurance estimation

Risk assessment

Risk analysis

Surveys and Questionnaire for workers

Determination of the limits of the production system (e.g. equipment , machinery)

Severity of harm estimation

Risk estimation

R=PxM

Risk evaluation

Yes Has the risk been adequately reduced ?

End

R=PxM

No

Maintenace strategies, models and methods (see Chapter 9)

Protective measures taken by the designer

Step 2 Safeguarding and complementary protective measures Step 3 Information for use

Protective measures taken by the users

Organization Additional safeguards Use of personal protective equipment Training

Risk

Risk reduction

Step 1 Inherently safe design measures

Level of Risk

Laws, Standards & specs

Residual risk after designer’s protective measures

Residual risk after designer and user’s protective measures

Fig. 3.4 Flowchart of risk assessment in a production system

weak; negligible, catastrophic). Risks are classified in a descending order of criticality, i. e., according to the level of emergency associated with the intervention of the safety manager or employer.

2. Semiqualitative approach. Both M and P are now ranked into categories according to prearranged scales of values (e. g., from 0 to 9). In this case too the safety manager can determine the priority of the intervention according to this scale.

60

3 Safety and Risk Assessment

Table 3.2 Technical Committee, occupational health and safety area Technical Committee

Title

CEN/TC 70 CEN/TC 93 CEN/TC 122 CEN/TC 126 CEN/TC 137 CEN/TC 191 CEN/TC 192 CEN/TC 211 CEN/TC 231

Manual means of fire fighting equipment Ladders Ergonomics Acoustic properties of building products and of buildings Assessment of workplace exposure Fixed firefighting systems Fire service equipment Acoustics Mechanical vibration and shock

Table 3.3 Technical Committee, personal and protective equipment area Technical Committee

Title

CEN/TC 79 CEN/TC 85 CEN/TC 158 CEN/TC 159 CEN/TC 160 CEN/TC 161 CEN/TC 162

Respiratory protective devices Eye protective equipment Head protection Hearing protectors Protection against falls from height including working belts Foot and leg protectors Protective clothing including hand and arm protection and lifejackets

3. Quantitative approach. For M several mathematical models are applied in order to quantify the outcomes of events such as explosion, fire, and leakage of pollutants, while P is evaluated by reliability models and techniques as described in Chaps. 5, 6, and 8.

a part of it, e. g., by adopting more reliable components, or operating on its connections, or modifying the operative conditions, or planning a different exploitation of the system. For this purpose, maintenance plays a fundamental role for the support of planning, execution, and control activities. 2. Protection strategy. It aims to reduce M , mainly by interventions on the system in order to protect any exposed subject and to reduce the outcomes of the event. In the case of individual protection, the employer must provide some protective devices, such as earphones, gloves, shoes, and overalls, capable of protecting the individual operator from specific hazards. Some devices are, of course, capable of reducing M for a group of people, or a community, in the same environment: e. g., acoustic baffles for noise reduction, fire-extinguisher devices3 such as hydrants, fire doors, and every solution to create compartments.4 In exchange, in such a situation it is possible to have some operators deliberately without individual protection. 3. Mixed strategy. A combination of the previous strategies.

It is worth noting that these three approaches are quite different in objectivity, accuracy, and, last but not least, cost. The last one is particularly expensive and time-consuming with regard to applicable results. In general, for the safety manager both qualitative and semiqualitative approaches, essentially by means of a checklist, are likely in the case of conventional or specific risks, while the quantitative approach cannot be rejected in the case of great risks having catastrophic effects on goods, people, and the environment (as the explosion of a nuclear reactor).

3.5 Protective and Preventive Actions According to the previous definition of R as a combination of M and P , three alternative strategies are applicable to reduce the risk: 1. Prevention strategy. It aims to reduce P , mainly by changing the configuration of the system or

3 4

“Active” devices “Passive” devices

3.5 Protective and Preventive Actions

61

Table 3.4 CEN/TC Ergonomics, standards published since 2008. Part 1

Standard

Title

Standard

Title

EN 1005-1:2001

Safety of machinery - Human physical performance - Part 1: Terms and definions

EN ISO 13406-2:2001

EN 1005-2:2003

Safety of machinery - Human physical performance - Part 2: Manual handling of machinery and component parts of machinery

Erg. requirements for work with visual displays based on flat panels - Part 2: Erg. requirements for flat panel displays (ISO 13406-2:2001)

EN ISO 13407:1999

Human-centred design processes for interacve systems (ISO 13407:1999)

EN ISO 13731:2001

Erg.s of the thermal environment Vocabulary and symbols (ISO 13731:2001)

EN ISO 13732-1:2008

Erg.s of the thermal environment Methods for the assessment of human responses to contact with surfaces - Part 1: Hot surfaces (ISO 13732-1:2006)

EN ISO 13732-3:2008

Erg.s of the thermal environment Methods for the assessment of human responses to contact with surfaces - Part 3: Cold surfaces (ISO 13732-3:2005)

EN ISO 14505-2:2006

Erg.s of the thermal environment Evaluaon of thermal environments in vehicles - Part 2: Determinaon of equivalent temperature (ISO 14505-2:2006)

EN ISO 14505-3:2006

Erg.s of the thermal environment Evaluaon of the thermal environment in vehicles - Part 3: Evaluaon of thermal comfort using human subjects (ISO 145053:2006)

EN ISO 14738:2008

Safety of machinery - Anthropometric requirements for the design of workstaons at machinery (ISO 14738:2002, including Cor 1:2003 and Cor 2:2005)

EN ISO 14915-1:2002

Soware Erg.s for mulmedia user interfaces - Part 1: Design principles and framework (ISO 14915-1:2002)

EN 1005-3:2002

Safety of machinery - Human physical performance - Part 3: Recommended force limits for machinery operaon

EN 1005-4:2005

Safety of machinery - Human physical performance - Part 4: Evaluaon of working postures and movements in relaon to machinery

EN 1005-5:2007

Safety of machinery - Human physical performance - Part 5: Risk assessment for repeve handling at high frequency

EN 13861:2002

Safety of machinery - Guidance for the applicaon of Erg.s standards in the design of machinery

EN 13921:2007

Personal protecve equipment - Erg. principles

EN 27243:1993

Hot environments - Esmaon of the heat stress on working man, based on the WBGTindex (wet bulb globe temperature) (ISO 7243:1989)

EN 547-1:1996+A1:2008

Safety of machinery - Human body measurements - Part 1: Principles for determining the dimensions required for openings for whole body access into machinery

EN 981:1996+A1:2008

Safety of machinery - System of auditory and visual danger and informaon signals

EN ISO 10075-1:2000

Erg. principles related to mental work-load Part 1: General terms and definions (ISO 10075:1991)

EN ISO 20685:2005

3-D scanning methodologies for internaonally compable anthropometric databases (ISO 20685:2005)

EN ISO 10075-2:2000

Erg. principles related to mental workload Part 2: Design principles (ISO 100752:1996)

EN ISO 6385:2004

Erg. principles in the design of work systems (ISO 6385:2004)

EN ISO 7250:1997

EN ISO 10075-3:2004

Erg. principles related to mental workload Part 3: Principles and requirements concerning methods for measuring and assessing mental workload (ISO 100753:2004)

Basic human body measurements for technological design (ISO 7250:1996)

EN ISO 7726:2001

Erg.s of the thermal environment Instruments for measuring physical quanes (ISO 7726:1998)

EN ISO 7730:2005

Erg.s of the thermal environment Analycal determinaon and interpretaon of thermal comfort using calculaon of the PMV and PPD indices and local thermal comfort criteria (ISO 7730:2005)

EN ISO 7731:2008

Erg.s - Danger signals for public and work areas - Auditory danger signals (ISO 7731:2003)

EN ISO 7933:2004

Erg.s of the thermal environment Analycal determinaon and interpretaon of heat stress using calculaon of the predicted heat strain (ISO 7933:2004)

EN ISO 8996:2004

Erg.s of the thermal environment Determinaon of metabolic rate (ISO 8996:2004)

EN ISO 9241-110:2006

Erg.s of human-system interacon - Part 110: Dialogue principles (ISO 9241110:2006)

EN ISO 10551:2001

Erg.s of the thermal environment Assessment of the influence of the thermal environment using subjecve judgement scales (ISO 10551:1995)

EN ISO 11064-1:2000

Erg. design of control centres - Part 1: Principles for the design of control centres (ISO 11064-1:2000)

EN ISO 11064-2:2000

Erg. design of control centres - Part 2: Principles for the arrangement of control suites (ISO 11064-2:2000)

EN ISO 11064-3:1999

Erg. design of control centres - Part 3: Control room layout (ISO 11064-3:1999)

EN ISO 11064-3:1999/AC:2002 Erg. design of control centres - Part 3: Control room layout (ISO 110643:1999/Cor.1:2002)

62

3 Safety and Risk Assessment

Table 3.5 CEN/TC Ergonomics, standards published since 2008. Part 2

Standard

Title

Standard

Title

EN 547-2:1996+A1:2008

Safety of machinery - Human body measurements - Part 2: Principles for determining the dimensions required for access openings Safety of machinery - Human body measurements - Part 3: Anthropometric data Safety of machinery - Erg. design principles Part 1: Terminology and general principles

EN ISO 14915-2:2003

Soware Erg.s for mulmedia user interfaces - Part 2: Mulmedia navigaon and control (ISO 14915-2:2003)

EN ISO 14915-3:2002

Soware Erg.s for mulmedia user interfaces - Part 3: Media selecon and combinaon (ISO 14915-3:2002) Erg.s of the thermal environment - Risk assessment strategy for the prevenon of stress or discomfort in thermal working condions (ISO 15265:2004)

EN 547-3:1996+A1:2008

EN 614-1:2006

EN ISO 15265:2004

EN 614-2:2000+A1:2008

Safety of machinery - Erg. design principles Part 2: Interacons between the design of machinery and work tasks

EN ISO 15535:2006

General requirements for establishing anthropometric databases (ISO 15535:2006)

EN 842:1996+A1:2008

Safety of machinery - Visual danger signals General requirements, design and tesng

EN ISO 15536-1:2008

Erg.s - Computer manikins and body templates - Part 1: General requirements (ISO 15536-1:2005)

EN 894-1:1997

Safety of machinery - Erg.s requirements for the design of displays and control actuators - Part 1: General principles for human interacons with displays and control actuators

EN ISO 15536-2:2007

Erg.s - Computer manikins and body templates - Part 2: Verificaon of funcons and validaon of dimensions for computer manikin systems (ISO 15536-2:2007)

EN 894-2:1997

Safety of machinery - Erg.s requirements for the design of displays and control actuators - Part 2: Displays

EN ISO 15537:2004

EN 894-3:2000

Safety of machinery - Erg.s requirements for the design of displays and control actuators - Part 3: Control actuators

EN ISO 15743:2008

Principles for selecng and using test persons for tesng anthropometric aspects of industrial products and designs (ISO 15537:2004) Erg.s of the thermal environment - Cold workplaces - Risk assessment and management (ISO 15743:2008)

EN ISO 11064-4:2004

Erg. design of control centres - Part 4: Layout and dimensions of workstaons (ISO 11064-4:2004)

EN ISO 9241-151:2008

Erg.s of human-system interacon - Part 151: Guidance on World Wide Web user interfaces (ISO 9241-151:2008)

EN ISO 11064-5:2008

Erg. design of control centres - Part 5: Displays and controls (ISO 11064-5:2008)

EN ISO 9241-171:2008

EN ISO 11064-6:2005

Erg. design of control centres - Part 6: Environmental requirements for control centres (ISO 11064-6:2005)

EN ISO 9241-400:2007

Erg.s of human-system interacon - Part 171: Guidance on soware accessibility (ISO 9241-171:2008) Erg.s of human-system interacon - Part 400: Principles and requirements for physical input devices (ISO 9241-400:2007)

EN ISO 11064-7:2006

Erg. design of control centres - Part 7: Principles for the evaluaon of control centres (ISO 11064-7:2006)

EN ISO 9241-410:2008

Erg.s of human-system interacon - Part 410: Design criteria for physical input devices (ISO 9241-410:2008)

EN ISO 11079:2007

Erg.s of the thermal environment Determinaon and interpretaon of cold stress when using required clothing insulaon (IREQ) and local cooling effects (ISO 11079:2007) Erg.s of the thermal environment Principles and applicaon of relevant Internaonal Standards (ISO 11399:1995)

EN ISO 9886:2004

Erg.s - Evaluaon of thermal strain by physiological measurements (ISO 9886:2004)

EN ISO 9920:2007

Erg.s of the thermal environment Esmaon of thermal insulaon and water vapour resistance of a clothing ensemble (ISO 9920:2007)

EN ISO 9921:2003

Erg.s - Assessment of speech communicaon (ISO 9921:2003)

EN ISO 11399:2000

EN ISO 12894:2001

EN ISO 13406-1:1999

Erg.s of the thermal environment - Medical supervision of individuals exposed to extreme hot or cold environments (ISO 12894:2001) Erg. requirements for work with visual display based on flat panels - Part 1: Introducon (ISO 13406-1:1999)

3.7 Standards and Specifications

Every solution adopted for reducing R has its own cost to be evaluated in conjunction with the effectiveness of the technical solutions, and to be compared with the currently available budget. In detail, laws and technical regulations for safety in production systems always suggest performing activities with special attention to the budget and according to the following policies: • • • •

Removal of hazard and risk. Preventive interventions for the reduction of P . Preventive interventions for the community. Individual preventive interventions. These solutions are not too expensive for the employer and can be applied immediately.

3.6 Risk Assessment, Risk Reduction, and Maintenance In conclusion, the most important steps of the procedure for risk assessment are summarized in Fig. 3.4 in the form of a self-explanatory flowchart. In particular, the role of models and methods for reliability evaluation and maintenance is clearly emphasized. On the importance of an integrated approach to health and safety management, risk assessment, and maintenance planning and execution, see the research report by Wintle et al. (2001). This study, commissioned by the Health and Safety Executive, proposes a plant integrity management based on riskbased inspection. This is an integrated approach to risk

63

assessment and maintenance planning, as discussed at the end of Chap. 9. Ad hoc rules for planning inspections to reduce risks of failures and improve safety and health, reduce costs by repair or replacement of deteriorating equipment in the best time and eliminating ineffective inspections.

3.7 Standards and Specifications The sector of interest in safety engineering is called “health and safety” and mainly operates in two different areas: 1. Occupational health and safety. It is linked with a large number of standardization fields such as machinery, pressure equipment, personal protective equipment, transport, and electrotechnical matters. 2. Personal protective equipment. The aim of this area is to meet the health and safety requirements of the directive for personal protective equipment (89/686/EEC). The main issues developed by the technical committee for the first area are reported in Table 3.2. Similarly, Table 3.3 presents the list of the main issues for the second area. Tables 3.4 and 3.5 report the list of standards belonging to the Technical Committee CEN/TC 122 Ergonomics and published since 2008.

4

Introduction to Maintenance in Production Systems

Contents 4.1

Maintenance and Maintenance Management . . . . 65

4.2

The Production Process and the Maintenance Process . . . . . . . . . . . . . . . . . . 66

4.3

Maintenance and Integration . . . . . . . . . . . . . . . . . . 69

4.4

Maintenance Workflow . . . . . . . . . . . . . . . . . . . . . . . 70

4.5

Maintenance Engineering Frameworks . . . . . . . . . 70

4.6

Reliability-Centered Maintenance . . . . . . . . . . . . . . 72

4.7

Total Productive Maintenance . . . . . . . . . . . . . . . . . 4.7.1 Introduction to TPM . . . . . . . . . . . . . . . . . . . . . 4.7.2 The Concept of TPM . . . . . . . . . . . . . . . . . . . . 4.7.3 TPM Operating Instruments . . . . . . . . . . . . . . 4.7.4 From Tradition to TPM: A Difficult Transition

4.8

Maintenance Status Survey . . . . . . . . . . . . . . . . . . . . 80

4.9

Maintenance Outsourcing and Contracts . . . . . . . 83

73 73 74 75 76

“Maintenance is the combination of all technical, administrative and managerial actions during the life cycle of an item intended to retain it in, or restore it to, a state in which it can perform the required function” (EN 13306:2001 Maintenance terminology). This chapter examines the fundamental definitions concerning maintenance, and discusses the maintenance question in product manufacturing companies or service suppliers. Emphasis is placed on integrating maintenance with the other activities of a company (e. g., production, R&D, quality assurance, purchasing). In conclusion, a survey on the status of maintenance in industrial companies and several observations about maintenance outsourcing are discussed.

4.1 Maintenance and Maintenance Management The life cycle of a generic component in a production system is firstly characterized by periods of uptime when the element is working correctly, i. e., in nominal conditions, secondly by periods of time when it is working but not as expected in the conditions, and thirdly by periods when it stops working altogether owing to a breakdown occurring and the subsequent repair work still having to be completed. Figure 4.1 shows this behavior. In general, the item1 (plant, component, system, equipment, etc.) is supposed to be subject to failures, and to a time-dependent process of degradation. The item can also be repaired by a restoration activity. Both failure and repair times are random variables. Nevertheless, there are different types of failures, repairs, and components/systems; in particular, Chap. 5 distinguishes repairable from nonrepairable items. Maintenance is the function that monitors and keeps plant, equipment, and facilities working. It must design, organize, carry out, and check the work to guarantee nominal functioning of the item during working times “T i ” (uptimes) and to minimize stopping intervals (downtimes) caused by breakdowns or by the resulting repairs. Maintenance management, as illustrated by the framework shown in Fig. 4.2, is made up of all activi1 The standard EN 13306:2001 defines the item as any part, component, device, subsystem, functional unit, equipment, or system that can be considered individually.

R. Manzini, A. Regattieri, H. Pham, E. Ferrari, Maintenance for Industrial Systems © Springer 2010

65

66

4 Introduction to Maintenance in Production Systems

0

preventive maintenance, condition-based maintenance, and corrective maintenance as discussed in Chap. 9, where several analytical models and methods are applied and compared. Maintenance planning is the activity of planning maintenance actions, e. g., inspection, replacement, overhaul, and repair, as properly defined in Chap. 9. In particular, maintenance planning schedules interventions over time, and identifies and allocates necessary resources for the implementation of strategies. Obviously, planning is followed by the execution of maintenance actions and also by the control and supervision of the production systems: on-site, i. e., at the location where the item is used, on-line, i. e., during the time that the item is used, remotely, i. e., without physical access to the item, etc. Maintenance strategies and planning can be properly updated on the basis of the feedback data extracted from the item performances. All these activities have to be properly supported by a maintenance support system made up of resources, services, and management.4 The configuration of such a support system depends on many factors, such as the complexity of maintenance tasks, the skill of the personnel, and availability of the facilities, and is therefore a very critical issue in maintenance management.

T1

τ1

T2

τ2

time

Ti : working time in nominal conditions (uptime) τi: failure time or not nominal working time or reparation time

Fig. 4.1 Life cycle of a component in a production system

ties that determine the maintenance objectives, strategies, and responsibilities2 and implement them by: • maintenance planning; • maintenance control and supervision; • improvement of methods in the organization. The objectives assigned for the maintenance activities can include key performance indicators3 such as reliability, availability, mean time to repair, number of failures, and maintenance costs, properly defined in the following chapters. Consequently, some exemplifying objectives are as follows: improve availability, retain health, safety and environmental preservation, and reduce maintenance costs. Four main classes of objectives are distinguished in the literature (Cheunusa et al. 2004): 1. Loss of production, as an indirect cost. A few examples are the minimization of breakdowns, downtime, rework, inventory, spare parts, overtime, and accidents. 2. Maintenance direct cost. Cost reduction by extension of the useful life of the assets. 3. Volume. This class mainly deals with the following objectives: • Improve reliability and availability; • Improve plant performance; • Support new market opportunities. 4. Price by the product quality increase. The first two classes reduce costs, while the remaining two increase revenues. All classes contribute to maximizing the profit. Maintenance strategies are different types of tasks including actions, procedures, resources, and time. These activities have to be carried out in accordance with established time schedules to guarantee maintenance targets. Some examples are represented by 2

See footnote 4. See also the European Standard EN 15341:2007 Maintenance – maintenance key performance indicators.

3

4.2 The Production Process and the Maintenance Process In modern production systems, the product, or the service, and the maintenance requirements are major outputs: that is to say, in parallel with the production process is the maintenance process. Maintenance is a system whose activities are carried out in synergy with those of the production systems. Figure 4.3 shows the relationship between different objectives relating to these processes. Production systems usually convert inputs (raw materials, energy, workload, etc.) into a product that satisfies customer needs. The mainte4

The European Standard CEN/TR 15628:2007 Maintenance – qualification of maintenance personnel classifies three different categories of maintenance personnel: the European Maintenance Technician, the European Maintenance Supervisor, and the European Maintenance Manager. All categories are characterized in terms of competences and responsibilities.

4.2 The Production Process and the Maintenance Process

67

Maintenance support system Planning

Objectives

Execution

Strategies

Services

Management

Control & supervision

Feedback

Resources

Responsabilities

Input

Improvements and revisions

Fig. 4.2 Maintenance management

INPUT

PRODUCTION PROCESS

OUTPUT

FEEDBACK

Fig. 4.3 Production and maintenance processes. (Duffuaa et al. 1999)

PRODUCTION CAPACITY

nance system, as a mix of know-how, labor, and spare parts, together with other resources aims to maintain equipment in a good working order, i. e., able to provide the appropriate level of production capacity. In a maintenance system, feedback control, planning, and organization activities are very critical and strategic issues. The first of these deals with the production system and control of maintenance activity (e. g., workload emission, spare parts management). Consequently, various actions must be taken to control production and maintenance activities and to resolve breakdowns. Moreover, these activities must be planned in advance whenever possible. Clearly the first aim of maintenance action in downtime periods, during an unexpected breakdown, is to put the plant back into working order: the planning phase is skipped and

MAINTENANCE PROCESS

MAINTENANCE NEED

the maintenance work is carried out as soon as possible. This is breakdown/corrective maintenance. In this situation the maintenance work must be completed quickly, or must be postponed until the next stop, simply leaving the system to run till the next scheduled recondition. In this second case, the definitive maintenance work is scheduled in a previously planned stop period. Maintenance activities are so numerous and complex that they require effective management and wellstructured organization. The starting point is the synchronized control of the production system that not only involves monitoring equipment but also maintenance control, planning, and organization, with a lot of subactivities. This is illustrated in Fig. 4.4 and summarized as follows:

68

4 Introduction to Maintenance in Production Systems PLANNING MAINTENANCE PHILOSOPHY MAINTENANCE LOAD FORECASTING MAINTENANCE CAPACITY MAINTENANCE ORGANIZATION

INPUT FACILITIES LABOR EQUIPMENT SPARES MANAGEMENT

ORGANIZATION JOB DESIGN TIME STANDARDS WORK MEASUREMENT PROJECT MANAGEMENT

MAINTENANCE PROCESS

OUTPUT OPERATIONAL MACHINES EQUIPMENT

FEEDBACK CONTROL

MONITORING PLANT CONTROL WORK CONTROL INVENTORY CONTROL COST CONTROL QUALITY CONTROL

Fig. 4.4 Characteristics of the maintenance process. (Duffuaa et al. 1999)

• Plant control. Control of system performance reliability and collection of on-field data for breakdowns and repair processes by the application of sensors or human checks. • Work control. The maintenance workload is influenced by the maintenance strategy adopted and is supported by well-designed control of the workload based on an effective reporting system. • Inventory control. This activity deals with spare parts management and with all the tools and equipment used in maintenance work. • Cost control. Maintenance usually consumes large amounts of money. There are two fundamental cost factors: the direct cost of investment, i. e., investment in production resources (e. g., plant, equipment, employees), and indirect costs caused by lack of production. It is extremely important to have an effective and continuous cost control process. • Quality control. The main aim of quality assurance of a process or a product is to measure several variables representing a range of specifications, as stated by the Six Sigma quality strategies, for example, and policies applied to production/logistic system management and optimization. The check and control process of the production system generates a large amount of useful data for planning the maintenance work. In particular, during the maintenance planning process it is necessary to assume some decisions involving:

• Maintenance philosophy. Several maintenance policies have been developed by practitioners and are discussed in the literature (see Chap. 9). Since no strategy is significantly more effective than the others, this problem usually deals with the identification of the best mix of strategies and policies in order to obtain the best global result (e. g., minimization of production costs). • Maintenance load forecasting and capacity. Maintenance requires the simultaneous use of several resources (e. g., manpower, spare parts, equipment). Consequently, the load forecasting process is essential to obtain the desired level of maintenance system performance. Critical aspects of maintenance capacity include the identification of the optimum number of craftsmen and their skills and the maintenance of the required tools. After the control and planning of maintenance processes has been carried out, the next step is to design the maintenance system correctly. This requires the integration of several aspects: • Job design. A variety of complicated tasks, called “jobs,” are usually required to maintain a production system. Each job must be designed correctly. The most important instrument for job design and management is the maintenance work order (illustrated in detail in Chap. 7): it contains all the details of the work required, e. g., its location, and all the skills and tools required. The work order is the

4.3 Maintenance and Integration

69

main instrument used in monitoring, planning, and reporting all maintenance activities. Moreover, in maintenance job design the evaluation of the duration of a generic activity is an extremely critical issue. To measure and estimate this duration, method time measurement and the Maynard operation sequence technique are two examples of effective decision-supporting tools. • Work measurement. Each maintenance job requires various resources and generates costs. The target of the workload analysis is to evaluate and control these costs. The ultimate aim of the maintenance process is to minimize the total cost of the production system. • Project management. Maintenance activities are frequently part of a general development plan for the production system. Project management techniques are very useful in supporting the maintenance planning activities and effecting maintenance work (Gantt charts, critical path methods, program evaluation review technique, heuristics for project scheduling and sequencing).

Maintenance activities can provide a significant contribution to meeting the set of the productivity targets for a system, as illustrated in Chap. 1. However, maintenance requires a great deal of time, considerable knowledge, and it also consumes a great deal of money. Consequently, the choice of the “best maintenance level” contains to be a hidden trade-off problem. Since performance maximization of the entire production system is the final goal of a maintenance system, the right approach and the most appropriate working instruments depend on the characteristics of the particular real-world instance examined. Before evaluating this trade-off, one needs to understand that companies often run the risk of underestimating the importance of maintenance, thus highlighting how important the introduction of a new managerial and organizational culture taking care of this issue is. Maintenance activities produce good results only if they are integrated with the other corporate functions, and particularly with the following activities (Chuenusa et al. 2004):

In conclusion, the monitoring phase is the starting point of all maintenance activities. In particular, the performance measurement of a production system can be effectively supported by reliability and availability theory and evaluation.

• • • • •

4.3 Maintenance and Integration In addition to performing maintenance work, “maintenance” must have a place in the design activity and in supporting the management decision-making in the company. For example, this applies in spare parts fulfillment and management, knowledge management, and other areas. Maintenance procedures affect different organizational levels in a company, and have several particularly important implications: • Financial. Production plants lock up a great deal of capital, and the related investment must be repaid. • Technological. Process and product (or service) quality are directly related to the state of the plant and production system maintenance. • Economic. Failures and defects reduce profits. • Social and legal. In poor conditions equipment and facilities can produce pollution and cause both accidents and safety problems.

strategic planning design; production planning; workload management; quality assurance and control; material purchasing management and material management; • human resource allocation and management; • administration and cost accounting; • information technology management. In particular, production planning, quality assurance and control, material purchasing and management, and human resource management influence maintenance the most. Production planning. A continuous flow of material guarantees that production systems will perform excellently, but this goal can only be achieved by the perfect integration of maintenance services and production planning: the shared aim is to make sure that the production system is always available. Quality assurance and control. High levels of process and product quality reduce the scrap rate and improve the customer service level. Furthermore, quality products do not need reworking activity or continuous measurement of the production processes. In conclusion, quality outcomes are the result of an effective integration of both maintenance and quality functions of

70

production systems that make products and/or supply services. Material purchasing. Equipment availability and continuous operability of the production system strongly depend on the availability of spare parts. As a result, the spare parts forecasting question is very critical in production system management and optimization (see Chap. 11). Firstly, maintenance must define the specifications of the spare parts required for functioning of the production system, then the purchasing department of the company must buy the spare parts under the best financial terms and conditions available, and finally maintenance personnel must check and either accept or reject incoming deliveries of materials. Human resources. Great care must be taken in appointing maintenance personnel since human resource skills and knowledge play a fundamental role in developing an effective maintenance division and in minimizing production costs. Two fundamental activities to apply the most appropriate maintenance policies are data collection and management. Consequently, the link between maintenance and the information technology system is one of the most important targets of a production system.

4.4 Maintenance Workflow The maintenance of a production system is strongly related to a set of activities and procedures to cope with for an effective management. The European standard EN 13460 proposes the maintenance workflow with its main activities and documents as reported in Fig. 4.5. The maintenance planning and execution system (Fig. 4.2) is supported by a maintenance information system, properly illustrated in Chap. 7. The main areas of information systems require the following information modules: • Work list and inventory, containing all technical and functional data of parts, components, plants, and resources in general. Also data regarding methods, costs, and times are collected and managed in this area. • Maintenance planning, dealing with frequency, procedure, and technical specifications of each item. • Scheduling and resource management. • Requests of interventions.

4 Introduction to Maintenance in Production Systems

• Work orders, i. e., authorizations and instructions for intervention. • Spare parts monitoring and management. • Cost reporting and controlling. • Inspection record and periodic inspections. • Reliability evaluation tools. The workflow presented is strongly based on a series of tools, approaches, and methodologies (e. g., reliability theory, maintenance policy models, spare parts management) that are properly discussed in the next chapters. For example, the control and supervision phase requires a continuous calculation of key performance indexes for a robust analysis of the status and above all the design of optimizing policies such as preventive interventions, inspections, or the optimal management of spare parts. The planning and scheduling and the execution phases are devoted to applying these policies in practice.

4.5 Maintenance Engineering Frameworks The previously introduced workflow is an output of the evolution of the maintenance concept since the end of the Second World War. As failure is a not eliminable occurrence, the first maintenance activity developed, called “breakdown or reactive maintenance,” was clearly devoted to the restoring of equipment. From the 1950s plant managers were encouraged to develop programs to prevent damage, according to the new trend of “preventive maintenance.” Although it helped to reduce the downtime, it was an expensive alternative. Parts were replaced on a time basis, while they could have lasted longer; a lot of unnecessary man-hours were also spent, thus resulting in an excess of activities, resulting in an increase of total costs in many cases. The problem is still therefore the determination of the optimal level of preventive activities. The monitoring of the real conditions of equipment can permit a calibration of the deadlines for preventive interventions: this is the main strategy of the “on condition monitoring policy,” introduced in the 1990s as the natural evolution of the preventive one. Currently, many companies are still coping with this evolution, from the extensive use of corrective maintenance to the introduction of significant preventive and on condition activities. The optimal mix of policies is strongly

4.5 Maintenance Engineering Frameworks

71

Planning & scheduling Input documents

Execution

Procedures, technical data, operational manual, component list, etc.

Work order, tools, spare parts, manpower resources, operational procedures

Control & supervision

History records, control procedures, laws and regulations

Preparatory phase: item conception, designing, manufacturing, assembly, etc.

Carry out the work

Work planning Production of reports

Closure of the work order

Work scheduling Reports analysis

Work order release & assignment

Output documents

Work order, planning sheet, scheduling sheet,etc.

History records, KPI

Reports, feedback documents and improment proposal

Fig. 4.5 The maintenance workflow. KPI key performance index

dependent on the real application, but several studies in the literature stated the “20-40-40” rule: 20% of corrective strategy and 40% both for preventive and on condition strategies. Anyway, the relevance of the maintenance question requires a systematic approach and a wide perspective involving not only the best mix of maintenance policies but every factor that has an impact on the global cost of a production system. For example, the human contribution, the maintenance information system, and the spare parts management are several important features to be managed in order to achieve excellence. During the past few years several conceptual frameworks for maintenance modeling and management following this “total approach” were developed. In particular, some fundamentals about reliability-centered

maintenance (RCM) and total productive maintenance (TPM) are briefly discussed. RCM is a systematic engineering process to determine what to do in order to ensure that the physical assets continue to behave as users wish. In other words, RCM supports the definition of a complete maintenance regime. The main tools and models traditionally related to the RCM approach are illustrated in the following chapters in this book. They regard maintenance as the way to maintain the functions of the machinery a user may require in a defined operating context. It enables the machinery stakeholders to monitor, assess, predict, and generally know how their physical assets work. TPM, firstly a Japanese idea, is a proactive and systematic engineering approach that essentially aims to prevent any kind of slack before occurrence. It em-

72

phasizes the importance of people, a “can do” and “continuous improvement” philosophy, and the importance of production and maintenance staff working together. The following sections discuss the main topics, problems, models, and methods dealing with maintenance in general. The authors do not have a preferred philosophy or a preferred approach to maintenance. The models, methods, numerical examples, and applications can support the manager or the practitioner of modern production systems to implement the “approach of the moment” when he/she knows the main pillars which define it. That is why we pick RAMS engineering, whose “reliability,” “availability,” “maintainability,” and “safety” are the basic keywords describing the content of this book, especially if we think of quality as part of them (see Chap. 1).

4.6 Reliability-Centered Maintenance The RCM process is known as a “reliability by design” based approach and is reliability-centered because its programs aim to achieve the inherent safety and reliability capabilities of a piece of equipment at a minimum cost. The fundamental goal of RCM is to give the equipment the opportunity to reach the maximum level of reliability that is consistent with the safety, environmental, operational, and profit goals of the organization. This is allowed by addressing the basic causes of system failures and ensuring that there are organizational activities designed to prevent them, predict them, or mitigate the business impact of the functional failures associated with them. The RCM approach is based on several basic steps for each asset: 1. Identification of the expected functions of the equipment to be used. Every facility is designed and built to produce some desired outputs. To achieve this goal the equipment operates some functions, usually grouped in two categories. The main, or primary, functions, e. g., velocity, quality, and safety, are necessary for the correct operation of the equipment, and therefore are strictly related to the reason why the asset has been installed. The second category includes the support functions expressing desirable conditions. The loss of these functions usually does not compromise the output,

4 Introduction to Maintenance in Production Systems

e. g., comfort, effectiveness, and noise, but only the way to get it. 2. Identification of the components of the system with their related failure modes. It is important to note that for the RCM approach any unsatisfactory condition is equivalent to a failure. By this definition it is possible to fix the concept of a failed but still working piece of equipment. Many programs for condition monitoring do not achieve their desired output because the people involved in the program often do not identify a failure as soon as an unsatisfactory condition has been detected. 3. Failure causes analysis. Identification and classification of faults and failures. The goal of this step is the determination of the causes for each functional failure. The cause may be the failure of a piece of equipment or a part of it, or sometimes a failure in some human activity as well. Improper operation and improper maintenance are likely to be the causes of failures. An effective tool to develop this analysis is fault tree analysis, discussed in Chap. 8. 4. Failure effects and consequences analysis. Failure effects analysis is a step-by-step approach devoted to studying the consequences of each failure. When a failure occurs, many different things resulting in different impacts on the equipment, hence on the company business, can happen. Every company fixes its targets for profitability, safety performance, environmental performance, and operational performance. Each failure has a different impact on the business performance, and for the RCM team it is important to evaluate the corresponding consequences, from lack of or minor effects to the total collapse of the business or, in extreme cases, the loss of lives. Failure modes and effects analysis and failure mode, effects, and criticality analysis, as discussed in Chap. 8, are two very interesting tools for an easy approach to this step. After this first phase devoted to “knowledge,” RCM provides some actions, divided into two categories too, dealing with failures. In particular: 5. Proactive tasks, i. e., preventive and/or predictive, i. e., on condition, maintenance tasks. Especially in the case of relevant consequences, something must be done to prevent or predict the failures, or at least to reduce their impact. The proactive tasks are practically the aforementioned preventive and

4.7 Total Productive Maintenance

on condition maintenance policies. It can be stated that the RCM framework, scheduling restoration, discard, and on-condition tasks, is based on the same fundamental concept expressed in Sect. 4.5. Scheduled restoration involves the remanufacturing of a component or an assembly at or before a specified age limit, regardless of its condition at that time. Similarly, scheduled discard implies rejecting an item at or before a specified life limit, regardless of its condition at that time. According to the on condition tasks, items keep providing their service since they meet the desired performance standards. An action is inspired by a requirement, whose evaluation can result in a big deal. Most failures provide warnings about their imminent occurrence. These warnings, or potential failures, are defined as recognizable physical conditions suggesting that a functional failure is about to occur or is in progress. The analysis of these warnings and the correlation with the probability of failure is still a current and significant problem. Chapter 9 deals with the techniques supporting the optimization of proactive tasks. 6. Default actions, i. e., failure-finding, redesign, and run-to-failure, when it is not possible to identify a proactive task. The appropriate default action can be decided according to the consequence of the failure. If there are no proactive tasks capable of reducing the operational consequences, the first default decision can be considered as “do nothing,” i. e., running until the failure occurs for successive corrective interventions. If the restoration cost is too high, a redesign might of course be required. If proactive tasks to improve safety or to reduce environmental risks to an acceptable level cannot be found, the equipment must be redesigned or the process/system where it is employed must be modified. In conclusion, the RCM method provides the last step principally devoted to the monitoring of implementation. 7. Implement and refine the maintenance plan. The RCM approach requires continuous monitoring of its procedure. The maintenance plan must be constantly reviewed taking into account how pieces of equipment evolve and react. The RCM maintenance plan properly requires a cross-functional team constituted of maintenance, operations, and

73

engineering personnel having a thorough understanding of the asset and a clear identification of the risks and profits of the company. Several models and methods useful for implementation of the previously mentioned decision steps are discussed in the following chapters. In particular, Chap. 8 introduces failure modes and effects analysis and failure mode, effects, and criticality analysis techniques for the identification of failure events and the criticality analysis, while some analytical planning models for preventive maintenance actions and inspections are discussed and applied in Chap. 9.

4.7 Total Productive Maintenance A few sections of this book are devoted to this conceptual maintenance framework, currently a reference for a lot of companies. TPM is a people-centered methodology, generally considered as a critical add-on to the “lean manufacturing” production philosophy.

4.7.1 Introduction to TPM The importance of the maintenance function has increased because it has a fundamental role in keeping and increasing the availability, product quality, safety requirements, and plant cost-effectiveness levels. Maintenance costs constitute an important part of the operating budget of manufacturing firms. During the 1960s the concept of TPM was developed in Japan in response to this problem. TPM is a manufacturing program designed primarily to maximize the effectiveness of equipment throughout its entire life by the participation and motivation of the entire workforce (Nakajima 1988). This approach provides a synergistic relationship among all the company’s functions, but particularly between production and maintenance, for continuous improvement of product quality, operational efficiency, capacity assurance, and safety. According to this vision, the word “total” in TPM may assume three meanings:

74

1. TPM pursues the total effectiveness such as economic efficiency and profitability. 2. TPM provides a total maintenance approach mainly including corrective, preventive, and on condition policies and other techniques. 3. TPM needs the total participation of all employees and involves every level and function in the organization, from the top executive to the production operator on the floor. There is a lot of documentation about the benefits arising from the adoption of TPM. Many papers, such as Koelsch (1993), Ferrari et al. (2001, 2002), Eti et al. (2004), Chan et. al (2005), and Gosavi (2006), told of similar success stories of companies that reduced breakdown labor rates, setup times, and production losses very significantly by TPM, thus avoiding costs per maintenance unit. TPM implementation presents several opportunities but also some threats, as discussed in the following sections together with some operative suggestions.

4.7.2 The Concept of TPM TPM is an evolution of the “preventive maintenance approach.” In the early 1960s in some Japanese companies (e. g., the famous Nippondenso) maintenance became a problem as soon as the demand for personnel dedicated to maintenance increased. The management decided to assign the routine maintenance of equipment directly to the operators, thus creating one of the pillars of TPM: the concept of autonomous maintenance. The maintenance personnel took up only important or difficult maintenance interventions, and at the same time suggested some solutions to improve reliability. This approach was completed over the years. At the moment, TPM is universally defined as a productive maintenance technique that is made up of a set of activities to be performed by every operator in order to get zero defects. From a general point, the main targets of TPM are: • maximum efficiency of the plant; • an accurate definition of the plan for preventive maintenance; • a diffusion of relevance of maintenance; • diffusion of workers’ participation, at any level; • organization of small groups of people for enhanced management of problems.

4 Introduction to Maintenance in Production Systems

TPM is based on several fundamental steps, generally called “pillars of TPM,” hereafter discussed briefly. (i) Deletion of causes of losses in productivity. Usually six fundamental causes are expected: 1. Time losses due to: (a) Breakdowns: failures of components require corrective interventions or restoration activities with eventual utilization of spare parts. (b) Setup activities: setting up means a series of operations such as attachment, adjustment, trial processing, readjustment, measurement, production, and finally the ability to produce excellent products. A large amount of time is spent in product-change adjustments until the production of the new item is completely satisfactory. 2. Speed losses due to: (a) Micro-stops: minor and idling stops, usually very short and difficult to trace, when production is interrupted by a temporary malfunction or when a machine is idling. (b) Speed reduction from nominal value: This is due to a misalignment between expected and actual speed or, less frequently, to inadequate technological standards. Sometimes the speed is reduced because of quality or mechanical problems, but there are also cases where the standard speed is not used because it will shorten the service life of the equipment. 3. Defects due to: (a) Equipment starting: some start-up phases (e. g., after periodic repairs, long-time stoppage, holidays, or lunch breaks) may have problems resulting in loss of time, production volume, and costs. (b) Quality defects: volume losses due to defects and reworks, and time losses arising from the time required to repair defective products to turn them into excellent products. (ii) Creation of a program of autonomous maintenance (AM) (maintenance by workers). Operators perform simple maintenance tasks, while more value added activities and technical repairs are performed by skilled maintenance people. Operators

4.7 Total Productive Maintenance

are responsible for upkeep of their equipment to prevent it from degradation. (iii) Plans of preventive and on condition maintenance for maintenance division (on staff position). The maintenance personnel plays a new role in performing only the nonconventional interventions and, above all, in developing activities, e. g., preventive activities, on condition monitoring systems, and plant design modifications, to increase the equipment reliability and safety. (iv) Advance in workers’ capability to provide maintenance. Training plays a crucial role in TPM application. It aims to have multiskilled and wellmotivated people eager to come to work and perform all the required functions effectively and independently. The goal is to create a factory full of experts. Education is continuously provided to operators and maintenance workers, in order to upgrade their skill. Employees should be trained to achieve the four phases of the educational process: do not know, know the theory but cannot do, can do but cannot teach, can do and also teach. (v) Plant/equipment management system. Equipment must be managed considering several aspects: the phase in and the warm-up phase, the normal operating time, and the phase out. Spare parts, design modifications, and continuous improvement are to be pursued with determination. Production and maintenance departments are engaged to develop policies and systematic approaches to achieve these targets. In conclusion, the core of the TPM approach deals with the new role of operators and maintenance workers. Operators and maintenance personnel must reach mutual understanding and share responsibility for equipment. A cooperative effort is required: operators develop the routine maintenance activities, and in particular the following:

75

The maintenance personnel is instead focused on tasks mostly requiring technical expertise and more sophisticated techniques for advanced manufacturing. In particular: • providing technical support for the AM activities; • restoring deterioration thoroughly and accurately, using inspections, condition monitoring, and overhaul; • clarifying operating standards by tracing design weaknesses and making appropriate improvements; • enhancing maintenance skills for checkups, condition monitoring, inspections, and overhaul. TPM introduces a vision significantly different from that of the preventive maintenance approach. The goal of TPM is the improvement of production efficiency to its maximum extent. Its purpose is to maximize the efficiency of production systems in an overall manner, also involving the human factor. In contrast, the preventive maintenance approach is centered on equipment, the target is the maximum efficiency. The preventive maintenance approach considers the fundamental role of the maintenance department and its activities, whereas TPM consists of small-group activities where all members, usually including managers, participate and work jointly on a self-discipline basis.

4.7.3 TPM Operating Instruments In addition to the well-known reliability theory, based on reliability, maintainability, and availability, TPM introduces a rather extended vision of a new synthetic indicator of analysis called “overall equipment effectiveness” (OEE), taking into account availability, quality, and performance efficiency. In particular, OEE D availability  production efficiency  rate of quality D A  PE  RQ;

• maintaining basic equipment conditions (cleaning, lubrication, bolting); • maintaining operating conditions (proper operation and visual inspection); • discovering deterioration, mainly by visual inspection and early identification of signs of abnormalities during operation; • enhancing skills such as equipment operation, setup, and adjustment, as well as visual inspection.

where uptime ; uptime C downtime theoretical cycle time ; PE D actual cycle time total products  defectives RQ D : total products AD

76

Any improvement process requires the measurement of performance. The choice of the appropriate metrics is a relevant purpose. OEE is a combination of operation maintenance, equipment management, and available resources expressing the “global” approach of TPM best. The goal of TPM is to maximize equipment effectiveness and the OEE is used as a measure of this parameter. Factors affecting OEE are not equally important in every situation and different weights should be assigned according to the specific application, as stated by several authors (Dal et al. 2000; Ferrari et al. 2001). The fine-tuning process of OEE can vary across different business sectors and industries. Generally speaking, a world-class OEE is 0.80–0.85, roughly multiplying an availability rate of about 0.92–0.94, a production efficiency rate of about 0.90–0.92, and a quality rate of about 0.98–0.99. By this new parameter the contributions of the most relevant causes of production losses, in terms of time losses, speed losses, and defects, can be seen: that is why OEE appears as a profitable instrument for TPM implementation.

4.7.4 From Tradition to TPM: A Difficult Transition The new vision introduced as TPM, with its concepts such as autonomous maintenance and instruments such as OEE, is certainly a big opportunity for a global consideration of maintenance but, at the same time, it has some threats. In spite of the continuous improvement observed over recent years, the tradition is still strong and therefore there is not a great disposition for those techniques that directly involve the workers. The principal difficulties are encountered in the area of the organizational change involving people. A cultural shortage can spread the misunderstanding that the TPM method requires production employees to work more, thus reducing the number of maintenance people. However, there are no binding elements for TPM application, but a tenable method for its gradual and smooth application must be found. The proposition of an implementation methodology for TPM, firstly as a new philosophy and successively as a new operational system, is extremely important.

4 Introduction to Maintenance in Production Systems

4.7.4.1 The Proposed Method Workers from any level in the factory have to be gradually but constantly involved in the implementation of TPM, basically made of five main steps: 1. Knowledge diffusion and creation of a structure for project management. For good application of TPM, “top-down” involvement is fundamental, especially in order to get the required change in mentality. For this reason it is necessary to carry on the training and education, both by theoretical sessions and practical simulation, before the on-field implementation. It is furthermore necessary to create a unit dedicated to TPM in order to pursue design activities and development control of the project. 2. Pilot line choice. The TPM technique represents a set of general prescriptions but it could require big changes and adaptations, especially in the western world. The selection of a pilot plant, or a line, to test the TPM approach with and to bring about some adjustments could be the right move for maximum limitation of problems and for better “calibration” of the system to the real situation. 3. Analysis of the de facto situation. At the starting phase and before continuing the TPM application, it is absolutely necessary to recover both technical and economic information, related to the performance parameters and to the costs of the maintenance system respectively, about the pilot line. In this phase it is useful to apply the reliability theory (i. e., mean time before failure, mean time to repair, and failure rate  – see Chap. 5) and the synthetic parameter OEE. 4. Criticality determination and proposition for improvements. The analysis of the starting situation allows one to underline criticalities, suggesting some possible improvements and solutions for the next steps. Obviously, the management procedures must be “lined up” with TPM feeling and consequently must be based on autonomous maintenance, small groups, and increase of workers’ competence. In this phase it is very important to keep the personnel continuously informed about the developing status, e. g., by explanation panels. 5. Economic evaluation of proposed developments and extension of analysis. Generally, the previous steps lead to some modifications, both technical and managerial, each of them to be valued by a cost– benefit balance before the application in practice.

4.7 Total Productive Maintenance

The real application of this method to the pilot line requires a warm-up period but after the following transitory period the methodology can be extended to other lines or plants of the factory. The proposed method is applied to an important company, a world leader in its business sectors, with very encouraging results, as presented in the following case study.

77

-

Targets definition Intervention priority choice Advance status check

TPM

-

OEE analysis Big losses analysis Solutions elaboration

Work

-

Applicative solutions

Project

4.7.4.2 Alfa Spa Case History The proposed procedure has been applied in the factory of a world leader, Alfa, in the manufacturing of plants for the metallurgical sector. Before the TPM project, Alfa approached maintenance in a conventional way based on a corrective system with some agreements linked to productive maintenance. The most significant points of the general procedure can be briefly traced as follows: Knowledge diffusion and creation of a structure for project management. For the right application and a consistent result of the project, it appears very important to spread the knowledge and the participation among workers, at any level in the factory. That is why the prime activity consisted in training and educational courses, with different levels and targets, and theoretical lessons about TPM targets and methods, fundamentally for top managers, and “operative” lessons and workshops for direct workers were both organized. After this alignment of knowledge, the creation of a structure for TPM management is important. In the case of Alfa, this organization is made up of three levels and three different teams; in particular: 1. Project team, with: • • • • •

plant director (team leader); workshop manager; manufacturing manager; maintenance manager; quality director.

2. TPM team, with: • • • • •

manufacturing manager (team leader); maintenance manager; workshop delegate; manufacturing delegate; quality control delegate.

3. Work team, with workers and maintenance people, and past members of the TPM team.

Fig. 4.6 Total productive maintenance team responsibilities. OEE overall equipment effectiveness

The corresponding responsibilities for each team are briefly reported in Fig. 4.6. Choice of the pilot line. A key factor for TPM success is the gradual application of the project. The implementation must start from a pilot line, from which it is possible to evidence the specific problems and specialties and, as a consequence, to adjust the TPM concepts and methods ahead of a global application. A boring unit made up of four machines, briefly from mac_1 to mac_4, a very capital intensive device with very big problems concerning maintenance, is the pilot line for the Alfa case. Analysis of the de facto situation. A deep analysis of the real situation is an inalienable starting point. It is very important to trace the situation of maintenance activities from both technical and economic aspects. In Alfa maintenance, especially for the pilot line, was centered on corrective and preventive policies performed by a maintenance division, eventually integrated with external suppliers. Figures 4.7 and 4.8 report for each machine the time per year dedicated to maintenance activities divided into internal and external interventions. For example, in 2007 mac_1 required 876 h for maintenance activities, of these 68.6% in corrective interventions with a significant contribution by external suppliers (40.0% of the number of hours). In parallel, some typical parameters for reliability evaluation are extracted from the maintenance database under the hypothesis of constant failure rates (Table 4.1). In TPM the OEE index enables one to express some different managerial aspects of the plant simultaneously. Still from the maintenance database of the factory, whose relevance is discussed in Chap. 7, the OEE

78

4 Introduction to Maintenance in Production Systems

Fig. 4.9 OEE performance – mac_2 Fig. 4.7 Distribution of maintenance activities (preventive– corrective)

Fig. 4.8 external)

Distribution of maintenance activities (internal–

values are calculated weekly. Figure 4.9 shows an extraction of the OEE index for mac_2 in the period from 1 February 2008 to 8 April 2008. This OEE index can be partitioned into its elements, such as availability, production efficiency, and rate of quality (Fig. 4.10). In particular, Fig. 4.11 aims to focus the setup and start-up times for mac_2 in the same period. Figure 4.12 shows a report concerning the different maintenance policies applied to mac_2. Criticality determination and proposition for improvements. The OEE parameter with its factors enables one to focus on the most significant causes of production losses. In particular for Alfa, for fundamental causes are underlined: setups, maintenance in-

terventions, management problems, i. e., absence of workers and shortage of materials, and technical problems, such as nonconformity of tool and materials. These criticalities assume a different relevance for mac_2: as reported in Fig. 4.13, setups and maintenance interventions represent the major important causes of production losses. Some remedial activities must follow the previous analysis in order for us to delete or to reduce constraints and distortions. The fundamental principles are automaintenance, small group activities, and participation of workers, but more in detail the proposed solution is as follows: a different management of setup activities, some modifications of the plant for the reduction of the failure rate, a total revision of preventive and predictive maintenance planning, and a remanagement of the staff of the maintenance division. It is very important to make all the workforce aware of the current situation. An informative panel, placed in the middle of the pilot line, reporting the OEE trend together with criticalities detected, proposed solutions, and final goals is very useful for the diffusion of knowledge. Economic evaluation of proposed developments and extension of the analysis. Before the application of the solutions picked out in the previous steps an economic survey is absolutely prescribed. Each solution has to be subjected to a cost–benefit estimation for a payback period analysis of investment. For example, the evaluation of the economic impact of a new procedure for the work cycle and tool management (June 2008 euro–dollar exchange rate) is briefly reported:

Table 4.1 Reliability parameters for 2007

mac_1 mac_2 mac_3 mac_4

MTTF (days)

MTTR (h)

 (days1 )

5.35 3.07 5.92 4.51

7.45 4.76 6.34 9.34

0.19 0.33 0.17 0.22

MTTF mean time to failure, MTTR mean time to repair

– – – –

Starting investment US$ 62,750; Annual investment US$ 3,750; Annual savings US$ 67,300; Payback period around 11 months.

The investment is mainly concentrated on personnel training and, for a minor fraction, on equipment use-

4.7 Total Productive Maintenance

Fig. 4.10 OEE factors – mac_2

Fig. 4.11 Setup and start-up activities (in hours) – mac_2

Fig. 4.12 Maintenance activities (in hours) – mac_2

79

80

4 Introduction to Maintenance in Production Systems 0

0.4

0.55

0.75 0.85

1.00

World class Excellence Proacve Emerging Reacve

Fig. 4.14 Assessment score

Fig. 4.13 Impact of mac_2 criticalities (period from 1 February 2008 to 8 April 2008)

ful to facilitate the operators in their automaintenance activity. The annual investment includes training owing to personnel turnover and spare parts for the TPM equipment. Savings are fundamentally due to the increase in production time, hence in revenue, and in product quality, i. e., defect reduction. The job satisfaction concerning a TPM project is very strictly related to the direct participation of workers, and that is why it is very important to plan a good and serious educational program at any level in the factory. Moreover, as previously stated, TPM aims at a gradual improvement by small, but continuous, steps: Alfa decided to extend the TPM system to all the other production lines.

4.8 Maintenance Status Survey Several studies devoted to the assessment of maintenance organization and strategies implemented by companies around the world have been reported in the literature. Smith (2003) developed a benchmark study of more than 170 assessments over a broad spectrum of plant and facility types. The study investigated the situation of maintenance in the companies in three different areas: the organization of maintenance, maintenance process support, and finally the support in the operative procedures, including maintenance engineering techniques and work planning and control. Each factor was evaluated according to an assessment scale from 0.00 to 1.00, as reported in Fig. 4.14. Tables 4.2–4.4 summarize the results. With reference to the first area “organization”, the diffusion of the maintenance principles and the level

of the target clarification were further singled out by the author. The presence of a master plan, with its own budget controlled by the management, related to the maintenance question, was another important feature investigated. Figures 4.15–4.17 summarize the results of the analysis: companies have insight into the importance of maintenance in a sufficient way but often face this question without a formal master plan and a systematic approach. As discussed in Chap. 1, an effective maintenance process has to be supported with scheduling and supervision of the designed subprocesses. Training of personnel, dedicated software, and, in general, information technology are important resources. Smith states a significant use of information technology, e. g., CMMS discussed in Chap. 7. Moreover, the training of personnel is sufficiently implemented, whereas scheduling and the required coordination of support are insufficient. This is further evidence of the organizational deficiency usually found in companies facing the maintenance question. Table 4.6 and Fig. 4.19 report as a whole how companies evaluate their preventive maintenance system by themselves. The last group of factors explored by Smith is the implementation of procedures, techniques, and methods for the application of the maintenance principles. On average, the situation is not positive. All the factors have a score in the reactive zone, and in particular work measurement and work planning are very critical. An interesting paper by the maintenance provider Corrigo (2007) included 142 assessments in companies from different sectors. The survey focused on the application of the preventive maintenance solutions and related factors. The inadequate situation is above all due to poor reporting after the interventions and consequently to information supporting the preventive

4.8 Maintenance Status Survey

81

Table 4.2 Assessment score: maintenance organization Scores

Governing principles

Objective clarification

Master plan

Budgetary control

Management control

lowest average highest median

0.000 0.468 0.925 0.475

0.040 0.388 0.880 0.360

0.000 0.279 0.960 0.160

0.100 0.526 1.000 0.500

0.033 0.471 0.900 0.433

Table 4.3 Assessment score: maintenance process support Scores

Training

supervision

Scheduling and coordination

Computer support

lowest average highest median

0.000 0.468 0.925 0.475

0.040 0.388 0.880 0.360

0.000 0.279 0.960 0.160

0.100 0.526 1.000 0.500

Fig. 4.15 Survey results: maintenance organization

Fig. 4.16 Survey results: maintenance process support

maintenance scheduling. Table 4.5 and Fig. 4.18 indicate that the preventive maintenance activities are usually scheduled and documented with significant support from automated system, but at the same time

interventions appear to be found mainly on an experience basis, with a very poor contribution from historical and reliability data not properly traced and stored in the database.

82

4 Introduction to Maintenance in Production Systems

Table 4.4 Assessment score: maintenance procedures Scores lowest average highest median

Maintenance engineering Prev/pred maintenance Work planning Work measurement Material support and control 0.1 0.4 0.9 0.4

0.1 0.5 0.9 0.5

0.0 0.4 0.9 0.3

0.0 0.3 0.7 0.2

0.1 0.6 0.9 0.6

Table 4.5 Preventive maintenance factors benchmark

Maintenance tasks scheduled and documented PM scheduling supported by an automated system Asset condition and history available before PM scheduling Full reporting of PM tasks executed

yes

no

59% 53% 34% 43%

41% 47% 66% 57%

PM preventive maintenance

Fig. 4.17 Survey results: maintenance procedures

Fig. 4.18 Preventive maintenance factors benchmark. PM preventive maintenance

Most of the companies had a lack of perception about preventive tasks, and only for 9% of the sample was the preventive policy optimal.

These surveys are clearly restricted to limitations in the sample size, industrial sectors, and geographical areas, but anyway a significant conclusion can

4.9 Maintenance Outsourcing and Contracts

83

Table 4.6 Overall self-rating of preventive maintenance situation Optimal

Sufficient

In place but insufficient

Non existent

9%

36%

36%

19%

Global evaluation of PM management system

Fig. 4.19 Overall self-rating of preventive maintenance

be drawn: maintenance practitioners apply good practices, although without full comprehension of the corresponding benefits. The importance of the maintenance management facility in manufacturing systems is increasing rapidly as many organizations aim to become world-class companies. Companies must respond to global competitive pressure by seeking to increase their productivity also by pursing an effective and efficient maintenance program. The crucial involvement of the management is fundamental to give guidance and direction to the maintenance function.

4.9 Maintenance Outsourcing and Contracts In the past few years many companies opted to outsource their “noncore” business activities, thus creating a discussion about what is “core” and what is “noncore.” This is a highly subjective process, often ending when a personal opinion has the upper hand over another personal opinion. For companies such as several service suppliers, e. g., airlines, railways, and amusement parks, maintenance is a primary business area, but in general, and above all for manufacturers, maintenance can be considered a noncore business aspect. In spite of this, the outsourcing of maintenance activities has strongly increased in the last few years. This is not a trivial choice, first of all in fixing what has to be outsourced. The maintenance management

process discussed in the following chapter involves, in general, three macro-activities: data collection, analysis and application of maintenance engineering techniques, and the execution of interventions. Companies often prefer to outsource the executive phase, while developing the remaining steps in-house. This is typical, e. g., when the external contractors support the in-house workforce during work-intensive periods, or during major shutdowns or overhauls. This can be considered as a minimalist approach. As an alternative, companies can outsource the planning in addition to the executive phase. In this case, only for preventive and on condition tasks of course, the external contractor decides how and when, but the outsourcing organization retains control over what is to be done. The global approach is to outsource all the activities. In this instance, every part of the agreement must be structured around the achievement of desired outcomes in terms of equipment performance. In other words, companies “buy” the performance reliability levels. In every situation there are advantages and disadvantages, and the most appropriate approach will depend on the particular case. Manufacturers using external maintenance providers can reduce the cost of the maintenance division, or at least they turn fixed costs into variable costs. The providers offer their services to many clients at a very convenient price, thus exploiting the scale effect, and the clients can find more competences in the external personnel than in their own operators, with better performance as a consequence. In conclusion, an effective provider can raise the technical performance of the equipment, paying continuous attention to costs, usually with a slight reduction. The rating process of the provider is a very complex task, because only few actors are well skilled and organized to provide a systematic and effective contribution. This remark is less significant when only the executive phase is outsourced, but in contrast is absolutely fundamental when manufacturers assign all their maintenance to an external provider.

84

Another limiting factor for maintenance outsourcing deals with the competences: to externalize completely the maintenance activities means to lose every related technical and organizational competence in a short time. This can result in some difficulties in the relations with the provider, or mainly in recovering this competence in the future. The challenge in maintenance outsourcing is that manufacturers and providers, also referred to as “contractors,” are independent and usually make decisions based on their own economic interests. Without coordination, their policies may not be compatible or may not lead to optimal system performance. An effective maintenance contract represents an instrument to ensure that manufacturers and contractors have the common target of system efficiency, in terms of performance and costs. The recent European standard EN 13269:2006 presents a useful guideline for the preparation of the maintenance contract. In particular, on the side of the contractor the standards are: • supplying the resources of personnel, material, and equipment; • Preparing a work program and carry out the work; • providing the management required to control the program and the workforce at every stage; • submitting claims for payment; • management of possible contract changes. On the side of the company the standard actions are: • budgeting and validation of the maintenance contractor’s claims for payment; • agreeing with any variation to the contract; • quality assurance requirement and overall management; • verifying that the maintenance performed complies with the requirement of the contract. This book can properly support the reader also in acquiring the basic knowledge for preparing a contract. The third approach mentioned at the beginning of this section, usually called “maintenance global service,” requires a very accurate definition of the cooperation between contractor and client. They have contrasting attitudes: providers are usually involved in limiting their costs and manufacturers are more concerned with the uptimes of the equipment. Anyway, success comes only when strong partnering arrange-

4 Introduction to Maintenance in Production Systems

contract fee

bonus area

0

technical performance

malus area 0 Fig. 4.20 Bonus–malus concept in maintenance contract

ments and cooperative relationships between contractor and client exist. Experimental evidence has demonstrated that an incentive-based contract improves the maintenance operations: usually a mix of equipment uptime, or availability, target levels, and a bonus–malus percentage on the extra profit eventually generated are fixed. Figure 4.20 shows a typical bonus–malus solution: when the providers generate the targeted technical performance for the equipment, the contracted fee is paid. An extra fee is paid in the case of better performance, a penalty is due in case of worse results. Through some recent diffusion of maintenance outsourcing, the outsourcing organization has to address many critical issues in the transition to the new arrangements. Among these are matters such as: • The personnel. Which will be retained by the organization, which will be employed by the contractor, which will be let go? • The drawings. Who is responsible for ensuring that drawings are kept up to date, who will be the custodian of site drawings? • The computer systems. Will the contractor have access to the client’s computerized maintenance management system (see Chap. 7)? Will the contractor maintain its own computerized maintenance records? Who is responsible for ensuring that all the data in the computerized maintenance management systems are accurate? • Materials management (spare parts and tools). Will the contractor provide his own materials, or will the client provide these? Another critical issue to be addressed before the contract is concluded, is how to manage the rescission of

4.9 Maintenance Outsourcing and Contracts

an existing contract. In particular, an agreement has to be reached regarding the duties and obligations of the outgoing contractor in handing over to the incoming contractor (or the client organization, should it decide to bring maintenance back in-house).

85

In conclusion, it is not worth taking the decision to outsource the maintenance activity with a light heart. The potential advantages are very significant and interesting, but a careful consideration of all major issues is vital for a good final result.

5

Basic Statistics and Introduction to Reliability

Contents 5.1

Introduction to Reliability . . . . . . . . . . . . . . . . . . . . 88

5.2

Components and Systems in Reliability . . . . . . . . . 88

5.3

Basic Statistics in Reliability Engineering . . . . . . . 89

5.4

Time to Failure and Time to Repair . . . . . . . . . . . . 90

5.5

Probability Distribution Function . . . . . . . . . . . . . . 90

5.6

Repairable and Nonrepairable Systems . . . . . . . . . 91

5.7

The Reliability Function – R(t) . . . . . . . . . . . . . . . . . 91

5.8

Hazard Rate Function . . . . . . . . . . . . . . . . . . . . . . . . 92 5.8.1 Hazard Rate Profiles . . . . . . . . . . . . . . . . . . . . . 94 5.8.2 Mean Time to Failure . . . . . . . . . . . . . . . . . . . . 95

5.9

Stochastic Repair Process . . . . . . . . . . . . . . . . . . . . . 95

5.10 Parametric Probability Density Functions . . . . . . . 97 5.10.1 Constant Failure Rate Model: The Exponential Distribution . . . . . . . . . . . . . 97 5.10.2 Exponential Distribution. Numerical example 99 5.10.3 The Normal and Lognormal Distributions . . . 103 5.10.4 Normal and Lognormal Distributions. Numerical example . . . . . . . . . . . . . . . . . . . . . 106 5.10.5 The Weibull Distribution . . . . . . . . . . . . . . . . . 110 5.10.6 Weibull Distribution. Numerical Example . . . 112 5.11 Repairable Components/Systems: The Renewal Process and Availability A(t) . . . . . . 113 5.12 Applications and Case Studies . . . . . . . . . . . . . . . . . 117 5.12.1 Application 1 – Nonrepairable Components . 117 5.12.2 Application 2 – Repairable System . . . . . . . . 122

Billions of dollars are currently spent producing high-technology products and services in a variety of production systems operating in different manufacturing and service sectors (e. g., aviation, automotive industry, software development, banks and financial

companies, health care). Most of these products are very complex and sophisticated owing to the number of functions and components (many systems are made of millions of parts). A good example is the largest passenger airliner in the world, the Airbus A380, also known as the “Superjumbo,” with an operating range of approximately 15;200 km, sufficient to fly directly from New York City to Hong Kong. The generic part of this very complex product can be characterized by life cycle and failure behavior, but also by repair behavior in case of failure detection, and in the presence/absence of a maintenance strategy, e. g., based on replacement and/or inspection or preventive action. Moreover, the failure and repair behavior of the generic part of the system can be directly or indirectly associated with thousands of different safety implications and/or quality expectations and performance measurements, which simultaneously deal with passengers, buildings, environment, and communities of people. In particular, reliability can be defined as the probability that a component (or system) will perform a required function for a given period of time when used under specific operating conditions. Another important basic definition is that of availability, which is the probability that a component (system) is performing its required function at a given point in time when used under specific operating conditions. Finally, maintainability is the probability that a failed component (system) will be restored (or repaired) to a specified condition within a period of time when maintenance is carried out in accordance with prescribed procedures. These definitions mean that the improvement, measurement, and control of software reliability and avail-

R. Manzini, A. Regattieri, H. Pham, E. Ferrari, Maintenance for Industrial Systems © Springer 2010

87

88

ability to support the operability of production systems are very important issues. In fact, most system outages and machine crashes are generated by malfunction of the software management system. The aim of this chapter is to introduce the reader to the definition, measurement, management, and control of the main reliability parameters that form the bases for modeling and evaluating activities in complex production systems.

5.1 Introduction to Reliability Reliability has become a very frequently used term during the last 10 years, not only used by engineers and practitioners but also by shop and superstore assistants who justify the price and performance of a product by stressing quality, reliability, warranty, and customer service if failures occur, etc. In particular, this term is implicit in the thought processes of modern society, from the housewife choosing a model of washing machine to the engineers who design the product and guarantee its performance. In doing this, engineers also consider the implications of the warranty and repair costs, a significant proportion of which is composed of the spare parts management costs (i. e., fulfillment, inventory management, replacement, etc.). As briefly introduced in Chap. 3, the importance of measuring reliability is closely related to risk determination and control: the generic risk event is related to the quantification of a probability, i. e., the reliability, and simultaneously the magnitude of the consequences. The importance of reliability also finds justification in the continuous quality control and improvement of the products/services, process, and production systems, and safety requirements and expectations: the more complex the product is, the larger the number of laws and regulations the product must comply with. For example, the previously mentioned Airbus A380 must meet an extremely large number of standards and obtain certification, mainly from the Federal Aviation Administration in the USA and the European Aviation Safety Agency. Reliability, quality, safety, warranty, etc. are very important keywords often used without respecting the original and correct meaning. Consequently, the main aim of this book is to provide the reader with the abil-

5 Basic Statistics and Introduction to Reliability

ity to marry correct notation with a set of definitions, appropriately supported by a set of effective decisionmaking methods and models. The identification of a universal notation used by most users, producers, designers, and practitioners would represent a revolution in customer and consumer expectations of products and services, guaranteeing benefits for all actors in the supply chain. When expectations are clearly defined, ambitious, and also shared by a group of people, all advantages can be shared with costs consequently reduced, and the performance of the production system simultaneously improved. Reliability management can be considered the fuel and energy of the most pure, natural, and valued face of competition providing significant incentives for self-improvement. This chapter explains reliability evaluation and management, which are then discussed in more detail in Chaps. 6–8. It introduces the basic statistical definitions, measurements, and models. It is organized as follows. Section 5.2 discusses the difference between the concept of components and systems in reliability engineering. Sections 5.3–5.10 present the fundamentals of the statistical inference and estimation with particular emphasis on the standard probability distribution functions and stochastic process evaluation. In particular, Sect. 5.10 presents several parametric statistical distributions and numerical examples. Section 5.11 introduces availability for repairable components. Finally, Sect. 5.12 presents two significant applications in which the basic reliability parameters are determined using the models and methods illustrated in this chapter.

5.2 Components and Systems in Reliability The aim of reliability theory is to study the failure behavior of components, such as parts of a production system, and the failure behavior of complex systems in order to guarantee that they function correctly during a period when they are in operation. In general, the production system analyzed is made of more than one part, which is in turn composed of several components that perform various functions. From the point of view of reliability, a component is a generic entity (e. g., a tool, a machine, an item of equipment, a part of the equipment) whose failure behavior (and eventu-

5.3 Basic Statistics in Reliability Engineering

ally repair behavior) is known and can be modeled accurately by evaluating a pool of statistical parameters. These are generally time-based and evaluated by ad hoc investigation of failure and repair events in different operating conditions (reliability evaluation models are properly illustrated and applied in Chap. 6). The system is an entity composed of more than one component, whose failure behavior can be evaluated using knowledge of the failure and repair behavior of its basic components. In other words, reliability evaluation of a system can be based on an analysis of the behavior of its components and their logical and physical connections. This analysis is supported by the effective models and methods presented in Chap. 8. In particular, the approach to the evaluation proposed in Chap. 8 attempts to bypass direct quantification of the system’s statistical parameters by implementing an ad hoc investigation that is very expensive in terms of time and money. In fact, the so-called ad hoc investigation is sometimes a destructive task requiring simultaneous analysis of a large and statistically significant number of equal entities (i. e., systems) operating under common conditions. In conclusion, a reliability system is an entity whose failure and/or repair behaviors are not known and whose complexity usually requires one to adopt effective models to support production system reliability evaluation to be based on the basic reliability and maintainability parameters of the components in the system. Finally, a part of a production system is a component when its reliability parameterization is well known, but it is a system when a reliability evaluation and prediction analysis has to be conducted with its components’ basic failure and repair behaviors and parameters.

89

follows:

nA ; (5.1) n where nA is the number of occurrences (chances) of event A in a period of time T and n is the number of occurrences (chances) in T . In other words, event A is a set of outcomes (a subset) to which a probability p.A/ is assigned. The following equations represent two main properties of random events: p.A/ D

N D 1; p.A/ C p.A/

(5.2)

p.;/ D 0;

(5.3)

where AN is the negation of event A and ; is an event without outcomes, i. e., a set without elements. In particular, the failure event is a random occurrence characterized by a probability function that measures the chance of the event occurring in accordance with a specific set of operating conditions. Similarly, repair activity can be modeled by a probability function measurement of the occurrence of the random repair process. A random process, sometimes called a “stochastic process,” is the counterpart in probability theory to a deterministic process and deterministic system. Reliability theory mainly refers to stochastic processes and to the basic statistics briefly introduced and discussed in the current section and in the following chapters to demonstrate the proposed and applied reliability and maintenance analytical models, which are the subject of this book. The conditional probability is the probability of an event A occurring given the occurrence of another event B, as follows: p.A=B/ D

5.3 Basic Statistics in Reliability Engineering In terms of reliability engineering, a failure or a repair can be described as a random event. A random event A can be characterized by the probability of the event occurring. The probability p.A/ is the likelihood or chance that A is either the case or will happen in the future. It is represented by a real number ranging from 0 to 1. p.A/ generally refers to a period of time T as

p.A \ B/ ; p.B/

(5.4)

where A \ B is the intersection of events A and B. Consequently, p.A \ B/ D p.A=B/  p.B/:

(5.5)

A and B are statistically independent in the case where p.A=B/ D p.A/; (5.6) p.A \ B/ D p.A/  p.B/:

90

5 Basic Statistics and Introduction to Reliability

Considering three statistically independent events, Y p.i /: p.A \ B \ C / D p.A/  p.B/  p.C / D i DA;B;C

(5.7) Two events are mutually (or statistically) exclusive in the case of p.A \ B/ D 0; A \ B D ;:

(5.8)

Another useful property in probability analysis and reliability evaluation is the probability of the union of events: p.A [ B/ D p.A/ C p.B/  p.A \ B/;

(5.9)

where A [ B is the union of events A and B. Now considering three independent events A, B, and C , p.A [ B [ C / D p.A/ C p.B/ C p.C /

The underlying general hypothesis is that the generic component is subject to time cycles composed of a functioning period followed by a nonfunctioning period. These periods are separated by the stochastic failure event.

5.5 Probability Distribution Function These random events can be related to probability distributions that describe the values and the probabilities of these events occurring. The values must cover all possible outcomes of the event, while the total amount of the probabilities must sum to 1 exactly. The probability density function represents a probability distribution in terms of an integral. In particular, a probability distribution has density f , where f is a nonnegative integrable function R ! R, so the probability of the interval [a, b] is given by Zb

 p.A/  p.B/  p.A/  p.C /  p.B/  p.C /

P .a  X  b/ D

(5.12)

a

C p.A/  p.B/  p.C /: (5.10) In the case where the events are mutually exclusive, [  X p Ai D p.Ai /; (5.11) i

f .x/ dx

i

where Ai is a generic random event.

for any two numbers a and b, where X is a generic random variable (e. g., ttf and ttr). The following is a very important property common to every probability density function and all random variables (i. e., probability distributions): Z1 f .x/ dx D 1:

(5.13)

1

5.4 Time to Failure and Time to Repair Failure of a product or component (system) is a stochastic process. Consequently, the so-called time to failure (ttf1 ), i. e., the time between the starting instant of time (the functioning starting time) of a component (system) and the failure instant of time, is a random variable often attributed to the “useful life.” The value of this variable is closely related to the component (system) operating conditions. The variable of time between failure occurring and the component (system) being returned to service is another random variable known as time to repair (ttr2 ). 1 2

Sometimes abbreviated as TTF Sometimes abbreviated as TTR

The definition of the cumulative distribution function F .y/ is Zy F .y/ D P .X  y/ D

f .x/ dx:

(5.14)

1

A probability distribution has a density function if and only if its cumulative distribution function is absolutecontinuous. In this case F is differentiable almost everywhere, and its derivative can be used as a probability density: dF .x/ : (5.15) f .x/ D dx A probability distribution is called “continuous” if its cumulative distribution function is a continuous

5.7 The Reliability Function – R(t)

91

Functioning period

Functioning period

Up Repair instant of time

Down Failure event

t

Fig. 5.1 Component (system) subject to failure and repair events

function. If the distribution of variable X is continuous, then X is called a “continuous random variable,” where pŒX D a D 0; (5.16) where a is a real number. A probability distribution is called discrete if it is characterized by a probability mass function, which is a function that provides the probability that a discrete random variable is exactly equal to a value. Thus, the distribution of a random variable X is discrete, and X is then called a discrete random variable if X p.X D u/ D 1; (5.17) u

where u is a feasible generic value of X . The distributions of discrete random variables do not have a density function.

5.6 Repairable and Nonrepairable Systems Reliability theory distinguishes nonrepairable from repairable entities (i. e., systems or components). When a failure occurs, an entity is nonrepairable if it is not possible to bring it back into service (i. e., function), which is to say its ttr is infinite. When a failure occurs, a component is repairable if it can be made to function again, as illustrated in Fig. 5.1. Nonrepairable equipment is a special class of repairable entities with infinite ttr. Different models are used to evaluate the reliability of repairable and nonrepairable systems. In particular, the reliability R.T /,

defined as the ability of a system or component to perform its required functions under stated conditions for a specified period of time T , is a probability function appropriate for nonrepairable entities. The equivalent quantity defined for repairable components or systems is the availability A.t/, which is a measure of the degree to which an item of equipment is operable in a generic instant of time t. In other words, the availability is the probability that the system is operating at a specified time t. Sections 5.7 and 5.8 examine the basic models and properties of nonrepairable components and systems, while the stochastic repair process is introduced in Sect. 5.9. The diagram in Fig. 5.2 illustrates a simplified failed nonrepairable component/system (the repair activity is forbidden). This is the two-state diagram of a nonrepairable component/system. The hypotheses adopted to model and manage this class of production system are that: • There are only two states for the generic component/system: “in order” (state 0) and “out of order” (state 1). Consequently, no “gray” conditions of functioning exist, i. e., different configurations of the system which differ from the “white” state 0 (the system is functioning perfectly) and the “black” state 1 (the system is not working at all). • The transition from state 0 to state 1 is instantaneous.

5.7 The Reliability Function – R(t) The ttf of a production component or system is generally a random variable due to several factors, most

92

5 Basic Statistics and Introduction to Reliability

of the period of time T :

Failure

R.T / D 1  F .T /

Out of order

In order

ZT f .x/ dx D 1 

D1 Fig. 5.2 Two-state diagram of a nonrepairable component/system

of which are not controllable. In the case of a continuous ttf and in the presence of a probability density function representing the distribution of the random values, identifying the parametric and statistical functions (e. g., exponential, lognormal, normal, logistic, loglogistic) which best fit the values could be useful. Equation 5.14 is the cumulative distribution of the random variable x, where f .x/ is the probability density function. This function is also known as the not conditional failure rate, i. e., a measurement of the failure rate assuming the component (system) is functioning at the instant of time t0 D 0. Formally, f .t/ is defined as f .t/ dt D P .t  ttf  t C dt/: (5.18) Equation 5.18 can also be directly obtained from Eq. 5.15. Equation 5.14 defines the so-called cumulative function of a generic random variable. This function is called the “failure probability function” in the case of a ttf random variable and is defined by a component (or a system) working under stated operating conditions through a related period of time T , called “mission time.” This period of time is the time horizon during which the component/system’s probabilistic failure behavior is quantified. Also called “survival function,” reliability can be defined as the probability that a component (or system) will perform a required function for a given period of time T (i. e., over a period of time) if used under stated operating conditions. It is formally defined as Z1 R.T / D P .ttf  T / D

f .x/ dx;

(5.19)

T

where f .t/ is the probability density function of the ttf random variable and T is the mission time. In other words, it measures the probability that the component/system will not fail before the conclusion

ZT

1

f .x/ dx;

(5.20)

0

where F .T / is the failure probability function and ttf is the failure random variable which belongs to the range Œ0; C1/. The reliability function of a component/system usually refers to t (i. e., the independent variable) as a generic instant of time that clearly identifies the mission time as (5.21) T D t  t0 assuming the component/system is functioning at the starting operating time t0 , generally equal to 0.

5.8 Hazard Rate Function The failure rate or hazard rate function (t) is an instantaneous rate of failure, and as a conditional probability referring to a point in time t is defined as follows: .t/t D P .t  ttf  t C t ncomponent-system functioning in t/ D P .t  ttf  t C t=ttf  t/:

(5.22)

Figure 5.3 illustrates the difference between the reliability function and the hazard rate in relation to t and T D t  t0 . What is the difference between f .t/ and (t)? As a “nonconditional failure rate,” f .t/ refers to the component/system being in function at point t0 D 0, and is a measurement of failure velocity. As a “conditional failure rate,” (t) differs from f .t/ because it refers to the functioning of the component/system at point t and is another failure velocity, assuming that the component/system is functioning in t. Equation 5.22 can be rewritten as follows: .t/t D P .t  ttf  t C tnttf  t/ D

R.t/  R.t C t/ : R.t/

(5.23)

5.8 Hazard Rate Function

93

λ(t) T

t + Δt τ

t0= 0

Fig. 5.3 Reliability R.T /, failure rate (t ), point in time t , and time mission T

t R(T) • Nh .t/ is the number of “healthy” components at time point t.

From Eq. 5.23, R.t/  R.t C t/ : R.t/t

.t/ D

(5.24)

By these assumptions, Nh .t/ D N  Nf .t/;   Nf .t/ D 1: lim t !1 N

In more detail, R.t/  R.t C t/ t !0 R.t/t f .t/ 1 dR.t/ D : D R.t/ dt R.t/

.t/ D lim

(5.25)

R.t Z /

.t/ dt D 0

  dR.t/ :  R.t/

(5.26)

R.0/D1

0 R.t/ D exp @

Zt

1 .x/ dx A ;

F .t/ D 1  exp @

Zt

Nh .t/ N  Nf .t/ D N N

(5.31)

and N  Nh .t/ Nf .t/ D D 1  R.t/: N N

(5.32)

 Nf .t C t/  Nf .t/ t !0 Nt   NF .t C t/  NF .t/ D lim t !0 Nt dF .t/ dR.t/ D D : (5.33) dt dt 

(5.27)

0

0

R.t/ D

F .t/ D

Then,

(5.30)

The expressions of the reliability and probability function are, respectively,

Consequently, a hazard function can be written as Zt

(5.29)

1 .x/ dx A ;

(5.28)

0

which are, respectively, the general expression of the reliability function and the probability distribution function defined for the period of time T D t  0. Now a simplified model3 of the reliability function based on the following assumptions is introduced: • N is number of identical and nonrepairable components start operating in t0 D 0, i. e., assuming the components are functioning (i. e., state 0, “up” in Fig. 5.1); • Nf .t/ is the number of “failed” components at time point t;

f .t/ D lim

Equation 5.33 is a well known property in statistics but assumes a special value in reliability theory because it links the R.t/ to the density function, f .t/, of the ttf random variable. Similarly,   Nf .t C t/  Nf .t/ .t/ D lim t !0 Nh .t/t   f .t/ NF .t C t/  NF .t/ D lim D t !0 NR.t/t R.t/ dR.t/ 1 D ; dt R.t/ (5.34)

3

Reliability evaluation models based on statistics are properly illustrated in Chap. 6.

which is identical to the previously given Eq. 5.25.

94

5 Basic Statistics and Introduction to Reliability λ(t)

λ(τ)

t

Fig. 5.5 Linear bathtub curve

t Fig. 5.4 Bathtub curve of the hazard rate function

5.8.1 Hazard Rate Profiles Figure 5.4 presents the well-known bathtub-curve hazard rate. It is a parametric rate function that identifies the failure behaviors of components/systems subject to a running-in period and a stress/strain period, as is typical, e. g., in parts production for mechanical applications. In particular, Fig. 5.4 reveals three different periods during the life cycle of a generic component/system: 1. Running in period (also called “run-in” or “infant mortality”). During the period of time the hazard function generally decreases while the operating time is running. 2. Service life period (also called “design life”). This is the lifetime expected, or the acceptable period of time in use. The hazard function is sometimes assumed to be constant during this period of time. 3. Subject to wear period (also called “wear out”). Degradation of the component/system accelerates, consequently the probability of failure occurring increases. The analytical model of a parametric and linear bathtub curve is introduced to model the random failure behavior of a production component/system as follows: 8 c0 ˆ 0t  c0  c1 t C ; ˆ ˆ c1 < c0 (5.35) .t/ D ; < t  t0 ˆ c1 ˆ ˆ : c2 .t  t0 / C ; t0 < t:

From Eq. 5.35 the expression of the reliability R.t/ is    8 c0 t2 ˆ ˆexp  .c0 C /t  c1 ; 0t  ˆ ˆ 2 c1 ˆ ˆ ˆ ˆ    ˆ 2 ˆ c0 ˆ 0:

(5.51)

The cumulative function and the mean function are quantified, respectively, as follows: Zx F .x/ D

f .x/ dx D 1  ex ;

(5.52)

1

4.8 4.4 4 3.6 3.2 2.8 λ = 0.25 λ= 1 λ= 5

5.10 Parametric Probability Density Functions

f(x)

2.4 2

1.6 1.2 0.8 0.4 0

0

0.5

1

1.5

2

2.5

3

x

Fig. 5.8 Exponential distribution, density function M.x/ D f4I 1I 0:2g 1 0.9 0.8 0.7 λ = 0.25 λ= 1 λ= 5

0.6

F(x)

This section presents a set of probability density functions presented in the literature that are used to determine the probability of failure and repair events occurring. These are parametric functions based on a small number of parameters whose values unequivocally identify a probability function and the stochastic behavior of a random event. There are several effective statistical methods of identifying the best parameterization of a generic density function in order to model a stochastic process. Some of these evaluating models and methods are presented and applied in the next chapter, and are supported by several commercial tools developed for both statistical and reliability evaluation.

0.5 0.4 0.3

5.10.1 Constant Failure Rate Model: The Exponential Distribution The models discussed in this section are based on the so-called exponential probability distribution. In par-

0.2 0.1 0

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

x

Fig. 5.9 Exponential distribution, cumulated function

5

98

5 Basic Statistics and Introduction to Reliability

Table 5.2 Stochastic failure process. Main definitions and properties of nonrepairable components Hazard function (t ) x and t are the random variable ttf t >0

Constant hazard rate function .t / D 

R.t / C F .t / D 1 R.0/ D 1

F .0/ D 0

R.1/ D 0

F .1/ D 1

f .t / D

dF .t / dt

f .t / dt D F .t C dt /  F .t / Zt F .t / D

f .x/ dx 0

R.t / D

Z1 f .x/ dx t

f .t / f .t / D .1  F .t // R.t / 1 0 Zt f .t / D .t / exp @ .x/ dx A

.t / D

0

0 F .t / D 1  exp @ 0 R.t / D exp @

Zt

f .t / D  et

1 .x/ dx A

0

Zt

.t / D 

F .t / D 1  et

1

.x/ dx A

R.t / D et

0

Z1 Z1 MTTF D xf .x/ dx D R.t / dt 0

MTTF D

1 

0

ttf time to failure

C1 Z 1 M.x/ D Œxf .x/ dx D : 

(5.53)

1

From Eq. 5.53 the mean value, i. e., the expected value, is constant. Consequently, in the case of a random failure process and an exponential distribution of values, the MTTF is constant and equal to the inverse of the constant hazard function. Figure 5.8 illustrates the trend of the exponential density function f .x/ for different values of constant hazard rate . Similarly, Fig. 5.9 shows the trend of the cumulative function F .x/.

When the distribution of failures is exponential, the following equations are obtained for reliability R.t/, failure probability F .t/, and nonconditional failure rate f .t/: R.t/ D et ; F .t/ D 1  e f .t/ D

t

(5.54) ;

(5.55)

dF .t/ D  et : dt

(5.56)

Similarly, for a random repair process in which ttr is exponentially distributed, maintainability G.t/ and not

5.10 Parametric Probability Density Functions

99

Table 5.3 Stochastic repair process. Main definitions and properties of repairable components Constant repair rate function .t / D 

Repair rate function (t ) x and t are the random variable ttr t >0 G.0/ D 0 G.1/ D 1 dG.t / dt g.t / dt D G.t C dt /  G.t / g.t / D

Zt G.t / D

g.x/ dx 0

.t / D

g.t / Œ1  G.t / 0

.t / D 

g.t / D .t / exp @ 0 G.t / D 1  exp @

Zt 0

Zt

1 .x/ dx A

g.t / D  et

1 .x/ dx A

G.t / D 1  et

0

Z1 MTTR D xg.x/ dx

MTTR D

1 

0

ttr time to repair Table 5.4 Time to failure (ttf) in minutes of an electronic component 12,571.02 16,566.82 18,433.96 18,741.88 11,35.786 19,025.89 19,556.63 22,477.93 27,838.93 32,185.33

52,492.86 53,197.55 56,094.05 56,539.05 32,290.36 56,788.96 56,878.74 57,106.58 57,541.64 58,470.93

76,739.5 77,284.16 77,656.09 82,304.53 63,034.87 82,733.7 83,145.33 83,336.68 92,298.63 97,400.47

141,107.7 142,527.9 145,527.7 148,483.6 97,443.35 150,747.2 151,409.6 152,489 154,131.8 155,809.6

221,538.8 246,367.7 257,147.7 257,335.3 158,096.7 278,000.5 279,977 285,308.8 290,657 295,666.3

conditional repair rate g.t/ are defined as G.t/ D 1  et ; g.t/ D

dG.t/ D  et : dt

(5.57)

2,321.06 6,340.624 7,007.418 10,591.91 10,743.09 11,695.93 7,201.37 7,433.18 8,352.128 9,512.557

36,523.39 36,727.35 38,415.69 48,893.78 49,081.61 51,812.46 41,429.79 42,878.09 44,267.55 44,415.77

64,559.04 65,590.31 67,692.19 73,302.27 74,263.19 76,394.68 68,527.89 69,292.39 69,720.86 71,725.63

97,914.57 101,450.9 104,813.9 134,817.2 134,993.7 138,521.1 106,475.2 109,851.8 120,703.4 128,467.2

159,237.6 161,166.7 163,365.4 192,251.1 198,138.9 216,529.9 164,287.1 165,079.1 180,107.8 189,962.4

Table 5.3 presents the summarizing analytical models for repairable components/systems in both the absence (first column) and the presence (second column) of an exponential distribution of ttr.

(5.58)

Table 5.2 reports the main definitions and properties of the stochastic failure process of nonrepairable components/systems for both the generic item (first column) and for items whose density function is assumed to be exponential (second column).

5.10.2 Exponential Distribution. Numerical example Table 5.4 presents the ttf of a sample of 100 electronic components produced by a company in the USA. The

100

5 Basic Statistics and Introduction to Reliability

ReliaSoft Weibull++ 7 - www.ReliaSoft.com

Failure Timeline FS Timeline Failure

0.000

60000.000

120000.000

180000.000

240000.000

300000.000

Time, (t) λ = 1.1340Ε−5, ρ = −0.9062

Fig. 5.10 Failure timeline. ReliaSoft® software

ReliaSoft Weibull++ 7 - www.ReliaSoft.com

TTF Histogram 8.000E-6

FS Histogram Pdf Line Failures

6.400E-6

Value

4.800E-6

3.200E-6

1.600E-6

0.000

0-29453.... 29453.05... 58906.10... 88359.15... 117812.2... 147265.2... 176718.3... 206171.3... 235624.4... 265077.4... 294530.5...

Period λ = 1.1340Ε −5, ρ = −0.9062

Fig. 5.11 ttf histogram. ReliaSoft® software

5.10 Parametric Probability Density Functions

101

ReliaSoft Weibull++ 7-www.ReliaSoft.com

Probability - Exponential

99.900

Probability-Exponential Folio2\Data 1 Exponential-1P RRX SRM MED FM F=100/S=0 Data Points Probability Line

Reliability, R(t)

50.000

10.000 5.000

1.000 0.500

0.100 0.000

60000.000

120000.000

180000.000

240000.000

300000.000

Time, (t) λ = 1.1340Ε−5, ρ = −0.9062

Fig. 5.12 Probability plot, exponential distribution. ReliaSoft® software

generic variable time relates to the use of the component and is expressed in minutes. Figure 5.10 presents the failure timeline, i. e., the graphical collection and representation of failures according to the available ttf, while Fig. 5.11 shows the related histogram from which it is possible to identify a possible parametric distribution of the random values. Figure 5.12 presents the so-called probability plot, which is a graphical technique for assessing whether or not a data set follows a given distribution. In particular, the data are plotted against a theoretical (in other words a parametric) distribution so that the points approximate a straight line. Departure from this straight line indicates departure from the specified distribution. Furthermore, conducted with the support of ReliaSoft® reliability software and illustrated in Fig. 5.12, the proposed analysis assesses whether or not the ttf values follow an exponential distribution. The following chapter discusses the ability of a generic parametric distribution to best fit an available set of stochastic data in order to develop the reliability evaluation models and methods useful to practitioners. In fact, the probability plots can be

generated for different competing parametric distributions to identify which provides the best fit, and the probability plot generating the highest correlation coefficient is the best choice since it generates the straightest probability plot. The plot illustrated in Fig. 5.12 shows that there seems to be good correlation between the available ttf and an exponential distribution, which is supported by the estimate of the cumulative distribution function F .t/, i. e., the failure probability function, i. e., unreliability, as reported in Fig. 5.13. Similarly, Fig. 5.14 presents the estimated reliability function, i. e., the survival function R.t/. The estimated value of failure rate is .t/ D 1:13  105 min1 : Figure 5.15 presents the related trend of the estimated probability density function f .t/ and the constant failure rate (t). Figure 5.16 completes the illustration of this numerical application. It is the result of a nonparametric reliability evaluation based on the estimation of a set of lower and upper bounds for the reliability R.t/. This analysis is illustrated and discussed in the next chapter.

102

5 Basic Statistics and Introduction to Reliability

ReliaSoft Weibull++ 7 - www.ReliaSoft.com

Unreliability vs Time Plot

1.000

Unreliability Folio2\Data 1 Exponential-1P RRX SRM MED FM F=100/S=0 Data Points Unreliability Line

Unreliability, F(t)=1-R(t)

0.800

0.600

0.400

0.200

0.000 0.000

140000.000

280000.000

420000.000

560000.000

700000.000

Time, (t) λ = 1.1340Ε−5, ρ = −0.9062

Fig. 5.13 Unreliability, exponential distribution. ReliaSoft® software

ReliaSoft Weibull++ 7 - www.ReliaSoft.com

Reliability vs Time Plot 1.000

Reliability Folio2\Data 1 Exponential-1P RRX SRM MED FM F=100/S=0 Data Points Reliability Line

Reliability, R(t)=1-F(t)

0.800

0.600

0.400

0.200

0.000 0.000

140000.000

280000.000

420000.000

560000.000

Time, (t) λ = 1.1340Ε−5, ρ = −0.9062

Fig. 5.14 Reliability, exponential distribution. ReliaSoft® software

700000.000

5.10 Parametric Probability Density Functions

103 ReliaSoft Weibull++ 7 - www.ReliaSoft.com

ReliaSoft Weibull++ 7 - www.ReliaSoft.com

Probability Density Function Pdf

1.600E-5

Folio2\Data 1 Exponential-1P RRX SRM MED FM F=100/S=0 Pdf Line

f(t)

1.200E-5

8.000E-6

2.000E-5

Failure Rate

1.600E-5

Folio2\Data 1 Exponential-1P RRX SRM MED FM F=100/S=0 Failure Rate Line

1.200E-5

8.000E-6

4.000E-6

4.000E-6

0.000 0.000

Failure Rate, f(t)/R(t)

Failure Rate vs Time Plot

2.000E-5

60000.000

120000.000

180000.000

240000.000

300000.000

0.000 0.000

60000.000

120000.000

Time, (t)

180000.000

240000.000

300000.000

Time, (t) λ = 1.1340Ε−5, ρ = −0.9062

λ = 1.1340Ε−5, ρ = −0.9062

Fig. 5.15 Probability density function and failure rate. ReliaSoft® software

ReliaSoft Weibull++ 7 - www.ReliaSoft.com

Reliability vs Time 1.000

Kaplan-Meier Reliability Lower Bound Upper Bound

0.800

Reliability

0.600

0.400

0.200

0.000 0.000

60000.000

120000.000

180000.000

240000.000

300000.000

Time, (t)

Fig. 5.16 Nonparametric evaluation. ReliaSoft® software

5.10.3 The Normal and Lognormal Distributions Two useful time-dependent statistical models are frequently applied in reliability theory. The normal probability density function is a continuous and parametric

distribution defined as follows:   .x  /2 1 exp  ; f .x/ D p 2 2  2

(5.59)

where  and  are two parameters, respectively, equal to the mean and the standard deviation of the random variable x.

104

5 Basic Statistics and Introduction to Reliability 1.1

The following models quantify the cumulative function and the mean function: F .x/ D 1

0.9

   x 1 ; 1 C erf f .x/ dx D p 2  2

0.7

F(x)

Zx

(5.60)

C1 Z M.x/ D Œxf .x/ dx D ;

μ=1, σ =0.2 0.5

μ=1, σ =0.4 μ=1, σ =0.8

(5.61)

0.3

1

The integral in Eq. 5.62 cannot be evaluated in closed form in terms of an elementary function (differential algebra), but it can be evaluated by expanding the integrand in a Taylor series as follows: ! 1 n x Y x 2 2 X : (5.63) erf.x/ D p i  nD0 2n C 1 i D1

0.1 -2

-1

-0.1

0

1

2

3

x

Fig. 5.18 Normal distribution, cumulative function.  D 1,  D f0:2; 0:4; 0:8g 40 35 30

f(x)/[1-F(x)]

where erf(x) is the error function (also called the “Gauss error function”). Erf(x) is a nonelementary function because it is not built from a finite number of exponential functions, logarithms, constants, one variable, and root (mathematics) of equations by function composition and combinations using the four arithmetic operations (+ –  /. In particular, it is defined as 8 2 Rx t 2 ˆ ˆ erf.x/ D p e dt ˆ ˆ <  0 (5.62) ˆ ˆ 2 x 2 ˆd ˆ : erf.x/ D p e : dx 

25 μ=1, σ=0.2

20

μ=1, σ=0.4 μ=1, σ=0.8

15 10 5

-1.5

-1

-0.5

0

0

0.5

1

1.5

2

2.5

x

Fig. 5.19 Normal distribution, rate (x).  D 1,  D f0:2; 0:4; 0:8g

2.2

1.7

1.2

f(x)

μ=1, σ=0.2 μ=1, σ=0.4 μ=1, σ=0.8 0.7

0.2

-1

0

1

2

3

-0.3

x

Fig. 5.17 Normal distribution, density function.  D 1,  D f0:2; 0:4; 0:8g

Figure 5.17 illustrates the trend of the normal density function f .x/ for different values of the standard deviation , assuming  D 1. Similarly, Fig. 5.18 presents the trend of the cumulative function F .x/, which is the failure probability function in the case where the variable x is the ttf. Figure 5.19 presents the values of (x) obtained by applying Eq. 5.25. Figure 5.20 presents the trend of f .x/ and F .x/ for different values of  and . The lognormal distribution is the probability distribution of a random variable whose logarithm is a normal distribution. The probability density function is

5.10 Parametric Probability Density Functions

105

2.2

1.1

0.9

1.7

0.7

1.2

f(x)

F(x)

μ=1, σ=0.2 μ=2, σ=0.4

0.5

μ=3, σ=0.8

0.7 0.3

0.2

0.1

0

1

2

-0.3

3

4

5

6

7

0

1

2

3

-0.1

x

4

5

6

7

x

Fig. 5.20 Normal distribution, density function and cumulative function

defined as follows: ! 8 ˆ 1 Œln.x/  2 0:

0.8 0.7 0.6

(5.64)

0.5 μ=1, σ =0.2

f(x)

0.4

The cumulative distribution is F .x/ D

f .x/ dx D

Zx f .x/ dx

x>0

1

0.1

0

  1 1 ln.x/   : D C erf p 2 2  2

0 2

4

-0,1

6

8

10

x

Fig. 5.21 Lognormal distribution, density function.  D 1,  D f0:2; 0:4; 0:8g

C1 C1 Z Z D M.x/ D Œxf .x/ dx Œxf .x/ dx x>0

  2 : D exp  C 2

0

(5.65)

The mean function, i. e., the expected value, is

1

μ=1, σ =0.8

0.2

1.1

0.9

0

(5.66)

Figures 5.21 and 5.22 illustrate the trend of the density function f .x/ and the cumulative function F .x/ for different parameterizations of the analytical model. Figure 5.23 presents the values of the rate obtained by applying Eq. 5.25. The lognormal distribution is generally used to model the stochastic repair process that is characterized by the previously introduced random variable time to repair (ttr). In particular, Figs. 5.24–5.26 illustrate the trend of the most significant functions that describe the repair process, assuming a lognormal distribution of a set of ttr values (represented by the dots

0.7

F(x)

Zx

μ=1, σ =0.4 0.3

μ=1, σ=0.2

0.5

μ=1, σ=0.4 μ=1, σ=0.8

0.3

0.1 0 -0.1

2

4

6

8

10

x

Fig. 5.22 Lognormal distribution, cumulative function.  D 1,  D f0:2; 0:4; 0:8g

106

5 Basic Statistics and Introduction to Reliability

4

1.0 0.9

3.5

0.8 0.7

3

G(x)

0.6 2.5 μ=1, σ=0.2 2

λ(x)

0.4 0.3

μ=1, σ=0.4

0.2

μ=1, σ=0.8

1.5

0.5

0.1 0.0

1

0

1

2

3

4

5

6

7

4

5

6

7

x

0.5

Fig. 5.25 Repair process, G.t /

0 0

2

4

6

8

10

12

x

Fig. 5.23 Lognormal distribution, .x/.  D 1,  D f0:2; 0:4; 0:8g

0.9 0.8 0.7

g(x)/[1-G(x)]

0.7 0.6

g(x)

0.5 0.4

0.6 0.5 0.4 0.3 0.2 0.1

0.3

0.0 0.2

0

1

2

0.1

3

x

Fig. 5.26 Repair process, (t )

0.0 0

1

2

3

4

5

6

7

x

Fig. 5.24 Repair process, g.t /

in figure). These functions are: • the density function of the ttr variable g.t/, also called “nonconditional repair rate”; • the cumulative function G.t/, also called “maintainability”; • the conditional repair rate (t).

5.10.4 Normal and Lognormal Distributions. Numerical example The failure timeline of the stochastic failure process for a mechanical component, for which a sample of

100 ttf is available, is reported in Fig. 5.27. The frequency distribution of values is illustrated in the histogram shown in Fig. 5.28. Figures 5.29 and 5.30 present the result of a parametric evaluation of the probability plot and reliability measures assuming a normal distribution of random values. Similarly, Figs. 5.31 and 5.32 present the results obtained assuming a lognormal distribution of random values. Both parametric evaluation analyses seem to fit the random variables effectively. Nevertheless, the statistical distributions (normal and lognormal) differ and so do the estimated values of the reliability parameters when one of them is assumed. In-depth analysis using ad hoc “goodness of the fit” models is introduced in the next chapter.

5.10 Parametric Probability Density Functions ReliaSoft Weibull++ 7 - www.ReliaSoft.com

107

Failures Timeline FS Timeline Failure

0.000

1.000

2.000

3.000

4.000

5.000

Time, (t) β = 8.8740, η = 3.2201, ρ = 0.9885

Fig. 5.27 Failure timeline. ReliaSoft® software

ReliaSoft Weibull++ 7 - www.ReliaSoft.com 2.000

TTF Histogram FS Histogram Pdf Line Failures

1.600

Value

1.200

0.800

0.400

0.000 0-0.2 0.2-0.4 0.4-0.6 0.6-0.8 0.8-1.0 1.0-1.2 1.2-1.4 1.4-1.6 1.6-1.8 1.8-2.0 2.0-2.2 2.2-2.4 2.4-2.6 2.6-2.8 2.8-3.0 3.0-3.2 3.2-3.4 3.4-3.6 3.6-3.8 3.8-4.0 4.0-4.2 4.2-4.4 4.4-4.6 4.6-4.8 4.8-5.0

Period β = 8.8740, η = 3.2201, ρ = 0.9885

Fig. 5.28 ttf histogram. ReliaSoft® software

108

5 Basic Statistics and Introduction to Reliability ReliaSoft Weibull++ 7 - www.ReliaSoft.com

Probability - Normal distribution

99.900

Probability-Normal (Normal-2P)

Unreliability, F(t)

Folio1\Data 1 Normal-2P RRX SRM MED FM F=100/S=0 Data Points Probability Line

50.000

10.000 5.000

1.000 0.500 0.100 1.000

Fig. 5.29 Probability plot, normal distribution. ReliaSoft® software

2.000

3.000

4.000

ReliaSoft W eibull++ 7 - www.ReliaSoft.com

Unreliability vs Time Plot distribution

Reliability vs Time Plot distribution

1.000

1.000

Unreliability (Normal-2P) Folio1\Data 1 Normal-2P RRX SRM MED FM F=100/S=0 Data Points Unreliability Line

0.600

0.400

Reliability (Normal-2P) Folio1\Data 1 Normal-2P RRX SRM MED FM F=100/S=0 Data Points Reliability Line

0.800

Reliability, R(t)=1-F(t)

0.800

Unreliability, F(t)=1-R(t)

6.000

μ = 3.1380, σ = 0.4668, ρ = 0.9581

ReliaSoft Weibull++ 7 - www.ReliaSoft.com

0.200

0.000 0.000

5.000

Time, (t)

0.600

0.400

0.200

1.000

2.000

3.000

4.000

0.000 0.000

5.000

1.000

2.000

3.000

4.000

μ = 3.1380, σ = 0.4668, ρ = 0.9581

μ = 3.1380, σ = 0.4668, ρ = 0.9581

ReliaSoft Weibull++ 7 - www.ReliaSoft.com

ReliaSoft Weibull++ 7 - www.ReliaSoft.com

Failure Rate vs Time Plot distribution

Probability Density Function distribution 0.900

20.000

Pdf (Normal-2P)

Failure Rate (Normal-2P)

Folio1\Data 1 Normal-2P RRX SRM MED FM F=100/S=0 Pdf Line

Folio1\Data 1 Normal-2P RRX SRM MED FM F=100/S=0 Failure Rate Line

0.720

Failure Rate, f(t)/R(t)

16.000

f(t)

0.540

0.360

0.180

0.000 0.200

12.000

8.000

4.000

1.360

2.520

3.680

4.840

6.000

0.000 0.200

1.360

2.520

3.680

Time, (t)

Time, (t) μ = 3.1380, σ = 0.4668, ρ = 0.9581

5.000

Time, (t)

Time, (t)

μ = 3.1380, σ = 0.4668, ρ = 0.9581

Fig. 5.30 F .t /, R.t /, f .t /, and (t ). Normal distribution. ReliaSoft® software

4.840

6.000

5.10 Parametric Probability Density Functions

109

ReliaSoft Weibull++ 7 - www.ReliaSoft.com

Probability - Lognormal distribution

99.900

Probability-Lognormal (Lognormal-2P)

Unreliability, F(t)

Folio1\Data 1 Lognormal-2P RRX SRM MED FM F=100/S=0 Data Points Probability Line

50.000

10.000 5.000

1.000 0.500 0.100 1.000

Fig. 5.31 Probability plot, lognormal distribution. ReliaSoft® software

10.000

Time, (t)

μ = 1.1053, σ = 0.1429, ρ = 0.9907

ReliaSoft Weibull++ 7 - www.ReliaSoft.com

ReliaSoft Weibull++ 7 - www.ReliaSoft.com

Probability Density Function distribution

Failure Rate vs Time Plot distribution

1.000

7.000

Pdf (Lognormal-2P)

Failure Rate (Lognormal-2P)

Folio1\Data 1 Lognormal-2P RRX SRM MED FM F=100/S=0 Pdf Line

Folio1\Data 1 Lognormal-2P RRX SRM MED FM F=100/S=0 Failure Rate Line 5.600

Failure Rate. f(t)/R(t)

0.800

f (t)

0.600

0.400

0.200

4.200

2.800

1.400

0.000 0.000

2.000

4.000

Time, (t)

6.000

8.000

0.000 0.000

10.000

μ = 1.1053, σ = 0.1429, ρ = 0.9907

16.000

48.000

64.000

80.000

Reliability vs Time Plot distribution

Unreliability vs Time Plot distribution 1.000

Unreliability (Lognormal-2P)

0.800

Folio1\Data 1 Lognormal-2P RRX SRM MED FM F=100/S=0 Data Points Unreliability Line

Reliability, R(t)=1-F(t)

Unreliability, F(t)=1-R(t)

Time, (t)

ReliaSoft Weibull++ 7 - www.ReliaSoft.com

ReliaSoft Weibull++ 7 - www.ReliaSoft.com

0.600

0.400

1.000

Reliability (Lognormal-2P)

0.800

Folio1\Data 1 Lognormal-2P RRX SRM MED FM F=100/S=0 Data Points Reliability Line

0.600

0.400

0.200

0.200

0.000 0.000

32.000

μ = 1.1053, σ = 0.1429, ρ = 0.9907

1.000

μ = 1.1053, σ = 0.1429, ρ = 0.9907

2.000

Time, (t)

3.000

4.000

5.000

0.000 0.000

1.000

μ = 1.1053, σ = 0.1429, ρ = 0.9907

Fig. 5.32 F .t /, R.t /, f .t / and (t ). Lognormal distribution. ReliaSoft® software

2.000

Time, (t)

3.000

4.000

5.000

110

5.10.5 The Weibull Distribution This is a time-dependent failure model and one of the most useful parametric distributions in reliability engineering. The Weibull density function f .x/ is defined as follows:     8   0; where a is a scale parameter4 and b is a shape parameter5 . b is called a “shape parameter” because: • b < 1 implies infant mortality, i. e., high mortality of infants typical of both electronic and mechanical systems. This is why, before the products are delivered, several of the components are subject to acceptance tests known as “burn-in” and stress screening so that infant mortality is bypassed. Hazard rate declines with age. • b D 1 implies random failures, i. e., failure modes are “ageless” and the probability density function is an exponential in which  D 1=a. • 1 < b < 4 implies early wear out. The cost of unplanned failure for this component is generally higher than the cost of planned failure. Consequently, there is an optimal replacement time that minimizes the global cost. • b  4 implies old age and rapid wear out. The probability density function is somewhat symmetrical and similar to a normal distribution. Typical failure modes are stress corrosion, material properties, erosions, etc. These components require inspection and corrective action. Waloddi Weibull (1887–1979) introduced the “B10” life, which is the age at which 10% of the “bearings” fail and can be directly read from the Weibull plot. For example, some manufacturers use B10 life for design requirements, some use lower values (e. g., B0.1 for serious failures or B0.01 for catastrophic failures). The cumulative function of the Weibull probability distribution is     x b : (5.68) F .x/ D 1  exp  a 4 5

Sometimes represented by ˛ Sometimes represented by ˇ

5 Basic Statistics and Introduction to Reliability

The mean function is C1 C1 Z Z M.x/ D Œxf .x/ dx D Œxf .x/ dx x>0

1

  1 ; D a 1 C b

0

(5.69)

where  .x/ is the gamma function defined as Z1  .X / D

y x1 ey dy:

(5.70)

0

Table 5.5 presents the value of the gamma function for different values of the variable x. From Eq. 5.25, function (x) is .x/ D

b  x b1 : a a

(5.71)

Figures 5.33–5.39 illustrate the trend of the density function f .x/, cumulative function F .x/, and rate function (x) for different combinations of parameters a and b. In particular, Figs. 5.33–5.35 assume b D 1 and different values of a. Similarly Figs. 5.36–5.38 illustrate the obtained values for different shape parameters given a equal to 2. Figure 5.39 presents a zoom of Fig. 5.37 based on a different scale for the function (x/ 2 Œ0I 3 : The equations for reliability and maintainability in the case of a ttf or a ttr random variable distributed in accordance with a Weibull probability distribution are the following: "   # 8 b ˆ 0; where T is the mission time defined on the ttf stochastic variable in agreement with the definition introduced in Eq. 5.19. Then, "   # 8 b ˆ 0;

ln[ .x/]

2.968879 2.252713 1.827814 1.524064 1.288023 1.095798 0.934581 0.796678 0.677087 0.572365 0.480031 0.398234 0.325552 0.260867 0.203281 0.152060 0.106595 0.066376 0.030969 0.000000

x

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00

19.470085 9.513508 6.220273 4.590844 3.625610 2.991569 2.546147 2.218160 1.968136 1.772454 1.616124 1.489192 1.384795 1.298055 1.225417 1.164230 1.112484 1.068629 1.031453 1.000000

 .x/

Table 5.5 Gamma function

1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.95 2.00

x

 .x/ 0.973504 0.951351 0.933041 0.918169 0.906402 0.897471 0.891151 0.887264 0.885661 0.886227 0.888868 0.893515 0.900117 0.908639 0.919063 0.931384 0.945611 0.961766 0.979881 1.000000

ln[ .x/] 0.026853 0.049872 0.069306 0.085374 0.098272 0.108175 0.115241 0.119613 0.121421 0.120782 0.117806 0.112592 0.105231 0.095808 0.084401 0.071084 0.055924 0.038984 0.020324 0.000000 2.05 2.10 2.15 2.20 2.25 2.30 2.35 2.40 2.45 2.50 2.55 2.60 2.65 2.70 2.75 2.80 2.85 2.90 2.95 3.00

x 0.021937 0.045438 0.070456 0.096947 0.124872 0.154189 0.184864 0.216859 0.250143 0.284683 0.320449 0.357412 0.395545 0.434821 0.475215 0.516703 0.559262 0.602870 0.647505 0.693147

ln[ .x/] 1.022179 1.046486 1.072997 1.101802 1.133003 1.166712 1.203054 1.242169 1.284209 1.329340 1.377746 1.429625 1.485193 1.544686 1.608359 1.676491 1.749381 1.827355 1.910767 2.000000

 .x/ 3.05 3.10 3.15 3.20 3.25 3.30 3.35 3.40 3.45 3.50 3.55 3.60 3.65 3.70 3.75 3.80 3.85 3.90 3.95 4.00

x 0.739777. 0.787375 0.835924 0.885405 0.935802 0.987099 1.039279 1.092328 1.146231 1.200974 1.256542 1.312923 1.370104 1.428072 1.486816 1.546322 1.606581 1.667580 1.729310 1.791759

ln[ .x/] 2.095468 2.197620 2.306944 2.423965 2.549257 2.683437 2.827178 2.981206 3.146312 3.323351 3.513252 3.717024 3.935761 4.170652 4.422988 4.694174 4.985735 5.299330 5.636763 6.000000

 .x/

4.05 4.10 4.15 4.20 4.25 4.30 4.35 4.40 4.45 4.50 4.55 4.60 4.65 4.70 4.75 4.80 4.85 4.90 4.95 5.00

x

1.854918 1.918777 1.983326 2.048556 2.114457 2.181021 2.248239 2.316103 2.384605 2.453737 2.523490 2.593857 2.664831 2.736405 2.808571 2.881323 2.954654 3.028557 3.103026 3.178054

ln[ .x/]

6.391177 6.812623 7.266873 7.756690 8.285085 8.855343 9.471046 10.136102 10.854777 11.631728 12.472045 13.381286 14.365527 15.431412 16.586207 17.837862 19.195079 20.667386 22.265216 24.000000

 .x/

5.10 Parametric Probability Density Functions 111

112

5 Basic Statistics and Introduction to Reliability Weibull F(x), a=2

Weibull f(x), b=1 2.5

1 0.8

1.5

a=4 a=2 a=1 a=0.5

1

F(x)

f(x)

2

b=4 b=2 b=1 b=0.5

0.6 0.4

0.5

0.2 0 0

2

4

6

8

10

0

x

0

Fig. 5.33 Weibull distribution, density function. b D 1

2

4

6

8

10

x

Weibull F(x), b = 1

Fig. 5.37 Weibull distribution, cumulative function. a D 2

1.2 1 0.8 F(x)

a=4

0.6

a=2 a=1

0.4

a=0.5

0.2 0 0

1

2

3

4

5

6

7

8

9

10

x

Fig. 5.34 Weibull distribution, cumulative function. b D 1

Fig. 5.38 Weibull distribution, (x). a D 2

Weibull λ(x), b=1 2.5

f(x)/[1-F(x)]

2 a=4 a=2 a=1 a=0.5

1.5 1 0.5 0 0

2

4

6

8

10

x

Fig. 5.35 Weibull distribution, failure rate (x). b D 1

Fig. 5.39 Weibull distribution, (x). a D 2, zoom

where T is the mission repair time defined on the ttr random variable in agreement with Eq. 5.48 for repairable components.

5.10.6 Weibull Distribution. Numerical Example Fig. 5.36 Weibull distribution, density function. a D 2

This section presents the statistical evaluation of the reliability parameters for the random ttf values intro-

5.11 Repairable Components/Systems: The Renewal Process and Availability A(t)

duced in Sect. 5.4, assuming a Weibull parametric distribution of values. In particular, Fig. 5.40 presents the Weibull probability plot, while Fig. 5.41 presents F .t/, R.t/, estimated f .t/, and (t). From a qualitative point of view, the graphical trends obtained for f .t/, F .t/, and R.t/ seem to be similar to those previously illustrated, assuming a normal or a lognormal distribution of random values. But their failure rate trends and values differ very much. This justifies the importance of the parametric reliability evaluation process discussed in the next chapter.

5.11 Repairable Components/Systems: The Renewal Process and Availability A(t) The first group of definitions, models, and properties previously discussed refer to “nonrepairable” components and the second group refer to “repairable” components in the stochastic repair process. This section introduces useful new definitions and models to characterize repairable components/systems subject to function, failure, and repair (FFR) cycles as illustrated in Fig. 5.1. A very important measurement of reliability for repairable components is the nonconditional hazard rate w(t) defined for the range t 2 Œt0 D 0; C1/. In fact, a generic repairable entity is subject to FFR cycles. Consequently, a nonconditional hazard rate f .t/ as introduced for nonrepairable entities (see Sects. 5.7 and 5.8) cannot be identified. f .t/ is the density function of the unique random variable ttf defined for a nonrepairable component subject to a degradation process to failure. In other words, while the failure event is unique for nonrepairable items, a repairable component exposed to FFR cycles is subject to several degradation processes to failure during its life cycle, starting from the point in time t0 D 0 as illustrated in Fig. 5.1. In particular, for a given repairable component which starts to function in t0 D 0, w.t/ quantifies the rate, i. e., the velocity, to failure at time point t as follows: P .t  ttQf  t C dtncomponent is in state of function in t D t0 / D w.t/ dt;

(5.74)

113

where ttQf is a random variable defined in the range Œt0 D 0; C1Œ. From Eq. 5.74, w.t/ measures the probability of the repairable component failing in the range Œt; t C dt. The variable ttQf differs from the traditional time to failure variable ttf introduced in Sect. 5.4. Moreover, ttf can also be defined for a repairable component subject to FFR cycles: it represents the period of time from a generic starting point time ti and the following time t when a failure occurs. ti can be equal to t0 or immediately follow the conclusion of the restoration process of a repaired item. Consequently, a set of different time to failure random variables can be defined for a repairable item, which strongly depends on the operating conditions during the generic cycle and the state of function and health of the component after the previous restoration. Similarly, different time to repair random variables can be defined for a repairable item subject to FFR cycles. The generic stochastic repair process obviously depends upon the state of failure and the operating conditions under the repair activity. The ttQf variable is used in the following chapters to demonstrate theoretical and analytical relationships, and in practice is substituted by a set of ttf and ttr random variables defined according to the previous assumptions. The following measurement of reliability is called “expected number of failures” (ENF) and quantifies the number of failures in a period of time T D Œt1 ; t2  for a repairable component/system subject to FFR cycles: Zt2 W .t1 ; t2 / D w.t/ dt : (5.75) t1

Figure 5.42 illustrates the trend of the ENF for the generic period of time [0,t], distinguishing a repairable from a nonrepairable component. The ENF for a nonrepairable item corresponds to the failure probability F .t/, i. e., the cumulative of the density function f .t/ defined for the ttf random variable. The availability is one of the most significant statistical measures defined for a repairable component subject to FFR cycles. The system operates until it fails, after which it is repaired and returned in its original operating condition. This is the so-called renewal process, which is a sequence of independent and not negative random variables. A renewal occurs when a unit

114

5 Basic Statistics and Introduction to Reliability ReliaSoft Weibull++ 7 - www.ReliaSoft.com

Probability - Weibull distribution

99.900

Probability-Weibull Folio1\Data 1 Weibull-2P RRX SRM MED FM F=100/S=0 Data Points Probability Line

90.000

Unreliability, F(t)

50.000

10.000 5.000

1.000 0.500

0.100 1.000

Fig. 5.40 Probability plot, Weibull distribution. ReliaSoft® software

β = 8.8740, η = 3.2201, ρ = 0.9885

ReliaSoft Weibull++ 7 - www.ReliaSoft.com

ReliaSoft Weibull++ 7 - www.ReliaSoft.com

Unreliability vs Time Plot Weibull

Reliability vs Time Plot - Weibull distribution

1.000

Unreliability

0.800

Folio1\Data 1 Weibull-2P RRX SRM MED FM F=100/S=0 Data Points Unreliability Line

Reliability, R(t)=1-F(t)

Unreliability, F(t)=1-R(t)

10.000

Time, (t)

0.600

0.400

1.000

Reliability

0.800

Folio1\Data 1 Weibull-2P RRX SRM MED FM F=100/S=0 Data Points Reliability Line

0.600

0.400

0.200

0.200

0.000 0.000

1.000

2.000

3.000

4.000

0.000 0.000

5.000

1.000

2.000

3.000

4.000

β = 8.8740, η = 3.2201, ρ = 0.9885

β = 8.8740, η = 3.2201, ρ = 0.9885

ReliaSoft Weibull++ 7 - www.ReliaSoft.com

ReliaSoft Weibull++ 7 - www.ReliaSoft.com

Probability Density Function f(t) - Weibull distribution 2.000

Failure Rate vs Time Plot 20000.000

Pdf

Failure Rate

Folio1\Data 1 Weibull-2P RRX SRM MED FM F=100/S=0 Pdf Line

Folio1\Data 1 Weibull-2P RRX SRM MED FM F=100/S=0 Failure Rate Line

1.600

Failure Rate, f(t)/R(t)

16000.000

f(t)

1.200

0.800

0.400

0.000 0.000

12000.000

8000.000

4000.000

2.000

4.000

6.000

8.000

10.000

0.000 0.000

Time, (t) β = 8.8740, η = 3.2201, ρ = 0.9885

5.000

Time, (t)

Time, (t)

2.000

4.000

6.000

Time, (t) β = 8.8740, η = 3.2201, ρ = 0.9885

Fig. 5.41 F .t /, R.t /, f .t /, and (t ). Weibull distribution. ReliaSoft® software

8.000

10.000

5.11 Repairable Components/Systems: The Renewal Process and Availability A(t)

115

The availability can be also defined over an interval of time T D t  t0 as follows:

W(0,t) repairable

N /D 1 A.T T

ZT A.t/ dt ;

(5.78)

t0

1

W(0,t) not repairable= F(t)

t

where A.t/ is the point availability in t. The availability in Eq. 5.78 is the so-called mean availability. The steady-state availability is the following: A.1/ D lim A.t/;

Fig. 5.42 Expected number of failures for repairable and nonrepairable components

t !1

(5.79)

where A.t/ is the point availability in t. Other definitions of availability refer to the following very simplified expression: fails and is restored to work. Its availability is the probability that it is performing the required function at a given point of time t when it is operating and maintained under specific conditions. In other words, A.t/ measures the capability of the component/system as the probability of being operational at a given time t: A.t/ D P .component is operating in time t/: (5.76) The literature presents several definitions of availability that mainly depend on which types of downtimes are chosen for analysis. In particular, the instantaneous or point availability A.t/ is the probability that a component/system is operational at any random time t. In other words, it is the sum of two contributions: 1. R.t/, the reliability of the component/system; Zt R.t  x/m.x/ dx; 2. 0

where m.x/ is the renewal density function of the system because the repairable component has a failure distribution and a repair distribution.6 The point availability is Zt R.t  x/m.x/ dx:

A.t/ D R.t/ C

(5.77)

0

6

The so-called renewal theory is properly discussed in Chap. 9.

AD

UT ; UT C DT

(5.80)

where UT is the component/system uptime and DT is the component/system downtime. In particular, Eq. 5.80 can be quantified by only assuming the corrective downtime for DT, or the total amount of downtime (corrective, preventive, inspection, etc.). The unavailability Q.t/ is the complementary function of A.t/, and a statistical measure of nonoperability of the component system at t: Q.t/ D 1  A.t/:

(5.81)

Given a repairable component subject to FFR cycles and assuming constant hazard and repair rates, i. e., .t/ D  and .t/ D , the simplified expressions of availability and unavailability are A.t/ D

  C e.C/t ; C C

(5.82)

Q.t/ D

   1  e.C/t ; C

(5.83)

where  is the constant hazard rate and  is the constant repair rate. The demonstration of Eqs. 5.82 and 5.83 now follows. In agreement with the previously introduced two-state diagram (see Fig. 5.2), it is assumed the state of function of a generic repairable component is 0 and the state of nonfunction is 1.

116

5 Basic Statistics and Introduction to Reliability

Then the following notation that defines four basic events is assumed: E0 .t/ the component is functioning at time t; E1 .t/ the component is not functioning at time t; Ef .t/ the component fails at time [t,t C dt]; Er .t/ the component is repaired in time [t,t C dt]. Consequently, the probability associated with event E0 .t/ is the availability of the component at time point t: A.t/. Finally, two further definitions are: (t) failure rate of the component in t; (t) repair rate of the component in t: Using these definitions, one obtains the following basic equation: E0 .t C t/ D E0 .t/EN f .t/ C E1 .t/Er .t/ D E0 .t/ Œ1  .t/t C E1 .t/.t/t: (5.84) From Eq. 5.84, 8 dE0 .t/ E0 .t C t/  E0 .t/ ˆ ˆ D lim ˆ ˆ t !0 dt t ˆ ˆ ˆ  ˆ ˆ E0 .t/ Œ1  .t/t  ˆ ˆ D lim ˆ ˆ ˆ t !0 t ˆ ˆ <  E1 .t/.t/t  E0 .t/ C ˆ ˆ t ˆ ˆ ˆ ˆ .t/ D .t/t E ˆ f ˆ ˆ ˆ ˆ ˆ Er .t/ D .t/t ˆ ˆ ˆ ˆ :E .t/ D 1  E .t/: 1 0 (5.85) The following derivative equation is obtained: .t/t  A.t/.t/t  A.t/.t/t dA.t/ D lim t !0 dt t D .t/  A.t/ Œ.t/ C .t/ : (5.86) Now assuming failure and repair rates to be constant, A.t / Z

A.0/D1

dA.t/ D   A.t/ . C /

Zt dt D t;

(5.87)

0

ˇ ˇA.t / ˇ ˇ ˇ 1 ln Œ  A.t/ . C /ˇ D t: ˇ C ˇ 1

(5.88)

Solving Eq. 5.88, one obtains 1 1 ln Π A.t/ . C / C ln./ D t: C C (5.89) Applying the properties of logarithms, one gets 



Π A.t/. C / exp ln 

D exp Œt. C / ; (5.90)

A.t/ D

  C e.C/t ; C C

(5.91)

thus demonstrating Eq. 5.82. The same result can be obtained by applying the Markov chains technique, as illustrated in Chap. 8, which discusses reliability models for dependent components/systems. The asymptotic values of availability and unavailability are     .C/t C e A.1/ D lim t !1  C  C MTTF  D ; (5.92) D C MTTR C MTTF 

i  h 1  e.C/t t !1  C  MTTR  D : D C MTTF C MTTR



Q.1/ D lim

(5.93)

Table 5.6 reports the main definitions and properties related to repairable components subject to failure and repair processes. The second column includes the results obtained assuming an infinite MTTR, i. e., a repair rate equal to 0: although the repairable component becomes a nonrepairable item, the analytical models do not change. In particular, 8 ˆ 4–5 years > 5 years No answer

4.8 5.7 16.2 12.4 12.4 6.7 25.7 16.2

support autonomously, without sharing any data with the other parts of the company, such as purchase office and administration. Table 7.2 points out the typical support offered by commercial CMMS: a database of interventions and the management of the scheduling of preventive actions. In general, the spare parts management is well supported and employed by users. Commercial CMMS packages usually do not support any model to optimize maintenance policies and to support maintenance engineering choices. Table 7.3 underlines a full commitment of maintenance directors and planners, while maintenance workers are less involved in the use of CMMS. Because of the scarce integration between maintenance and production, the

202

7 Maintenance Information System and Failure Rate Prediction

Table 7.6 Reasons for CMMS choice Reason

Most important (%)

Second most important (%)

Don’t know Integration with other commercial software General functionality and features Ease of use Price General reputation of software and its vendor Compatibility with previous CMMS Compatibility with operating system Availability of training Availability of local support It uses the latest technology Speed of system response Ease of implementation Integration with other technical software Availability in local language version Other/not applicable

22.9 15.2 9.5 8.6 6.7 3.8 3.8 2.9 1.9 1.0 1.0 1.0 1.0 0.0 0.0 21.0

21.0 7.6 13.3 3.8 6.7 8.6 1.9 2.9 1.0 6.7 3.8 1.9 1.0 1.9 1.0 17.1

Table 7.7 CMMS success factors Factor

Most important (%)

Second most important (%)

Senior management commitment Effective training Choosing the right CMMS Effective change management CMMS vendor support Adequate budget Focus on business benefits Effective BPR Effective project management Consultant support

46.9 37.5 31.3 31.3 21.9 18.8 15.6 15.6 15.6 12.5

53.1 53.1 21.9 15.6 6.3 25.0 28.1 25.0 15.6 6.3

BPR business process reengineering

production personnel is rarely aware of the potentiality of a CMMS. Another interesting study was developed by the Plant Maintenance Resource Center (PMRS 2004) of Booragoon (Australia). In this case, a sample of 105 companies from several sectors (automotive, petroleum, food and beverage, transport) in the USA (29.5%), Australia (10.5%), the UK (6.7%), and Canada (5.7%) was investigated. The study was particularly devoted to analyzing the reason for the choice of CMMS. These companies generally had in their trading staff more than ten people, (84.8%, and in particular 47.6% had more than 100). CMMS was present in the 81.9% of the sample, and 13.3% of CMMS was developed in-house, while the first seven commercial packages had about 60% penetration (Table 7.4). Most of the systems analyzed had been in

place in recent years, but a significant proportion had been in place for at least 5 years or more (Table 7.5). The analysis of the factors that influence the software selection is very interesting. A great number of maintenance managers who replied to this question were not aware of the reasons driving this process. Anyhow, the most commonly stated reasons were general functionality and features and integration with other commercial software, as summarized in Table 7.6. In addition, some other factors, such as the possibility to handle enormous amounts of data, the commonality with tools adopted in other divisions of the company, or a convenient price, were considered. The Plant Maintenance Resource Center research points out the importance of a senior management commitment, an effective change in the management, and valuable training (Table 7.7).

7.4 CMMS Implementation: Procedure and Experimental Evidence Table 7.8 “Hot” factors Factor

Percentage

Effective training Effective BPR Effective change management Choosing the right CMMS Senior management commitment Effective project management Adequate budget Focus on business benefits CMMS vendor support Consultant support Other/not applicable

19.0 15.2 11.4 8.6 7.6 4.8 4.8 1.9 2.9 1.0 22.9

As reported in Table 7.8, training is the activity with the biggest potential improvement, but a lot of effort and time was also paid to an effective business process reengineering. The commitment of senior management and an effective change in management are very popular factors: in other words, the success of the CMMS implementation is related to a significant change in mentality firstly of top management and secondly of workers. The last important question deals with the benefits accrued from the CMMS implementation. The results in Table 7.9 report the prevalence of “don’t know/not applicable,” including people who currently do not use CMMS. The most important benefits concern the possibility to improve the control of technical activities, such as maintenance history, planning and scheduling of interventions and spare parts, and the related costs. There is not a clear vision about benefits concerning the reliability and availability of equipment, thus confirming a weak approach to optimization strategies: current CMMS systems are considered overall

203

as large databases useful for data classification and management. The work by O’Hanlon (2005) confirms the difficulties in the implementation process of a CMMS system. The investigation involved more than 600 companies all over the world and focused on the expected return of investment due to introduction of CMMS. Fifty-seven percent of companies declared missing the expected return of investment, 4% had no idea about the expected return of investment, and for only 39% was the investment successful. This low percentage of successful investments is mainly due to an incomplete implementation of CMMS. In particular, CMMS is often considered as a formal attainment requiring time and resources without positive impacts on the maintenance work. Consequently, interventions are partially registered in the database and with great time delay, spare parts are managed in an informal manner without the CMMS support, and data elaborations by CMMS (i. e., mean time to failure, mean time to repair calculus) are not used to support maintenance policies. This situation is clearly reported in Figs. 7.12 and 7.13. The return of the investment associated with a CMMS system can be seriously compromised by discontinuous training. Companies often invest their time and money in a CMMS system without supporting this choice through training of new personnel, updating the software through new releases, and “maintaining” the CMMS during the “go live” years. Figure 7.14 shows how much companies reserve for updating the system and the correspondent training on average per year. The CMMS impact is strongly related to a massive use of its potentiality: every critical asset must be registered, all the interventions must be recorded, the spare parts must be fully managed with the dedicated

Table 7.9 CMMS benefits Benefit

Significant (%)

Some (%)

None (%)

Don’t know/not applicable (%)

Improved cost control Improved maintenance history Improved maintenance planning Improved maintenance scheduling Improved spare parts control Improved equipment reliability Improved equipment availability Reductions in materials costs Reductions in other costs Reductions in labor costs

35.2 30.5 30.5 28.6 21.9 13.3 9.5 11.4 8.6 5.7

23.8 37.1 36.2 39.0 35.2 41.0 37.1 32.4 36.2 32.4

16.2 9.5 8.6 6.7 12.4 15.2 21.9 22.9 23.8 29.5

24.8 22.9 24.3 25.7 30.5 30.5 31.4 33.3 31.4 32.4

204

7 Maintenance Information System and Failure Rate Prediction [no answer] 1%

7.5 Failure Rate Prediction [all] 19%

[ 250,000 $/year 100,000–250,000 $/year 75,000–100,000 $/year 50,000–75,000 $/year 25,000–50,000 $/year 10,000–25,000 $/year < 10,000 $/ year no answer

Fig. 7.14 Yearly investment devoted to CMMS update and training

[ 0:90

0.10 0.72 1.0

7.5 Failure Rate Prediction Environmental factor – lamps (MIL-HDBK-

Environment

E

Ground benign (GB / Ground fixed (GF / Ground mobile (GM / Naval sheltered (NS / Naval unsheltered (NU / Airborne inhabited cargo (AIC / Airborne inhabited fighter (AIF / ...

1.0 2.0 3.0 3.0 4.0 4.0 4.0 ...

12

λb (failures/millions hs)

Table 7.26 217F2)

211

10 8 6 4 2 0 0

5

10

15

20

25

30

35

40

45

50

supply voltage (V)

Fig. 7.18 Base failure rate predictions under different supply voltages – pp series lamp

• p D 4:5  3:3  0:1  3:0 D 4:455 failures/106 h for lamp 2. All the devices in the circuit have a serial placement: the failure of a single component compromises all the system. The predicted failure rate of the entire circuit is therefore the sum of the different contributions of the predicted failure rates of the components: X system D components : Table 7.27 summarizes the results. The final predicted failure rate for this part of the signaling system of an automatic cutting machine is 67:462  106 h1 . In the case of a continuous variation of parameters, the MIL-HDBK-217F2 standard provides some equations to estimate the failure rates. This very interesting feature allows a kind of sensitivity analysis for the failure rate under varying conditions. For example, the MIL-HDBK-217F2 standard suggests for lamps the following law devoted to b evaluation in order to take into consideration the effect of the supply voltage Vr : b D 0:074Vr1:29

(failures/1;66 h):

(7.7)

Fig. 7.19 Failure rate predictions under different supply voltages and environments

It is possible therefore to investigate the variations of failure rate as a function of voltage. For example, the supply voltage of lamp 1 belonging to the pp series in the electronic circuit in Fig. 7.17 runs from 4 to 48 V, and its base failure rate can change according to this variation as represented in Fig. 7.18. Figure 7.19 presents the failure rate for the lamp of the pp series under different supply voltages and

Table 7.27 Failure rate predictions – signaling system electronic circuit (MIL-HDBK-217F2) Name

MIL-STD-217 category

Part number

Failure rate (failures/106 h)

Connector Fuse Switch 1 Switch 2 Selector Lamp 1 Lamp 2

15.2 Connectors, socket 22.1 Fuses 14.1 Switches 14.1 Switches 14.1 Switches 20.1 Lamps 20.1 Lamps

xc102 whsk 20 qa1304 qa1304 ff56 pp24-60 ght24-56

0.013 0.080 6.009 6.009 6.346 44.550 4.455

Total

67.462

212

environmental conditions. Application and utilization factors are fixed, i. e., U equal to 1.0 and A equal to 3.3.

7.5.2.3 GIDEP and FARADA GIDEP is a cooperative effort to exchange research, development, design, testing, acquisition, and logistics information among the government and participant industries. The objective of GIDEP is to improve the availability of information for the total quality management of critical materials. This goal includes improving reliability, maintainability, and cost of ownership while reducing or eliminating the use of critical resources for redundant testing and avoiding the use of known problem or discontinued parts and materials. GIDEP was born in 1959 as the Interservice Data Exchange Program (IDEP), a mutual agreement created by the Army, Navy, and Air Force in an effort to reduce duplicate qualification and environmental testing carried on for the military services by various contractors on the same parts, components, and materials. Initially IDEP covered only the military equipment and in a second stage it was expanded to include other types of data and information and others participants according to the requirements of the US defense industries. The program was renamed GIDEP to reflect the makeup of its participants and its evolution. In the early 1960 the data were collected, cataloged, analyzed, and published in a series of books known as the FARADA handbooks. Recently, several technical modernizations were made, with particular reference to the connection to automated data mining systems. At the time of writing, the GIDEP database contains five major data areas: • Engineering data. Information in engineering data covers a broad range of technical reports related to parts, components, materials, processes, systems, and subsystems applicable to all the engineering and technical disciplines. Soldering technology, best manufacturing practices, and value engineering reports are also contained in this data area. • Product information data. The product Information data include the diminishing manufacturing sources and material shortages notices, product change notices, and product information notices. • Failure experience data. This part of the database contains information about important failures and

7 Maintenance Information System and Failure Rate Prediction

their consequences. Failure experience data include the well-known ALERTs problem advisories and agency action notices. • Reliability–maintainability data. The reliability– maintainability data contain failure rate, failure mode, replacement rate, and mean time to repair data on parts, components, and subsystems. Some information is also in the failure experience data section. This is the core of the database when the problem is the failure rate prediction. The FARADA handbook is derived from this section. • Metrology data. This part contains the calibration procedures and technical manuals for test and measurement equipment. GIDEP data are accessible through a series of menus. Every document required is downloadable electronically. Data about new products are continually being assessed and are available according to the analysis and recommendations of the Data Committee.

7.5.2.4 RADC Nonelectronic Reliability Notebook In early 1980, RADC, New York State, USA, was engaged by the Air Force Agency to increase knowledge of the reliability performance of nonelectronic components in avionic equipment. At first, RADC developed methodologies to test components, thus introducing the “testability engineering principles.” Afterwards RADC published reliability handbooks containing failure data and reliability methods pertaining to a variety of applications. Its objective was the collection, analysis, and presentation of nonelectronic component failure data and the presentation of analytical methods forming the state of the art in nonelectronic reliability analysis. Topics include applicable statistical methods for nonelectronic reliability; reliability specifications; special application methods for reliability prediction; part failure characteristics; reliability demonstration tests. The last available version of this handbook is RADC-TR-85-194 distributed in 1985. The above-mentioned approaches, i. e., MIL-STD217, GIDEP, and RADC, are still applied to estimate figures for the predicted reliability of products. Many studies (Economou 2004) have indicated that their predictions are not concordant. Usually, MIL-HDBK217F2 is conservative and the actual value is several

7.5 Failure Rate Prediction

213

times better than the one predicted. The databases are built through information collected in the field and provided by supplier or users; since field failures depend on the specific application, these data are not representative for every situation. During the last few years, effort has mainly been devoted to enlarging the information in the database considering more influencing parameters: starting from MIL-STD-217 several other sources of reliability information have been developed, such as FIDES 2004, Telcordia SR-332, Naval Surface Warfare Center (NSWC) NSWC Handbook of Reliability Prediction Procedures for Mechanical Equipment, RDF 2000/2003, and the China 299B Electronic Reliability Prediction standard.

7.5.2.5 FIDES 2004 This approach has been developed since 2004 by a group of French companies working in the aeronautic and defense sector. It is based on the physics of failures method and supported by the analysis of test data and field returns. The FIDES approach provides models for components considering technological and physical factors, precise consideration of the mission profile, consideration of mechanical and thermal overstress, and the possibility of distinguishing the failure rate of a specific supplier of a component. Moreover, it takes into account failures linked to development, production, field operation, or maintenance processes. In synthesis, the failure rate predicted by the FIDES method is related to three parameters:  D phis man proc :

(7.8)

phis is the physical failure rate. It is calculated using the base failure rate, usually represented by 0 and provided in tables, corrected by several factors, such as thermal conditions, electrical stresses, and humidity. man is a factor considering the quality level surrounding the part. Usually, the value is linked to specific certifications by the supplier of the components. proc is a factor linked to the characteristics of the realized process. In order to determine this value, a set of questions are provided. The FIDES approach is consistent with MILHDBK-217F2 (Marin and Pollard 2005) and it is usually less conservative, its failure rate being close to the observed rate.

7.5.2.6 Telcordia (Bellcore) SR-332 Telcordia is the new name of Bellcore Company (Bell Communications Research, a spin-off of AT&T Bell Labs). Bellcore previously referred to MIL-HDBK217 for its reliability predictions, and subsequently modified this model to reflect the field experience more exactly, thus developing in 1985 the Bellcore reliability prediction procedure, still applied to commercial electronic products. Many commercial electronic product companies are now choosing to use the Bellcore handbook for their reliability predictions. Typically this approach is useful to provide predictions for devices, units, or serial systems constituted by commercial electronic products. The information requested is the physical design data, the installation’s parameters, and the boundary conditions (e. g., temperature, vibrations).

7.5.2.7 NSWC Mechanical Reliability Prediction (US Navy Standard NSWC 06/LE1) Since 1992 the US Navy has dealt with the reliability prediction problem through its NSWC. The NSWC Handbook of Reliability Prediction Procedures for Mechanical Equipment contains 23 chapters of information with equations, engineering tables, and procedures for estimating the reliability of a mechanical design for the intended operating environment. The NSWC 06/LE1 standard is particularly devoted to mechanical components. Handbook procedures are used to determine the reliability of fundamental components such as springs, bearings, seals, and gaskets. These component applications are then expanded to subassemblies such as valves, actuators, and pumps and then to the system level. Equations in the handbook include parameters for material properties, operating conditions, and stress levels at each equipment indenture level, providing a full reliability, maintainability, and availability analysis at the system, assembly, and component indenture levels.

7.5.2.8 IEC 62380 (RDF 2000/2003 UTEC 80810 Method) The IEC 62380 module supports reliability prediction methods based on the European Reliability Predic-

214

tion Standard. This standard is directly derived from a French standard published by the Union Technique de L’Electricite in 2000. The standard evolved and became the European Standard for Reliability Prediction (IEC 62380). It includes most of the same components as MIL-HDBK-217, mainly therefore electronic devices. As this standard becomes more widely used, it could become the international successor to the US MIL-HDBK-217. Since it is difficult to evaluate the environmental factor, IEC 62380 uses equipment mission profiles and thermal cycling for evaluation. IEC 62380 provides complex models that can handle permanent working, on/off cycling, and sleeping applications. Its unique approach and methodology has gained worldwide recognition. IEC 62380 is a significant step forward in reliability prediction when compared with older reliability standards. It makes equipment reliability optimization studies easier to carry out, thanks to the introduction of influence factors. The reliability data contained in the IEC 62380 handbook are derived from field data concerning electronic equipment operating in these environments: • ground; stationary; weather-protected (equipment for stationary use on the ground in weatherprotected locations, operating permanently or otherwise); • ground; stationary; non-weather-protected (equipment for stationary use on the ground in nonweather-protected locations); • airborne, inhabited, cargo (equipment used in an aircraft, benign conditions); • ground; nonstationary; moderate (equipment for nonstationary use on the ground in moderate conditions of use). In conclusion, the latest version provides: • failure rate calculation at component, block, and system levels; • unavailability calculation at the system level; • repairable system calculation; • component and block i factors (see MIL-STD217 equations).

7.5.2.9 China 299B Electronic Reliability Prediction The China 299B standard is a reliability prediction approach based on the internationally recognized method

7 Maintenance Information System and Failure Rate Prediction

of calculating electronic equipment reliability given in the Chinese Military Standard GJB/z 299B. This standard uses a series of models, also very complicated, for various categories of electronic, electrical, and electromechanical components to predict failure rates that are affected by environmental conditions, quality levels, stress conditions, and various other parameters. The procedure requires a hierarchy process associating components, often not so user-friendly.

7.6 Remote Maintenance/Telemaintenance In this manuscript the authors strongly sustain the need for a “continuous” check of the equipment conditions, as a prerequisite to applying advanced maintenance policies (i. e., preventive and on condition). In the last few years, from this important issue of the technological evolution companies have been able to gain advantages: sensors, data capture systems, and the data transfer systems permit automatic data collection from the field. The integration of the automatic data collection and the CMMS database is a natural evolution of the system, suggesting very interesting advantages in terms of completeness of data and consumption of resources (i. e., workload and money). Moreover, in several cases the maintenance interventions are executed remotely thanks to remote control of actuators. This approach is generally called “remote maintenance” or “telemaintenance.” Early studies and applications have been developed in high-risk sectors, such as nuclear and chemical. The research linked to the International Thermonuclear Experimental Reactor (Haange 1995) is very interesting. Afterwards, the remote maintenance was extended to “capital-intensive” industrial sectors. General Electric can be considered a pioneer for proactive maintenance in large power plants (Rosi and Salemme 2001; Rotival et al. 2001). At the moment, the technology allows an extension of the remote control principle to small and medium-sized plants, thus opening enormous possibilities to plant managers, plant suppliers, and external companies for a global service. In summary, the technological resources (e. g., sensors, data management systems, actuators), the Internet, and other communication technologies can give or facilitate:

7.6 Remote Maintenance/Telemaintenance

215

• remote monitoring and as a consequence the analysis of degradation of plants; • notification of faults; • remote maintenance intervention (in particular, on the logical controller of the plant); • help on-line and remote counseling in real time; • management of spare parts; • education of personnel and continuous training. It is important to underline that the Internet and remote signaling are very powerful instruments also for off-line, which is not strictly linked to production flow, functions, having continuous development and a great impact in maintenance systems. Figure 7.20 gives a general representation of a telemaintenance system. Quick response and integration are the main advantages permitted by the automatic remote control, and practically their consequences lead to a significant reduction of cost. In particular, it is possible to build quickly a database for failures by the concentration of recorded data in some locations, even very far from each other. With this information a set of optimization algorithms and different approaches are usable, from simple ones to very complicated ones, such as expert systems or neural networks. Moreover, this centralized and continuously updated source of data guarantees maximum flexibility and real-time diffusion of knowledge. The absence of data sharing in industrial organizations is often a great problem. In this new vision each modification in the management system of main-

tenance data is very quick and easy: in fact, it is firstly based on the centralized master system and only secondarily on remote and local slaves. This new approach offers relevant possibilities about integration between users and suppliers of plants. This innovative link allows rapid interventions, maybe directly remotely, and can limit intermediate levels of maintenance structure, with maintenance engineers and local technicians. The heavy exchange of data that is usually realized between the customer and the supplier of equipment can be simplified by means of on-line counseling: e. g., remote training both in the starting phase and in the work phase, remote management of spare parts, and technical support and placing of purchase orders. Now that the potentiality of telemaintenance has been underlined some observations about the actors could be interesting. The evolution of the industrial market and the increasing costs of manpower are pushing companies to the delocalization of plants. In this situation, remote maintenance service can be an “owner resource,” totally managed by the enterprise. On the other hand, also in a “localized” case, many companies use external services for maintenance. From this point of view, remote strategies are very significant instruments. In fact, a lot of maintenance global service suppliers, with specific skills in different sectors, such as packaging machines, petroleum, and food and beverages, could be interested in offering their services to a set of similar plants owned by different companies around the world. These companies can use the high-level competences developed by

Fleet of machines/equipment

Database Algorithms Support ….

Control parameter adjustment Performance feedback Maintenance history Telemaintenance and diagnosis Spare parts Training

Real-time controller Production management

Remote site

Internet, LAN, WAN

Production/service site

Fig. 7.20 Remote maintenance system structure

216

7 Maintenance Information System and Failure Rate Prediction

Fig. 7.21 Peeling line scheme

outsourcers in different plants, and this is very crucial, especially during the start-up phases. Equipment suppliers can achieve concentration and scale economy, even offering their service 24 h=day with very competitive costs. It must be remembered that in a global service condition customers buy a fixed level of availability and productivity of plants. Plant supplier is the third category that can take advantage of remote maintenance, making it a not marginal factor: providing skills and competences to the plant customer in a rapid and economic way could turn into a strategic competitive advantage. Moreover, by punctual control of an installed fleet it is important to keep in mind that suppliers have strategic feedback, useful for addressing the research and the development of new products. Industrial experience shows that some criticalities are actually linked to remote maintenance and for this reason researchers will have a great job to overcome them in the future. Primarily, some observations about the measuring system must be underlined. The fundamental question is the definition, for each plant, of the most important parameters to take under control and to send. This choice, usually among temperatures, velocities, vibrations, torques, and electrical intensities, masks a determination of models linking the states of the plant to these parameters. In this perspective, research appears very long and interesting. Anyway, as the net of sensors will expand following the same increasing trend of recorded information, its management will turn into a very complex task. Sensors must transmit robust and reliable data, and actually we can use algorithms for the validation of field signals. In this following interpretative phase the human contribution is still desirable. Use of remote transmission systems, the Internet, and LANs involve questions about protocol standardization, security of data, and precompression techniques in order to make data transmission less onerous. The electronic and information technology sectors must provide suitable methods and instruments. In addition to this “technical question,” there are political and psychological criticalities. First, plant

users are still suspicious of maintenance systems based on remote suggestions. Second, the same plant suppliers are still reluctant to install sensors on machines. In this perspective, the last industrial positive results will surely be a great impulse.

7.6.1 Case Study This is an application of remote maintenance to a “peeling line” for wood panel manufacturing. In particular, the company is European leader for plywood panel production. The plant considered is located in northern Italy and started its production in February 2004, while the supplier is a great north European company. In 2006 the wood panel manufacturer accepted the supplier’s offer to adopt remote control and maintenance, management of spare parts, and continuous training of personnel by the Internet. Figure 7.21 shows the scheme of the plant and Fig. 7.22 presents a photograph of the exit section of the peeler. The plant works 16 h=day with two shifts, and has a cost per hour of about $1,500. Telecontrol required

Fig. 7.22 Product exiting from peeler. (Courtesy of Reni Ettore Spa)

7.6 Remote Maintenance/Telemaintenance

217

Table 7.28 Comparison between traditional maintenance and remote maintenance Traditional maintenance (2005)

Remote maintenance (2006)

Total hours available per year Production losses (h)a Production losses ($)a Corrective interventions by supplier (n/b Corrective interventions by supplier ($)b Corrective interventions by wood manufacturer (n/b Corrective interventions by wood manufacturer ($)b Preventive interventions by supplier .n/b Preventive interventions by supplier ($)b Preventive interventions by wood manufacturer .n/b Preventive interventions by wood manufacturer ($)b Remote interventions by supplier .n/c Total spare parts costs ($)d

5,198 287.3 (5.5%) 430,950 26 70,345 23 7,540 4 7,778 5 2,220 0 405,68 559,401

5,185 145.2 (28%) 217,100 5 19,874 16 2,350 2 13,520 20 8,952 6 42,550 304,346   255; 055

Production losses ($)a Maintenance policies total costs Spare parts total costs

430,950 87,883 40,568

217,100 44,696 304,346

a

Due only to corrective and preventive maintenance Excluding spare parts costs c Cost is included in the annual fee d Spare parts used in corrective and preventive interventions b

the introduction of a management system for the signal based on Sinumerik© technology by Siemens and the installation of new sensors. The fundamental variables under control are angular velocities of shafts, temperatures, intensity of currents, and vibrations, both for machines and the working environment. The interventions on hardware were realized during the 2005 winter stoppage, and the correspondent cash flow was about $ 130,200 .$ 1 D ¤ 0:98/. For this service of remote maintenance the supplier requested an annual fee of about $ 9,300 for remote counseling, training, and ordering of spare parts. In 2006 the new system worked, and Table 7.28 matches the most relevant maintenance factors for the traditional system (2005) and the new system (2006). A great recovery in hours worked, and therefore in costs, due to production losses can be observed immediately. At the same time the total cost of maintenance policies is decreased, and costs for spare parts are not changed much. Continuous remote control of the plant on more than one opportunity permitted an intervention, during unproductive time, before the failure. The possibility to use the great competences of the supplier in real time with very competitive costs (the largest fraction of supplier contri-

butions was only in a remote way) reduces downtimes in a significant manner. Finally, it must be noted that this system enabled the training of personnel, still in progress, and a continuous alignment between the technological improvements of the plant by operators. In conclusion, telemaintenance is a very powerful resource that can open great perspectives for industrial/service systems. Not only manufacturers, but also services industries can take advantage with remote control diagnosis and maintenance, both for users and suppliers. Experimental evidence shows the wide applicability of this technique: increasing availability and reducing costs are gained by punctual and continuous equipment monitoring, a rationalization of maintenance interventions, and low-cost management of spare parts and training. A large part of the technologies required to provide remote maintenance is available. Progress in sensors, protocols, and compression methods is desirable, but first and foremost a more intensive diffusion of the remote concept is needed. Very significant initial results of real applications surely will represent a great impulse.

8

Effects Analysis and Reliability Modeling of Complex Production Systems

Contents 8.1

Introduction to Failure Modes Analysis and Reliability Evaluation . . . . . . . . . . . . . . . . . . . . . 220

8.2

Failure Modes and Effects Analysis . . . . . . . . . . . . 8.2.1 Product Analysis . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Failure Mode, Effects, and Causes Analysis . 8.2.3 Risk Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.4 Corrective Action Planning . . . . . . . . . . . . . . . 8.2.5 FMEA Concluding Remarks . . . . . . . . . . . . . .

220 221 222 222 225 229

8.3

Failure Mode, Effects, and Criticality Analysis . . 8.3.1 Qualitative FMECA . . . . . . . . . . . . . . . . . . . . . 8.3.2 Quantitative FMECA . . . . . . . . . . . . . . . . . . . . 8.3.3 Numerical Examples . . . . . . . . . . . . . . . . . . . .

229 231 231 232

8.4

Introduction to Fault Tree Analysis . . . . . . . . . . . . . 236

8.5

Qualitative FTA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.1 Fault Tree Construction Guidelines . . . . . . . . 8.5.2 Numerical Example 1. Fault Tree Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.3 Boolean Algebra and Application to FTA . . . 8.5.4 Qualitative FTA: A Numerical Example . . . .

8.6

8.7

8.8

239 239 240 241 242

Quantitative FTA . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.1 Quantitative FTA, Numerical Example 1 . . . . 8.6.2 Quantitative FTA, Numerical Example 2 . . . . 8.6.3 Numerical Example. Quantitative Analysis in the Presence of a Mix of Statistical Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . .

244 248 252

Application 1 – FTA . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7.1 Fault Tree Construction . . . . . . . . . . . . . . . . . . 8.7.2 Qualitative FTA and Standards-Based Reliability Prediction . . . . . . . . . . . . . . . . . . . . 8.7.3 Quantitative FTA . . . . . . . . . . . . . . . . . . . . . . .

263 264

Application 2 – FTA in a Waste to Energy System 8.8.1 Introduction to Waste Treatment . . . . . . . . . . . 8.8.2 Case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8.3 Emissions and Externalities: Literature Review . . . . . . . . . . . . . . . . . . . . . . .

277 277 278

254

266 269

279

8.8.4 SNCR Plant . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8.5 SNCR Plant. Reliability Prediction and Evaluation Model . . . . . . . . . . . . . . . . . . . 8.8.6 Qualitative FTA Evaluation . . . . . . . . . . . . . . . 8.8.7 NOx Emissions: Quantitative FTA Evaluation 8.8.8 Criticality Analysis . . . . . . . . . . . . . . . . . . . . . . 8.8.9 Spare Parts Availability, What-If Analysis . . 8.8.10 System Modifications for ENF Reduction and Effects Analysis . . . . . . . . . . . . . . . . . . . . . 8.9

Markov Analysis and Time-Dependent Components/Systems . . . . . . . . . . . . . . . . . . . . . . . . . 8.9.1 Redundant Parallel Systems . . . . . . . . . . . . . . 8.9.2 Parallel System with Repairable Components 8.9.3 Standby Parallel Systems . . . . . . . . . . . . . . . . .

280 281 283 287 292 295 300 301 302 304 306

8.10 Common Mode Failures and Common Causes . . . 309 8.10.1 Unavailability of a System Subject to Common Causes . . . . . . . . . . . . . . . . . . . . . 310 8.10.2 Numerical Example, Dependent Event . . . . . 311

Given a complex system made of thousands of parts and components, such as an Airbus A380, a flexible manufacturing system, an item of health-care equipment (e. g., a radiation machine, a cardiograph), a particle accelerator, etc., there are several modes in which the system does not function properly, i. e., in accordance with specifications. The first problem is the identification of all these modes, even the rarest and most hidden ones, especially if the safety of people and the environment could be compromised. The second problem is the identification of the minimal conditions which can bring a system into one of its possible states of “not function” (i. e., failures). What about the number of failure events, the downtime, the uptime, and the availability of a complex system given a period of time T ? How can the performance of a system be improved? How can the exter-

R. Manzini, A. Regattieri, H. Pham, E. Ferrari, Maintenance for Industrial Systems © Springer 2010

219

220

8 Effects Analysis and Reliability Modeling of Complex Production Systems

nalities generated by a piece of equipment be reduced for a given reliability system configuration? A very critical problem deals with the treatment of dependency among failure and repair events for the basic components of the system under investigation. The Markov chain technique can effectively support the modeling activity of such a production system. The models and methods proposed and exemplified in this chapter will support the introduction of costbased optimization models for planning and executing the maintenance actions and the spare parts fulfillment and management, as properly discussed in the following chapters.

8.1 Introduction to Failure Modes Analysis and Reliability Evaluation The objective of this chapter is the introduction to models and methods supporting the production system designer and the safety and/or maintenance manager to identify how subsystems and components could fail and what are the corresponding effects on the whole system, and to quantify the reliability parameters for complex systems. A system is complex when it is made of physical and logical combinations of several primary components, a lot of basic items whose failure and repair behaviors are known in terms of reliability performance indexes, e. g., failure rate, expected number of failures (ENF), and the mean time to repair (MTTR). This chapter is organized as follows: firstly some models and tools i. e., failure modes and effects analysis (FMEA) and failure mode, effects, and criticality analysis (FMECA) for the identification of failure modes and causes are illustrated and exemplified; afterwards fault tree analysis (FTA) is introduced and applied to several significant examples; and, finally, Markov chain modeling is illustrated and applied.

8.2 Failure Modes and Effects Analysis FMEA is a systematic inductive technique designed to identify the potential failure modes for a product or a process, to assess the risk associated with those failure modes, to rank the issues in terms of importance,

and to identify and carry the correspondent corrective actions out. The final goal is to anticipate problems and minimize their occurrence and impact. Practically, the target is to prioritize the failure modes (product or process) by an index usually called “risk priority number” (RPN) which is very useful in designing activities to reduce the criticalities. FMEAs are often referred to by type, such as design FMEA (DFMEA) and process FMEA (PFMEA). DFMEA is focused on the product, the failure modes and their causes being related to product functions and components. The primary objective is to uncover the potential failures associated with the product that could cause malfunctions, safety hazards for the user, or shortened product life. Ideally the DFMEA should be conducted throughout the entire product design process, from the preliminary design until the product goes into production, with an iterative procedure. PFMEA examines how failures in manufacturing and assembly processes can affect operation and quality of a product or service. PFMEA indicates what can be done to prevent potential process failures prior to the first production run. Ideally the PFMEA should be conducted throughout the process design phase. Overall, FMEA is intended to be a dynamic and iterative process where practitioners review and update the analysis as new information becomes available, corrective actions are implemented, design phases progress, etc. FMEA requires different skills; hence, it is absolutely necessary to build an FMEA group usually organized and conducted by a FMEA process owner. This group may include representatives from the following areas: product design, testing, materials, suppliers/OEM, manufacturing and assembling, quality, and field service. The project leader plays a fundamental role in defining the rules and the organization of work. FMEA can represent a very powerful approach but in compliance with rules and personnel commitment, otherwise FMEA is only a time-consuming activity. There are several guidelines and standards for the requirements of FMEA as well as the recommended reporting format. Some of the main published standards for this type of analysis include: • MIL-STD-1629A; • J1739 from the Society of Automotive Engineers for the automotive industry;

8.2 Failure Modes and Effects Analysis

221

• AIAG FMEA-3 from the Automotive Industry Action Group for the automotive industry; • ARP5580 from the Society of Automotive Engineers for nonautomotive applications; • IEC 812 from the International Electrotechnical Commission; • BS 5760 from the British Standards Institution. In addition, many industries and companies have developed their own procedures to meet the specific requirements of their products/processes. The standards are slightly different, but the core of the FMEA procedure is the same: 1. 2. 3. 4. 5.

life application dealing with a fundamental part of a drink vending machine: the distribution valve system. These automatic machines for the preparation of various drinks are normally equipped with a multiway valve used for supplying water or steam to different collecting vessels, according to the drink required. The multiway valve is exposed to considerable stresses due to temperatures and pressures, and usually its behavior can significantly influence the total reliability of the machine (Fig. 8.1).

8.2.1 Product Analysis

FMEA group formation and rule sharing; product or process analysis; FMECA; risk evaluation; corrective action planning.

In the following pages, the DFMEA procedure (MILSTD-1629A standard) is detailed by means of a real-

The FMEA team must analyze the machine (in general, the system) with the goal to define the system structure having its subsystems and components placed at different hierarchical levels. This structure, usually in a top-down form, represents a very useful permanent reference when the system is very com-

Distribution valve subsystem Subsystem 1 Switch A Distribution valve subsystem Drink vending machine

Subsystem 3

Valve actuator Discharge pipe

……….. Subsystem 4

Fig. 8.1 Distribution valve subsystem, drink vending machine

………..

222

8 Effects Analysis and Reliability Modeling of Complex Production Systems

plex. This subsection arrangement is usually generated according to the different functions performed by subsystems, such as supply electrical energy, storage data, and sound recording. Normally each subsystem performs a single function. In this phase the analysis can usually require a lot of information, such as design drawings, description and operation documentation, and supplier information. In the real case discussed, the system has several subsystems, but the focus is on the distribution valve subsystem (item code 1100). Its “critical” components are an electrical switch (switch A), the valve actuator, and the discharge pipe: it is very important in this phase to concentrate the analysis on a small group of components having a strong impact on reliability. Machines have hundreds or thousands of items, and a thorough investigation is not applicable.

8.2.2 Failure Mode, Effects, and Causes Analysis Failures may potentially occur for each subsystem or function, resulting in several effects such as loss of production, no entrance of people, and absence of lighting. Usually each failure, or failure mode, can have several causes. A basic step of the FMEA procedure is the definition of the sequence of failure modes, effects, and causes. Typically data are arranged into a structured standard worksheet or a hierarchical diagram, as reported in Figs. 8.2 and 8.3, respectively (distribution valve subsystem – drink vending machine example). Switch A (item 1100.1), whose main function is to allow the distribution valve to supply the beverage, has three principal failure modes: oxidation, mechanical break, and pin disconnection from the connector. Columns 1 and 2 in the worksheet shown in Fig. 8.2 show, respectively, the item and its correspondent failure modes. Speaking about effects, one can distinguish among different categories: a local effect (FMEA worksheet, column 3), i. e., strictly concerning the item analyzed, a next-higher-level effect (FMEA worksheet, column 4), i. e., involving items set on the next-higher assembly level, and an end effect (FMEA worksheet, column 5), the most important in the FMEA.

Each failure mode can have different causes as reported in column 8 of the FMEA worksheet. Considering oxidation as a failure mode for switch A, the end effect is a difficult supply of beverage and the causes of oxidation can be a loss of water and steam and a problem with gaskets (tear and wear). Several FMEA styles (e. g., MIL-STD-1629A) potentially provide a failure detection method and a compensating provisions action (FMEA worksheet, columns 10 and 11). This supplementary information is very useful when corrective actions are investigated and implemented.

8.2.3 Risk Evaluation The core task of the FMEA is the evaluation of risks associated with the potential problems identified through the failure modes identification and analysis. The purpose of FMEA is to take actions in order to eliminate or reduce failures, starting with the highestpriority ones. It may be used to evaluate risk management priorities for mitigating known threat vulnerabilities. FMEA helps to select some remedial actions by reducing the cumulative impacts of life-cycle consequences resulting from a system failure. The risk of each failure is called “risk priority number” (RPN) and it is expressed by the product of severity (S ), occurrence (O), and detection (D). For a generic cause of failure i , RPNi D Si Oi Di :

(8.1)

Severity (Si ) is the amount of harm or damage the failure effect may cause to people or equipment. This parameter is rated following a qualitative scale. From the MIL-STD-1629A standard the correspondent magnitudes range from 1 to 4 as expressed in Table 8.1; this rate is reported in column 7 of the FMEA worksheet in Fig. 8.2. Occurrence (Oi ) is the rate stating the likelihood of occurrence for each cause of failure. The probability of occurrence ranges from extremely unlikely to frequent. Also in this case the evaluation is qualitative but it is clearly linked to the failure rate. This concept will be stressed later on when we speak about criticality analysis. From the MIL-STD-1629A standard the classification of occurrence is expressed in Table 8.2.

difficult supply of beverage

gaskets tear

water loss

Fig. 8.2 FMEA worksheet. RPN risk priority number, distribution valve subsystem

superficial cut

water loss

no supply of beverage

wear of internal crown

no supply of beverage

disconnection from connector

End Effects

(6)

no supply of beverage

Next Higher Level

(5)

mechanical break

Local Effects

(4)

difficult supply of beverage

Mission Phase/Operational Mode

(3)

oxidation

Failure Modes

(2)

1100.3 - discharge pipe keep clean supply system disconnection from chassis

1100.2 - valve actuator beverages supply

1100.1 - switch A supply permission

Item/Function

(1)

Causes

Sev - Si supply incorrect

assembly incorrect

vibrations from pump

thermal stress

pipe occlusion (residuals)

superficial treatment failed

assembly incorrect

assembly incorrect

presswork incorrect

normal use of disposal

assembly incorrect

vibrations from pump

vibrations from pump

wear

gaskets wear

gaskets tear

steam or water loss by valve

( 8)

(7) (10)

Failure Detection Method

(9)

Compensating Provisions

(11)

(13)

RPN

(12)

8.2 Failure Modes and Effects Analysis 223

Det - Di

Occ - Oi

224

8 Effects Analysis and Reliability Modeling of Complex Production Systems

1.1.1.1 - steam or water loss by valve (joint loosening) 1.1 - oxidation

1.1.1 - no supply of beverage

1.2 - mechanical break

1.2.1 - no supply of beverage

1.3 - disconnection from connector

1.3.1 - no supply of beverage

1.1 - wear of internal crown

1.1.1 - no supply of beverage

1.1.1.2 - gaskets tear 1.1.1.3 - gaskets wear

1.2.1.1 - wear 1100.1 - switch A 1.2.1.2 - vibrations from pump 1.3.1.1 - vibrations from pump 1.3.1.2 - assembly incorrect 1.1.1.1 - normal use of disposal 1.1.1.2 - presswork incorrect 1.1.1.3 - assembly incorrect 1.2.1.1 - assembly incorrect

1100 - distribution valve

1100.2 - valve actuator

1.2 - gaskets tear

1.2.1 - no supply of beverage

1.1 - disconnection from chassis

1.1.1 - water loss

1.2.1.2 - superficial treatment failed 1.1.1.1 - pipe occlusion (residuals) 1.1.1.2 - thermal stress

1100.3 - discharge pipe

1.1.1.3 - vibrations from pump

Fig. 8.3 FMEA diagram, distribution valve subsystem

In the FMEA worksheet (Fig. 8.2) this rate is posted in column 9. Detection (Di / is the likelihood that the failure will be detected. This parameter introduces an important point of view, often not considered in the classic magnitude-effect analysis. The difficulty of failure detection can represent a significant problem increasing the total criticality of a cause of a failure characterized by average severity and occurrence. Table 8.3 shows the criteria adopted for detection evaluation and the correspondent qualitative numerical ranking. Column 12 of the FMEA worksheet collects this ratio. The scales adopted by MIL-STD-1629A and presented here are only a model: various textbooks and

manuals addressing FMEA, or the standards adopted by major industries provide several rating scales, with the possibility for the team to create/modify them in order to fit the specific analysis. The basic concept remains to rate the failure risk by RPN. High values of RPN reveal critical causes of failure. The sum of the RPNi for a lower level (i. e., subsystem, subassembly, components) is the overall RPN for the upper level, up to the entire product. Considering to the distribution valve example, and in particular to switch A and its first failure mode (i. e., oxidation), the correspondent severity level is near the maximum (rate 3 – critical) because in this condition the customer has significant difficulties to obtain the

8.2 Failure Modes and Effects Analysis

225

Table 8.1 Severity rating scale (MIL-STD-1629A) Rate

Description

Criteria

1

Category IV – minor

2

Category III – marginal

3

Category II – critical

4

Category I – catastrophic

A failure not serious enough to cause injury, property damage, or system damage, but which will result in unscheduled maintenance or repair A failure which may cause minor injury, minor property damage, or minor system damage which will result in delay or loss of availability or mission degradation A failure which may cause severe injury, major property damage, or major system damage which will result in mission loss A failure which may cause death or weapon system loss (i. e., aircraft, tank, missile, ship, etc.)

Table 8.2 Occurrence rating scale (MIL-STD-1629A) Rate

Description

Criteria

1

Level E – extremely unlikely

2

Level D – remote

3

Level C – occasional

4

Level B – reasonably probable

5

Level A – frequent

Probability of occurrence is essentially zero during the item operating time interval. A single failure mode probability of occurrence is less than 0.001 of the overall probability of failure during the item operating time An unlikely probability of occurrence during the item operating time interval. A single failure mode probability of occurrence is more than 0.001 but less than 0.01 of the overall probability of failure during the item operating time An occasional probability of occurrence during the item operating time interval. A single failure mode probability of occurrence is more than 0.01 but less than 0.10 of the overall probability of failure during the item operating time A moderate probability of occurrence during the item operating time interval. A single failure mode probability of occurrence is more than 0.10 but less than 0.20 of the overall probability of failure during the item operating time A high probability of occurrence during the item operating time interval. A single failure mode probability greater than 0.20 of the overall probability of failure during the item operating time interval

Table 8.3 Detection rating scale (MIL-STD-1629A) Rate 1 2 3 4 5 6 7 8 9 10

Description

Criteria

Almost certain

Current controls almost always will detect the failure. Reliable detection controls are known and used in similar processes Very high likelihood current controls will detect the failure Good likelihood current controls will detect the failure Moderately high likelihood current controls will detect the failure Medium likelihood current controls will detect the failure Low likelihood current controls will detect the failure Slight likelihood current controls will detect the failure Very slight likelihood current controls will detect the failure Remote likelihood current controls will detect the failure No known controls available to detect the failure

Very high High Moderately high Medium Low Slight Very slight Remote Almost impossible

drink. The three causes of failure detected have an average value of probability of occurrence, but the higher level of probability is assigned to the wear of gaskets (ranked 4 in the occurrence scale), a cause linked to the natural use of the machine. All the above-mentioned causes are relatively easy to detect; the wear of gaskets is the higher level of criticality (ranked 5 – medium) in this case too. The result of the iteration of this approach to other components is the risk evaluation summarized in Fig. 8.4.

8.2.4 Corrective Action Planning The risk evaluation is the starting point for the design and the execution of corrective actions. The goal of FMEA is to anticipate potential problems and to perform activities in order to reduce and/or remove risks. RPN permits the interventions to be prioritized. It is worth remembering that RPN ratings are related to a specific analysis. A crossover comparison of some RPN values among different applications (product or process) is in fact meaningless.

difficult supply of beverage

gaskets tear

water loss

Fig. 8.4 Risk evaluation for distribution valve system (MIL-STD-1629A standard)

superficial cut

water loss

no supply of beverage

wear of internal crown

no supply of beverage

disconnection from connector

End Effects

(6)

no supply of beverage

Next Higher Level

(5)

mechanical break

Local Effects

(4)

difficult supply of beverage

Mission Phase/Operational Mode

(3)

oxidation

Failure Modes

(2)

1100.3 - discharge pipe keep clean supply system disconnection from chassis

1100.2 - valve actuator beverages supply

1100.1 - switch A supply permission

Item/Function

(1)

2

2

3

4

4

4

1

supply incorrect

4

vibrations from pump

4

2

thermal stress

assembly incorrect

3

2

superficial treatment failed

pipe occlusion (residuals)

2

4

assembly incorrect assembly incorrect

2

2

assembly incorrect

presswork incorrect

3

vibrations from pump

4

2

vibrations from pump

normal use of disposal

3

4

gaskets wear wear

3

gaskets tear

Failure Detection Method

3

(10)

(9)

Occ - Oi

steam or water loss by valve

Causes

Sev - Si 3

(8)

(7)

Compensating Provisions

(11)

5

5

6

4

9

10

3

3

7

2

2

9

8

2

5

4

10

40

48

16

54

68

18

48

56

32

16

108

64

24

60

36

18

Det - Di 2

(13)

RPN

(12)

226 8 Effects Analysis and Reliability Modeling of Complex Production Systems

8.2 Failure Modes and Effects Analysis

227

The RPN analysis recommends corrective actions focused on reducing a single factor or more than one factor. Usually the FMEA team provides a new level of RPN, the so-called revised RPN, to be compared with the initial RPN. The FMEA team must spend time analyzing the RPNi configuration. Typical instruments are the Pareto chart of RPNi , the occurrence–severity matrix, the causes by occurrence analysis, and the effects analysis. Application of these tools with reference to the distribution valve example is shown in Figs. 8.5–8.8. The most critical cause of failure has RPNi D 108, which corresponds to Si D 4, Oi D 3, and Di D 9 due to vibrations from the pump as a result of the disconnection of switch A from the connector. Others critical issues engage switch A and pump vibrations: in particular, a mechanical break is possible (RPNi D 64, Si D 4, Oi D 2, and Di D 8). Switch A has a very high occurrence among the greatest RPN values. Its problems are fundamentally due to pump vibrations and gaskets. The occurrence–severity matrix is another interesting tool for the risk assessment. The user can set three different regions on the two-dimensional space severity (on x-axes) and occurrence (on y-axes) by the

Causes Ranked by Initial RPN 120 108 100

Cause RPN

80

64 60 60

60

56

54 48 48 40

40

36 32 24

20

18 18

16 16 10

0

1

2

3

4

5

6

7

8

9

definition of high and low levels. The matrix gives a prompt idea about the criticality of the causes of failure. The analysis can be completed by other studies such as the causes by occurrence (Fig. 8.7) and the effects classification (Fig. 8.8). In conclusion, the analysis of RPNi allows one to prioritize some corrective actions usually linked to the product design. For the distribution valve case, the FMEA team decided to improve the first four criticalities sorted by the Pareto analysis of RPN. As mentioned, the more critical problems deal with the vibrations induced by the pump and the resistance and the retaining of valve gaskets. In particular, several corrective actions are defined: • A rubber bumper insertion in the fixing system between the pump and the chassis to reduce the vibrations induced on other components (i. e., switch A and discharge pipe). The responsibility is shared by the mechanical design division and the procurement division. The activity starts on 1 November 2008 and the due date is fixed at 1 June 2009. • A new switch design with mechanical redundancy to increase the availability of disposal. The responsibility is shared by the mechanical design division and the procurement division. The activity starts on 1 November 2008 and the due date is fixed at 1 June 2009. • A new connection system to avoid disconnection of electrical pins. The quality assurance division must guarantee the study and the procurement division must search for a new effective supplier. The starting date is 1 November 2008 and the new system must work before 1 April 2009. • A new material or new treatment for gaskets. At the same time a new profile is needed for the gasket to avoid tearing. The mechanical design division must develop the new profile, and the quality assurance division executes the experiments to validate new materials and a new profile. The procurement division must search for new suppliers. The activity starts on 1 November 2008 and the due date is 1 April 2009.

10 11 12 13 14 15 16 17

Cause

Fig. 8.5 Pareto analysis of initial RPN, distribution valve subsystem

The corrective actions provided have a significant potential effect on the criticality of the distribution valve, as confirmed by the 50% decrease of the criticality

228

8 Effects Analysis and Reliability Modeling of Complex Production Systems Occurrence/Severity Matrix (Initial Ratings) 5

Low Priority Line High Priority Line

Occurrence

4

x2

x2

3

x2

x2

Severity limit HIGH: 7 LOW: 3 Occurrence limit HIGH: 6 LOW: 4 2

x2

x3

1

0

0

1

2

3

4

Severity

High-priority causes: Normal use of disposal Assembly incorrect Gaskets wear Wear Vibrations from pump

(Item: 1100.2 - valve actuator) (Item: 1100.2 - valve actuator) (Item: 1100.1 - switch A) (Item: 1100.1 - switch A) (Item: 1100.1 - switch A)

Sev = 4, Sev = 4, Sev = 3, Sev = 4, Sev = 4,

Occ = 4 Occ = 4 Occ = 4 Occ = 3 Occ = 3

Medium-priority causes: Assembly incorrect Steam or water loss by valve Gaskets tear Presswork incorrect Vibrations from pump Assembly incorrect Assembly incorrect Superficial treatment failed Pipe occlusion (residuals) Thermic stress Vibrations from pump

(Item: 1100.3 - discharge pipe) (Item: 1100.1 - switch A) (Item: 1100.1 - switch A) (Item: 1100.2 - valve actuator) (Item: 1100.1 - switch A) (Item: 1100.1 - switch A) (Item: 1100.2 - valve actuator) (Item: 1100.2 - valve actuator) (Item: 1100.3 - discharge pipe) (Item: 1100.3 - discharge pipe) (Item: 1100.3 - discharge pipe)

Sev = 2, Sev = 3, Sev = 3, Sev = 4, Sev = 4, Sev = 4, Sev = 3, Sev = 3, Sev = 2, Sev = 2, Sev = 2,

Occ = 4 Occ = 3 Occ = 3 Occ = 2 Occ = 2 Occ = 2 Occ = 2 Occ = 2 Occ = 3 Occ = 2 Occ = 4

Low-priority causes: Supply incorrect

(Item: 1100.3 - discharge pipe)

Sev = 2, Occ = 1

Fig. 8.6 Occurrence–severity matrix, distribution valve subsystem

of the “original” causes at least. The FMEA procedure suggests a calculus of the new levels of severity, occurrence, and detection parameters (so-called revised) and in conclusion a new revised RPN is available.

Clearly, both the initial RPN and the revised RPN are based on an estimation of their factors, no mathematical models, or something similar supporting these evaluations. Figure 8.9 shows the action plan and the comparison between RPN values.

8.3 Failure Mode, Effects, and Criticality Analysis

229

Fig. 8.7 Causes by occurrence (distribution valve system)

Fig. 8.8 Effects classification (distribution valve system)

8.2.5 FMEA Concluding Remarks FMEA is a well-known qualitative reliability method. It is devoted both to the product and to the process analysis. It provides a systematic approach requiring all known or suspected potential failures to be considered. Usually the analysis directly results in actions to reduce failures and anyhow includes a follow-up system and reevaluation of potential causes of reliability problems. By paying attention to the customer point of view, it permits a tangible improvement of product and process reliability. Since FMEA represents a valid support to the design review provided by EN ISO 9001 and gives immediacy to the problem’s revision procedures, it should be approached together with the design phase as a whole. Some difficulties are of course related to its application. In particular, FMEA is a time-consuming process with very complex tasks taking hours or days to complete the process; it accounts for every cause of problems as a single event, and the combinations of events are captured as a single initiating event. Moreover, the process relies on recruiting the right participants and the personnel involved must be truthful about the respective activities. Nevertheless, it is worth mentioning some complications due to human error, some-

times overlooked because of the limited possibility of examination. Finally, it is important to remember that FMEA is only a qualitative procedure based on different scales of attributes such as severity, occurrence, and detection of failures, whose evaluations are dependent on the team involved. Just to overcome this last criticism, FMECA was developed as an extension of FMEA. The fundamental feature of FMECA is the introduction of the criticality factor, which is an effort to evaluate the criticality of the components on a quantitative basis instead of the qualitative approach adopted by FMEA.

8.3 Failure Mode, Effects, and Criticality Analysis FMECA differs from FMEA in investigating the criticality of failure in detail. This process systematically determines functions, functional failures, and failure modes of the production system, i. e., the equipment, with particular attention to the related effects, severity, and frequency of failure effects. A fundamental reference for the FMECA is represented by the MIL-STD-1629A standard. It provides two levels of criticality analysis: the qualitative and the quantitative FMECA.

Failure Modes

no supply of beverage

disconnection from connector

superficial cut

keep clean disconnection supply from chassis system

gaskets tear

water loss

water loss

difficult supply of beverage

beverages wear of internal no supply of supply crown beverage

no supply of beverage

difficult supply of beverage

End Effects

mechanical break

supply oxidation permission

Function

Si

24 64 rubber bumper insertion under pump fixing system - switch with mechanical redundancy

56 48 18

16 32

4 1

2 assembly incorrect supply incorrect

5 5

4 6

40 10

16 48 rubber bumper insertion under pump fixing system

54

10 60 developing and testing new material or new treatment - new profile for gasket to reveal tear

2 4

2

superficial treatment failed

7 3 3

9

2 4 2

presswork incorrect assembly incorrect 3 assembly incorrect

2 2

3

2 4

assembly incorrect 4 normal use of disposal

Corrective action description

36 60 developing and testing new material or new treatment - new profile for gasket to reveal tear

18

RPNi

9 108 rubber bumper insertion under pump fixing system - new connection system

2 pipe occlusion (residuals) thermal stress vibrations from pump

3

2 8

4 5

3 4

3 2

2

Di

3

4 vibrations from pump

4 wear vibrations from pump

3 steam or water loss by valve gaskets tear gaskets wear

Causes

Oi

Fig. 8.9 Planned corrective actions and revised RPN, distribution valve subsystem

1100.3 discharge pipe

1100.2 valve actuator

1100.1 switch A

item

quality assurance / mech. design / procurement

quality assurance / mech. design / procurement

quality assurance / procurement

mech. design / procurement

quality assurance / mech. design / procurement

Responsability

2008-11-01

2008-10-01

2008-11-01

2008-11-01

2008-10-01

Planned start date

2009-04-01

2009-02-01

2009-04-01

2009-06-01

2009-02-01

Completion Date

Sr 2

3

4

4

3

3

1

1

1

2

4

8

9

8

3

Dr

revised Or

initial

RPNr 24

24

36

32

18

230 8 Effects Analysis and Reliability Modeling of Complex Production Systems

8.3 Failure Mode, Effects, and Criticality Analysis

8.3.1 Qualitative FMECA The qualitative FMECA approach is a direct followup of the FMEA result. The target is to assign a priority to the failure modes and to group them in different “classes of criticalities,” usually three, according to a qualitative criticality matrix including the parameters severity and occurrence. The first factor can be evaluated by four different levels, from minor to catastrophic, as used for FMEA (Table 8.1). In the same way, the occurrence of the second factor is evaluated according to a qualitative scale ranging from extremely unlikely to frequent, as in FMEA (Table 8.2). Each failure mode is classified into the matrix depending on its own evaluations, usually indicated as Si and Oi for severity and occurrence, respectively. The most critical failure modes are revealed immediately, since three areas of criticalities, low, medium, and strong as in Fig. 8.10, are provided as a standard. The relative position of each failure mode with respect to the position of the “best” and “worst” categories gives a qualitative idea of its corresponding criticality level. The qualitative FMECA applied to the example of the distribution valve system is summarized by the criticality matrix in Fig. 8.11. Comparing some failure modes, the oxidation of switch A contacts, the wear of the internal crown of the valve actuator, and the mechanical break of the switch are very critical, while the disconnection of the discharge pipe from chassis failure mode has a medium level of criticality.

231

On one hand, the simplicity of the approach makes it suitable as a preliminary activity in order to drive the qualitative FMECA; however, on the other hand, it is sometimes very hard to estimate the qualitative evaluations of factors in a significant way.

8.3.2 Quantitative FMECA This approach is based on a quantitative procedure representing the most rigorous method currently available. The fundamental goal is the development of a numerical expression of the item criticality. Considering an item having c significant components, the correspondent item criticality is IC D

c X

CCi ;

(8.2)

i D1

where CCi is the criticality of component i defined as CCi D

m X

FMCij ;

(8.3)

j D1

where m is the number of failure modes for component i and FMCij is the failure mode criticality of failure mode j for component i . Each failure mode is characterized by a criticality value derived from FMCij D CUi .t  /  RUij  PLij ;

(8.4)

Level A frequent

Occurrence

Level B reasonably probable

strong

Level C occasional

medium

Level D remote

low

Level E extremely unlikely

Fig. 8.10 Criticality matrix and criticality regions

IV - minor

III - marginal Severity

II - crical

I - catastrophic

232

8 Effects Analysis and Reliability Modeling of Complex Production Systems

PROBABILITY OF OCCURRENCE LEVEL ( INCREASING LEVEL OF PROBABILITY ----------> )

(HIGH)

(LOW)

Level A frequent Level B reasonably probable Level C occasional

- disconnection from chassis

- oxidation

- gaskets tear

- mechanical break

Level D remote

- disconnection from connector

Level E extremely unlikely

- superficial cut

Category IV - minor

- wear of internal crown

Category III - marginal

Category II - critical

Category I - catastrophic

SEVERITY CLASSIFICATION (INCREASING LEVEL OF SEVERITY ------>)

Fig. 8.11 Criticality matrix, distribution valve system

where t  is the operating time, CUi .t  / is the unreliability of component i at operating time t  , RUij is the ratio of unreliability of failure mode j for component i , and PLij is the probability of loss of function, due to the failure mode j for component i . As shown in Eq. 8.4, for each failure mode the criticality is the product of three numerical factors. The first one, CUi .t  /, is common for all the failure modes of the same component, and represents the unreliability of the component at the operating time t  , thus disclosing a bridge between the quantitative FMECA and the theory of reliability. The definition of the component unreliability requires the operating time setting and the evaluation of the time-dependent failure distributions through well-known mathematical approaches, e. g., Weibull and exponential, as discussed in Chaps. 5 and 6. The ratio of unreliability RUij of the failure mode j is the probability that the component failure will be due to the considered failure mode j ; it is the percentage of failures, among all the failures allowed for the component, that will be caused by the given mode. It is important to note that the total percentage assigned to all modes must be obviously equal to 100%: m X j D1

RUij D 1

(8.5)

The probability of loss PLij is the probability of the loss of function at the occurrence of the considered failure mode j . This value is often equal to 1, because the failure gives rise to a complete loss of functionality of the component. In conclusion, the quantitative FMECA requires a procedure based on several steps: • definition of the reliability statistical distribution for different components of each item; • definition of an analysis operating time; • identification of the part of unreliability assigned to each potential failure mode; • rating of the probability of loss of function resulting from each failure mode that may occur; • calculation of the criticality for each component; • calculation of total item criticality by the sum of previous calculated criticalities. The final results are numerical evaluations of item criticalities which represent the starting points for a critical analysis and for the corrective action plan.

8.3.3 Numerical Examples We now present two numerical examples. Consider an item X, composed of two components A and B. The experimental evidence permits

8.3 Failure Mode, Effects, and Criticality Analysis

233

Table 8.4 Statistical distribution of reliability of components A and B f .t / Component A Component B

Exponential Normal

Table 8.5 Statistical distribution of reliability of components of the distribution valve system

Parameters

f .t / 1

.t / D 0:000207 h  D 6:578 h  D 1:211 h

an evaluation of their reliability performance, summarized in Table 8.4. Setting the operating time t  D 6;000 h, the correspondent unreliabilities of the two components are CUA D FA .6000/ D 0:712; CUB D FB .6000/ D 0:316; Consider component A responsible for a generic function, named “function A,” and two failure modes, named “failure mode A.1” and “failure mode A.2,” generating, respectively, two causes named “cause A.1.1” and “cause A.1.2” and a single cause A.2.1. Failure mode A.1 is responsible for 60% of the failures of component A, then the remaining 40% is due to failure mode A.2. Failure mode A.1 gives rise to a complete loss of function A, while the probability of loss of function for failure mode A.2 is about 90%. Focusing on failure modes, FMCA;1 D CUA  RUA;1  PLA;1 D 0:712  0:6  1 D 0:427; FMCA;2 D CUA  RUA;2  PLA;2 D 0:712  0:4  0:9 D 0:256: Then the criticality of component A is CCA D FMCA;1 C FMCA;2 D 0:427 C 0:256 D 0:683: Similarly for component B the criticality is CCB D 0:269. In conclusion, item X has a criticality defined by the sum of the criticalities of its components:

Switch A

Normal

Valve actuator Discharge pipe

Exponential Weibull

 D 752 h  D 321 h .t / D 0:001 h1 ˇ D 2:766 D 2;463 h

Now consider the application of the distribution valve system, the significant components are switch A (ID 1100.1), the valve actuator (ID 1100.2), and the discharge pipe (ID 1100.3). For each of them the failure statistical distributions are defined in Table 8.5. The operating time is set to 1;000 h; for a drink vending machine, having an average operating of about 4 hours per day, this time represents more or less 1 year of work, that is the time between two consequent overhaul interventions. Figure 8.13 shows the final result of the quantitative FMECA approach. The results of the quantitative FMECA have different levels of detail: the criticality index can be defined for a single failure mode, or for a single component, i. e., groups of failure modes, or finally for a single item, i. e., groups of components. This feature allows a complete top-down analysis for the research of the most critical items of a product, its most critical components, and their related failure modes. In spite of this, a very effective corrective action plan can be developed. The distribution valve system has a criticality index of 1.289 fundamentally due to the criticality of switch A (0.689) and of the valve actuator (0.533). The discharge pipe has a secondary effect on the criticality of the entire item (Table 8.6). Analyzing the criticality of failure modes, the oxidation of contacts, the mechanical break for switch A, and the wear of the internal crown for the valve ac-

Table 8.6 Distribution valve criticality and component criticalities Criticality

ICX D CCA C CCB D 0:683 C 0:269 D 0:952: Figure 8.12 presents a typical worksheet used for the quantitative FMECA populated with the data of the previous example referred to item X.

Parameters

1100 – distribution valve 1100.1 – switch A 1100.2 – valve actuator 1100.3 – discharge pipe

1.289 0.689 0.533 0.077

234

8 Effects Analysis and Reliability Modeling of Complex Production Systems

Fig. 8.12 Quantitative failure mode, effects, and criticality analysis (FMECA) worksheet (item X example)

Table 8.7 Failure mode criticalities for the distribution valve system Failure modes and causes Wear of internal crown – Normal use of disposal – Presswork incorrect – Assembly incorrect Mechanical break – Wear – Vibrations from pump Oxidation – Steam or water loss by valve – Gaskets tear – Gaskets wear Disconnection from chassis – Pipe occlusion (residuals) – Thermal stress – Vibrations from pump Disconnection from connector – Vibrations from pump – Assembly incorrect Gaskets tear – Assembly incorrect – Superficial treatment failed Superficial cut – Assembly incorrect – Supply incorrect

Mode criticality 0.491

0.351

0.281

0.071

0.047

0.042

0.006

tuator are clearly very critical modes. The remaining modes have marginal criticalities. In conclusion, the product designers must focus their attention on the causes of these critical modes, listed in Table 8.7. The characteristic numerical approach of the quantitative FMECA allows a robust comparison in terms

of criticalities among different items of a product, and moreover gives priority to the corrective actions to be taken, ranking the failure modes and the related causes. It is important to note that this robustness is paid for, on the other hand, in terms of the time spent collecting data and developing the calculus of criticality factors. Moreover, the quantitative FMECA also requires some subjective assumptions; in particular, the unreliability ratio of failure mode j for component i RUij and the probability of loss of failure mode j for component i PLij depend on personal evaluations by the engineers, the technicians, and the practitioners who will develop the analysis. For this reason, some authors consider FMEA and in particular FMECA very effective instruments in the product/process design phase, but suggest their use exclusively for a comparison among the different failure modes or/and the components of a single product or process. In the case of a cross-check of the results among different products or processes, these methods reach their limits. Another typical result of the quantitative approach is the quantitative criticality matrix. It represents a hybrid matrix mixing the severity evaluation and the criticality value of each failure mode. As well as the FMEA criticality matrix, it usually individuates three zones characterized by different levels of criticality. Figure 8.14 shows the quantitative criticality matrix for the distribution valve system.

Fig. 8.13 Quantitative FMECA worksheet for the distribution valve system

8.3 Failure Mode, Effects, and Criticality Analysis 235

236

8 Effects Analysis and Reliability Modeling of Complex Production Systems

CRITICALITY NUMBER (Cr) ( INCREASING LEVEL OF CRITICALITY ------ > )

(HIGH)

(LOW)

0.491

- wear of internal crown

0.351

- mechanical break

- oxidation

0.281

- disconnection from chassis

0.071

0.047

- disconnection from connector

- gaskets tear

0.042

- superficial cut

0.006 Category IV - minor

Category III - marginal

Category II - critical

Category I - catastrophic

SEVERITY CLASSIFICATION (INCREASING LEVEL OF SEVERITY ------ >)

Fig. 8.14 Quantitative FMECA matrix for the distribution valve system

8.4 Introduction to Fault Tree Analysis FTA is a systematic technique which is used to acquire information on a system, in the case of normal behavior but, in particular, in the presence of a failure, in order to support the very complex decision-making process during the design stage as well as its managing and controlling activities. This process generally involves people dealing with the system, from suppliers to customers passing through managers and employees working daily within the system. This analysis can also support the decision-making process developed by safety and maintenance engineers who plan and organize preventive and/or breakdown maintenance and monitoring activities on the production systems. The fault tree is a deductive system analysis by which the analyst postulates that the system could fail in a certain way and attempts to find out how the system or its components could contribute to this failure. Born as a qualitative model, it turned into a quantitative tool: for this reason in this chapter qualitative and quantitative analyses are distinguished and applied to trivial academic examples and some industrial case studies. A fault tree is a whole set of entities called “gates” addressing the bottom-up transmission of fault logic. These gates represent the relationships of events for the occurrence of a higher event, called “father event.” The higher event is the output of the gate, while the

events at a lower level, also called “sons of the father,” are the input. Figure 8.15 reports a list of main typologies of events, gates, and transfers. Figure 8.16 shows a list of gates available in the commercial Relex® Reliability software. Figure 8.17 illustrates a FTA applied to an elevator, here referred to as a particular production system. The top event “passenger injury which occurs in an elevator” is analyzed by Relex® Reliability software. In general, the top event is the result of different combinations of basic events identified for the components of the system. The behavior of every element in the system is known in terms of failures and repairs, and it can be modeled by the usual parameters coming from the reliability evaluating activities. With reference to the failure rate, two kinds of components can be mainly distinguished: passive and active components. A passive, or quasi-static, component transmits a signal, e. g., a current or a force: the failure rates are below 104 per demand, i. e., about 3  107 h1 . An active component causes or modifies a signal above this value. Usually there are 3 orders of magnitude between these rate values. In the case of failure of an active component, e. g., a switch in an electrical circuit, a hydraulic pump, or a valve regulating the fluid flow in a piping system, the output signal could be incorrect or absent, while the failure of a passive component, e. g., an electric wire in a circuit or a pipe in a piping system, can result in a no-signal transmission.

8.4 Introduction to Fault Tree Analysis Symbol

237

Name

Description

Basic event

A fault event which does not require further development

Top event

This event is related to a failure mode of the production system. The aim of a FTA is the characterization of this event

Conditioning event

It specifies the condition and/or the restrictions applied to a logic gate (e.g., a P-AND gate)

Intermediate event

It occurs because of one or more former causes acting through logic gates

AND gate

Output fault occurs if all input faults occur

OR gate

Output fault occurs if at least one of the input faults occurs

XOR gate (exclusive OR gate)

Output fault occurs if solely one of the input faults occurs

P-AND gate (priority AND gate)

It is a special case of an AND gate. Output fault occurs if all of the input fails in a specific sequence, stated by a conditioning event

INHIBIT gate

The output is caused by a single input if only it is conditional, i.e., under the condition specified by the conditioning event

Transfer IN

It points out that the tree is developed further at the transfer OUT

Transfer OUT

It shows the portion of the tree that has to be attached to the related transfer IN

Fig. 8.15 Main gates, events and transfers in a fault tree analysis (FTA)

238

8 Effects Analysis and Reliability Modeling of Complex Production Systems

In order to introduce the reader to the meaning and use of a fault tree, Fig. 8.18 illustrates a two-input OR gate, where A and B are the input events and C is the output. By Eq. 5.9 the probability of event C can be expressed as follows: P .C / D P .A/ C P .B/  P .A \ B/ D P .A/ C P .B/  P .A/P .BnA/:

(8.6)

Equation 8.6 can be properly modified in accordance with the following hypotheses: 1. A and B are mutually exclusive events: ( P .A \ B/ D 0; P .C / D P .A/ C P .B/: Fig. 8.16 Gate list in Relex® Reliability software

Fig. 8.17 FTA, passenger injury in an elevator (Relex® Reliability software)

8.5 Qualitative FTA

239

8.5.1 Fault Tree Construction Guidelines Before the introduction of the main notation and properties of Boolean algebra, a few guidelines for the construction of a fault and its application to a production system, with a previously identified top event, could be useful. It is a top-down process of analysis starting from the top event defined for the system, or a generic part (subsystem) of the system: Fig. 8.18 OR gate

2. A and B are independent events: ( P .B=A/ D P .B/; P .C / D P .A/ C P .B/  P .A/P .B/: 3. Event B is completely dependent on event A: ( P .B=A/ D 1;

1. Identification of a more detailed event. The generic event or input is substituted by a new and more detailed output event, as in Fig. 8.19. 2. Classification. The generic input event is analyzed in depth by the identification of two, or more, basic and alternatives configurations, e. g., cases 1 and 2 in Fig. 8.20. This identification is based on a process of classification applied to the input event and the introduction of an OR gate which classifies the available configuration (and/or failure) modes of the starting event, as illustrated in Fig. 8.20.

P .C / D P .A/ C P .B/  P .A/ D P .B/: Figure 8.17 reports the value of unavailability, or failure probability, for every basic event or combination; e. g., the failure probability for the basic Event11 “controller failure” is Q D 0:00741239, while for Gate5 “door close failure” Q D 0:00989076. The determination of these measures of unavailability, accomplished by ENF values, MTTR values, etc., is the result of the so-called quantitative FTA, properly illustrated and exemplified in Sect. 8.6. The next section presents the “qualitative” FTA, whose aim is the identification of the so-called cut sets, which are the minimal combinations of primary failure components/events causing the top event of the production system.

Abstract event

More detailed event

Fig. 8.19 A more detailed event

Event

8.5 Qualitative FTA The objective of this section is to identify the minimal cut sets (MCS) of a fault tree defined for a specific top event in a production system. A MCS is an intersection of “primary,” or “basic,” events essential for the top event: if a single failure in the cut set does not occur, there is no top event failure. The identification of cut sets can be effectively supported by the application of the Boolean algebra, whose basic notation and properties are introduced below.

+

Case A

Case B

Fig. 8.20 Classification of failure modes

240

8 Effects Analysis and Reliability Modeling of Complex Production Systems

+

Event

Cause A

Cause B

Fig. 8.21 Identification of distinct causes

x

Event

Absence of protection/ protective action

Hazard/failure event

Fig. 8.22 Absence of protection/protective action

8.5.2 Numerical Example 1. Fault Tree Construction Figure 8.24 presents a pumping system supplying cooling water for temperature control of a reactor and the related tank pressure. In particular, given the catastrophic top event “reactor explosion” and knowing the reliability performance indexes for a set of basic components, Fig. 8.25 shows a fault constructed according to the previously illustrated guidelines. The breakage of valves V1 and V2, of pumps P1 and P2, of processor PR, and the absence of electric power PW are the failure basic events defined for the system. Only supply line 2, exactly like line 1, is considered in the fault tree. The proposed fault tree corresponds to the hypothesis of redundant pumping lines in parallel, i. e., the cooling service is ensured by a single line at least. If the two circuits are both required simultaneously to supply the reactor’s demand, an OR gate replaces the AND gate, and the fault tree changes as illustrated in Fig. 8.26.

x

Event

3. Identification of distinct causes. Some different causes for the generic failure event are identified, and an OR gate is introduced, as in Fig. 8.21. The generic cause is capable of generating the failure event. 4. Failure event and absence of protection. A generic failure event is coupled with the absence of protection or a protective action (see Fig. 8.22). An AND gate is introduced. 5. Concurrent causes. The generic failure event occurs only in the case of concurrent causes, as exemplified in Fig. 8.23.

Cause 1

Cause 2

Fig. 8.23 Concurrent causes

Fig. 8.24 Pressure control in a chemical reactor

8.5 Qualitative FTA

241

No cold water from line 2

Reactor explosion

+

Pressure tank rupture

Internal overpression V2 does not open

P2 does not work

+

Cold water supplier fails

X

V2 broken

PR broken No electric power

No cold water from line 1

+

Temperature out of control

No cold water from line 2

P2 broken

No electric power

Fig. 8.25 Fault tree construction. AND gate, configuration A

8.5.3 Boolean Algebra and Application to FTA The Boolean algebra, or “algebra of events,” is particularly useful for conducting a FTA from both a qualitative and a quantitative point of view. In particular, this algebra supports the designer and manager of a production system in answering to this critical question: What are the basic/primary events causing the defined top event for the production system? Given a production system and a top event related to the system function, it is possible to construct a fault tree. The Boolean algebra materially supports the application of reducing and simplifying properties to obtain an equivalent fault tree (EFT), as a result of different MCS.

Boolean algebra is the algebra of two values introduce by George Boole, a British mathematician and philosopher of the nineteenth century. These values are usually taken to be 0 and 1, corresponding to false and true. In particular, given a generic event A, a Boolean variable XA can be defined as follows: ( 0 if event A does not occur (8.7) XA D 1 if event A occurs. Tables 8.8 and 8.9 refer to the main properties and rules of the Boolean algebra, useful for conducting a FTA and in particular for obtaining the EFT. The significance and validity of the Boolean rules can be checked by the application of Venn diagrams. An EFT is a tree made of two levels: level 0 identifies the top event and level 1 the set of MCS, as il-

242

8 Effects Analysis and Reliability Modeling of Complex Production Systems

No cold water from line 2

Reactor explosion

+

Pressure tank rupture

Internal overpression V2 does not open

P2 does not work

+

Cold water supplier fails

+

Temperature out of control

+

V2 broken

PR broken No electric power

No cold water from line 1

P2 broken

No electric power

No cold water from line 2

Fig. 8.26 Fault tree construction. OR gate, configuration B

lustrated in Fig. 8.27. A MCS defines a failure mode of the top event, because it is a smaller combination of component failures capable of causing the top event, if all component failures occur. A generic MCS can be represented by the fault tree in Fig. 8.28. The application of the Boolean properties previously illustrated allows one to express the MCS for the top event in an EFT as follows: TOP D

n X i D1

MCSi D

mi n Y X i D1

 Cij ;

(8.8)

j D1

where MCSi is the MCS i for the top event, n is the number of MCS, mi is the number of primary events in MCS i , and Cij is primary event j for MCS i . Every algebraic operation in Eq. 8.8 is executed in accordance with Boolean definitions and properties, as illustrated below.

It is possible to rank the MCS according to their size, thus weighting the relevance of a failure; moreover, it could be useful to conduct a quantitative evaluation of a fault tree in order to properly identify the system’s criticalities, as illustrated below.

8.5.4 Qualitative FTA: A Numerical Example This numerical example refers to the system represented in Fig. 8.24, which is useful for identifying the MCS, given the top event “reactor explosion.” In Sect. 8.5.2 two different reliability configurations, A and B, were considered, but in this case the FTA applies to configuration A made up of two redundant lines for cooling water in parallel. Figure 8.30 presents

8.5 Qualitative FTA

243

Table 8.8 Boolean algebra and Venn diagrams Event

Venn diagrams

Boolean algebra

U

A

A

XA

U

AN

Boolean variable

Complement or negation XAN D XN A D 1  XA

A

Disjunction ˚ U

A [ B or A C B A

XA[B D XA ˚ XB D

B

a

Xi

iDA;B

D 1  .1  XA /.1  XB /

Conjunction ˝ U

A \ B or A  B A

B

XA\B D XA ˝ XB Y D Xi D XA XB iDA;B

˚ Boolean sum, ˝ Boolean product

Table 8.9 Rules of Boolean algebra Events domain

Boolean algebra

Operation with

A[;D A

X A C ; D XA C 0 D XA

events ; and U

A\;D ;

X A  ; D XA C 0 D ; D 0

U [ADU

XU C X A D X U D 1 XU  X A D X A

Complementation

U \ADA AN \ A D ;

Commutative law

A[B DB [A

X A C XB D XB C XA

AB DB A

X A  XB D XB  XA

A [ .B [ C / D .A [ B/ [ C

XA C .XB C XC / D .XA C XB / C XC

A \ .B \ C / D .A \ B/ \ C

XA .XB XC / D .XA XB /XC

A \ .B [ C / D .A \ B/ [ .A \ C /

XA .XB C XC / D .XA XB / C .XA XC /

Associative law Distributive law

XAN  XA D 0

A [ .B \ C / D .A [ B/ \ .A [ C /

XA C .XB XC / D .XA C XB /.XA C XC /

Law of absorption

A [ .A \ B/ D A

XA C .XA XB / D XA

A \ .A \ B/ D A \ B

XA .XA XB / D XA XB

Idempotent Law

A[ADA

X A C XA D XA

A\ADA

X A  XA D XA

244

8 Effects Analysis and Reliability Modeling of Complex Production Systems Top event

Level 0

OR

Level 1

Cut set n

Cut set 1

Cut set 2 Cut set i

Fig. 8.27 Equivalent fault tree (EFT)

the EFT resulting from the application of the qualitative evaluation of the fault tree in Fig. 8.29, in accordance with the following expression: AND gate

TOP D Œ.V1 C PR/ C .P1 C PW/  Œ.V2 C PR/ C .P2 C PW/ D V1  V2 C V1  PR C V1  P2 C V1  PW C PR  V2 C PR C PR  P2 C PR  PW C P1  V2 C P1  PR C P1  P2 C P1  PW C PW  V2 C PW  PR C PW  P2 C PW D V1  V2 C V1  P2 C P1  V2 law of absorption

C P1  P2 C PR C PW D

5 X

MCSi :

i D1

Level 1

CSi -Cut set i

AND

Cimi

Ci1 Ci 2

Cij

Fig. 8.28 EFT and generic cut set

Level 2

On a whole there are five MCS, two on five of cardinality 1, i. e., including only one basic event (PR and PW) and the remaining three of cardinality 2 (V1  V2; V1  P2; P1  V2; P1  P2). Figures 8.29 and 8.30 are both based on the introduction of a few “mirrored blocks.” A mirrored block is an event repeated more than once in the system: e. g., the basic event “no electric power” is repeated four times and it certainly represents a very critical component for the system, especially in the case of a great value of failure rate .t/. Figure 8.31 reports the equivalent reliability block diagram generated by the fault tree in Fig. 8.29 and made up of two parallel and identical subsystems corresponding to the inputs of the AND gate in Fig. 8.25. Similarly, Fig. 8.32 presents the equivalent reliability block diagram generated by the EFT in Fig. 8.30. Figure 8.33 presents the fault tree generated for the not redundant configuration B, where the two lines are both necessary to properly control the reactor temperature level. In this special configuration there are six cut sets of cardinality 1, because every basic event is critical. Figure 8.34 lists the cut sets obtained by the qualitative analysis applied to the system in configuration B.

8.6 Quantitative FTA The aim of quantitative FTA is the determination of some reliability and probabilistic parameters, mainly referred to the top event declared for the production system investigated. This analysis can be performed

8.6 Quantitative FTA

245

Top

AND

OR

OR

OR

V1 broken

OR

OR

No electric power

P1 broken

PR broken

No electric power

V2 broken

No electric power

OR

PR broken

P2 broken

Fig. 8.29 FTA, “reactor explosion.” Configuration A – “redundancy.” ReliaSoft® software Top

OR

AND

V1 broken

V2 broken

AND

P1 broken

P2 broken

AND

AND

P1 broken

V2 broken

P2 broken

AND

V1 broken

AND

PR broken

PW broken ®

Fig. 8.30 Qualitative fault tree evaluation. EFT. Configuration A – “redundancy.” ReliaSoft software

V2 broken

No electric power

PR broken

P2 broken

No electric power

Extra Starting Block

Extra Ending Block

V1 broken

No electric power

PR broken

P1 broken

No electric power

Fig. 8.31 Equivalent reliability block diagram, “reactor explosion.” Configuration A – “redundancy.” ReliaSoft® software

No electric power

246

8 Effects Analysis and Reliability Modeling of Complex Production Systems

V2 broken

P2 broken

Extra Starting Block

V2 broken

Extra Node3 V1 broken

Extra Node4 P1 broken

V1 broken Extra Node5

P1 broken

PR broken

PW broken

P2 broken

Fig. 8.32 Equivalent reliability block diagram by the EFT. Configuration A – “redundancy.” ReliaSoft® software

Top

OR

OR

OR

OR

V1 broken

OR

OR

No electric power

PR broken

P1 broken

No electric power

V2 broken

No electric power

OR

PR broken

P2 broken

No electric power

Fig. 8.33 FTA, “reactor explosion.” Configuration B – “no redundancy.” ReliaSoft® software

OR

once MCS have been identified. It is a sequential evaluation which firstly determines the failure probability for the components, then the MCS, and finally the probabilities for the system, given the top event. The main equations for the determination of these probabilities are give as follows:

Fig. 8.34 Qualitative fault tree evaluation. EFT. Configuration B – “no redundancy.” ReliaSoft® software

• Component failure probability. Generally, for any component, or basic primary event, a constant failure rate per hour is assumed, and any timedependent effect is ignored. If a generic component is considered, it could be necessary to distinguish a “standby failure rate” from an “operating failure rate”: as a consequence, the proper failure rate has to be coupled to the proper time period, standby

Top

V1 broken

V2 broken

P1 broken

P2 broken

PR broken

PW broken

8.6 Quantitative FTA

247

time t or operating time t, respectively. The component failure probability, which mainly refers to the nonrepairable items, is Fj .t/ D Fj;s .ts / C Œ1  Fj;s .ts /Fj;o .to /;

(8.9)

where s is the standby phase, ts is the ready (i. e., standby) time period, o is the operating phase, and to is the operating time period. Assuming an exponential distribution for the random variable t, one can approximate the cumulated value F .t/ by its first-order term, when t < 0:1, as follows: F .t/ Š t;

(8.10)

where  is the conditional and constant rate defined for the variable t. In particular, if t is the time to failure (ttf), then F .t/ is the failure probability function (unreliability) and  is the constant failure rate. For repairable failures the constant asymptotic unavailability of a component is quantified by qj

D

Dconstant Dconstant

MTTR     D Š ; C    (8.11)

• Component failure occurrence rate. This rate is defined for both repairable and nonrepairable components or systems. For nonrepairable items it is defined as w.t/ D f .t/ D  et ;

(8.15)

where f .t/ is the probability density function of the ttf. For both unrepairable and repairable failures .t/ is a reasonable approximation of this rate. • Failure occurrence rate of a cut set given a TOP event. A MCS failure occurs at time t to t C t if all components except one are down at time t, and the other component fails at time t to t C t. Consequently, X Y wj .t/ qk .t/; (8.16) wCSi .t/ D j 2CSi

k¤j k;j 2CSi

where wj .t/ is the failure rate of component j in MCS i . • ENF for a cut set. The ENF for a cut set CSi on a time period T is ENFCSi .T / D WCSi .0; T / D WCSi .T / ZT

where  is the repair rate. • Failure probability and unavailability of a cut set given a top event. The general model for the evaluation of cut set unavailability, equivalent to failure probability, is Y qCSi .t/ D qj .t/; (8.12)

D

wCSi .t/ dt 0

D

ZT  X 0

wj .t/

j 2CSi

Y k¤j k;j 2CSi

(8.17)

j 2CSi

where CSi is cut set i and qj .t/ is the unavailability of component j which belongs to CSi . • Unavailability of the system given a top event. a Y qCSi .t/ D 1  Œ1  qCSi .t/: QS .t/ D i

i

(8.13)

where T is the time period. • ENF of a system on a time period T, given a top event. ENF.T / D WS .T / n\ o X D WCSi .T /  Pr E.CSi / i

A simplified equation quantifying the unavailability of the system is X QS .t/ Š qCSi .t/: (8.14) i

 qk .t/ dt;



X

i

WCSi .T /;

(8.18)

i

where Prf: : :g is the failure probability and E.CSi / is the failure event defined for cut set i .

248

8 Effects Analysis and Reliability Modeling of Complex Production Systems

For the system the ENF is generally quantified by the following expression: X ENF.T / Š WCSi : (8.19)

TOP

i

• Virtual MTTR of a system given a top event. The following equations quantify the MTTR for the production system, given a top event: 8 QS .T / ˆ ˆ < MTTRS Š ws .T / (8.20) ˆ Ws .T / ˆ : ws .T / D ; T

G1

where ws .T / is the average estimated failure rate for the system.

G2

C G3

G4

G5

8.6.1 Quantitative FTA, Numerical Example 1 The fault tree reported in Fig. 8.35 relates to a repairable system and five repairable components, or basic events, A, B, C, D, and E, having well-known failure and repair behaviors. The analyst needs to quantify the unavailability, the ENF, and the MTTR of the system for a given top event and assuming a period of time T equal to 8;000 h. Table 8.10 presents the values of the failure and repair rates assuming an exponential distribution, i. e., random failure and repair durations, for ttf and the time to repair (ttr). By the application of the Boolean algebra, three MCS can be identified, each made up of two basic components: TOP D AB C ABE C ABD C ABC C EC C CD D AB C EC C CD: The quantitative analysis of the fault tree is found on the values of availability and unavailability for each basic component illustrated in Table 8.11. In particular, the unavailability has been quantified by the application both of the simplified model in Eq. 8.11, as reported in the fourth column in Table 8.11, and the exact exponential analytical model illustrated in Chap. 5 (Eq. 5.83) as reported in the fifth column in Table 8.11. The reliability of the component, representing the survival function of the item to the first failure, has been quantified by the application of the simplified model [see Eq. 8.10 for the failure probability function F .t/], as reported in the sixth column in Table 8.11, and of

A

B

A

B

D G6

E

D

Fig. 8.35 Fault tree, numerical example 1

the exact model (see Eq. 5.27), as reported in the seventh column in Table 8.11. Sometimes the simplified analytical models previously introduced are not applicable, as demonstrated by the value 2.4 assumed by the reliability for component C, while for other applications, such as for basic event D, the exact and simplified values of reliability significantly differ. A similar consideration can be made for the estimated values of availability.

Table 8.10 Reliability parameters, numerical example 1 Basic event

 (h1 )

 (h1 )

A B C D E

2  105 105 3  104 104 105

102 5  102 0 5  102 0

8.6 Quantitative FTA

249

Table 8.11 Reliability and availability evaluation, numerical example 1 Basic event A B C D E

MTTF (h)

MTTR (h)

  MTTR

Œ=. C /Œ1  exp.. C /t /

T

1  exp.  T /

50,000 100,000 3,333.333 10,000 100,000

100 20 1 20 1

0.002 0.0002

0.0020 0.0002 0.9093 0.0020 0.0769

0.16 0.08 2.4 0.8 0.08

0.148 0.077 0.909 0.551 0.077

0.002

MTTF mean time to failure, MTTR mean time to repair

Assuming the hypothesis of statistical independence between basic events related to the component of the system, the unavailabilities of the cut sets are qAB D qA qB Š 0:002  0:0002 Š 4  107 ;

where B is the MTTR of component B and A is the MTTR of component A, in accordance with the opportunity to apply the simplified analytical models of the unavailability. Similarly, for the other cut sets,

qE C D qE qC Š 0:910  0:08 Š 0:073; 3

qCD D qC qD Š 0:910  0:002 Š 1:82  10 : By application of Eq. 8.13, the unavailability of the system is a QS .8;000 h/ D qCSi .t/ i

D 1  .1  4  107 /.1  0:073/  .1  1:82  103 / Š 0:0747: If the simplified Eq. 8.14 is applied, X QS .8;000 h/ D qCSi .t/ Š 0:0748: In order to quantify the ENF of the system, Eq. 8.17 has been applied for each cut set: 8;000 Z

X i

0

WEC .T / Š

ŒC qE .t/ C E qC .t/ dt 0

ZT Š E

wi .t/

Y

 qj .t/ dt;

j ¤i

i. e.,

0

  1 C t T 1 je j 0 C C T 2 Š E T C C 2   1 C T 1 Š E T C .e  1/ C C T 2 C 2  1 4 .e8;000310  1/ Š 105 8;000 C 3  104  C 0:5  3  104  8;0002

ZT WCD .T / Š

ŒD qC .t/ C C qD .t/ dt 0

ZT Š D 0

ZT WAB .0; 8;000/ Š

ŒA qB .t/ C B qA .t/ dt 0 8;000 Z

Š

Œ.1  eC t / C C t  dt

Š 0:146 failures;

i

WCS .0; 8;000/ D

ZT

ŒA B B C B A A  dt 0

Š ŒA B B C B A A   8;000 Š 1:92  104 failures;

Œ.1  eC t / C C D  dt

  1 C t T je j 0 C  C D T Š D T C C   1 C T .e  1/ C C D T Š D T C C  1 4 4 .e8000310  1/ Š 10 8;000 C 3  104  1 C 3  104  8;000 5  102 Š 0:502 failures:

250

8 Effects Analysis and Reliability Modeling of Complex Production Systems

As a consequence, given the top event and assuming a period of time of 8;000 h, the ENF for the system is X WCSi Š 0:648 failures: ENF.T / Š

The failure probability of the system FS .T / is FS .8;000 h/ D

a

FCSi .T /

i

i

D 1  .1  0:0114/.1  0:07/

Now it is possible to quantify the MTTR of the system by the application of the Eq. 8.20: 0:648 WS .T / D Š 8:1  105 h1 wS Š T 8000

 .1  0:501/ Š 0:541; which is very similar to the “simplified” value:

and QS .T / QS .T / Š s .T / wS .T / 0:0748 Š Š 923:5 h: 8:1  105

FS .T D 8;000 h/ Š

MTTRS D

If the analyst has to quantify the failure probability of the repairable system considering the first failure, it is useful to evaluate the failure probabilities for the cut sets as follows: FAB .T / D FA .T /FB .T / Š 0:148  0:077 Š 0:0114; FE C .T / D qE C D FE .T /FC .T / Š 0:910  0:077 Š 0:070; FCD .T / D FC .T /FD .T / Š 0:910  0:551 Š 0:501: ReliaSoft BlockSim 7 - www.ReliaSoft.com

Figures 8.36 and 8.37 present the results obtained by the application of the Monte Carlo simulation analysis on the system for T D 8;000 h. In particular, Fig. 8.36 shows the up/down diagram obtained for components/events A–E and their contributions. Component C is clearly nonrepairable, but fortunately it is not a cut set and the system is always repairable within 8;000 h. For T longer than 8;000 h, the system can reach a state of nonrepairable failure owing to the simultaneous failure of the nonrepairable components E and C, as illustrated in Fig. 8.37. Finally, Fig. 8.38 presents the histogram of the expected failures.

Block Up/Down

Operating Time Time Under Repair

E

D

C

B

A

System

1600.000

3200.000

4800.000

FCSi .T / Š 0:582:

i

State

0.000

X

6400.000

Time, (t)

Fig. 8.36 Block up/down analysis, T D 8;000 h. ReliaSoft® software

8000.000

8.6 Quantitative FTA

251

ReliaSoft BlockSim 7 - www.ReliaSoft.com

Block Up/Down State Operating Time Time Under Repair

E

C

System

0.000

10000.000

20000.000

30000.000

40000.000

50000.000

Time, (t)

Fig. 8.37 Block up/down analysis, T D 5;000 h. ReliaSoft® software

ReliaSoft BlockSim 7 - www.ReliaSoft.com

Block Expected Failures 0.912

RS FCI 100%

50%

0.730 0% 5 Item(s)

0.547

0.365

0.182

0.000

C

D

A

E

Fig. 8.38 Component expected failures. ReliaSoft® software

B

252

8 Effects Analysis and Reliability Modeling of Complex Production Systems

8.6.2 Quantitative FTA, Numerical Example 2 The FTA is applied in this case to the system previously described in Sect. 8.5.2, whose cut sets were illustrated in Sect. 8.5.4.

8.6.2.1 System Configuration A The analytical evaluation of the reliability and the failure rate for the system, given a top event and assuming the redundant configuration A, is as follows: RS .t/ D RP2 .t/  Relectric power .t/  RPR .t/  RV2 .t/ C RP1 .t/  Relectric power .t/  RPR .t/  RV1 .t/  RP2 .t/  RV1 .t/  Relectric power .t/  RPR .t/  RP1 .t/  RV2 .t/; S .t/ D P1 broken .t/RV1 .t/Relectric power .t/RPR .t/ C PR broken .t/RV1 .t/Relectric power .t/RP1 .t/ C no electric power .t/RV1 .t/RPR .t/RP1 .t/ C V1 broken .t/Relectric power .t/RPR .t/RP1 .t/ C V2 broken .t/RP2 .t/Relectric power .t/RPR .t/ C PR broken .t/RP2 .t/Relectric power .t/RV2 .t/ C PR broken .t/RP2 .t/Relectric power .t/RV2 .t/ C no electric power .t/RP2 .t/RPR .t/RV2 .t/ C P2 broken .t/Relectric power .t/RPR .t/RV2 .t/  V2 broken .t/RP2 .t/RV1 .t/Relectric power .t/  RPR .t/RP1 .t/  V1 broken .t/RP2 .t/RV2 .t/Relectric power .t/  RPR .t/RP1 .t/  P1 broken .t/RP2 .t/RV1 .t/Relectric power .t/  RPR .t/RV2 .t/  P2 broken .t/RP1 .t/RV1 .t/Relectric power .t/  RPR .t/RV2 .t/  PR broken .t/RP2 .t/RV1 .t/Relectric power .t/  RV2 .t/RP1 .t/  no electric power .t/RP2 .t/RV1 .t/RV2 .t/  RPR .t/RP1 .t/:

A quantitative analysis based on different scenarios is illustrated next for configuration A and exponential distributions of ttf and ttr random variables. Table 8.12 reports the values of ttf and ttr assumed for the basic components in the system illustrated in Fig. 8.24. Given the top event “reactor explosion,” Fig. 8.39 shows the trends of F .t/, R.t/, f .t/, and .t/ as a function of time t for system configuration A; as a consequence, the components and the system, subject to the top event, are supposed to be not repairable. These trends also illustrate the top event for the system in the case of repairable components, but considering the so-called first failure top event as catastrophic. From the reliability importance analysis in Fig. 8.40, the most critical component is the electric power supplier, whose “absence of power” event is very critical because of its great failure rate and the cardinality 1 of the corresponding cut set. The same conclusion is supported by the static reliability importance analysis for time t D 4;000 h and t D 8;000 h, as reported in Fig. 8.41. Figures 8.42–8.45 present the results of a dynamic Monte Carlo simulation analysis for a period T of 50;000 h, assuming the hypothesis of repairable components. It is worth noting in Fig. 8.42 that each time the electric power supply fails, the system fails too. Figure 8.43 presents the trend of the system failures NF.t/ cumulated from t0 D 0 to the generic time point t. Figure 8.44 shows the expected downing events for the set of components, or basic events, and, finally, Fig. 8.45 shows the point availability A.t/. 8.6.2.2 System Configuration B Considering the not redundant configuration B, the analytical evaluation of reliability functions RS .t/ and S .t/ results in the following: RS .t/ D RV1 .t/Relectric power .t/RPR .t/  RP1 .t/RV2 .t/RP2 .t/; S .t/ D V1 broken .t/ C no electric power .t/ C PR broken .t/ C P1 broken .t/ C V2 broken .t/ C P2 broken .t/: As for configuration A, Figs. 8.46–8.48 illustrate the results for configuration B, assuming the failure and repair probability distributions listed in Table 8.12.

8.6 Quantitative FTA

253

Table 8.12 Constant failure and repair rates. Configuration A .t / D 

1=.t / D 1= D MTTR

3  105 h1 3  104 h1 104 h1 106 h1

25 h 18 h 15 h 30 h

Component P1 , P2 pumps PW electric power supplier V1 , V2 valves PR processor

ReliaSoft BlockSim 7 - www.ReliaSoft.com

ReliaSoft BlockSim 7 - www.ReliaSoft.com

Unreliability vs Time

1.000

Reliability vs Time

1.000

Unreliability

Reliability

Fault Tree1 Unreliability Line

Fault Tree1 Reliability Line

0.800

Reliability, R(t)

Unreliability, F(t)=1-R(t)

0.800

0.600

0.400

0.200

0.000 0.000

0.600

0.400

0.200

4000.000

8000.000

12000.000

16000.000

0.000 0.000

20000.000

4000.000

8000.000

Time, (t) ReliaSoft BlockSim 7 - www.ReliaSoft.com

12000.000

16000.000

ReliaSoft BlockSim 7 - www.ReliaSoft.com

Probability Density Function

4.000E-4

Failure Rate vs Time

5.000E-4

Pdf

Failure Rate

Fault Tree1 Pdf Line

Fault Tree1 Failure Rate Line

4.400E-4

Failure Rate, f(t)/R(t)

3.200E-4

f(t)

2.400E-4

1.600E-4

8.000E-5

0.000 0.000

20000.000

Time, (t)

3.800E-4

3.200E-4

2.600E-4

4000.000

8000.000

12000.000

16000.000

2.000E-4 0.000

20000.000

4000.000

8000.000

Time, (t)

12000.000

16000.000

20000.000

Time, (t)

®

Fig. 8.39 Event “reactor explosion,” configuration A. F .t /, R.t /, f .t /, .t /. ReliaSoft software ReliaSoft BlockSim 7 - www.ReliaSoft.com

ReliaSoft BlockSim 7 - www.ReliaSoft.com

Reliability Importance vs Time

0.200

Reliability Importance Value

0.160

0.120

0.080

0.040

0.000 0.000

Importance Fault Tree1 Extra Starting Block Extra Ending Block V1 broken no electric power PR broken P1 broken no electric power V2 broken no electric power PR broken P2 broken No electric power

0.800

Reliability Importance Value

Fault Tree1 Extra Starting Block Extra Ending Block V1 broken no electric power PR broken P1 broken no electric power V2 broken no electric power PR broken P2 broken No electric power

Reliability Importance vs Time

1.000

Importance

0.600

0.400

0.200

4000.000

8000.000

12000.000

Time, (t)

16000.000

20000.000

0.000 0.000

4000.000

8000.000

12000.000

Time, (t)

Fig. 8.40 Event “reactor explosion,” configuration A. Reliability importance. ReliaSoft® software

16000.000

20000.000

254

8 Effects Analysis and Reliability Modeling of Complex Production Systems

ReliaSoft BlockSim 7 - www.ReliaSoft.com

ReliaSoft BlockSim 7 - www.ReliaSoft.com

Static Reliability Importance

Static Reliability Importance

0.832

0.577

Reliability

Reliability

100%

100%

50%

50%

0.462 0% 5 Item(s)

0.499

0.333

Reliability Importance Value

Reliability Importance Value

0.666

0.166

0.000

0% 5 Item(s)

0.346

0.231

0.115

no electric power

PR broken

V1 broken

V2 broken

0.000

P2 broken

no electric power

Time = 4000

PR broken

V2 broken

V1 broken

P2 broken

Time = 8000

Fig. 8.41 Event “reactor explosion,” configuration A. Static reliability importance, t D 4;000 h and t D 8;000 h. ReliaSoft® software

ReliaSoft BlockSim 7 - www.ReliaSoft.com

Block Up/Down State

V2 broken

V1 broken

PR broken

P2 broken

P1 broken

no electric power

System

0.000

10000.000

20000.000

30000.000

40000.000

50000.000

Time, (t)

Fig. 8.42 Event “reactor explosion,” repairable components, configuration A. Simulation analysis. Up/down diagram. ReliaSoft® software

From the reliability importance analysis in Fig. 8.47, the most critical component is the electric power supplier, whose “absence of power” event is very critical because of its great failure rate and the cardinality 1 of the corresponding cut set. Fig. 8.48 presents the result of a static reliability importance analysis for t D 4;000 h and t D 8;000 h.

8.6.3 Numerical Example. Quantitative Analysis in the Presence of a Mix of Statistical Distributions This numerical example rejects the assumption of constant failure rates, and the probability distributions for ttf and ttr vary as reported in Table 8.13.

8.6 Quantitative FTA

255

ReliaSoft BlockSim 7 - www.ReliaSoft.com

System Failures

20.000

System Failures Fault Tree1 System Failures

System Failures(t)

16.000

12.000

8.000

4.000

0.000 0.000

10000.000

20000.000

30000.000

40000.000

50000.000

Time, (t)

Fig. 8.43 Event “reactor explosion,” repairable components, configuration A. Simulation analysis. System failures. ReliaSoft® software ReliaSoft BlockSim 7 - www.ReliaSoft.com

Block Expected Downing Events 14.928

RS DECI 100%

50%

11.942 0% 6 Item(s)

8.956

5.971

2.986

0.000

no electric po...

V2 broken

V1 broken

P1 broken

P2 broken

PR broken

Fig. 8.44 Event “reactor explosion,” repairable components, configuration A. Expected downing events. ReliaSoft® software

8.6.3.1 System Configuration A Given the top event “reactor explosion,” Fig. 8.49 shows the trends of F .t/, R.t/, f .t/, and .t/ as a function of time t for system configuration A; as

a consequence, these trends can support the determination and analysis of the first failure process assuming the system is not repairable, i. e., in the case of a failure catastrophic event and repairable components (see Table 8.13). In particular, assuming a mission time T

256

8 Effects Analysis and Reliability Modeling of Complex Production Systems

ReliaSoft BlockSim 7 - www.ReliaSoft.com

Point Availability vs Time

1.000

Availability Fault Tree1 Point Availability Line

Availability, A(t)

0.998

0.996

0.994

0.992

0.990 0.000

10000.000

20000.000

30000.000

40000.000

50000.000

Time, (t)

Fig. 8.45 Event “reactor explosion,” configuration A. Availability A.t /. ReliaSoft® software ReliaSoft BlockSim 7 - www.ReliaSoft.com

ReliaSoft BlockSim 7 - www.ReliaSoft.com

Unreliability vs Time

Reliability vs Time

1.000

1.000

Unreliability

Reliability

Fault Tree1 Unreliability Line

Fault Tree1 Reliability Line

0.800

Reliability, R(t)

Unreliability, F(t)=1-R(t)

0.800

0.600

0.400

0.200

0.000 0.000

0.600

0.400

0.200

1800.000

3600.000

5400.000

7200.000

0.000 0.000

9000.000

1800.000

Time, (t)

3600.000

5400.000

7200.000

ReliaSoft BlockSim 7 - www.ReliaSoft.com

ReliaSoft BlockSim 7 - www.ReliaSoft.com

Probability Density Function

Block Failure Rate vs Time

6.000E-4

6.000E-4

Pdf

Failure Rate

Fault Tree1 Pdf Line

Fault Tree1 System no electric power P1 broken P2 broken PR broken V1 broken V2 broken

4.800E-4

Failure Rate, f(t)/R(t)

4.804E-4

f(t)

3.608E-4

2.412E-4

1.216E-4

2.000E-6 0.000

9000.000

Time, (t)

3.600E-4

2.400E-4

1.200E-4

1800.000

3600.000

5400.000

Time, (t)

7200.000

9000.000

0.000 0.000

1000000.000

2.000E+6

3.000E+6

Time, (t)

Fig. 8.46 Event “reactor explosion,” configuration B. F .t /, R.t /, f .t /, (t /. ReliaSoft® software

4.000E+6

5.000E+6

8.6 Quantitative FTA

257

ReliaSoft BlockSim 7 - www.ReliaSoft.com

Reliability Importance vs Time 1.000

Importance Fault Tree1 V1 broken P1 broken V2 broken P2 broken no electric power no electric power No electric power no electric power PR broken PR broken

Reliability Importance Value

0.800

0.600

0.400

0.200

0.000 0.000

1800.000

3600.000

5400.000

7200.000

9000.000

Time, (t)

Fig. 8.47 Event “reactor explosion,” configuration B. Reliability importance. ReliaSoft® software

ReliaSoft BlockSim 7 - www.ReliaSoft.com

ReliaSoft BlockSim 7 - www.ReliaSoft.com

Static Reliability Importance

Static Reliability Importance

0.352

0.124

Reliability

Reliability

100%

100%

50%

50%

Reliability Importance Value

0% 5 Item(s)

0.074

0.050

Reliability Importance Value

0.282

0.099

0.211

0.141

0.070

0.025

0.000

0% 5 Item(s)

no electric power

V1 broken

V2 broken

P2 broken

0.000

P1 broken

no electric power

V1 broken

Time = 8000

V2 broken

P2 broken

Time = 4000

Fig. 8.48 Event “reactor explosion,” configuration B. Static reliability importance. ReliaSoft® software

Table 8.13 Mix of failure and repair distributions. Configuration A Component

Process

Distribution

Parameter 1

Parameter 2

P1, P2 pumps

Failure Repair Failure Repair Failure Repair Failure Repair

Weibull Lognormal Exponential Exponential Weibull Lognormal Exponential Exponential

1=a D 33;333 h  D 25 h  D 3  104 h1 MTTR D 18 h 1=a D 1;000 h  D 15 h  D 106 h1 MTTR D 30 h

b D 1:5 3h

PW electric power supplier V1, V2 valves PR processor

b D 1:5 0:5 h

P1 broken

258

8 Effects Analysis and Reliability Modeling of Complex Production Systems

ReliaSoft BlockSim 7 - www.ReliaSoft.com

ReliaSoft BlockSim 7 - www.ReliaSoft.com

Unreliability vs Time

Reliability vs Time

1.000

1.000

Unreliability

Reliability

Fault Tree1 Unreliability Line

Fault Tree1 Reliability Line

0.800

Reliability, R(t)

Unreliability, F(t)=1-R(t)

0.800

0.600

0.400

0.200

0.000 0.000

0.600

0.400

0.200

600.000

1200.000

1800.000

2400.000

0.000 0.000

3000.000

600.000

1200.000

Time, (t)

1800.000

2400.000

3000.000

Time, (t)

ReliaSoft BlockSim 7 - www.ReliaSoft.com

ReliaSoft BlockSim 7 - www.ReliaSoft.com

Probability Density Function

Failure Rate vs Time

8.000E-4

0.003

Pdf

Failure Rate

Fault Tree1 Pdf Line

Fault Tree1 Failure Rate Line

0.002

Failure Rate, f(t)/R(t)

6.400E-4

f(t)

4.800E-4

3.200E-4

1.600E-4

0.000 0.000

0.002

0.001

7.600E-4

600.000

1200.000

1800.000

2400.000

3000.000

2.000E-4 0.000

600.000

1200.000

Time, (t)

1800.000

2400.000

3000.000

Time, (t)

Fig. 8.49 Event “reactor explosion,” configuration A and mix of distributions.F .t /, R.t /, f .t /, .t /. ReliaSoft® software

ReliaSoft BlockSim 7 - www.ReliaSoft.com

Reliability Importance vs Time

1.000

Importance Fault Tree1 Extra Starting Block Extra Ending Block no electric power no electric power No electric power PR broken PR broken no electric power V2 broken V1 broken P1 broken P2 broken

Reliability Importance Value

0.800

0.600

0.400

0.200

0.000 0.000

600.000

1200.000

1800.000

2400.000

3000.000

Time, (t)

Fig. 8.50 Event “reactor explosion,” configuration A and mix of distributions. Reliability importance. ReliaSoft® software

8.6 Quantitative FTA ReliaSoft BlockSim 7 - www.ReliaSoft.com

259 ReliaSoft BlockSim 7 - www.ReliaSoft.com

Static Reliability Importance

0.860

Static Reliability Importance

0.507

Reliability

Reliability 100%

100%

50%

50%

0.406 0% 5 Item(s)

0.516

0.344

0.172

Reliability Importance Value

Reliability Importance Value

0.688

0% 5 Item(s)

0.304

0.203

0.101

0.000

no electric power

PR broken

V1 broken

V2 broken

0.000

P2 broken

V2 broken

V1 broken

Time = 600

no electric power

PR broken

P1 broken

Time = 1200

Fig. 8.51 Event “reactor explosion,” configuration A and mix of distributions. Static reliability importance, t D 600 h and t D 1;200 h. ReliaSoft® software

ReliaSoft BlockSim 7 - www.ReliaSoft.com

Failure Rate vs Time 0.005

Failure Rate Fault Tree1 Failure Rate Line

Failure Rate, f(t)/R(t)

0.004

0.003

0.002

0.001

2.000E-4 0.000

400.000

800.000

1200.000

1600.000

2000.000

Time, (t)

Fig. 8.52 Event “reactor explosion,” configuration B and mix of distributions .t /. ReliaSoft® software

of about 3;000 h, the system certainly fails as clearly illustrated by the unreliability function, i. e., the failure probability function. Figure 8.50 shows the results of the reliability importance analysis conducted by ReliaSoft® software: the most critical component is the electric power supply before t about 1;200 h, while later valves V1 and V2 reveal themselves as the most important components in terms of reliability. The same conclusion is supported by the static reliability importance analysis illustrated in Fig. 8.51.

8.6.3.2 System Configuration B Given the top event “reactor explosion,” Figs. 8.52 and 8.53 present the failure rate .t/ and the reliability importance function for the repairable system in configuration B, made up of components subject to random failure and repair processes with different probability distributions, as listed in Table 8.13.

260

8 Effects Analysis and Reliability Modeling of Complex Production Systems

ReliaSoft BlockSim 7 - www.ReliaSoft.com

Reliability Importance vs Time 1.000

Importance Fault Tree1 no electric power no electric power No electric power PR broken PR broken no electric power V2 broken V1 broken P1 broken P2 broken

Reliability Importance Value

0.800

0.600

0.400

0.200

0.000 0.000

400.000

800.000

1200.000

1600.000

2000.000

Time, (t)

Fig. 8.53 Event “reactor explosion,” configuration B and mix of distributions. Reliability importance. ReliaSoft® software ReliaSoft BlockSim 7 - www.ReliaSoft.com

Block Expected Downing Events

0.992

RS DECI 100%

50%

0.793 0% 6 Item(s)

0.595

0.397

0.198

0.000

V2 broken

V1 broken

no electric po...

P2 broken

P1 broken

PR broken

Fig. 8.54 Event “reactor explosion,” configuration A and repairable components. Expected downing events. Simulation, t D 3;000 h. ReliaSoft® software

8.6.3.3 Monte Carlo Simulation The following results relate to the application of the Monte Carlo dynamic simulation of system configuration A, whose top event is the same as in the numerical example illustrated in Sect. 8.5.4 (see also

Figs. 8.29 and 8.30), assuming the hypothesis of repairable components and a mix of random variables ttf and ttr (see Table 8.13). Figure 8.54 presents the expected values of downing events related to the components of the repairable system and assuming t D 3;000 h.

8.6 Quantitative FTA

261

ReliaSoft BlockSim 7 - www.ReliaSoft.com

ReliaSoft BlockSim 7 - www.ReliaSoft.com

Block Up/Down

Block Up/Down

State

State Operating Time Time Under Repair

V2 broken

V1 broken

V1 broken

PR broken

PR broken

P2 broken

P2 broken

P1 broken

P1 broken

no electric power

no electric power

System

0.000

Operating Time Time Under Repair

V2 broken

System

600.000

1200.000

1800.000

2400.000

3000.000

0.000

600.000

1200.000

Time, (t)

1800.000

2400.000

3000.000

Time, (t)

Fig. 8.55 Event “reactor explosion,” configuration A and repairable components. Up/down dynamic analysis. ReliaSoft® software ReliaSoft BlockSim 7 - www.ReliaSoft.com

System Failures

2.000

System Failures Fault Tree1 System Failures

System Failures(t)

1.600

1.200

0.800

0.400

0.000 0.000

600.000

1200.000

1800.000

2400.000

3000.000

Time, (t)

Fig. 8.56 Event “reactor explosion,” configuration A and repairable components. Simulation analysis: system failures, t D Œ0; 3;000 h. ReliaSoft® software

Figure 8.55 presents the up/down (i. e., 0/1) diagrams obtained by two different simulation runs of the repairable system. In the first diagram the system fails twice because of the failure events for the electric power supply. A third time relates to the failure of valve V1 (very close to time point t D 1;800 h) following the failure of valve V2 in accordance with the existence of the cut set V1V2. In the second diagram the system fails when the failure of valve V2 occurs, given a previous failure of valve V1. Figure 8.56 presents the trend of the system failures for t belonging to the range Œ0; 3;000 h. This is the

result of a specific simulation run of the system and the top event. Figure 8.57 reports the measure of the downing event criticality index for the components, or basic events, of the system, given the “reactor explosion” top event. Figure 8.58 presents the values of the point availability A.t/ for the system subject to the top event, i. e., the probability that the system is operational at a given time in accordance with the so-called alternating renewal process made up of ttf and ttr stochastic processes. In particular, it is useful to remember that A.t/ is the probability that the system is up at time t.

262

8 Effects Analysis and Reliability Modeling of Complex Production Systems

ReliaSoft BlockSim 7 - www.ReliaSoft.com

RS DECI 36.265

Availability 100%

50%

29.012 0% 5 Item(s)

21.759

14.506

7.253

0.000

V2 broken

V1 broken

no electric power

P1 broken

P2 broken

Fig. 8.57 Event “reactor explosion,” configuration A and repairable components. Downing event criticality index (DECI), t D 3;000 h. ReliaSoft® software ReliaSoft BlockSim 7 - www.ReliaSoft.com

Availability and Reliability vs Time

1.000 Fault Tree1 Point Availability Line Point Reliability Line

0.800

A(t), R(t)

0.600

0.400

0.200

0.000 0.000

600.000

1200.000

1800.000

2400.000

3000.000

Time, (t)

Fig. 8.58 Point system availability A.t / and reliability R.t /, configuration A and repairable components. Simulation analysis. ReliaSoft® software

In other words, during the simulation analysis a special counter would be required in order to get this value at t  . This counter is incremented by one every time the system is up at t  considering the whole set of simulations runs; thus, the point availability at t  is

the number of times the system is up at t  divided by the number of simulation runs in the dynamic analysis. Figure 8.58 also reports the value of the point reliability R.t/ obtained in the same way as for A.t/, i. e., by means of several runs of dynamic simulation: this is

8.7 Application 1 – FTA

263

the probability that the nonrepairable system has not failed by time t.

The hot water produced by the boiler is pumped by a force pump, called a “boiler pump,” along a primary loop of piping; some thermic and hydraulic drops are obviously encountered. The hydraulic circuit is completed by a secondary loop, when the two heat exchangers in the controlled zone are fed by the same boiler, but it is possible to double the secondary loop (loop1 and loop2) in order to feed the fan-coils by two distinct and independent boiler systems. Each secondary loop is supported by its own pump. The generic loop associated with a boiler is made of two subloops, one for each exchanger. The environmental temperature is controlled by adjusting the hot water flow by means of automatic valves, one for each secondary loop, and a zone valve (mixing three-way valve) for each exchanger and for each loop. As a consequence, in the case of two fan-coils and two boilers, four valves are required. The boiler pump as well as every pump

8.7 Application 1 – FTA This application deals with the FTA conducted on a heating plant for a 160-m2 public lounge. The system, conventionally split into a hydronic device for warm water and a heating device based on water temperatures and thermic energy conservation, has three main components, as illustrated in the functional simplified block scheme of Fig. 8.59: the boiler, the distribution system (pumps, collectors, valves, etc.), and the heat exchangers. In particular, two fan-coils are fed in a redundant configuration, i. e., the heating system is supposed to be capable when at least one fan-coil is operating.

8 O100 mm

sub-loop 2.1

sub-loop 2.2

5 sub-loop 2.1

sub-loop 1.1

10 6 9 1

sub-loop 1.1 sub-loop 2.1

3

natural gas

water supply 11

2

4

primary loop 7 return

supply

secondary loop 1 Legend - Thermal system Code Block/component name 1 Boiler 2 Pump 1 3 Thermal sensor 4 Hydraulic disjunctor 5 3-way zone valve 6 Fan-coil 1

Legend - Thermal system Code Block/component name 7 Gas adducon valve 8 Gas burner system 9 Boiler circulaon pump 10 Flow fan temperature sensor 1 11 Control valve loop 1

Fig. 8.59 Functional block scheme of the thermic system

264

8 Effects Analysis and Reliability Modeling of Complex Production Systems

on the secondary loops operate according to the simultaneous and integrated control of sensors, such as a thermic sensor for each subloop, a flow fan temperature sensor for each fan, and an environmental sensor. Some other critical components playing a significant role complete the generic FTA: • a boiler system with natural gas adduction and combustion gas evacuation; • two fan-coils; • the electric power supply system; • the water supply system with a hydraulic pipe adduction; • the piping system, i. e., the piping distribution network; • the hydraulic disjunctors, as many as the secondary loops, for the right mix of hot and cold water in the primary and secondary loops.

The hydraulic circuit has to be filled up with water at the start-up, and later the water recirculates in the system when it is working. A refill is sometimes required in order to compensate for some water leaks.

8.7.1 Fault Tree Construction Assuming the situation “no thermic comfort” as the top event for the heating plant or “thermic system,” one can develop some different fault trees in accordance with different system configurations and hypotheses. These trees are made of the four basic “subtrees” illustrated in Figs. 8.60–8.63, representing the events of absence of hot water within the two available fans as follows:

no warm H2O supp.1

1.1

cold water

hydraulic disjunctor 1

3-way zone valve closed 1.1

no discharge head - loop 1.1

not operating boiler no electric power

piping rupture loop 1.1

zone valve 1.1 rupture

not operating pump 1

not operating circulation pump

no natural gas

valve automatic control 1

gas burner rupture 1

no drive c.v. no gas supply

gas adduction valve 1 closed

no drive c.p.

boiler pump 1 broken

no drive pump

pump 1 broken

no control

no electric power

no electric power flow fan n°1 temperature sensor

environmental thermal sensor

Fig. 8.60 Fault tree construction, subloop 1.1

Control Valve rupture loop1

no control p.

no electric power

no control

thermal sensor n°1.1 not operative environmental thermal sensor environmental thermal sensor

thermal sensor n° 1.1 not operative

thermal sensor n° 1.1 not operative

8.7 Application 1 – FTA

265

2.1

no warm H2O supp.2

cold water

hydraulic disjunctor 2

3-way zone valve closed 2.1

no discharge head - loop 2.1

not operating boiler no electric power

piping rupture loop 2.1

zone valve 2.1 rupture

not operating pump 2

not operating circulation pump

no natural gas

valve automatic control 2

gas burner rupture 2

no drive c.v. no gas supply

gas adduction valve 2 closed

no drive c.p.

no drive pump

no electric power

Control Valve rupture loop2

boiler pump 2 broken

pump 2 broken no electric power

no control

no control

no electric power flow fan n°1 temperature sensor

no control p.

thermal sensor n° 2.1 not operative environmental thermal sensor environmental thermal sensor

environmental thermal sensor

thermal sensor n° 2.1 not operative

thermal sensor n° 2.1 not operative

Fig. 8.61 Fault tree construction, subloop 2.1

• Transfer out block 1.1. It refers to subloop boiler 1, and no hot water on fan 1. • Transfer out block 2.1. It refers to subloop boiler 2 (i. e., in the case of the existence of a ond boiler), and no hot water on fan 1. • Transfer out block 1.2. It refers to subloop boiler 1, and no hot water on fan 2. • Transfer out block 2.2. It refers to subloop boiler 2 (i. e., in the case of the existence of a ond boiler), and no hot water on fan 2.

1.1, 2.1, sec1.2, 2.2, sec-

Every tree configuration, on five configurations A, B, C, D, and E proposed, has been generated and analyzed from both a qualitative and a quantitative point of view as follows: • Configuration A – one boiler and fan-coil redundancy and fill water (Fig. 8.64.) There is only a single boiler and two redundant fan-coils, i. e., there is

one secondary loop made of two subloops, one for each fan. It is supposed the system requires the water supplier to be operative, i. e., in a state of function. • Configuration B – one boiler and fan-coil redundancy (Fig. 8.65): There is only a single boiler and two redundant fan-coils, i. e., there is one secondary loop made of two subloops, one for each fan. It is also supposed the system does not require the water supplier to be operative because the piping has already been filled. • Configuration C – one boiler and no fan-coil redundancy (Fig. 8.66). There is only a single boiler and two fan coils, both necessary to guarantee thermic comfort. It is also supposed the system does not require the water supplier to be operative because the piping has already been filled.

266

8 Effects Analysis and Reliability Modeling of Complex Production Systems

1.2

no warm H2O supp.1

cold water

hydraulic disjunctor 1

3-way zone valve closed 1.2

no discharge head - loop 1.2

not operating boiler 1 no electric power

piping rupture loop 1.2

zone valve 1.2 rupture

not operating pump 1

not operating circulation pump

no natural gas

valve automatic control

gas burner rupture 1

no drive c.v. no gas supply

gas adduction valve 1 closed

no drive c.p.

boiler pump 1 broken no drive pump 1

Control Valve rupture loop1

pump 1 broken

no control

no electric power

no electric power no electric power

flow fan n°2 temperature sensor

environmental thermal sensor

no control

no control p.

thermal sensor n°1.2 not operative environmental thermal sensor environmental thermal sensor

thermal sensor n° 1.2 not operative

thermal sensor n° 1.2 not operative

Fig. 8.62 Fault tree construction, subloop 1.2

• Configuration D – two boilers and fan-coil redundancy and fill water (Figs. 8.67 and 8.68). There are two alternative boilers (i. e., one is redundant) and two redundant fan-coils. It is also supposed the system requires the water supplier to be operative because the piping network could be empty. • Configuration E – two boilers and fan-coil redundancy (Fig. 8.69). There are two alternative boilers (i. e., one is redundant) and two redundant fan-coils. It is also supposed the production system does not require the water supplier to be operative because the piping network is already filled (both primary and secondary loops).

8.7.2 Qualitative FTA and Standards-Based Reliability Prediction The generic fault tree previously illustrated is made up of several blocks, many of which are primary blocks/events related to the components of the system investigated. Many blocks are mirrors of a few primary events, such as the so-called no electric power, the rupture on the “environmental thermic sensor,” and the “no gas supply” event related to the natural gas supply system. The generic event mirror of a basic/primary component can be represented by a “little square” near the block associated with the event. The

8.7 Application 1 – FTA

267

2.2

no warm H2O supp.2

cold water

hydraulic disjunctor 2

3-way zone valve closed 2.2

no discharge head - loop 2.2

not operating boiler 2

no electric power

piping rupture loop 2.2

zone valve 2.2 rupture

not operating pump 2

not operating circulation pump

no natural gas

no gas supply

gas adduction valve 2 closed

valve automatic control 2

gas burner rupture 2

no drive c.v. no drive c.p.

boiler pump 2 broken

Control Valve rupture loop2

pump 2 broken no drive pump 2

no electric power

no control no control

no electric power no electric power

flow fan n°2 temperature sensor

no control p.

thermal sensor n° 2.2 not operative

environmental thermal sensor

environmental thermal sensor

environmental thermal sensor

thermal sensor n° 2.2 not operative

thermal sensor n° 2.2 not operative

Fig. 8.63 Fault tree construction, subloop 2.2

event associated with a component is considered “basic/primary” in accordance with the availability of data related to the failure and repair random behaviors. In particular, Table 8.14 reports the failure rates of the basic events/components collected by a library reference of nonelectronic parts (see standards-based reliability database of predefined components MIL-217, NSWC98/LE1, etc.). Another trivial but significant consideration can be made. The presence of redundancies justifies the absence of AND gates in fault tree construction (e. g., only OR gates in configuration C). In particular, according to the previously introduced and discussed Boolean absorption laws, configuration C is as illustrated in Fig. 8.70. The number of MCS is 19, each one made up of a single member. Given the top event, the failure rate

of the system is S D pump 1 broken C piping rupture loop 1.1 C boiler pump 1 broken C thermal sensor 1.1 not operative C control valve rupture loop 1 C hydraulic disjunctor 1 C fan axial flow 1 C no electric power C flow fan 1 temperature sensor Cenvironmental thermal sensor C thermal sensor 1.2 not operative C piping rupture loop 1.2 C zone valve 1.2 rupture C fan axial flow 2 C flow fan 2 temperature sensor C zone valve 1.1 rupture C no gas supply C gas adduction valve 1 closed C gas burner rupture 1 : Tables 8.15 and 8.16 illustrate the configuration of the MCS identified by the qualitative analysis for the

268

8 Effects Analysis and Reliability Modeling of Complex Production Systems

No Thermal Comfort

no warm air

no warm air Fan 2

no warm air Fan 1

no supply H2O

no flow fan 1

flow fan n°1 temperature sensor

fan axial flow 1

no flow fan 2

no electric power

no water supply

rupture hydronic pipe adduction

1.1

flow fan n°2 temperature sensor

fan axial flow 2

no supply H2O

no electric power

no water supply

rupture hydronic pipe adduction

1.2

Fig. 8.64 One boiler and fan-coil redundancy and fill water. Configuration A

Table 8.14 Failure rates from standards-based reliability libraries. FT fault tree  (106 )

Number of components Configurations Configurations A, B and C – D and F – 1 boiler 2 boilers

Code

FT component

Other reference

1 2 3

Fan axial flow No electric power Flow fan temperature sensor Rupture hydronic pipe adduction No gas supply Boiler pump broken

Fancoil Electric power supplier Sensor transmitter temperature Piping water system

1.586 13.65 25.69

630,517 73,260 38,926

2 1 2

2 1 2

1.066

938,086

1

1

Gas supplier Pump hydraulic boiler feed Pump hydraulic Valve mixing 3-way Valve hydraulic gate Water supplier system Sensor temperature

50.7 0.4216

19,724 2,371,916

1 1

1 2

86.28 18.54 1.336 95.1 0.1053

11,590 53,937 74,8503 10,515 9,496,676

1 2 1 1 1

2 4 2 1 1

Valve automatic control

10.87

91,966

1

2

4 5 6 7 8 9 10 11 12

Pump broken Zone valve rupture Gas adduction valve No water supply Environmental thermal sensor Control valve rupture

MTTF (h)

8.7 Application 1 – FTA

269

No Thermal Comfort

no warm air

no warm air Fan 2

no warm air Fan 1

no flow fan 1 no flow fan 2

flow fan n°1 temperature sensor

fan axial flow 1

no electric power

1.1

flow fan n°2 temperature

fan axial flow 2

no electric power

1.2

Fig. 8.65 One boiler and fan-coil redundancy. Configuration B

available configurations. In particular, the number of cut sets is 36 for configuration A, 34 for configuration B, 19 for configuration C (as previously demonstrated), 414 for configuration D, and 412 for configuration E.

8.7.3 Quantitative FTA By the application of the analytical model illustrated in the previous sections of this chapter, it is possible to quantify the reliability parameters of the system, e. g., reliability RS .t/ and MTTF. Table 8.16 summarizes these values for the five system configurations previously illustrated. In particular, the reliability function has been quantified for t D 4;000 h and

t D 6;570 h, corresponding to an operating period of 1 year (i. e., 365 days per year and 18 h per day). The system is supposed to be nonrepairable and made up of nonrepairable components, and as a consequence these values refer to the first occurrence of the system failure event. In accordance with this hypothesis, the following sections illustrate some basic results obtained for the five system configurations previously introduced.

8.7.3.1 Configuration A – One Boiler and Fan-Coil Redundancy and Fill Water Figure 8.71 presents the failure probability function F .t/ (i. e., the unreliability), the reliability R.t/ (i. e., the survival function), the probability density func-

270

8 Effects Analysis and Reliability Modeling of Complex Production Systems

No Thermal Comfort

no WARM air

no warm air Fan 2

no warm air Fan 1

no flow fan 1 no flow fan 2

flow fan n°1 temperature sensor

fan axial flow 1

no electric power

flow fan n°2 temperature

1.1

fan axial flow 2

no electric power

1.2

Fig. 8.66 One boiler and no fan coil redundancy. Configuration C

tion f .t/, and the failure rate .t/ for the thermic system made up of one boiler and two redundant fan-coils. The hydraulic circuit could be empty. Figure 8.72 presents the results obtained by the static reliability importance analysis (see Chaps. 5 and 6) applied to the system for t D 4;000 h and t D 8;000 h. The most critical components are the water supply system, pump 1, the gas supply system, the electric power system, and the automatic control valve. This rank ordering list is confirmed by the time-dependent reliability importance analysis, whose main results are illustrated in Fig. 8.73, and whose most critical components have the highest values of the reliability importance value (in the vertical y-coordinate). Figure 8.74 compares the failure rate of the system S .t/ with the failure rates of the most critical components previously identified. Now, the reliability of two exemplifying cut sets is quantified as follows: Y qj .t/ qCSfpump 1 brokeng .t/ D j 2CSfpump 1 brokeng

D qpump 1 broken .t/ D 1  epump 1 broken t 6 t

D 1  e86:2810

qCS˚

fan axial_flow_1I zone_valve_1.2_rupture

D

.t/

Y

j 2CS˚ fan axial_flow_1I zone_valve_1.2_rupture

qj .t/

D qfan axial_flow_1 .t/  qzone_valve_1.2_rupture .t/ D Œ1  efan axial_flow_1 t Œ1  ezone_valve_1.2_rupture t 6

6

D Œ1  e1:586x10 tŒ1  e18:54x10 t

8.7.3.2 Configuration B – One Boiler and Fan-Coil Redundancy As previously applied to configuration A, Fig. 8.75 presents the failure probability function F .t/, the reliability R.t/, the probability density function f .t/, and the failure rate .t/ for the thermic system made up of one boiler and two redundant fan-coils, without requiring water from the water supplier system in this case. Figure 8.76 presents the results obtained by the static reliability importance analysis applied to the system for t D 4;000 h and t D 8;000 h. The most critical components are the same as for configuration A: the rank ordering list is confirmed by the time-dependent reliability importance analysis (see Fig. 8.77). Fig-

8.7 Application 1 – FTA

271

No Thermal Comfort

no warm air

no warm air Fan 2

no warm air Fan 1

no supply H2O

no flow fan 1

flow fan n°1 temperature sensor

fan axial flow 1

no flow fan 2

no electric power

no water supply

rupture hydronic pipe adduction

1.1

flow fan n°2 temperature sensor

AND

fan axial flow 2

2.1

no supply H2O

no electric power

no water supply

rupture hydronic pipe adduction

1.2

AND

2.2

Fig. 8.67 Two boilers and fan-coil redundancy and fill water. Configuration D

ure 8.78 compares the failure rate of the system S .t/ with the failure rates of the most critical components.

As previously demonstrated, the failure rate of the system is constant, i. e., the top event is random.

8.7.3.3 Configuration C – One Boiler and No Fan-Coil Redundancy

8.7.3.4 Configuration D – Two boilers and Fan-Coil Redundancy and Fill Water

Figure 8.79 presents the failure probability function F .t/, the reliability R.t/, the probability density function f .t/, and the failure rate .t/ for the system made up of one boiler and two fan-coils, all necessary to guarantee environmental thermic comfort, without requiring water from the water supply system. Figures 8.80–8.82 are similar to those introduced for configurations A and B. The most critical basic events/components are the failure of the pump, the gas supply system, the flow fan thermic sensors, and the subloop thermic sensors.

Figure 8.83 presents the failure probability function F .t/, the reliability R.t/, the probability density function f .t/, and the failure rate .t/ for the thermic system made up of two boilers and two redundant fan-coils. The hydraulic circuit could be empty. Figures 8.84–8.86 correspond to those introduced for the previous system configurations. The most critical basic events/components are the water supply system, the gas supply system, the electric power system, the hydronic pipe adduction (for the water supply system), and the environmental thermic sensor.

fan axial flow 1

no electric power

hydraulic disjunctor 1

no electric power

cold water

boiler pump 1 broken

environmental thermal sensor

flow fan n°1 temperature sensor

no control

no drive c.p.

no electric power

gas adduction valve 1 closed

not operating circulation pump

not operating boiler

no natural gas

no gas supply

zone valve 1.1 rupture

rupture hydronic pipe adduction

no warm H2O supp.1

no water supply

no supply H2O

thermal sensor n°1.1 not operative

gas burner rupture 1

piping rupture loop 1.1

pump 1 broken

thermal sensor n° 1.1 not operative

no control p.

environmental thermal sensor

no electric power

no drive pump

not operating pump 1

no discharge head - loop 1.1

no electric power

environmental thermal sensor

thermal sensor n° 1.1 not operative

Control Valve rupture loop1

no control

no drive c.v.

no electric power

hydraulic disjunctor 2

valve automatic control 1

3-way zone valve closed 2.1

AND

gas burner rupture 2

environmental thermal sensor

thermal sensor n° 2.1 not operative

boiler pump 2 broken

no control

no drive c.p.

flow fan n°1 temperature sensor

no electric power

gas adduction valve 2 closed

not operating circulation pump

not operating boiler

no natural gas

no gas supply

zone valve 2.1 rupture

cold water

no warm H2O supp.2

environmental thermal sensor

thermal sensor n° 2.1 not operative

no control p.

pump 2 broken

not operating pump 2

no drive pump

no electric power

piping rupture loop 2.1

fan axial flow 2

no discharge head - loop 2.1

flow fan n°2 temperature sensor

no flow fan 2

environmental thermal sensor

thermal sensor n° 2.1 not operative

no control

Control Valve rupture loop2

valve automatic control 2

no drive c.v.

no electric power

no electric power

no water supply

3-way zone valve closed 1.2

rupture hydronic pipe adduction

no supply H2O

Fig. 8.68 Thermic system, two boilers and fan-coil redundancy and fill water. Configuration D

3-way zone valve closed 1.1

flow fan n°1 temperature sensor

no flow fan 1

no warm air Fan 1

no warm air

No Thermal Comfort

no electric power

hydraulic disjunctor 1

gas adduction valve 1 closed

no control

gas burner rupture 1

thermal sensor n°1.2 not operative

boiler pump 1 broken

environmental thermal sensor

flow fan n°2 temperature sensor

no electric power

no drive c.p.

not operating circulation pump

not operating boiler 1

no natural gas

no gas supply

zone valve 1.2 rupture

cold water

no warm H2O supp.1

AND

environmental thermal sensor

thermal sensor n° 1.2 not operative

no control p.

pump 1 broken

not operating pump 1

no gas supply

no control

no drive c.v.

thermal sensor n° 1.2 not operative

Control Valve rupture loop1

environmental thermal sensor

flow fan n°2 temperature sensor

gas burner rupture 2

thermal sensor n° 2.2 not operative

boiler pump 2 broken

no control

no drive c.p.

no electric power

gas adduction valve 2 closed

not operating circulation pump

not operating boiler 2

cold water

no warm H2O supp.2

no natural gas

valve automatic control

zone valve 2.2 rupture

hydraulic disjunctor 2

environmental thermal sensor

no electric power

no electric power

no discharge head - loop 1.2

3-way zone valve closed 2.2

no drive pump 1

no electric power

piping rupture loop 1.2

no warm air Fan 2

no electric power

piping rupture loop 2.2

thermal sensor n° 2.2 not operative

no control p.

environmental thermal sensor

no drive pump 2

pump 2 broken

not operating pump 2

no discharge head - loop 2.2

no electric power

environmental thermal sensor

Control Valve rupture loop2

thermal sensor n° 2.2 not operative

no control

no drive c.v.

valve automatic control 2

272 8 Effects Analysis and Reliability Modeling of Complex Production Systems

8.7 Application 1 – FTA

273

No Thermal Comfort

no warm air

no warm air Fan 2

no warm air Fan 1

no flow fan 1 no flow fan 2

flow fan n°1 temperature sensor

fan axial flow 1

no electric power

flow fan n°2 temperature sensor

AND

1.1

fan axial flow 2

2.1

no electric power

AND

1.2

2.2

Fig. 8.69 Two boilers and fan-coil redundancy. Configuration E

No Thermal Comfort

no WARM air

flow fan n°1 temperature

fan axial flow 1

no electric power

zone valve 1.1 rupture

hydraulic disjunctor 1

Fig. 8.70 Configuration C, EFT

no gas supply

gas adduction valve 1

environmental thermal sensor

boiler pump 1 broken

gas burner rupture 1

piping rupture loop 1.1

thermal sensor n° 1.1

pump 1 broken

Control Valve rupture loop1

flow fan n°2 temperature

fan axial flow 2

zone valve 1.2 rupture

piping rupture loop 1.2

thermal sensor n° 1.2

pump 1 broken no electric power no gas supply gas adduction valve 1 closed gas burner rupture 1 boiler pump 1 broken Control Valve rupture loop1 hydraulic disjunctor 1 environmental thermal sensor piping rupture loop 1.1 and thermal sensor no. 1.2 not operative piping rupture loop 1.1 and fan axial flow 2 piping rupture loop 1.1 and flow fan no. 2 temperature sensor piping rupture loop 1.1 and piping rupture loop 1.2 piping rupture loop 1.1 and zone valve 1.2 rupture zone valve 1.1 rupture and thermal sensor no. 1.2 not operative zone valve 1.1 rupture and piping rupture loop 1.2 zone valve 1.1 rupture and zone valve 1.2 rupture zone valve 1.1 rupture and fan axial flow 2 zone valve 1.1 rupture and flow fan no. 2 temperature sensor thermal sensor no. 1.1 not operative and thermal sensor no. 1.2 not operative thermal sensor no. 1.1 not operative and piping rupture loop 1.2 thermal sensor no. 1.1 not operative and zone valve 1.2 rupture thermal sensor no. 1.1 not operative and fan axial flow 2 thermal sensor no. 1.1 not operative and flow fan no. 2 temperature sensor fan axial flow 1 and thermal sensor no. 1.2 not operative fan axial flow 1 and piping rupture loop 1.2 fan axial flow 1 and zone valve 1.2 rupture fan axial flow 1 and fan axial flow 2 fan axial flow 1 and flow fan no. 2 temperature sensor flow fan no. 1 temperature sensor and thermal sensor no. 1.2 not operative flow fan no. 1 temperature sensor and piping rupture loop 1.2 flow fan no. 1 temperature sensor and zone valve 1.2 rupture flow fan no. 1 temperature sensor and fan axial flow 2 flow fan no. 1temperature sensor and flow fan no. 2 temperature sensor

1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

environmental thermal sensor piping rupture loop 1.1 and thermal sensor no. 1.2 not operative

piping rupture loop 1.1 and fan axial flow 2 piping rupture loop 1.1 and flow fan no. 2 temperature sensor piping rupture loop 1.1 and piping rupture loop 1.2

piping rupture loop 1.1 and zone valve 1.2 rupture zone valve 1.1 rupture and thermal sensor no. 1.2 not operative zone valve 1.1 rupture and piping rupture loop 1.2 zone valve 1.1 rupture and zone valve 1.2 rupture

zone valve 1.1 rupture and fan axial flow 2

zone valve 1.1 rupture and flow fan no. 2 temperature sensor

thermal sensor no. 1.1 not operative and thermal sensor no. 1.2 not operative thermal sensor no. 1.1 not operative and piping rupture loop 1.2

thermal sensor no. 1.1 not operative and zone valve 1.2 rupture

thermal sensor no. 1.1 not operative and fan axial flow 2

thermal sensor no. 1.1 not operative and flow fan no. 2 temperature sensor fan axial flow 1 and thermal sensor no. 1.2 not operative fan axial flow 1 and piping rupture loop 1.2 fan axial flow 1 and zone valve 1.2 rupture fan axial flow 1 and fan axial flow 2

fan axial flow 1 and flow fan no. 2 temperature sensor

flow fan no. 1 temperature sensor and thermal sensor no. 1.2 not operative flow fan no. 1 temperature sensor and piping rupture loop 1.2 flow fan no. 1 temperature sensor and zone valve 1.2 rupture

flow fan no. 1 temperature sensor and fan axial flow 2 flow fan no. 1 temperature sensor and flow fan no. 2 temperature sensor

Configuration B

1 1 1 1 1 1 1 1 1 1

Cardinality

pump 1 broken no water supply rupture hydronic pipe adduction no electric power no gas supply gas adduction valve 1 closed gas burner rupture 1 boiler pump 1 broken Control Valve rupture loop 1 hydraulic disjunctor 1

Configuration A

Table 8.15 Minimal cut sets configuration, configurations A, B, and C

2 2

2

2

2 2 2 2

2

2

2

2

2

2

2

2 2 2 2

2 2 2

2 2

1 1 1 1 1 1 1 1 1 2

Cardinality

zone valve 1.2 rupture fan axial flow 2 flow fan no. 2 temperature sensor

environmental thermal sensor thermal sensor no. 1.2 not operative piping rupture loop 1.2

pump 1 broken piping rupture loop 1.1 no electric power zone valve 1.1 rupture no gas supply gas adduction valve 1 closed gas burner rupture 1 boiler pump 1 broken thermal sensor no. 1.1 not operative Control Valve rupture loop 1 hydraulic disjunctor 1 fan axial flow 1 flow fan no. 1 temperature sensor

Configuration C

1 1 1

1 1 1

1 1

1 1 1 1 1 1 1 1 1 1

Cardinality

274 8 Effects Analysis and Reliability Modeling of Complex Production Systems

8.7 Application 1 – FTA

275

Table 8.16 Minimal cut sets configuration, configurations D and E Configuration D no water supply rupture hydronic pipe adduction no electric power no gas supply environmental thermal sensor cut sets of cardinality 2 40 cut sets example: ...

Cardinality

Configuration E

Cardinality

1

no electric power

1

1 1 1 1

no gas supply environmental thermal sensor

1 1

2 2

cut sets of cardinality 3 289 cut sets example: pump 1 broken and piping rupture loop 2.1 and zone valve 2.2 rupture ... cut sets of cardinality 4 80 cut sets example: zone valve 1.1 rupture, zone valve 1.2 rupture, thermal sensor no. 2.1 not operative, piping rupture loop 2.2 piping rupture loop 1.1 and zone valve 2.1 rupture and zone valve 1.2 rupture and zone valve 2.2 rupture ...

3 3

4 4

Configuration A Configuration B Configuration C Configuration D Configuration E

0.3288 0.4831 0.2886 0.4492 0.6599

0.1534 0.2886 0.1299 0.2367 0.4453

cut sets of cardinality 3 288 cut sets example: Control Valve rupture loop 1 and zone valve 2.1 rupture and flow fan no. 2 temperature sensor ... cut sets of cardinality 4 81 cut sets example: ...

2 2

3 3

4

4

Table 8.17 System reliability parameters Reliability t D 4;000 t D 6;570

cut sets of cardinality 2 40 cut sets example: pump 1 broken and gas burner rupture 2 ...

MTTF 3,524 5,180 3,218 4,510 7,062

8.7.3.5 Configuration E – Two Boilers and Fan-Coil Redundancy Figure 8.87 presents the failure probability function F .t/, the reliability R.t/, the probability density function f .t/, and the failure rate .t/ for the thermic system, made up of two alternative boilers (i. e., one is redundant) and two redundant fan-coils, without requiring water supply. Figures 8.88–8.90 are similar to those introduced for the previous system configurations. The most critical basic events/components are the gas supply system, the electric power system, the

environmental thermic sensor, pump 1 and pump 2, and the control valve rupture event. Table 8.17 reports the values of reliability (t D 4;000 and 6,570) and MTTF for configurations A– E. In particular configuration E assumes the best values of reliability and MTTF if compared with the others.

8.7.3.6 Repairable System and Monte Carlo Simulation Now the system is supposed to be repairable and all basic components subject to very similar repair behaviors. Figure 8.91 presents the results of the evaluation of the probability distribution of the ttr values in accordance with the availability of a set of 100 historical values. In particular, by a normal distribution is detected with mean value 4.844 hours and standard deviation 1.104 hours. All components are supposed to be repairable in accordance to this statistical distribution.

276

8 Effects Analysis and Reliability Modeling of Complex Production Systems

ReliaSoft BlockSim 7 - www.ReliaSoft.com

ReliaSoft BlockSim 7 - www.ReliaSoft.com

Unreliability vs Time

1.000

Reliability vs Time

1.000

Unreliability

Reliability

Fault Tree1 Unreliability Line

Fault Tree1 Reliability Line

0.800

Reliability, R(t)

Unreliability, F(t)=1-R(t)

0.800

0.600

0.400

0.200

0.600

0.400

0.200

0.000 0.000

4000.000

8000.000

12000.000

16000.000

0.000 0.000

20000.000

4000.000

8000.000

Time, (t) ReliaSoft BlockSim 7 - www.ReliaSoft.com

12000.000

16000.000

20000.000

Time, (t) ReliaSoft BlockSim 7 - www.ReliaSoft.com

Probability Density Function

3.000E-4

Failure Rate vs Time

4.000E-4

Pdf

Failure Rate

Fault Tree1 Pdf Line

Fault Tree1 Failure Rate Line

3.400E-4

Failure Rate, f(t)/R(t)

2.401E-4

f(t)

1.802E-4

1.204E-4

6.048E-5

2.800E-4

2.200E-4

1.600E-4

6.000E-7 0.000

4000.000

8000.000

12000.000

16000.000

1.000E-4 0.000

20000.000

4000.000

8000.000

Time, (t)

12000.000

16000.000

20000.000

Time, (t)

®

Fig. 8.71 F .t /, R.t /, f .t /, and .t /. System configuration A. ReliaSoft software ReliaSoft BlockSim 7 - www.ReliaSoft.com

ReliaSoft BlockSim 7 - www.ReliaSoft.com

Static Reliability Importance

0.481

Static Reliability Importance

0.213

Reliability

Reliability

100%

100%

50%

50%

0.170 0% 5 Item(s)

0.289

0.192

0.096

0.000

Reliability Importance Value

Reliability Importance Value

0.385

0% 5 Item(s)

0.128

0.085

0.043

no water supply

pump 1 broken

no gas supply

no electric power Control Valve rup...

0.000

no water supply

Time = 4000

pump 1 broken

no gas supply

no electric power Control Valve rup...

Time = 8000

Fig. 8.72 Static reliability analysis. System configuration A. ReliaSoft® software

Figure 8.92 presents the system up/down diagram, within an operating time period of 10 years, corresponding 65,700 h, obtained by the application of the Monte Carlo simulation. Figure 8.93 presents the block up/down analysis obtained by the Monte Carlo dynamic evaluation ap-

plied to the most critical basic components/events of the failure tree. It can be stated that the mean availability is 0.9997, the point availability (for t D 65;700 h) is 1, the ENF is 4.15, the uptime is 65;679 h, and the corrective downtime is 20:17 h.

8.8 Application 2 – FTA in a Waste to Energy System

277

ReliaSoft BlockSim 7 - www.ReliaSoft.com

Reliability Importance vs Time 1.000

Importance Fault Tree1 Extra Starting Block Extra Ending Block no electric power no electric power no electric power no electric power no electric power no electric power no electric power flow fan n°1 temperature sensor flow fan n°2 temperature sensor no water supply no electric power rupture hydronic pipe adduction hydraulic disjunctor 1 no gas supply gas adduction valve 1 closed gas burner rupture 1 boiler pump 1 broken environmental thermal sensor environmental thermal sensor environmental thermal sensor pump 1 broken Control Valve rupture loop1 thermal sensor n° 1.1 not operative environmental thermal sensor thermal sensor n° 1.2 not operative thermal sensor n°1.1 not operative thermal sensor n°1.2 not operative fan axial flow 1 flow fan n°1 temperature sensor no water supply rupture hydronic pipe adduction no electric power zone valve 1.1 rupture gas adduction valve 1 closed no gas supply environmental thermal sensor boiler pump 1 broken piping rupture loop 1.1 thermal sensor n° 1.1 not operative pump 1 broken zone valve 1.2 rupture piping rupture loop 1.2

Reliability Importance Value

0.800

0.600

0.400

0.200

0.000 0.000

4000.000

8000.000

12000.000

16000.000

20000.000

Time, (t)

Fig. 8.73 Time-dependent reliability analysis. System configuration A. ReliaSoft® software

8.8 Application 2 – FTA in a Waste to Energy System This section introduces a case study including a costbased model for failure modes analysis, reliability prediction, and magnitude evaluation of a waste to energy (WtE) plant. The model pays particular attention to the economic determination and evaluation of the environmental effects, here called “externalities,” of those facilities dedicated to the thermic treatment of waste, in accordance with the adoption of different maintenance policies. In detail, after a short description of the incinerator object of the study, this section illustrates the FTA conducted on some critical subsystems of the WtE plant. A qualitative and quantitative evaluation of the solid waste incinerator is carried out and the results of these FTAs, as reported in Sects. 8.8.6 and 8.8.7, join in a cost-based prediction reliability model for the determination of the economic effects of the emissions, e. g., nitrogen oxides (NOx ) and carbon dioxide. This

model is based on the integration of a failure modes analysis, a reliability prediction analysis, and a “magnitude of consequences” evaluation, which takes inspiration from the large number of literature studies on the determination of the externalities in WtE plants.

8.8.1 Introduction to Waste Treatment An incinerator is a waste treatment technology for the thermic treatment of waste. By high-temperature combustion it transforms waste into thermic energy useful for the generation of electricity and/or for district heating. An incinerator also produces gaseous emissions in the atmosphere and residual ash. The incinerator represents one of the most popular alternative technology to landfilling and biological treatment of waste. It is particularly popular in countries such as Japan where land is a scarce resource, but several municipalities all over the world, such as

278

8 Effects Analysis and Reliability Modeling of Complex Production Systems

ReliaSoft BlockSim 7 - www.ReliaSoft.com

Block Failure Rate vs Time 4.000E-4

Failure Rate Fault Tree1 System Control Valve rupture loop1 no electric power no gas supply no water supply pump 1 broken

Failure Rate, f(t)/R(t)

3.200E-4

2.400E-4

1.600E-4

8.000E-5

0.000 0.000

100000.000

200000.000

300000.000

400000.000

500000.000

Time, (t)

Fig. 8.74 Failure rates of the system and of the most critical components. System configuration A. ReliaSoft® software

Hong Kong, Saugus in Massachusetts, USA, Brescia in Italy, London in the UK, and Tokyo in Japan, have adopted municipal solid waste incinerators. Table 8.18 presents a snapshot on WtE plants in Europe as of 2002. A WtE plant is equipped with high-efficiency furnaces and devices for continuous monitoring of emissions and air pollution control. There are various types of incinerator plants: • Simple incinerator made of a brick-lined cell, with a metal grate over a lower ash pit, and openings, called “clinkers,” for waste loading and refuse removal; often used for domestic heating. • Moving grate combustion. A grate enables the movement of waste through the combustion chamber. • Rotary kiln, made of a long, slightly inclined cylindrical tube along which refuse is continuously moved and spills out of the end through the clink-

ers. The system is made of some different sections where waste is dried, ignited, and completely burned. • Multiple/stepped heart. Waste is transported through the furnace by moving teeth mounted on a central rotating shaft. • Fluidized bed. An flow of air is forced through a bed of sand. The sand particles separate, enabling air to flow through; thus, a fluidized bed is created and fuel and waste can be introduced. The mass of waste, fuel, and sand is fully circulated through the furnace.

8.8.2 Case study The WtE plant considered, as reported in Table 8.19, has a plant capacity, or waste treatment capacity, of about 200 ton=day for 2;600 kcal=kg

8.8 Application 2 – FTA in a Waste to Energy System ReliaSoft BlockSim 7 - www.ReliaSoft.com

279 ReliaSoft BlockSim 7 - www.ReliaSoft.com

Unreliability vs Time

1.000

Reliability vs Time

1.000

Unreliability

Reliability Fault Tree1 Reliability Line

Fault Tree1 Unreliability Line

0.800

Reliability, R(t)=F(t)

Unreliability, F(t)=1-R(t)

0.800

0.600

0.400

0.200

0.600

0.400

0.200

0.000 0.000

6000.000

12000.000

18000.000

24000.000

0.000 0.000

30000.000

6000.000

ReliaSoft BlockSim 7 - www.ReliaSoft.com

12000.000

18000.000

24000.000

30000.000

Time, (t)

Time, (t) ReliaSoft BlockSim 7 - www.ReliaSoft.com

Probability Density Function

2.000E-4

Failure Rate vs Time

3.000E-4

Pdf

Failure Rate

Fault Tree1 Pdf Line

Fault Tree1 Failure Rate Line

1.600E-4

Failure Rate, f(t)/R(t)

2.400E-4

f(t)

1.201E-4

8.012E-5

4.016E-5

1.800E-4

1.200E-4

6.000E-5

2.000E-7 0.000

6000.000

12000.000

18000.000

24000.000

0.000 0.000

30000.000

6000.000

12000.000

18000.000

24000.000

30000.000

Time, (t)

Time, (t)

Fig. 8.75 F .t /, R.t /, f .t /, and .t /. System configuration B. ReliaSoft® software ReliaSoft BlockSim 7 - www.ReliaSoft.com

ReliaSoft BlockSim 7 - www.ReliaSoft.com

Static Reliability Importance 0.682

Static Reliability Importance 0.428

Reliability

Reliability

100%

100%

50%

50%

0.342 0% 5 Item(s)

0.409

0.273

0.136

Reliability Importance Value

Reliability Importance Value

0.546

0% 5 Item(s)

0.257

0.171

0.086

0.000

0.000 pump 1 broken

no gas supply

no electric power Control Valve rup... gas adduction val...

pump 1 broken

Time = 4000

no gas supply no electric power Control Valve rup... gas adduction val...

Time = 8000

Fig. 8.76 Static reliability analysis. System configuration B. ReliaSoft® software

of waste, resulting in 11;000 MWh=year of electric energy and 34;000 MWh=year of thermic energy produced, thus corresponding to 1:238 kWh for each kilogram of waste. The system supplies thermic energy for a community of about 2,600 families.

8.8.3 Emissions and Externalities: Literature Review Even incinerators are faced with environmental and health questions. An exemplifying list obtained from the literature mentions damage to buildings, forests,

280

8 Effects Analysis and Reliability Modeling of Complex Production Systems

ReliaSoft BlockSim 7 - www.ReliaSoft.com

Reliability Importance vs Time 1.000

Importance Fault Tree1 Extra Starting Block Extra Ending Block no electric power no electric power no electric power no electric power no electric power no electric power no electric power flow fan n°1 temperature sensor flow fan n°2 temperature sensor no electric power hydraulic disjunctor 1 no gas supply gas adduction valve 1 closed gas burner rupture 1 boiler pump 1 broken environmental thermal sensor environmental thermal sensor environmental thermal sensor pump 1 broken Control Valve rupture loop1 thermal sensor n° 1.1 not operative environmental thermal sensor thermal sensor n° 1.2 not operative thermal sensor n°1.1 not operative thermal sensor n°1.2 not operative fan axial flow 1 flow fan n°1 temperature sensor no electric power zone valve 1.1 rupture gas adduction valve 1 closed no gas supply environmental thermal sensor boiler pump 1 broken piping rupture loop 1.1 thermal sensor n° 1.1 not operative pump 1 broken zone valve 1.2 rupture piping rupture loop 1.2 no electric power gas burner rupture 1 flow fan n°2 temperature sensor fan axial flow 2

Reliability Importance Value

0.800

0.600

0.400

0.200

0.000 0.000

6000.000

12000.000

18000.000

24000.000

30000.000

Time, (t)

Fig. 8.77 Time-dependent reliability analysis. System configuration B. ReliaSoft® software

and agricultural yields; costs associated with transportation and logistics (e. g., vehicle emissions, congestion, accidents, noise); odor, dust, visual intrusion, etc. The magnitude of these effects strongly depends on the distance from the site, the type of waste, topography, prevailing wind directions, etc., and as a consequence the costs of externalities can range in a wide interval. According to EC Directives, published in 2000, NOx emissions, with about 70% of the total health costs, are the most critical externality generated by an incinerator. They are believed to aggravate asthmatic conditions, and react with the oxygen in the air to produce ozone, which is also an irritant, and eventually forming nitric acid when they are dissolved in water. When they are dissolved in atmospheric moisture, the result is acid rain, which can damage entire forest ecosystems. As illustrated in Table 8.20, costs associated with NOx vary very significantly in literature studies (Eshet et al. 2006), ranging from US$ 0.13 to US$ 18.6

per kilogram of NOx . This table presents economic unit values of all externalities associated with different emissions (CO2 , CH4 , NOx , PM10 , SO2 , etc.) for both landfill and incinerators. These economic unit values are quantified in dollar per kilogram of pollutant (at 2003 prices). Table 8.21 reports economic valuations in US dollars per ton of waste (2003 prices) for specific impacts (e. g., transportation, leachate) for incineration. The following analysis and results refer to the control and reduction of NOx emissions in the incinerator considered, with particular attention to the so-called selective noncatalytic reduction (SNCR) technology.

8.8.4 SNCR Plant Table 8.22 quantifies the annual cost of externalities associated with some critical emissions of the incinerator, in accordance with the economic unit values

8.8 Application 2 – FTA in a Waste to Energy System

281

ReliaSoft BlockSim 7 - www.ReliaSoft.com

Block Failure Rate vs Time 3.000E-4

Failure Rate Fault Tree1 System Control Valve rupture loop1 gas adduction valve 1 closed no electric power no gas supply pump 1 broken

Failure Rate, f(t)/R(t)

2.400E-4

1.800E-4

1.200E-4

6.000E-5

0.000 0.000

100000.000

200000.000

300000.000

400000.000

500000.000

Time, (t)

Fig. 8.78 Failure rates of the system and of the most critical components. System configuration B. ReliaSoft® software

collected from the literature (total average value in Table 8.20, last row). In particular, the emission of NOx represents about 33.5% of the admissible value of 85;619 kg=year (EC Directives); moreover, the related cost represents 99% of global social costs associated with pollutant emissions. In order to limit gas emissions in the atmosphere, and in particular the emissions of NOx , in accordance with the limits fixed by 2000/76/CE Directive, a SNCR plant has been recently introduced. The SNCR technology injects urea into the firebox of the boiler to react with the nitrogen oxides formed in the combustion process at a gas temperature between 1,600 and 2;100 ı F. This chemical reaction produces elemental nitrogen, carbon dioxide, and water. As a result of the introduction of the SNCR plant, the average value of NOx emissions decreased from 150 to 120 mg=Nm3 . This is the control parameter of the incineration process, and values greater than 200 mg=Nm3 , as declared by the manufacturer, can reveal anomalies.

Figure 8.94 illustrates the statistical distribution of NOx (mg=Nm3) emissions during a period of time T from June 2005 to February 2007, for the power plant considered. This analysis is based on more than 25,000 half-hour observations. A half-hourly observation gives the average value of 30 values registered each minute. Figure 8.95 reports the trend of half-hour values during the 20-month observation period. By an in-depth analysis of these values, for 12;185 h the NOx emissions did not pass the critical value of 200 mg=Nm3, while for 75 h the SNCR system did not function correctly. In particular, the emission values exceeded 235 mg=Nm3 for 4 h.

8.8.5 SNCR Plant. Reliability Prediction and Evaluation Model A FTA was implemented by Relex® Reliability software in order to investigate the minimal conditions

282

8 Effects Analysis and Reliability Modeling of Complex Production Systems

ReliaSoft BlockSim 7 - www.ReliaSoft.com

ReliaSoft BlockSim 7 - www.ReliaSoft.com

Unreliability vs Time

1.000

Reliability vs Time

1.000

Unreliability

Reliability

Fault Tree1 Unreliability Line

Fault Tree1 Reliability Line

0.800

Reliability, R(t)=F(t)

Unreliability, F(t)=1-R(t)

0.800

0.600

0.400

0.200

0.600

0.400

0.200

0.000 0.000

4000.000

8000.000

12000.000

16000.000

0.000 0.000

20000.000

4000.000

8000.000

12000.000

Time, (t) ReliaSoft BlockSim 7 - www.ReliaSoft.com

16000.000

20000.000

Time, (t) ReliaSoft BlockSim 7 - www.ReliaSoft.com

Probability Density Function

4.000E-4

Failure Rate vs Time

4.000E-4

Pdf

Failure Rate

Fault Tree1 Pdf Line

Fault Tree1 Failure Rate Line

3.600E-4

Failure Rate, f(t)/R(t)

3.201E-4

f(t)

2.402E-4

1.603E-4

8.040E-5

3.200E-4

2.800E-4

2.400E-4

5.000E-7 0.000

4000.000

8000.000

12000.000

16000.000

2.000E-4 0.000

20000.000

4000.000

Time, (t)

8000.000

12000.000

16000.000

20000.000

Time, (t)

Fig. 8.79 F .t /, R.t /, f .t /, and .t /. System configuration C. ReliaSoft® software

ReliaSoft BlockSim 7 - www.ReliaSoft.com

ReliaSoft BlockSim 7 - www.ReliaSoft.com

Static Reliability Importance

0.408

Static Reliability Importance

0.166

Reliability

Reliability

100%

100%

50%

50%

0.133 0% 5 Item(s)

0.245

0.163

0.082

0.000

Reliability Importance Value

Reliability Importance Value

0.326

0% 5 Item(s)

0.100

0.066

0.033

pump 1 broken

no gas supply

flow fan n°2 tem... thermal sensor n... flow fan n°1 tem...

0.000

pump 1 broken

Time = 4000

no gas supply thermal sensor n... flow fan n°1 tem... flow fan n°2 tem...

Time = 8000

®

Fig. 8.80 Static reliability analysis. System configuration C. ReliaSoft software

8.8 Application 2 – FTA in a Waste to Energy System ReliaSoft BlockSim 7 - www.ReliaSoft.com

283

Reliability Importance vs Time

1.000

Importance Fault Tree1 no electric power no electric power no electric power no electric power no electric power no electric power no electric power flow fan n°1 temperature sensor flow fan n°2 temperature sensor no electric power hydraulic disjunctor 1 no gas supply gas adduction valve 1 closed gas burner rupture 1 boiler pump 1 broken environmental thermal sensor environmental thermal sensor environmental thermal sensor pump 1 broken Control Valve rupture loop1 thermal sensor n° 1.1 not operative environmental thermal sensor thermal sensor n° 1.2 not operative thermal sensor n°1.1 not operative thermal sensor n°1.2 not operative fan axial flow 1 flow fan n°1 temperature sensor no electric power zone valve 1.1 rupture gas adduction valve 1 closed no gas supply environmental thermal sensor boiler pump 1 broken piping rupture loop 1.1 thermal sensor n° 1.1 not operative pump 1 broken zone valve 1.2 rupture piping rupture loop 1.2 no electric power gas burner rupture 1 flow fan n°2 temperature sensor fan axial flow 2 hydraulic disjunctor 1 Control Valve rupture loop1

Reliability Importance Value

0.800

0.600

0.400

0.200

0.000 0.000

4000.000

8000.000

12000.000

16000.000

20000.000

Time, (t)

Fig. 8.81 Time-dependent reliability analysis. System configuration C. ReliaSoft® software

which cause an incorrect functioning of the system identified by the top event “NOx emissions exceeding the threshold 200 mg=Nm3.” Figure 8.96 shows the fault tree obtained for the determination of the unavailability Q.t/ of the SNCR plant and the probability associated with the top event.

where AIR_SEC D VR1101_fail C AIR_fail level 3

D VR1101 C ELECRTRIC_fail

level 4

C AIR_fail D VR1101 C TT101  TT105

8.8.6 Qualitative FTA Evaluation This section illustrates the qualitative evaluation of the fault tree, given the top event “exceeding NOx 200 mg=Nm3 limit.” By applying the Boolean algebra, one can explain the top event explained as follows (see Fig. 8.96 for nomenclature): TOP D TCOMB C P_UREA level 1

D AIR_SEC C m_CIRCU C TKUREA

level 2

C m_DOSAGE C e_ELECTRIC C m_SUPPLY;

level 5

C AIR_fail; m_CIRCU D p_CIRCU C f_CIRCU level 3

D CX51005  CX51006

level 4

C DH51001  DH51002; m_DOSAGE D p_DOSAGE C f_DOSAGE level 3

D CX51008  CX51009

level 4

C DH51003  DH51004;

284

8 Effects Analysis and Reliability Modeling of Complex Production Systems

ReliaSoft BlockSim 7 - www.ReliaSoft.com

Block Failure Rate vs Time 4.000E-4

Failure Rate Fault Tree1 System flow fan n°1 temperature sensor flow fan n°2 temperature sensor no gas supply pump 1 broken thermal sensor n° 1.1 not operative

Failure Rate, f(t)/R(t)

3.220E-4

2.440E-4

1.660E-4

8.800E-5

1.000E-5 0.000

40000.000

80000.000

120000.000

160000.000

200000.000

Time, (t)

Fig. 8.82 Failure rates of the system and of the most critical components. System configuration C. ReliaSoft® software

m_SUPPLY D SPEARS_1  SPEARS_2 level 3

D .INJ51101L C INJ51102L

level 4

C INJ51103L/  .INJ51101H C INJ51102H C INJ51103H/ D INJ51101L  INJ51101H

level 5

C INJ51101L  INJ51102H C INJ51101L  INJ51103H C INJ51102L  INJ51101H C INJ51102L  INJ51102H C INJ51102L  INJ51103H C INJ51103L  INJ51101H C INJ51103L  INJ51102H C INJ51103L  INJ51103H:

Consequently, TOP D VR1101 C TT101  TT105 C AIR_fail C CX51005  CX51006 C DH51001  DH51002 C TKUREA C CX51008  CX51009 C DH51003  DH51004 C e_ELECTRIC C INJ51101L  INJ51101H C INJ51101L  INJ51102H C INJ51101L  INJ51103H C INJ51102L  INJ51101H C INJ51102L  INJ51102H C INJ51102L  INJ51103H C INJ51103L  INJ51101H C INJ51103L  INJ51102H C INJ51103L  INJ51103H:

8.8 Application 2 – FTA in a Waste to Energy System

285

ReliaSoft BlockSim 7 - www.ReliaSoft.com

ReliaSoft BlockSim 7 - www.ReliaSoft.com

Unreliability vs Time

1.000

Reliability vs Time

1.000

Unreliability

Reliability

Fault Tree1 Unreliability Line

Fault Tree1 Reliability Line

0.800

Reliability, R(t)=F(t)

Unreliability, F(t)=1-R(t)

0.800

0.600

0.400

0.200

0.600

0.400

0.200

0.000 0.000

4000.000

8000.000

12000.000

16000.000

0.000 0.000

20000.000

4000.000

8000.000

Time, (t)

12000.000

16000.000

20000.000

Time, (t)

ReliaSoft BlockSim 7 - www.ReliaSoft.com

ReliaSoft BlockSim 7 - www.ReliaSoft.com

Probability Density Function

2.000E-4

Failure Rate vs Time

4.000E-4

Pdf

Failure Rate

Fault Tree1 Pdf Line

Fault Tree1 Failure Rate Line

3.200E-4

Failure Rate, f(t)/R(t)

1.600E-4

f(t)

1.200E-4

8.000E-5

4.000E-5

2.400E-4

1.600E-4

8.000E-5

0.000 0.000

4000.000

8000.000

12000.000

16000.000

0.000 0.000

20000.000

4000.000

8000.000

Time, (t)

12000.000

16000.000

20000.000

Time, (t)

Fig. 8.83 F .t /, R.t /, f .t /, and .t /. System configuration D. ReliaSoft® software ReliaSoft BlockSim 7 - www.ReliaSoft.com

ReliaSoft BlockSim 7 - www.ReliaSoft.com

Static Reliability Importance

0.657

Static Reliability Importance

0.345

Reliability

Reliability

100%

100%

50%

50%

0.276 0% 5 Item(s)

0.394

0.263

0% 5 Item(s)

0.207

0.138

0.069

0.131

0.000

Reliability Importance Value

Reliability Importance Value

0.526

no water supply

no gas supply

no electric power rupture hydronic ... environmental th...

0.000

no water supply

no gas supply

no electric power rupture hydronic ... environmental th...

Time = 8000

Time = 4000

Fig. 8.84 Static reliability analysis. System configuration D. ReliaSoft® software

Filters DH, pumps CX, and spears INJ can be considered to be identical items, and consequently the analyst could be seduced into appling the absorption laws. The previous equation seems to change as follows:

TOP D VR1101 C TT101  TT105 C AIR_fail C TKUREA C e_ELECTRIC C INJ C CX C DH;

286

8 Effects Analysis and Reliability Modeling of Complex Production Systems

ReliaSoft BlockSim 7 - www.ReliaSoft.com

Reliability Importance vs Time

1.000

Importance Fault Tree1 Extra Starting Block Extra Ending Block no water supply rupture hydronic pipe adduction fan axial flow 1 no electric power rupture hydronic pipe adduction no water supply no electric power fan axial flow 2 hydraulic disjunctor 1 piping rupture loop 1.1 zone valve 1.1 rupture gas burner rupture 1 no electric power pump 1 broken boiler pump 1 broken Control Valve rupture loop1 thermal sensor n°1.1 not operative no electric power no electric power environmental thermal sensor thermal sensor n° 1.1 not operative no electric power zone valve 2.1 rupture no gas supply no electric power environmental thermal sensor flow fan n°1 temperature sensor flow fan n°1 temperature sensor flow fan n°1 temperature sensor thermal sensor n° 1.1 not operative thermal sensor n° 2.1 not operative environmental thermal sensor environmental thermal sensor no electric power no electric power piping rupture loop 2.1 environmental thermal sensor thermal sensor n° 2.1 not operative thermal sensor n° 2.1 not operative no electric power hydraulic disjunctor 1 zone valve 1.2 rupture

Reliability Importance Value

0.800

0.600

0.400

0.200

0.000 0.000

4000.000

8000.000

12000.000

16000.000

20000.000

Time, (t)

Fig. 8.85 Time-dependent reliability analysis. System configuration D. ReliaSoft® software

where CX D CX51008 D CX51008  CX51009 C CX51005  CX51006; DH D DH51001 D DH51001  DH51002 C DH51003  DH51004; INJ D INJ51101L D INJ51101L  INJ51101H

cardinality 1. Nevertheless this equation is not correct because the absorption laws can be applied only in the case when the same basic component event, i. e., the same item, is redundant in a Boolean equation. For example, if components DH51001 and DH51002 have the same failure behavior but they deal with distinct items, the following reduction is consequently false: DH D DH51001 D DH51001  DH51002

C INJ51101L  INJ51102H

C DH51003  DH51004:

C INJ51101L  INJ51103H C INJ51102L  INJ51101H C INJ51102L  INJ51102H C INJ51102L  INJ51103H C INJ51103L  INJ51101H C INJ51103L  INJ51102H C INJ51103L  INJ51103H: By the last equation eight cut sets are obtained, one of cardinality 2 (TT101  TT105) and the others of

In the same way the other reductions in the equation reported above are not feasible. The basic events involved are not mirror1 items. Similarly for the control of every critical emission and pollutant, e. g., HCl, CO, and SO2 , specific fault trees have been designed. Qualitative analyses for the determination of the MCS and quantitative anal1

The meaning of mirror event was illustrated at the beginning of this chapter.

8.8 Application 2 – FTA in a Waste to Energy System ReliaSoft BlockSim 7 - www.ReliaSoft.com

287

Block Failure Rate vs Time

4.000E-4

Failure Rate Fault Tree1 System environmental thermal sensor no electric power no gas supply no water supply rupture hydronic pipe adduction

Failure Rate, f(t)/R(t)

3.200E-4

2.400E-4

1.600E-4

8.000E-5

0.000 0.000

100000.000

200000.000

300000.000

400000.000

500000.000

Time, (t)

Fig. 8.86 Failure rates of the system and of the most critical components. System configuration D. ReliaSoft® software

yses for the determination of the reliability parameters, e. g., unavailability, ENF, and reliability function, which describe the correct and incorrect function of the system, have been implemented.

8.8.7 NOx Emissions: Quantitative FTA Evaluation This section summarizes the results obtained by the evaluation of the most important reliability parameters related to the system, given a specific top event “exceeding NOx limit.” For this purpose, Table 8.23 summarizes some significant parameters for the basic/primary components of the system which are involved in MCS previously identified. In particular,

assuming a length of the period of time T equal to 365 h, about 15 days, the approximated values of the unavailability by Eq. 8.11 and of the probability function F .T / by Eq. 8.10 are reported in Table 8.23, columns 4 and 5, respectively, while the exact value of F .T / is in the last column. In order to properly illustrate the correct quantitative evaluation of the fault trees in Figs. 8.96 and 8.100, the analysis is conducted on MCS assuming the same failure behavior for every component of the same kind, i. e., pumps, filters, and spears. Table 8.24 reports the unavailability by Eq. 8.12, the ENF by Eq. 8.17, the probability function, and the survival function for the generic cut set CSi . The following equation exemplifies the calculus of the ENF for the MCS made up of two temperature transmitters TT101 and TT105 (related to the cut set

288

8 Effects Analysis and Reliability Modeling of Complex Production Systems

ReliaSoft BlockSim 7 - www.ReliaSoft.com

ReliaSoft BlockSim 7 - www.ReliaSoft.com

Unreliability vs Time

1.000

Reliability vs Time

1.000

Unreliability

Reliability

Fault Tree1 Unreliability Line

Fault Tree1 Reliability Line

0.800

Reliability, R(t)=F(t)

Unreliability, F(t)=1-R(t)

0.800

0.600

0.400

0.200

0.600

0.400

0.200

0.000 0.000

6000.000

12000.000

18000.000

24000.000

0.000 0.000

30000.000

6000.000

12000.000

Time, (t) ReliaSoft BlockSim 7 - www.ReliaSoft.com

18000.000

24000.000

30000.000

Time, (t) ReliaSoft BlockSim 7 - www.ReliaSoft.com

Probability Density Function

1.000E-4

Failure Rate vs Time

3.000E-4

Pdf

Failure Rate Fault Tree1 Failure Rate Line

Fault Tree1 Pdf Line

2.500E-4

Failure Rate, f(t)/R(t)

8.010E-5

f(t)

6.020E-5

4.030E-5

2.040E-5

2.000E-4

1.500E-4

1.000E-4

5.000E-7 0.000

6000.000

12000.000

18000.000

24000.000

5.000E-5 0.000

30000.000

6000.000

Time, (t)

12000.000

18000.000

24000.000

30000.000

Time, (t)

Fig. 8.87 F .t /, R.t /, f .t /, and .t /. System configuration E. ReliaSoft® software

ReliaSoft BlockSim 7 - www.ReliaSoft.com

ReliaSoft BlockSim 7 - www.ReliaSoft.com

Static Reliability Importance

0.808

Static Reliability Importance

0.522

Reliability

Reliability

100%

100%

50%

50%

0.417 0% 5 Item(s)

0.485

0.323

0.162

0.000

Reliability Importance Value

Reliability Importance Value

0.647

0% 5 Item(s)

0.313

0.209

0.104

no gas supply

no electric power environmental th... pump 1 broken

pump 2 broken

0.000

no gas supply

Time = 4000

Fig. 8.88 Static reliability analysis. System configuration E. ReliaSoft® software

no electric power environmental th... pump 1 broken

Time = 8000

pump 2 broken

8.8 Application 2 – FTA in a Waste to Energy System

289

ReliaSoft BlockSim 7 - www.ReliaSoft.com

Reliability Importance vs Time 1.000

Importance Fault Tree1 Extra Starting Block Extra Ending Block fan axial flow 1 no electric power no electric power fan axial flow 2 hydraulic disjunctor 1 piping rupture loop 1.1 zone valve 1.1 rupture gas burner rupture 1 no electric power pump 1 broken boiler pump 1 broken Control Valve rupture loop1 thermal sensor n°1.1 not operative no electric power no electric power environmental thermal sensor thermal sensor n° 1.1 not operative no electric power zone valve 2.1 rupture no gas supply no electric power environmental thermal sensor flow fan n°1 temperature sensor flow fan n°1 temperature sensor flow fan n°1 temperature sensor thermal sensor n° 1.1 not operative thermal sensor n° 2.1 not operative environmental thermal sensor environmental thermal sensor no electric power no electric power piping rupture loop 2.1 environmental thermal sensor thermal sensor n° 2.1 not operative thermal sensor n° 2.1 not operative no electric power hydraulic disjunctor 1 zone valve 1.2 rupture gas adduction valve 1 closed no electric power piping rupture loop 1.2 flow fan n°2 temperature sensor

Reliability Importance Value

0.800

0.600

0.400

0.200

0.000 0.000

6000.000

12000.000

18000.000

24000.000

30000.000

Time, (t)

Fig. 8.89 Time-dependent reliability analysis. System configuration E. ReliaSoft® software

TT101TT105) on the ground of Eq. 8.17 for a period of time T D 365 h: ENFCSTT101TT105 .T D 365 h/ Z365 wCSTT101TT105 .t/ dt D

D 0

0

D 365.T T 101 T T 105 T T 105 C T T 105 T T 101 T T 101 /

X

Š 3:77  106 failures.

wj .t/

j 2CSTT101TT105



Y

 qk .t/ dt

k¤j k;j 2CSTT101TT105

Z365 D ŒT T 101 qT T 105 .t/ C T T 105 qT T 101 .t/ dt 0

1 T T 105 T T 101 D  1 T T 101

C T T 105 .T T 101 T T 101 / dt

0

Z365

D

T T 105 D 

Z365 ŒT T 101 .T T 105 T T 105 /

By the application of Eq. 8.13 for a period of time T D 15 days, a Y qCSi .T / D 1  Œ1  qCSi .T / QS .t D T / D i

i 4

D 7:410  10 X  qCSi .T / Š 7:412  104 : i

290

8 Effects Analysis and Reliability Modeling of Complex Production Systems

ReliaSoft BlockSim 7 - www.ReliaSoft.com

Block Failure Rate vs Time 2.500E-4

Failure Rate Fault Tree1 System environmental thermal sensor no electric power no gas supply pump 1 broken pump 2 broken

Failure Rate, f(t)/R(t)

2.000E-4

1.500E-4

1.000E-4

5.000E-5

0.000 0.000

100000.000

200000.000

300000.000

400000.000

500000.000

Time, (t)

Fig. 8.90 Failure rates of the system and of the most critical components. System configuration E. ReliaSoft® software

time to repair - ttr Evaluation Normal Mean StDev N

100

Percent

80

60

40

20

0 00 25 50 75 0 0 2 5 50 75 00 25 50 7 5 0 0 25 50 75 00 25 50 7 5 00 25 50 75 00 2. 2. 2. 2. 3 . 3 . 3. 3. 4. 4. 4. 4. 5 . 5 . 5. 5. 6. 6. 6. 6 . 7 . 7. 7. 7. 8.

ttr

Fig. 8.91 Time to repair (ttr) probability distribution evaluation

4.844 1.104 100

8.8 Application 2 – FTA in a Waste to Energy System ReliaSoft BlockSim 7 - www.ReliaSoft.com

291

System Up/Down State Operating Time Time Under Repair

System

0.000

13140.000

26280.000

39420.000

52560.000

65700.000

Time, (t)

Fig. 8.92 Repairable system. System up/down analysis. ReliaSoft® software

ReliaSoft BlockSim 7 - www.ReliaSoft.com

Block Up/Down State Operating Time Time Under Repair

pump 2 broken

pump 1 broken

no gas supply

no electric power

environmental thermal sensor

System

0.000

13140.000

26280.000

39420.000

52560.000

65700.000

Time, (t)

Fig. 8.93 Repairable system. Component up/down analysis. ReliaSoft® software

292

8 Effects Analysis and Reliability Modeling of Complex Production Systems

Table 8.18 Waste to energy (WtE) plants in Europe (2002) Country

Number of plants

Burned quantities (ton=year)

Austria Belgium Denmark France Germany UK Italy Norway Holland Portugal Spain Sweden Switzerland Hungary

2 18 32 112 60 3 50 4 11 2 8 19 31 1

406,700 2,652,000 3,136,000 11,965,800 16,787,400 1,071,000 3,488,776 273,000 4,412,000 933,800 1,070,300 2,344,000 3,150,700 420,000

Total

354

52,111,476

Table 8.19 Operative characteristic of the WtE plant, case study Operative characteristic Incinerator capacity. Waste quantities (nominal value considering 2 lines) Waste heat of combustion Smoke flow during gas purification Mean temperature of furnace Mean temperature of the postcombustion chamber Smoke temperature during cleaning Smoke temperature (ref. chimney) Vapor production Vapor pressure Overheated temperature Operation hours per year

Value

Unit of measure

8.33 (200) 10,868 (2,600) 50,400 1,000 980 230 170 28 10 300 8,000

ton=h (ton=day) kJ=kg (kcal=kg) Nm3 =h ı C ı C ı C ı C ton=h bar ı C h=year

Similarly, the failure probability function of the system for the period of time T is a Y FCSi .T / D 1  Œ1  FCSi .T / FS .t D T / D i

i

D 0:08373 

X

FCSi .T / Š 0:08665:

i

Applying Eq. 8.18, the ENF for the system is X WCSi Š 7:25  102 failures: ENF.T D 365 h/ Š i

Finally, the MTTR defined for the system, given the top event, can be quantified by the application of

Eq. 8.20: 8 7:412  104 QS .T / ˆ ˆ MTTR D Š Š 3:73 h ˆ S ˆ ws .T / 1:986  104 ˆ ˆ ˆ < 0:0725 Ws .T / ˆ D ws .T / D ˆ ˆ T 365 ˆ ˆ ˆ ˆ : Š 1:986  104 day1 :

8.8.8 Criticality Analysis Figure 8.97 presents a view of the criticality analysis conducted with Relex® Reliability software. There are three main measures to detect weak points in the

2001

2002

2002

2003

RDC and PIRA

Eunomia

AEA Technology

Dijkgraaf and Vollebergh

Total average value

2000

EU 0.0035

0.0238

0.034

0.0245–0.0257 (0.0251)

0.6242

0.379

0.4506–0.4892 (0.4694)

1999

Krewitt et al. 0.004–0.042 (0.023)

18.6

1998

6.8104

3.291

1.4–8.2 (4.2)

4.3–18.34 (11.32)

3.4–5.4 (5 EU)

18.05

1998

4.6–18 (11.3)

8

0.9–18 (9.45)

0.13

0.19

18.34

7.33 2.4–4.7 (3.55)

0.132–0.523 (0.3275)

0.132–0.523 (0.3275)

NOx

Rabl et al. (a) (average France) Rabl et al. (b)

0.053–2.223 (1.138)

0.124

0.086

2.69

0.051–0.2216 (0.1363)

0.0496–0.2216 (0.1356)

CH4

1997

0.0038–0.1339 (0.072)

0.023

0.023

0.004

0.04 0.004

0.0065–0.0496 (0.02805)

0.0017–0.0136 (0.00765)

Pollutant

CO2

ExternE (Spain–France)

UK

1998

1996

Enosh

Eyre EU

1996

EC (a) (German case)

1996

1995 1996

ECON EC (b) (average EU12)

1997

1994

Powell and Brisson

Rosendash

1993

CSERGE

EMC

Year

Study

36.156

1.7–22 (14)

24

12.8–17.4 (13 EU)

6.6–62.7 (34.6)

13.6

4.41–57 (30.7)

15

1.3–57 (29.5)

260

22.2

13.6

28.7

20.5 9.5–12.8 (11.15)

22.75

22.75

PM10

5.383

4.701

5.2

1

2.1–12.2 (7.15)

6–8.3 (6 EU)

13.4

12.2

4.21–15.3 (9.755)

7

1–15 (7.5)

0.38

0.42

7.3

2.1 3.1–7.3 (5.2)

0.392–0.68 (0.536)

0.392–0.68 (0.536)

SO2

0.1905

0.002–0.009 (0.0055)

0.002

0.045–1.583 (0.814)

0.124

0.007

CO

8.239–14.161 (11.2)

1.469

N2 O

Economic unit-values (US$=kg emission, $, 2003) calculated/used in estimates of landfill and incineration externalities (average in parenthesis)

Table 8.20 Economic unit values of emissions for incinerators (Eshet et al. 2006)

1.262

2.1

0.73

0.7

0.7

2.53

1.65

VOC

314.24

VCI

293

1916

1445

Heavy metals

0

3378

Leachate Pb C Cd C Hg

16,300,000

2,000,000

Dioxins

8.8 Application 2 – FTA in a Waste to Energy System 293

294

8 Effects Analysis and Reliability Modeling of Complex Production Systems

Table 8.21 Costs and benefits from incineration (US$=ton waste) (Eshet et al. 2006) Valuation results (costs and benefits) on emissions from incineration (US$=ton waste, $, 2003) Pullutant study

CO2

Tellus (1992) CSERGE et al. (1993)b Powell and Brisson (1994)b ECON (1995)c EC (1996) Enosh (1996) EMC (1996) Miranda and Hale (1997)d Rabl et al. (1998a) ExternE (1998) EC (2000a,b) Eunomia (2002) Dijkgraaf and Vollebergh (2003)e

NO2

Other conventional

Transportation

Energy recovery

1.1–10.72

1.64–3.3

0.17–1.64

6.88–23.6

1–5 5.77–19.8

1.1–10.72

1.85–4.08

0.368–0.567

10.99–15.04

(–)3.15–6.3

3.9

0.5–1 19.65–20.69 17.26

5–108 8.72–23.43

Total estimatea

28–171 1.3 10.09 1.65 5.17–31.5

8.55 8.55

2.51

0.97–1.68

Leachate (most ash)

0–115 0.05 0

22.62

12.3 15–92d (–)9–124 29.39–45.85 17.57

a

Each of the estimate is a sum of different components and not necessarily the sum of the values in the line. The ranges refer to rural and urban sites for UK and UK C ECE. c The rang presents different types of materials (left for glass and right for plastic). d The ranges refers to differences between countries. e Modern incinerator with energy recovery including calculation of chemicals and materials. b

Table 8.22 Annual emissions (year 2006) and annual costs (2003 prices) Pollutant PM10 CO COT HCl SO2 NOx

Total amount of annual emissions (kg)

Unit cost ($=kg)

Annual cost ($=year)

28 541 70 42 73 28,711

36.2 0.2 1.3 5.4 5.4 6.8

1,005 103 89 224 393 195,534

design and to put in light the most critical component failures for the system. They can assist in identifying the fault tree event whose upgrade is most likely to yield the greatest improvement in system performance. These measures are: • Birnbaum. It determines the maximum increase of the risk due to the failure event of a component in comparison with when the component is operating. This measure is very important because it allows one to rate how much the top gate probability changes when the unavailability of a basic event has changed; as a consequence, it is possible to rank the events according to the Birnbaum measure and to select those on which to concentrate the best efforts for improvement. The Birnbaum measure is defined

as follows: IB .A/ D P .TOPnA/  P .TOPnAN /; where A is the primary/basic event and TOP is the top event. • Criticality. The criticality importance measure of event A determines the probability that the top event, here assumed to have occurred, is due to the failure of component A: IC .A/ D IB .A/

P .A/ : P .TOP/

• Fussell–Vesely. Given that the system failed, the Fussell–Vesely measure determines the probability

8.8 Application 2 – FTA in a Waste to Energy System

295

Fig. 8.94 Distribution of NOx emission values. Year 2006 (25,091 half-hour observations)

Fig. 8.95 Half-hour values of NOx emissions (mg=Nm3 )

that component A contributed to this failure. In particular, it is the ratio of the probability of occurrence of any cut set containing event A and the probability of the top event. The Birnbaum importance measure considers only the conditional probability that event A is critical, while the criticality importance measure also takes into account the overall probability of the occurrence of the top event due to event A. According to this criticality analysis, the urea tank, electric equipment, and air secondary piping are the most critical parts.

8.8.9 Spare Parts Availability, What-If Analysis As illustrated in Fig. 8.96, the system unavailability for a period of time T equal to 365 h is 7:407  104 ; for a longer period of 1 year the availability of the

system, given the top event, is 0.9984, as reported in the second column of Table 8.25. This last value was obtained by the application of the Monte Carlo dynamic simulation with 10,000 repetitions, i. e., simulating the failures and repair events for 10,000 virtual production systems based on the same components/basic events parameterization. The point availability A.t/ at t D 8;760 h is about 0.9979, while the reliability is about 0.1735 for a mission period T (D t  t0 ) equal to 1 year. Other significant results, reported in Table 8.25, are the ENF, the mean time to first failure, and the annual downtime, which amounts to 13:74 h=year. This system configuration is called “optimistic” because it does not consider the lead times required to supply spare parts, such as valves and pumps, in the case of failures and corrective maintenance actions. In other words, the MTTR is based on the optimistic hypothesis of assured availability of every generic spare part, i. e., a fulfillment lead time equal to zero or an infinite number of spare parts in storage.

8 Effects Analysis and Reliability Modeling of Complex Production Systems

Fig. 8.96 FTA. Top event: Exceeding NOx limit. Relex® Reliability software

296

8.8 Application 2 – FTA in a Waste to Energy System

297

Table 8.23 Components’ basic reliability parameters, T D 365 h  (h1 )

 (h1 )

=

T

1  exp.T /

1:25  105 6:23  105 6:23  105 2:49  105 6:23  105 3:11  105 3:11  105 3:11  105 3:11  105 1:56  105 1:56  105 1:56  105 1:56  105 1:04  104 1:04  104 1:04  104 1:04  104 1:04  104 1:04  104 6:23  105

5:00  101 1:25  101 3.00 3.00 5:0  101 1:35  101 1:35  101 1:35  101 1:35  101 3.00 3.00 3.00 3.00 8:62  101 8:62  101 8:62  101 8:62  101 8:62  101 8:62  101 6:67  101

2:49  105 4:98  104 2:08  105 8:30  105 1:25  104 2:30  104 2:30  104 2:30  104 2:30  104 5:19  106 5:19  106 5:19  106 5:19  106 1:20  104 1:20  104 1:20  104 1:20  104 1:20  104 1:20  104 9:34  105

4:55  103 2:27  102 2:27  102 9:09  102 2:27  102 1:14  102 1:14  102 1:14  102 1:14  102 5:68  103 5:68  103 5:68  103 5:68  103 3:79  102 3:79  102 3:79  102 3:79  102 3:79  102 3:79  102 2:27  102

4:54  103 2:25  102 2:25  102 8:69  102 2:25  102 1:13  102 1:13  102 1:13  102 1:13  102 5:67  103 5:67  103 5:67  103 5:67  103 3:72  102 3:72  102 3:72  102 3:72  102 3:72  102 3:72  102 2:25  102

Component AIR_fail VR1101 TT101 TT105 TKUREA CX51005 CX51006 CX51008 CX51009 DH51001 DH51002 DH51003 DH51004 INJ51101H INJ51102H INJ51103H INJ51101L INJ51102L INJ51103L e_ELECTRIC

Table 8.24 MCS evaluation, T D 365 h Minimal cut set i VR1101 AIR_fail TKUREA e_ELECTRIC TT101  TT105 CX51005  CX51006 CX51008  CX51009 DH51001  DH51002 DH51003  DH51004 INJ51101L  INJ51101H INJ51101L  INJ51102H INJ51101L  INJ51103H INJ51102L  INJ51101H INJ51102L  INJ51102H INJ51102L  INJ51103H INJ51103L  INJ51101H INJ51103L  INJ51102H INJ51103L  INJ51103H

qCSi

WCSi 4

4:98  10 2:49  105 1:25  104 9:34  105 1:72  109 5:31  108 5:31  108 2:69  1011 2:69  1011 1:45  108 1:45  108 1:45  108 1:45  108 1:45  108 1:45  108 1:45  108 1:45  108 1:45  108

1  FCSi

FCSi 2

2:27  10 4:55  103 2:27  102 2:27  102 3:77  106 5:24  106 5:24  106 5:90  108 5:90  108 9:12  106 9:12  106 9:12  106 9:12  106 9:12  106 9:12  106 9:12  106 9:12  106 9:12  106

2

2:25  10 4:54  103 2:25  102 2:25  102 1:95  103 1:28  104 1:28  104 3:21  105 3:21  105 1:38  103 1:38  103 1:38  103 1:38  103 1:38  103 1:38  103 1:38  103 1:38  103 1:38  103

9:78  101 9:95  101 9:78  101 9:78  101 9:99  101 9:99  101 9:99  101 9:99  101 9:99  101 9:99  101 9:99  101 9:99  101 9:99  101 9:99  101 9:99  101 9:99  101 9:99  101 9:99  101

Table 8.25 also summarizes the predicted values of system reliability parameters for two different scenarios:

case of an exponential distribution of probability for ttr, and to about 29:21 h=year when ttr is constant, as reported in the last column of Table 8.25.

• Realistic operating scenario. The required supply lead time LTS is 2 weeks, corresponding to 10 working days or 15 operating days, or 360 h, for pumps and 1 day, or 24 h, for valves. The system downtime amounts to about 28:77 h=year in the

• Pessimistic operating scenario. Same hypotheses of the realistic scenario for pumps and valves, while for the other parts LTS is equal to 144 h, or 6 days. The system downtime amounts about to 203 h=year.

298

8 Effects Analysis and Reliability Modeling of Complex Production Systems

Fig. 8.97 Criticality analysis. Relex® Reliability software ReliaSoft BlockSim 7 - www.ReliaSoft.com

Block Up/Down State Operating Time Time Under Repair

VR1101

TKUREA

INJ51103L

e_ELECTRIC

Air_fail

System

0.000

1752.000

3504.000

5256.000

7008.000

8760.000

Time, (t)

Fig. 8.98 System up/down analysis, pessimistic configuration. Reliasoft® Reliability software

An exponential distribution of ttr random values is assumed and the MTTR for pumps is the value reported in Table 8.23 (MTTR D 1=) in the realistic scenario with 360 h in addition. A similar consideration applies for the MTTR defined for valves of S and for the other parts in case of a “pessimistic” scenario. Figure 8.98 shows the results of the up/down analysis obtained by Monte Carlo simulation applied to the

“pessimistic” system. Figure 8.99 presents the most critical components in terms of the number of failures in the same system configuration. The values obtained assuming the so-called realistic hypothesis agree with the results obtained by the analysis of the historical data of NOx emissions. The following equation can be applied in order to quantify the economic effects of externalities, in terms

8.8 Application 2 – FTA in a Waste to Energy System

299

ReliaSoft BlockSim 7 - www.ReliaSoft.com

Block Expected Failures 0.878

RS FCI 100%

50%

0.702 0% 5 Item(s)

0.527

0.351

0.176

0.000

INJ51103L

TKUREA

e_ELECTRIC

VR1101

Air_fail

Fig. 8.99 Expected failures, pessimistic configuration. Reliasoft® Reliability software

of euros per year, on the environment and on the community: MNOx D Q.CNOx ;failure  CNOx ;function /tfailure ; where MNOx is the extra emission quantity of NOx (mg/year) in comparison with the correct function of the system,Q is the air flow, i. e., 24;860 Nm3 =h, CNOx ;failure is the NOx emission concentration in the case of failure, i. e., 212:4 mg=Nm3, CNOx ;function is the NOx emission concentration in the case of correct function, i. e., 133:7 mg=Nm3 , and tfailure is the annual downtime of the system, given the top event.

Table 8.25 reports the economic impact for the system configurations/parameterizations evaluated, assuming a unit cost of the NOx emission equal to US$ 6.81 per kilogram (2003 prices; see Table 8.20). The results demonstrate that the estimated extra cost of externalities, due to an incorrect function of the system, amounts about to US$ 180,000 per year assuming the optimistic hypothesis and the first what-if scenario configuration, and to ¤ 2,700,000 per year in case of the pessimistic, but not realistic, scenario. It is worth noting how important it is to conduct a quantitative analysis more accurately and as realisti-

Table 8.25 Reliability parameters prediction, multiscenario analysis Optimistic T (h) Mean availability (all events) Point availability (all events) at 8;760 h Reliability (8;760 h) Expected number of failures (failures) MTTFF (h) System uptime (h) System downtime (h) NOx (kg) NOx externality costs (2003 US$=year) MTTFF mean time to first failure

8,760 0.9984 0.9979 0.1735 1.74 5,013.38 8,746.26 13.74 26,882 183,066

Spare parts availability scenarios Realistic Pessimistic Realistic MTTR constant 8,760 0.9967 0.9962 0.1663 1.77 4,885.94 8,731.23 28.77 56,286 383,308

8,760 0.9768 0.976 0.139 1.94 4,451.88 8,556.93 203.07 397,311 2,705,687

8,760 0.9967 0.996 0.1704 1.76 4,933.15 8,730.79 29.21 57,149 389,185

300

8 Effects Analysis and Reliability Modeling of Complex Production Systems

cally as possible, and to manage spare parts. For this purpose it could be useful to repeat the FTA assuming more realistic probabilistic distributions of ttr and ttf random variables, e. g., introducing a Weibull parametric distribution. Chapter 11 will opportunely discuss basic and innovative models and methods to optimize the management of critical spare parts, in accordance with the adoption of different maintenance strategies and actions.

ing a planning period T D 8;760 h; the total amount of the annual cost saving, due to the introduction of a second redundant valve, for three scenarios is: 1. Optimistic configuration, Costextern.,annual .opt./ D Cost2 valves .opt./  Cost1 valves .opt./ D 126;675  183;066 D US$ 56;391 per year .30:8%/: 2. Realistic configuration,

8.8.10 System Modifications for ENF Reduction and Effects Analysis This section exemplifies the impacts on reliability and costs associated with some modifications to the SNCR plant and to the strategies/rules for the control of NOx emissions. In particular, they deal with the introduction of two alternative management policies for the critical valve VR1101. Similar considerations could of course be applied to other parts and components of the system.

Costextern.,annual .real./ D Cost2 valves .real./  Cost1 valves .real./ D 156;288  383;308 D US$ 227;020 per year .59:2%/: 3. Pessimistic configuration, Costextern.,annual .pess./ D Cost2 valves .pess./  Cost1 valves .pess./ D 2;486;237  2;705;687 D US$ 219;450 per year .8:1%/:

8.8.10.1 A Redundant Valve In the case of insertion of a new redundant valve in a parallel configuration, the fault tree changes. Figure 8.100 shows this new situation, given the top event, assuming T D 365 h and the optimistic configuration of the system. In Table 8.26 the performance of the system and the related externality costs are compared for different configurations/parameterizations, assum-

It is worth noting that the redundant valve brings very important benefits from an environmental and social point of view; moreover, this introduction is very profitable, considering an annual investment cost of about $6,000. Similar considerations can be made, considering different system alternative and/or simultaneous modifications, with reference to other externality costs, such as the emissions of CO2 , CH4 , PM10 , SO2 , CO, and N2 O.

Table 8.26 Valve redundancy introduction, what-if analysis 1 vs. 2 valves T (h) Mean availability (all events) Point availability (all events) at 8;760 h Reliability (8;760 h) Expected number of failures (failures) MTTFF (h) System uptime (h) System downtime (h) NOx (kg) NOx externality costs (2003 US$=year)

Realistic – 1 valve 8,760 0.9967 0.9962 0.1663 1.77 4,885.94 8,731.23 28.77 56,286 383,308

Spare parts availability scenarios Realistic – Optimistic – 2 valves 2 valves 8,760 0.9987 0.9989 0.2918 1.219 7,126.994 8,748.2699 11.7301 22,950 156,288

8,760 0.9989 0.9987 0.3037 1.1986 7,297.2985 8,750.4925 9.5075 18,601 126,675

Pessimistic – 2 valves 8,760 0.9787 0.9808 0.2351 1.3988 6,105.6822 8,573.3966 186.6034 365,086 2,486,237

8.9 Markov Analysis and Time-Dependent Components/Systems

301

Fig. 8.100 System modification: valves VR1101 and VR1102

8.9 Markov Analysis and Time-Dependent Components/Systems

and past states are independent: P fXnC1 D xnXn D xn ; : : : ; X1 D x1 g D P fXnC1 D xnXn D xn g: (8.21)

Markov modeling and analysis are very useful in the presence of dependences among basic/primary events in a fault tree, in particular with standby redundancies and common causes. A Markov chain is a discretetime stochastic process complying with the so-called Markov property: given the present state of a system/component, its future states are independent of its past states. Alternatively stated, the present state description fully captures all the information that can influence the future evolution of the process. Thus, given the present, the future is conditionally independent of the past. In particular, at the generic time instant the system may change its state from the current state to another state, or it may remain in the same state, according to a certain probability distribution. These changes of state are called “transitions,” and the probabilities associated with various state changes are termed “transition probabilities.” Formally given a sequence of random variables X1 ; X2 ; X3 ; : : : with the Markov property, the future

The state space of the chain is the set of possible values assumed by Xi . Markov chains are often described by a directed graph, where the edges are labeled by the probabilities of going from one state to the other states, as illustrated in Fig. 8.101. In other words, considering a generic system, Si .ti / identifies the state Si of the system at the instant of time ti and Eq. 8.21 changes as follows: P fSnC1 .tn C t/nSn .tn /; Sn1 .tn1 /; : : : ; S1 .t1 /g D P fSnC1 .tn C t/nSn .tn /g D Pn;nC1 ; (8.22) where Pn;nC1 represents the transition from state n to state n C 1. The generic Markov chain can be modeled by a set of differential equations, in accordance with the notation introduced in Fig. 8.101. Given a state i for the system and transitions tk and tj , respectively, from

302

8 Effects Analysis and Reliability Modeling of Complex Production Systems

Fig. 8.102 Vertex sections in the graph representation of a Markov chain

Fig. 8.101 Markov chain and differential equation model

state i to state k and from state j to state i , Pi .t C t/ D Pi .t/.1  tk t/ C Pj .t/tj t: (8.23) Equation 8.23 can be explained as follows: Pi .t C t/  Pi .t/ dPi .t/ D lim t !0 dt t D Pj .t/tj  Pi .t/tk :

(8.24) Fig. 8.103 Markov chain for a parallel system and nonrepairable components

In general, dPi .t/ D dt

X

Pj .t/tj 

j 2fstate IN i g

X

Pi .t/tk

k2fstate OUT i g

(8.25) when X

Pj .t/ D 1:

(8.26)

j 2fstate of the system Sg

8.9.1 Redundant Parallel Systems A significant example of the Markov chain theory is its application to the reliability prediction for a system made of two components, A and B, in a parallel configuration. For each component, consider the two states of function f0; 1g, representing the state of function or of failure, respectively; typical notation is reported schematically in Fig. 8.102. Figure 8.103 presents the Markov chain model, based on a vertex made of three sections as in Fig. 8.102, for a parallel system made of nonrepairable components (A D B D 0).

By the application of Eq. 8.25, 8 dP .t/ 1 ˆ D P1 .t/.A C B / ˆ ˆ ˆ dt ˆ ˆ ˆ ˆ dP2 .t/ ˆ ˆ D P1 .t/A  P2 .t/B < dt ˆ dP3 .t/ ˆ ˆ D P1 .t/B  P3 .t/A ˆ ˆ ˆ ˆ dt ˆ ˆ ˆ : dP4 .t/ D P .t/ C P .t/ ; 2 B 3 A dt

(8.27)

considering the following starting conditions: ( P1 .0/ D 1 Pj .0/ D 0 8j ¤ 1; where 1, 2, etc. refer to states S1 , S2 , etc. (see Fig. 8.103). By the application of the Laplace transform, Z1 F .s/ D LŒy.t/ D 0

est y.t/ dt

(8.28)

8.9 Markov Analysis and Time-Dependent Components/Systems

and the following property   dy.t/ D sF .s/  y.t D 0C / L dt to Eq. 8.27,

(8.29)

.s/ D

LŒf .t/ D F .s/; 1 LŒ1 D ; s k LŒk D ; s 1 LŒt D 2 ; s 1 ; LŒ ekt  D sCk L1 ŒF .s/ D f .t/:

.s  a/P .s/ Q.s/

(8.34)

and a1 ; : : : ; an are nonmultiple roots of Q.s/ D 0. The roots obtained in Eq. 8.32 when Q.s/ D 0 are 8 a1 D .A C B / < .s  a1 /P .s/ : .s D a1 / D D 1: Q.s/ As a consequence,   1 P .s/ P1 .t/ D L D .a1 / ea1 t D e.A CB /t : Q.s/ (8.35) Exactly the same result can be obtained by the integration of the first term in Eq. 8.27:

As a consequence, it is useful to derive from Eq. 8.26 the following equation: 1  p1  p2  p3 : s

(8.31)

From Eqs. 8.30 and 8.31 the values of pi .s/ are 8 P1 .s/ 1 ˆ ˆ p1 .s/ D D ˆ ˆ ˆ s C .A C B / Q1 .s/ ˆ ˆ ˆ ˆ 1   A A ˆ ˆ p2 .s/ D p1 D ˆ ˆ s C B s C B s C .A C B / ˆ ˆ ˆ ˆ ˆ P2 .s/ ˆ ˆ D ˆ ˆ Q2 .s/ ˆ ˆ ˆ ˆ B B 1 ˆ ˆ ˆ < p3 .s/ D s C  p1 D s C  s C . C  / A

(8.33)

where (8.30)

Other general and useful analytical relationships and properties are

A

The inverse Laplace transform is then applied in accordance with the following property:   P .s/ D .a1 / ea1 t C .a2 / ea2 t L1 Q.s/ C    C .an / ean t ;

8 sp1  1 D p1 .A C B / ˆ ˆ ˆ < sp2 D p1 A  p2 B ˆ sp 3 D p1 B  p3 A ˆ ˆ : sp4 D p2 B C A p3 :

p4 D

303

A

B

P3 .s/ ˆ ˆ ˆ D ˆ ˆ Q 3 .s/ ˆ ˆ ˆ ˆ ˆ Œs C .A C B /.s C B /.s C A / ˆ ˆ ˆ ˆ s.s C B /.s C A / ˆ ˆ ˆ ˆ  A s.s C A /  B s.s C B / ˆ ˆ ˆ p4 .s/ D ˆ ˆ sŒs C .A C B /.s C B /.s C A / ˆ ˆ ˆ ˆ P4 .s/ ˆ ˆ D : : (8.32) Q4 .s/

8 dP1 .t/ ˆ D .1 C 2 / dt ˆ ˆ ˆ P1 .t/ ˆ ˆ ˆ ˆ ˆ P ˆ Z1 .t / Zt ˆ < dP1 .t/ D  .1 C 2 / dt P1 .t/ ˆ ˆ 0 P1 .0/ ˆ ˆ ˆ ˆ ˆ ˆ lnŒP1 .t/ D .1 C 2 /t ˆ ˆ ˆ : P1 .t/ D e.1 C2 /t :

(8.36)

This result is the well-known expression of the reliability of a serial system made of unrepairable components as illustrated in Sect. 6.4. In fact, in state 1 components A and B have to be in a state of function.

304

8 Effects Analysis and Reliability Modeling of Complex Production Systems

Similarly, we have the expression for P2 .t/: 8 a1 ˆ ˆ ˆ ˆ ˆ ˆ a2 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ .s D a1 / ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ < .s D a2 / ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ P2 .t/ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ :

D .A C B / D B D

.s  a1 /P2 .s/ Q2 .s/

D Œs C .A C B /

A 1 s C B s C .A C B /

ˇ A ˇˇ D D 1 s C B ˇsD.A CB / D

.s  a2 /P2 .s/ Q2 .s/

Fig. 8.104 Markov chain for a parallel system and repairable components

1 A s C B s C .A C B / ˇ ˇ A ˇ D1 D s C . C  / ˇ

D .s C B /

A

 D L1

P2 .s/ Q2 .s/

B

is the determination of P1 :

sDB



D .a1 / ea1 t C .a2 / ea2 t D eB t  e.A CB /t : (8.37)

In the same way we obtain P3 .t/ and P4 .t/: P3 .t/ D eA t  e.A CB /t ; P4 .t/ D 1  eA t  eB t C e.A CB /t : (8.38) By the calculus of 1P4 .t/ it is possible to evaluate the reliability of a parallel redundant system as illustrated in Sect. 6.5.

8.9.2 Parallel System with Repairable Components This section applies the Markov chain modeling to the analysis of a parallel system made up of repairable components, as illustrated in Fig. 8.104. In particular, it is assumed that it is not possible to return to state S2 or S3 , starting from S4 . The main aim of this analysis

8 dP1 .t/ ˆ ˆ D A P2 .t/ C B P3 .t/  .A C B /P1 .t/ ˆ ˆ ˆ dt ˆ ˆ ˆ ˆ ˆ dP2 .t/ ˆ ˆ D P1 .t/A  .B C A /P2 .t/ ˆ ˆ dt ˆ ˆ ˆ ˆ ˆ ˆ < dP3 .t/ D P1 .t/B  .A C B /P3 .t/ dt ˆ ˆ ˆ ˆ ˆ dP4 .t/ ˆ ˆ D P3 .t/A C P2 .t/B ˆ ˆ ˆ dt ˆ ˆ ˆ ˆ ˆ ˆ P1 .0/ D 1 ˆ ˆ ˆ ˆ : Pj .0/ D 0; j ¤ 1: (8.39) Applying Laplace transforms, 8 sp1  1 D A p2 C B p3  .A C B /p1 ˆ ˆ ˆ ˆ ˆ ˆ sp2 D A p1  .B C A /p2 < ˆ ˆ ˆ ˆ ˆ ˆ :

sp3 D B p1  .A C B /p3 p1 C p2 C p3 C p4 D

(8.40)

1 : s

As a consequence, p1 D

1 s C A C B 

A  A sCB CA



B  B sCA CB

: (8.41)

8.9 Markov Analysis and Time-Dependent Components/Systems

305

State S 1 - P1(t) 1 0.9 0.8

Probability

0.7 0.6 0.5 P1(t)

0.4 0.3 0.2 0.1 0 0

50000

100000

150000

200000

250000

300000

Unit of me

Fig. 8.105 Probability of the event “system in state S1 ”

Applying the inverse Laplace transform in the special case A D B D  and A D B D , P1 .t/ D L1 Œp1 .s/ p

1   2  C 12  C 12 2 C 6 C 2 p

  exp  12 t.3 C   2 +6 C 2 / D p 2 C 6 C 2

1p  2 C 6 C 2 C 12 .  / 2 p

  exp  12 t. 2 C 6 C 2 C 3 C / C p : 2 C 6 C 2 (8.42) Similarly, it is possible to quantify P2 .t/ and P3 .t/. Figure 8.105 presents the probability that the system is in state 1. In the case of repairable component A and/or component B and in the state of failure of both (see state S4 in Fig. 8.106), it could be useful to quantify the unavailability of the system, which is equal to the probability P4 .t/, i. e., the availability: A.t/ D 1  P4 .t/

(8.43)

Fig. 8.106 Markov chain for a parallel system and repairable components

The differential equation related to state S4 is dP4 .t/ D P3 .t/A C P2 .t/B  .A C B /P4 .t/: dt (8.44)

306

8 Effects Analysis and Reliability Modeling of Complex Production Systems

8.9.3 Standby Parallel Systems In this section different examples regarding repairable systems are illustrated in accordance with the new notation reported in Fig. 8.107. Figure 8.108 represents the Markov chain model of the standby parallel system when the generic component, in the standby state, is not subject to failures. This is the so-called cold standby parallel system. Similarly, Fig. 8.109 presents the Markov chain model of the system when the generic standby component C can fail, with failure rate 0C , during the “waiting time”: this is a “warm standby” parallel system.

8.9.3.1 Cold Standby

In the case of A D B D  and A D B D , 8 dP1 .t/ ˆ ˆ D P3 .t/  P1 .t/ ˆ ˆ dt ˆ ˆ ˆ ˆ ˆ dP2 .t/ ˆ ˆ D P4 .t/  P2 .t/ ˆ ˆ ˆ dt ˆ ˆ ˆ ˆ ˆ dP3 .t/ ˆ ˆ D P2 .t/ C P5 .t/  . C /P3 .t/ ˆ ˆ ˆ dt ˆ < dP4 .t/ D P1 .t/ C P5 .t/  . C /P4 .t/ ˆ ˆ dt ˆ ˆ ˆ ˆ ˆ ˆ dP5 .t/ ˆ ˆ D  ŒP3 .t/ C P4 .t/  2P5 .t/ ˆ ˆ dt ˆ ˆ ˆ ˆ ˆ ˆ P1 .t/ C P2 .t/ C P3 .t/ C P4 .t/ C P5 .t/ D 1 ˆ ˆ ˆ ˆ ˆ P1 .0/ D 1 ˆ ˆ : (8.46) Pj .0/ D 0; j ¤ 1:

In the cold standby parallel system (Fig. 8.108), 8 dP1 .t/ ˆ ˆ D B P3 .t/  P1 .t/A ˆ ˆ dt ˆ ˆ ˆ ˆ ˆ dP2 .t/ ˆ ˆ D A P4 .t/  P2 .t/B ˆ ˆ ˆ dt ˆ ˆ ˆ ˆ ˆ dP3 .t/ ˆ ˆ D B P2 .t/ C A P5 .t/  .A C B /P3 .t/ ˆ ˆ ˆ dt ˆ < dP4 .t/ D A P1 .t/ C B P5 .t/  .B C A /P4 .t/ ˆ ˆ dt ˆ ˆ ˆ ˆ ˆ ˆ dP5 .t/ ˆ ˆ D A P3 .t/ C B P4 .t/  .A C B /P5 .t/ ˆ ˆ dt ˆ ˆ ˆ ˆ ˆ ˆ P .t/ C P2 .t/ C P3 .t/ C P4 .t/ C P5 .t/ D 1 ˆ ˆ 1 ˆ ˆ ˆ ˆ P1 .0/ D 1 ˆ : Pj .0/ D 0; j ¤ 1: (8.45)

Fig. 8.107 Vertex sections in the graph representation of a Markov chain

Fig. 8.108 Markov chain for a parallel cold standby system

Fig. 8.109 Markov chain for a parallel warm standby system

8.9 Markov Analysis and Time-Dependent Components/Systems

It is now possible to define three new states for the system as follows: 8 ˆ < P0 .t/ D P1 .t/ C P2 .t/ (8.47) PI .t/ D P3 .t/ C P4 .t/ ˆ : PII .t/ D P5 .t/: Then, 8 dP0 .t/ dP1 .t/ dP2 .t/ ˆ ˆ D C ˆ ˆ ˆ dt dt dt ˆ ˆ ˆ ˆ D ŒP3 .t/ C P4 .t/  ŒP1 .t/ C P2 .t/ ˆ ˆ ˆ ˆ ˆ ˆ D PI .t/  P0 .t/ ˆ ˆ ˆ ˆ ˆ ˆ dP3 .t/ dP4 .t/ dPI .t/ ˆ ˆ ˆ < dt D dt C dt D P0 .t/ C 2PII .t/  . C /PI .t/ ˆ ˆ ˆ ˆ ˆ ˆ dP5 .t/ dPII .t/ ˆ ˆ D D PI .t/  2PII .t/ ˆ ˆ dt dt ˆ ˆ ˆ ˆ ˆ ˆ P0 .t/ C PI .t/ C PII .t/ D 1 ˆ ˆ ˆ ˆ ˆ PI .0/ D 1 ˆ ˆ ˆ : (8.48) Pj .0/ D 0; j ¤ I:

307

following: PII .t/ D p p . C 4/  . C 4/



p   exp  12 t.3 C 2/ cosh 12 t . C 4/ 2 p  .2 C 22 C / . C 4/ 

1 p 

1 2 exp  2 t.3 C 2/ sinh 2 t . C 4/  3  p .2 C 22 C / . C 4/

1 

1 p  3 exp  2 t.3 C 2/ sinh 2 t . C 4/ p ;  2 .2 C 22 C / . C 4/ (8.51) while the state of function is 1  PII .t/:

(8.52)

Figure 8.110 presents the trend of the probability PII .t/ assuming  D 104 (unit of time)1 and  D 103 (unit of time)1 .

8.9.3.2 Warm Standby By the application of Laplace transforms, 8 sp0  1 D pI  p0  ˆ ˆ ˆ < spI D p0   pI . C / C 2pII ˆ spII D pI  2pII ˆ ˆ : p0 C pI C pII D 1=s:

(8.49)

As a consequence, 8 .s 2 C 3s C s C 22 / ˆ ˆ ˆ p0 D ˆ ˆ s.s 2 C 2s C 3s C 2 C 2 C 22 / ˆ ˆ ˆ < .s C 2/ pI D 3 2 ˆ s C 2s C 3s 2 C 2s C 2 s C 22 s ˆ ˆ ˆ ˆ ˆ 2 ˆ ˆ : pII D : s.s 2 C 2s C 3s C 2 C 2 C 22 / (8.50) It is possible to quantify the probability of the system being in states S0 , SI , and SII , by the application of the inverse Laplace transform to p0 , pI , and pII . In particular, the state of not function is quantified by the

In the warm standby parallel system (Fig. 8.109), 8 dP1 .t/ ˆ ˆ D B P3 .t/  P1 .t/.A C 0B / ˆ ˆ dt ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ dP2 .t/ D A P4 .t/  P2 .t/.B C 0 / ˆ A ˆ ˆ ˆ dt ˆ ˆ ˆ ˆ ˆ dP3 .t/ ˆ ˆ D B P2 .t/ C 0B P1 .t/ C A P5 .t/ ˆ ˆ ˆ dt ˆ ˆ ˆ .A C B /P3 .t/ ˆ < dP4 .t/ D A P1 .t/ C 0B P2 .t/ C B P5 .t/ ˆ ˆ dt ˆ ˆ ˆ .B C A /P4 .t/ ˆ ˆ ˆ ˆ dP .t/ ˆ 5 ˆ ˆ D A P3 .t/ C B P4 .t/  .A C B /P5 .t/ ˆ ˆ dt ˆ ˆ ˆ ˆ ˆ ˆ ˆ P1 .t/ C P2 .t/ C P3 .t/ C P4 .t/ C P5 .t/ D 1 ˆ ˆ ˆ ˆ ˆ ˆ P1 .0/ D 1 ˆ ˆ ˆ : Pj .0/ D 0; j ¤ 1: (8.53)

308

8 Effects Analysis and Reliability Modeling of Complex Production Systems

State II - PII(t) 5.00E-03 4.50E-03 4.00E-03

Probability

3.50E-03 3.00E-03 2.50E-03 PII(t) 2.00E-03 1.50E-03 1.00E-03 5.00E-04 0.00E+00 0

2000

4000

6000

8000

10000

12000

14000

Unit of me

Fig. 8.110 Failure probability of the standby system (“system in state II”)

In the case of A D B D , A D B D , and 0A D 0B D 0 , 8 dP1 .t/ ˆ ˆ D B P3 .t/  P1 .t/.A C 0B / ˆ ˆ ˆ dt ˆ ˆ ˆ D P3 .t/  P1 .t/. C 0 / ˆ ˆ ˆ ˆ ˆ dP2 .t/ ˆ ˆ D A P4 .t/  P2 .t/.B C 0A / ˆ ˆ ˆ dt ˆ ˆ ˆ D P4 .t/  P2 .t/. C 0 / ˆ ˆ ˆ ˆ ˆ dP3 .t/ ˆ ˆ D B P2 .t/ C 0B P1 .t/ C A P5 .t/ ˆ ˆ ˆ dt ˆ ˆ .A C B /P3 .t/ ˆ ˆ ˆ ˆ ˆ ˆ D P2 .t/ C 0 P1 .t/ C P5 .t/ ˆ ˆ ˆ ˆ ˆ ˆ . C /P3 .t/ < dP4 .t/ ˆ D A P1 .t/ C 0B P2 .t/ C B P5 .t/ ˆ ˆ dt ˆ ˆ ˆ .B C A /P4 .t/ ˆ ˆ ˆ ˆ ˆ ˆ D P1 .t/ C 0 P2 .t/ C P5 .t/ ˆ ˆ ˆ ˆ ˆ ˆ . C /P4 .t/ ˆ ˆ ˆ ˆ ˆ dP5 .t/ ˆ ˆ D A P3 .t/ C B P4 .t/  .A C B /P5 .t/ ˆ ˆ dt ˆ ˆ ˆ ˆ D P3 .t/ C P4 .t/  2P5 .t/ ˆ ˆ ˆ ˆ ˆ P1 .t/ C P2 .t/ C P3 .t/ C P4 .t/ C P5 .t/ D 1 ˆ ˆ ˆ ˆ ˆ ˆ P .0/ D 1 ˆ ˆ 1 ˆ : (8.54) Pj .0/ D 0; j ¤ 1:

In particular, it is possible to define three new states for the system as follows: 8 ˆ < P0 .t/ D P1 .t/ C P2 .t/ (8.55) PI .t/ D P3 .t/ C P4 .t/ ˆ : PII .t/ D P5 .t/: The unavailability of the system is quantified by PII .t/: 8 dP0 .t/ ˆ ˆ D PI .t/  P0 .t/. C 0 / ˆ ˆ dt ˆ ˆ ˆ ˆ ˆ dPI .t/ ˆ ˆ D . C 0 /P0 .t/  . C /PI .t/ ˆ ˆ ˆ dt ˆ < C2PII .t/ (8.56) .t/ dP II ˆ ˆ D PI .t/  2PII .t/ ˆ ˆ dt ˆ ˆ ˆ ˆ P0 .t/ C PI .t/ C PII .t/ D 1 ˆ ˆ ˆ ˆ ˆ ˆ P0 .0/ D 1 ˆ : Pj .0/ D 0; j ¤ 0: By the application of Laplace transforms, 8 sp0  1 D pI  p0 . C 0 / ˆ ˆ ˆ < spI D p0 . C 0 /  pI . C / C 2pII ˆ spII D pI  2pII ˆ ˆ : (8.57) p0 C pI C pII D 1=s:

8.10 Common Mode Failures and Common Causes

309

Then, pI D

. C 0 /.s C 2/ : s 3 C 3s 2 C 2s 2 C 2s C 0 s 2 C20 s C s2 C s0 C 22 s

(8.58)

The probability of the system being in a state of function, but with a component under repair, can be quantified by the application of the inverse Laplace transform as follows: p . C 0 /. 02  20  C 2 C 4

PI .t/ D

 2 C 2 C 0 C 0 /

1 0  exp  2 t. C 3 C 2 p   02  20  C 2 C 4/ .22 C 20 C 2 C 2 C 0 / p Œ 02  20  C 2 C 4 C 2. C 0 / Œ.22 C 20 C 2 C 2 C 0 / . C 0 /.2 C 2 C 0 C 0 p C  02  20  C 2 C 4/ p

 exp  12 t. 02  20  C 2 C 4  C 0 C 3 C 2/

In particular, for the previously introduced differential equations, 8 dP0 .t/ ˆ ˆ D 0 D PI .t/  P0 .t/. C 0 / ˆ ˆ dt ˆ ˆ ˆ ˆ ˆ ˆ ˆ dPI .t/ D 0 D . C 0 /P0 .t/  . C /PI .t/ < dt C2PII .t/ ˆ ˆ ˆ .t/ dP ˆ II ˆ ˆ D 0 D PI .t/  2PII .t/ ˆ ˆ dt ˆ ˆ ˆ : P0 .t/ C PI .t/ C PII .t/ D 1; (8.62) i. e., 22 C 2. C 0 / C . C 0 / Œ. C 0 /2 22 C 2. C 0 / C . C 0 / PI .t ! 1/ D 2 2 . C 0 / 2 2 C 2. C 0 / C . C 0 / : PII .t ! 1/ D . C 0 / (8.63) P0 .t ! 1/ D 22

These results are true in the case of asymptotic values of availability and unavailability. The applications of  : the Markov chain modeling and analysis illustrated so far are a few examples of the power and effectiveness .22 C 20 C 2 C 2 C 0 / p 0 Œ 02  20  C 2 C 4 C 2. C  / of this set of tools. Other advanced applications are presented in the literature and are not subject of this Œ.22 C 20 C 2 C 2 C 0 / book. (8.59) Similarly, it is possible to quantify PII .t/ and P0 .t/. It could be useful to quantify, for each state j of the system, the probability Pj .t/ in the case of stationary conditions, i. e., dPj .t/ D 0: dt

(8.60)

As a consequence, the generic condition explained by the Eq. 8.25 becomes dPi .t/ D dt

X j 2fstate IN j g

D 0;

Pj .t/tj 

X

Pi .t/tk

k2fstate OUT j g

(8.61)

8.10 Common Mode Failures and Common Causes The assumption of independency of failures among different components within a production system is sometimes violated. Some components can share the same power source or external environmental conditions. This is the reason why in FTA, given a top event, it is possible to identify several identical basic events, and mirror events were properly introduced in the numerical examples illustrated in Sects. 8.5 and 8.7. How should we consider a MCS with two or more different basic components subject to common mode failures (also called “common causes”)?

310

8 Effects Analysis and Reliability Modeling of Complex Production Systems

μ1 0,0

For this purpose consider a MCS made of two components A and B, modeled as .i ; i / with i D 1; : : : ; n and n D 2, subject to a common cause modeled as .c; b/ and the following events:

1,0

λ1 μ2

λ2

C

μ2

C

1. There are no common cause events in .0; t/. 2. The last common cause event occurs in .u du; u, where u 2 .0; t/.

λ2

The MCS can be considered as a redundant parallel system of components A and B. As a consequence, the unavailability of the system is the result of two different contributions:

μ1 C

0,1

1,1

1. Hypothesis I. The system unavailability is the result of the application of Eq. 8.12 when components A and B are supposed to be in a state of function for t D 0, i. e., .0; 0/t D0. The probability of components A and B being in a state of failure in t is

λ1 Fig. 8.111 Markov chain, common cause and a 2-dim MCS

A common cause can be modeled as a repairable event based on constant failure and repair rates. In particular, if we call them c and b, respectively; the density function for the common cause event wc_cause .t/ by the application of Eq. 5.952 is wc_cause .t/ D

2 Y i D1

i .1  e.i Ci /t /: i C i

Equation 8.66 does not consider the event “no common cause in Œ0; t.” The probability of no common causes in the system during Œ0; t is quantified by the basic equation (Eq. 5.27) as follows:

2

c cb C expŒ.c C b/t: cCb cCb (8.64)

ect :

The asymptotic value is wc_cause .1/ D lim Œwc_cause .t/ D t !1

cb c Cb

Š if 1=cŠ1=b

(8.65) QI;S .t/ D ect

If a MCS is made up of two or more basic events subject to common causes, Eq. 8.12 cannot be applied. For example, in presence of a cut set made up of two repairable components subject to a common cause of rates .; / D .c; b/ it is possible to introduce the Markov chain as in Fig. 8.111.

i .1  e.i Ci /t /: i C i (8.68)

2. Hypothesis II. Assuming configuration .1; 1/u , Eq. 5.82 can be applied3 to quantify the probability of components A and B remaining in .1; 1/t : 2  Y i D1

See also Table 5.6.

2 Y i D1

8.10.1 Unavailability of a System Subject to Common Causes

2

(8.67)

As a consequence, the system unavailability assuming hypothesis I is

c:

The object of this section is to present an analytical model for the determination of the unavailability of a system with two or more components subject to common causes.

(8.66)

 i i .i Ci /.t u/ : C e i C i i C i (8.69)

Consequently, Eq. 8.69 differs from Eq. 5.82 because of the swapping of terms  and . 3

A new failure event is introduced: the failure rate is  and the repair rate is  and Eq. 5.82 is applied.

8.10 Common Mode Failures and Common Causes

311

The probability of a common cause event occurring in .u  du; u is wc_cause .u/ du

Š

Eq. 8.65

c du:

i D1

ec.t u/ :

QII;S .t/ D

c ec.t u/

i D1

0

C

i i C i

 i e.i Ci /.t u/ du: i C i (8.72)

As a consequence, the system unavailability, i. e., the probability of components A and B being in a state of failure in t is

(8.75) This is the result of the application of Eqs. 5.83 and 8.12.

8.10.2 Numerical Example, Dependent Event Consider the application illustrated in Sect. 8.6.1 and the hypothesis that there is a common cause between the basic components/events A and B. Then the value of c is supposed to be 0:2 year1 (five events per year). By the application of the Eq. 8.74 for T D 8;000 h, when the system operates 365 days per year and 24 h per day,4 the unavailability is QS .8;000/ D ect

D ect

i D1

Zt C

ce

i .1  e.i Ci /t / i C i cs

Y 2 i D1

C

n Y i D1

Zt C

ce 0

Y n i D1

C

 i i .i Ci /s ds C e i C i i C i

2  105 2  105 C 102 105 5 2  .1  e.210 C10 /8;000 / 5 10 C 5  102 5 8;000

i i C i

5 C5102 /8;000

 .1  e.10 Zt C 0

/

5 s

2:28  105 e2:2810

 2  105 102 C 2  105 C 102 2  105 C 102  5 2  . e.210 C10 /s /   5  102 105 C  105 C 5  102 105 C 5  102  5 2  . e.10 C510 /s / ds 



i .1  e.i Ci /t / i C i cs

i D1

i .1  e.i Ci /t / i C i

D e2:2810

In general, for a MCS made of n components subject to a common cause, QS .t/ D ect

ce

cs

Y n

0

 i .i Ci /s ds: C e i C i (8.73)

0

n Y i D1

Zt

QS .t/ D QI;S .t/ C QII;S .t/ 2 Y

i .1  e.i Ci /t /: i C i

(8.71)

By Eqs. 8.69–8.71 the probability of components A and B remaining in the state of failure .1; 1/ in t as in .1; 1/u , because it is subject to a common cause between .u  du; u, is Y 2

n Y

QS .t; c D 0/ D

(8.70)

Hypothesis II is based on the assumption that the last common cause occurs in .u du; u. In particular, the probability that the system stays in .1; 1/ during the period Œu; t can be quantified similarly to Eq. 8.67:

Zt

If c D 0,

i i C i

 i e.i Ci /s ds: i C i (8.74) 4

c D 0:2 year1 D 0:2=.24  365/ h1 2:28  105 .

312

8 Effects Analysis and Reliability Modeling of Complex Production Systems

D 3:9999  107 C 2:2850  105 8;000 Z



5 C2105 C102 C105 C5102 /s

e.2:2810

ds

0

D 3:9999  107 C 2:2850  105

1 6  102 .6102 /8;000  .1  e /

D 3:9999  107 C 2:2850  105

8;000 Z

0

2 s

e610

ds

D 3:81  104 :

(8.76)

This value differs from qAB quantified in Sect. 8.6.1 and also influences the system availability in T D 8;000 h.

9

Basic Models and Methods for Maintenance of Production Systems

Contents 9.1

Introduction to Analytical Models for Maintenance of Production Systems . . . . . . . . . 314 9.1.1 Inspection Versus Monitoring . . . . . . . . . . . . . 315

9.2

Maintenance Strategies . . . . . . . . . . . . . . . . . . . . . . . 315

9.3

Introduction to Preventive Maintenance Models . 318

9.4

Component Replacement . . . . . . . . . . . . . . . . . . . . . . 319 9.4.1 Time-Related Terms and Life Cycle Management . . . . . . . . . . . . . . 319 9.4.2 Numerical Example. Preventive Replacement and Cost Minimization . . . . . . . 320

9.5

9.6

9.7

Time-Based Preventive Replacement – Type I Replacement Model . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.1 Numerical Example. Type I Replacement Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.2 Numerical Example. Type I Model and Exponential Distribution of ttf . . . . . . . . . 9.5.3 Type I Replacement Model for Weibull distribution of ttf . . . . . . . . . . . . . . . . . . . . . . . . 9.5.4 The Golden Section Search Method . . . . . . . . 9.5.5 Numerical Example. Type I Model and the Golden Section Method . . . . . . . . . . . Time-Based Preventive Replacement Including Duration of Replacements . . . . . . . . . . . . 9.6.1 Numerical Example 1: Type I Replacement Model Including Durations Tp and Tf . . . . . . 9.6.2 Type I Model with Duration of Replacement for Weibull Distribution of ttf . 9.6.3 Numerical Example 2: Type I Model with Durations Tp and Tf . . . . . . . . . . . . . . . . . 9.6.4 Practical Shortcut to t p Determination . . . . . . Block Replacement Strategy – Type II . . . . . . . . . . 9.7.1 Renewal Process . . . . . . . . . . . . . . . . . . . . . . . . 9.7.2 Laplace Transformation: W(t) and w(t) . . . . . 9.7.3 Renewal Process and W(t) Determination, Numerical Example . . . . . . . . . . . . . . . . . . . . . 9.7.4 Numerical Example, Type II Model . . . . . . . .

323 324 325 326 326 328 333 333 335 335 335 339 340 341 341 343

9.7.5 Discrete Approach to W(t) . . . . . . . . . . . . . . . . 348 9.7.6 Numerical Examples . . . . . . . . . . . . . . . . . . . . 349 9.7.7 Practical Shortcut to W(t) and t p Determination . . . . . . . . . . . . . . . . . . . . . . . . . . 352 9.8

Maintenance Performance Measurement in Preventive Maintenance . . . . . . . . . . . . . . . . . . . . 353 9.8.1 Numerical Example . . . . . . . . . . . . . . . . . . . . . 354

9.9

Minimum Total Downtime . . . . . . . . . . . . . . . . . . . . 355 9.9.1 Type I – Minimum Downtime . . . . . . . . . . . . . 355 9.9.2 Type II – Downtime Minimization . . . . . . . . . 357

9.10 Group Replacement: The Lamp Replacement Problem . . . . . . . . . . . . . . 358 9.11 Preventive Maintenance Policies for Repairable Systems . . . . . . . . . . . . . . . . . . . . . . . 359 9.11.1 Type I Policy for Repairable Systems . . . . . . . . . . . . . . . . . . 360 9.11.2 Type II Policy for Repairable Systems . . . . . . 370 9.12 Replacement of Capital Equipment . . . . . . . . . . . . 372 9.12.1 Minimization of Total Cost . . . . . . . . . . . . . . . 372 9.12.2 Numerical Example . . . . . . . . . . . . . . . . . . . . . 372 9.13 Literature Discussion on Preventive Maintenance Strategies . . . . . . . . . . 372 9.14 Inspection Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373 9.15 Single Machine Inspection Model Based on a Constant Value of Conditional Probability Failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 9.15.1 Numerical Example 1, Elementary Inspection Model . . . . . . . . . . . . . 376 9.15.2 Numerical Example 2, Elementary Inspection Model . . . . . . . . . . . . . 377 9.16 Inspection Frequency Determination and Profit per Unit Time Maximization . . . . . . . . . 378 9.17 Inspection Frequency Determination and Downtime Minimization . . . . . . . . . . . . . . . . . . 380

R. Manzini, A. Regattieri, H. Pham, E. Ferrari, Maintenance for Industrial Systems © Springer 2010

313

314

9 Basic Models and Methods for Maintenance of Production Systems

9.18 Inspection Cycle Determination and Profit per Unit Time Maximization . . . . . . . . . 9.18.1 Exponential Distribution of ttf . . . . . . . . . . . . 9.18.2 Weibull Distribution of ttf . . . . . . . . . . . . . . . . 9.18.3 Numerical Example . . . . . . . . . . . . . . . . . . . . .

381 381 382 382

9.19 Single Machine Inspection Model Based on Total Cost per Unit Time Minimization . . . . . . 383 9.20 Single Machine Inspection Model Based on Minimal Repair and Cost Minimization . . . . . . 384 9.21 Inspection Model Based on Expected Availability per Unit Time Maximization . . . . . . . . 385 9.22 Group of Machines Inspection Model . . . . . . . . . . . 386 9.23 A Note on Inspection Strategies . . . . . . . . . . . . . . . . 387 9.24 Imperfect Maintenance . . . . . . . . . . . . . . . . . . . . . . . 388 9.24.1 Imperfect Preventive Maintenance p – q . . . . 388 9.25 Maintenance-Free Operating Period . . . . . . . . . . . . 390 9.25.1 Numerical Example (Kumar et al. 1999) . . . . 391 9.25.2 MFOPS and Weibull Distribution of ttf . . . . . 392 9.26 Opportunistic Maintenance Strategy . . . . . . . . . . . 393

“There is a proverb in this country which says prevention is better than cure,” interrupted Mr Vladimir, throwing himself into the arm-chair. (The Secret Agent by Joseph Conrad)

The European standard EN 13306 (Maintenance terminology) distinguishes two main types of maintenance, called “maintenance strategies,” as: • “Preventive maintenance ... carried out at predetermined intervals or according to prescribed criteria and intended to reduce the probability of failure or the degradation of the functioning of an item”; • “Corrective maintenance ... carried out after fault recognition and intended to put an item into a state in which it can perform a required function”. One of the most critical decisions for the analyst, i. e., the maintenance manager, is the determination of the items subject to preventive maintenance, then the time schedule or the number of units of use suitable for performing the maintenance actions. A famous proverb, also used a lot in television spots of a well-known toothpaste, is “prevention is better than cure,” known as “prevenire è meglio che curare” by the Italian authors of this book. This is the Hamlet-like

maintenance issue against the “outrageous fortune,” as William Shakespeare calls stochastic processes: “is it better to prevent or wait and see?: that is the question”. This is the question of strategy in maintenance management and this chapter introduces models, methods, and significant applications to support the choice of the best reply and reaction to it. Other very important questions deal with the identification of the production system’s performance and parameters subject to monitoring and inspection activities, monitoring or inspection?, deferred or immediate maintenance?, on-line or off-line maintenance?, on-site or off-site and/or remote maintenance? replacement or overhaul or rebuilding?

9.1 Introduction to Analytical Models for Maintenance of Production Systems Chapter 4 defined maintenance management as the set of “activities of the management that determine the maintenance objectives, strategies, and responsibilities and implement them by means such as maintenance planning, maintenance control and supervision, improvement of methods in the organization including economical aspects” (European standard EN 13306). This chapter aims to classify and illustrate the most significant maintenance strategies proposed in the literature and applied to production systems. The largest number of automotive companies suggest their customers, and sometimes force them, to plan a preventive maintenance action (sometimes called “voucher”) in accordance with an established time schedule (e. g., 1 year) and/or an established number of units of use (e. g., 20;000 km). Nevertheless, the car could be subject to unexpected downtimes and require corrective maintenance, or a “compliance test”, i. e., a test used to show whether or not a property of an item complies with the stated specifications, or a “function checkout,” etc. What is the best number of time units or units of use to schedule a preventive action? Finally it is necessary to remember that maintenance excellence is the result of maintenance decisions with technical, economic, and organizational implications. In particular, a very critical issue, not the subject of this book, is summarized by the following ques-

9.2 Maintenance Strategies

tion: What are the best procedures and resources cited in the definition of maintainability as “The ability of an item under given conditions of use, to be retained in, or restore to, a state in which it can perform a required function, when maintenance is performed under given conditions and using stated procedures and resources” (EN 13306:2001 Maintenance terminology)? An effective reply to this question differs from business to business, company to company, department to department of the same company, production system to production system of the same department, component to component of the same production system, etc. For this reason this is not the subject of this book but the analyst, which could be the reader of this book, has to be conscious of its existence and criticality. In order to introduce the reader to the most significant maintenance strategies, it is useful to cite the standard EN 13306:2001, which identifies two main kinds of strategies – preventive and corrective – whose definitions are reported at the beginning of this chapter. How many strategies exist? We think that an answer to this question does not exist, because it is possible to identify different conceptual frameworks useful for classifying strategies and actions in maintenance management. For this purpose we choose to illustrate the classification proposed by the European standards and specifications (see Fig. 9.1) and another framework proposed by the authors, inspired by the literature and introduced in Sect. 9.2. In particular the proposed framework is coherent with the models and methods illustrated and applied in this chapter. We now give a few definitions from EN 13306 to properly illustrate the framework reported in Fig. 9.1: • Condition based maintenance ... Preventive maintenance based on performance and/or parameter monitoring and the subsequent actions. ... monitoring may be scheduled, on request or continuous. • Predetermined maintenance ... Preventive maintenance carried out in accordance with established intervals of time or number of units of use but without previous condition investigation. • Deferred maintenance ... Corrective maintenance which is not immediately carried out after a fault detection but is delayed in accordance with given maintenance rules. • Immediate maintenance ... is carried out without delay after a fault has been detected to avoid unacceptable consequences.

315

9.1.1 Inspection Versus Monitoring The framework illustrated in Fig. 9.1 classifies the most important strategies which operatively are maintenance activities. The activities classified by the EN 13306 are inspection, monitoring, compliance test, function checkout, routine maintenance (e. g., cleaning, lubrication), overhaul, rebuilding, repair, fault diagnosis (the well-known troubleshooting), fault localization, improvement, and modification (i. e., change the function of an item). In particular EN 13306 also helps us to identify the most important differences between inspection and monitoring activities. Inspection is defined as “Check for conformity by measuring, observing, testing or gauging the relevant characteristics of an item. ... inspection can be carried out before, during or after other maintenance activity”; monitoring is defined as “Activity, performed either manually or automatically, intended to observe the actual state of an item ... used to evaluate any changes in the parameters of the item with time. ... continuous, over time interval or after a given number of operations. ... usually carried out in the operating state.” The remainder of this chapter is organized as follows. Section 9.2 presents the classification of the maintenance strategies adopted by the authors and a little bit different from that illustrated in Fig. 9.1. Sections 9.3–9.10 present different analytical models and several applications on preventive maintenance based on replacements. Section 9.11 presents preventive maintenance policies for repairable systems. Section 9.12 illustrates a model for planning the replacement of capital equipment. Sections 9.14–9.24 present analytical models for inspection maintenance. Section 9.25 introduces and exemplifies an important reliability measure: maintenance-free operating period. Finally, Sect. 9.26 discusses opportunistic maintenance.

9.2 Maintenance Strategies During last few decades academic researchers and practitioners of industrial companies developed several rules and techniques for planning and managing maintenance activities in production systems. These supporting decision-making models and methods can

316

9 Basic Models and Methods for Maintenance of Production Systems

Failure

Time between failures

Failure

Maintenance Before a detected fault

After a detected fault

Preventive maintenance

Condition-based maintenance

Scheduled, continuous or on request

Corrective maintenance

Predetermined maintenance

Scheduled

Deferred

Immediate

Fig. 9.1 Maintenance strategies overview, EN 13306:2001

be classified in accordance with one of the following maintenance philosophies: • Breakdown/corrective maintenance (CM). It is performed when the production system stops functioning correctly, i. e., in accordance with a set of known operating conditions. There are no planning activities to optimize equipment maintenance and support management decisions. This strategy is influenced by the spare parts fulfillment and management system adopted and the cost of a breakdown maintenance action obviously depends on the availability (unavailability) of spare parts necessary to perform the repair action. • Preventive maintenance (PM) (scheduled and unscheduled). It deals with planned actions performed to face and counteract potential failures on a component/system. Timing (i. e., frequency) and outcome of a preventive maintenance action have to be properly planned and optimized, maximizing the throughput of the production system and minimizing costs. It is supposed that a preventive maintenance strategy can be performed only with the continuous knowledge of system operating conditions, which can be correct (incorrect) when they respect (do not respect) a pool of predefined specifications in accordance with the definition of continuous monitoring action introduced in Sect. 9.1.1.

Several models and methods to support management and practitioners in planning and scheduling preventive maintenance activities have been presented in the literature. Some examples are represented by replacement and the adoption of the as good as new hypothesis, refurbishment, and overhaul (i. e., restoration). The first class of preventive maintenance is the socalled statistically and reliability based preventive maintenance, which mainly refers to the analysis of the equipment historical records. Two widely used approaches to this preventive maintenance planning strategy are the use-based preventive maintenance actions, performed on an hours run of the component/system basis, and the time-based preventive maintenance actions, performed on a calendar basis. These are also known as scheduled-basis preventive maintenance strategies. Another special class of preventive maintenance is the condition-based preventive maintenance (the so-called predictive maintenance or unscheduledbasis maintenance), which is carried out on the basis of the continuous monitoring and knowledge of the operating condition and performance of the equipment. In particular, a set of relevant system functions’ parameters is monitored on-line or off-line, detecting a deterioration or degradation in the functional performance of the component/system.

9.2 Maintenance Strategies

By the current definition, preventive maintenance requires continuous monitoring of the system. Obviously preventive maintenance actions need to be properly integrated with spare parts fulfillment and management decisions. • Replacement. This widely used maintenance strategy can be classified in two main classes of rules: planned replacement and replacement upon failure. The first class belongs to the family of preventive maintenance rules (the so-called preventive replacement) and is based on the determination and optimization of the best timing and outcome of the maintenance action as previously introduced (see the introduction to preventive maintenance) and discussed in detail later. Applying the replacement upon failure, the component/system is left to run until it fails. As a consequence, this second class of rules belongs to the breakdown/corrective maintenance strategy. Both replacement rules are significantly influenced by the spare parts fulfillment and management system adopted. • Inspection maintenance (IM). These maintenance actions firstly determine the state of the equipment and ad hoc models and methods try to identify the points in time at which these actions have to be performed. This strategy is also called “fault finding”: measurements and inspections can be properly planned in advanced, but restorative or preventive tasks (e. g., preventive replacement, failure repair, or replacement, overhaul) can not. The state of function of the system/component can be based on a set of indicators capable of describing the health of the system in accordance with a pool of specifications. As a consequence, inspection rules can be referred to the previously cited condition-based maintenance strategy, because the state of function of the equipment can depend exclusively upon one or more monitored and relevant conditions. The basic difference between condition-based preventive maintenance and condition-based inspection maintenance is that the first one needs a continuous monitoring activity of the production system to reduce downtimes/failure occurrences/events and to detect them when they occur, while condition-based inspection maintenance schedules fault-finding actions at specific points in time t to detect if the system is in a state of failure and eventually perform a maintenance action.

317

The primary aim of the inspection strategy is to make a system more reliable, but an inspection action costs money in terms of materials, wages, and loss of production owing to scheduled downtimes. For these reasons managers of production systems have to properly plan and schedule inspection maintenance actions capable of maximizing throughput and profit, and minimizing global production costs. • Condition-based maintenance. This strategy requires monitoring a relevant variable or a set of relevant variables that are closely related to equipment failure. As previously illustrated, conditionbased maintenance refers to models and rules which can belong to preventive maintenance (in the case of continuous monitoring of equipment parameters) or to inspection maintenance, when the state of the equipment is known only after an inspection activity that can be properly scheduled. In condition-based maintenance based on continuous monitoring, the decision refers to the value of a suitable diagnostic signal (e. g., operating/use times, structural parameter, cost indicator) associated with the item and equipment under consideration. As a consequence, a continuous condition-based maintenance is not a scheduled basis preventive maintenance (i. e., based on predetermined time intervals). Some examples of monitored parameters are related to equipment operations, e. g., vibration of machines, operating temperature, and noise, or to indirect measures of equipment function, e. g., product dimensions and quality levels. The first problem related to condition-based maintenance is the determination of the best set of parameters to be monitored and measures of system function. • Opportunistic maintenance. Maintenance actions are performed when the opportunity arises (such as during shutdown periods). • Overhaul. This strategy is based on maintenance actions for the restoration of a component/system to an acceptable condition. The action restores the equipment to a desired level of function. As a consequence, overhaul actions can belong to the class of preventive maintenance, e. g., the socalled time-based preventive maintenance, or to inspection maintenance (i. e., condition-based maintenance) in the case of detection of a degraded con-

318

9 Basic Models and Methods for Maintenance of Production Systems

dition or performance of the production system by performing an inspection action. • Design modification. This strategy deals with the introduction of modifications in system configurations and/or components in order to increase the reliability and the productivity of the production system. Figure 9.2 reports the classification of the main maintenance strategies whose analytical models and methods are illustrated and applied in examples and case studies presented in following sections. No maintenance philosophy or maintenance rule is better than the others. The efficacy is based on the operative context and conditions of the production system, which usually requires managers and practitioners to apply a combination (i. e., a mix) of different models and techniques. As a consequence, different strategies and rules need to be properly integrated in accordance with both preventive maintenance and inspection maintenance programs, whose tasks are grouped by periodicity (e. g., daily, weekly, based on the number of cycles), availability and skills of maintenance teams of workers (also called “maintenance crews”), and the availability of spare parts

and equipment necessary to perform the maintenance action. In complex production systems the planning activity of maintenance tasks needs to be properly supported by models and methods for finite capacity constraints scheduling and sequencing problems, whose significant and efficacy contributions are supported by operations research studies (e. g., Jeong et al. 2007; Tam et al. 2007; Oke and Charles-Owaba 2006).

9.3 Introduction to Preventive Maintenance Models Preventive maintenance is defined as a series of tasks, called “planned maintenance actions,” performed to face known causes of potential failures of a production system (i. e., a component or a piece of equipment). As previously introduced, there are two main categories of preventive maintenance: statistically and reliability based and condition-based (Fig. 9.2). In preventive maintenance the first critical question is to identify the tasks that should be performed to

Maintenance strategies

Design modification

Breakdown – corrective maintenance

Maintenance actions

Preventive maintenance (PM)

Opportunity maintenance (OP)

Conditionbased

Statistically and reliability based

Inspection maintenance (IM)

Condition-based maintenance

Overhault/restoration

Replacement or repair

Fig. 9.2 Classification of maintenance strategies

9.4 Component Replacement

prevent failures and reduce downtimes, i. e., select the components and subsystems of the production system subject to planned maintenance actions instead of corrective tasks in the presence of failures. The second level of decisions deals with planning and scheduling of maintenance actions. The following sections present a set of different models for supporting managers and practitioners in planning and scheduling preventive maintenance activities. These models belong to the statistically and reliability based class of preventive maintenance and in particular they deal with preventive component replacement decisions. The proposed analytical models and methods are accompanied by numerical examples and case studies. A list of notation used in preventive replacement models follows: Cp Cf f .t/ F .t/ R.t/ r.t/ W .t/ UEC

preventive replacement unit cost; corrective replacement unit cost; probability function of the variable time to failure (ttf) of the generic component; failure probability function; survival probability functionI failure rate function;1 expected number of failures in .0; t/; unit (i. e., per unit time) expected cost of replacement.

Since failure is unexpected, a failure replacement is more costly than a preventive replacement, i. e., Cf > Cp . This is true especially if a failure results in damage to the equipment, or to other production systems, and is accompanied by delays related to the organization of maintenance teams/crews, the fulfillment of spare parts, etc. A balance is required between the amount spent on the preventive replacements and the resulting benefits, i. e., the reduction of downtimes and in particular of failure replacements, which are more expensive than preventive replacement. Section 9.8 discusses performance measures of effectiveness of preventive maintenance , with particular attention to preventive replacement.

1

In the case of nonrepairable components/systems, the failure rate function is generally represented by .t /; see Chap. 5.

319

9.4 Component Replacement The replacement of parts and components of a production system can be a preventive maintenance action, whose first decision deals with the determination of which critical entities have to be preventively replaced and which components, subject to breakdown/corrective actions, should be left to run until they fail. The second decision refers to the determination of timing of actions capable of improving the availability and reliability of the system. Barlow and Hunter (1960) proposed two simple analytical models for the determination of the optimal replacement policy minimizing the operating cost of the production system: 1. age-based replacement policy, or time based preventive replacement, also called “type I policy”; 2. constant interval replacement policy, also called “type II policy” or “block replacement policy.” These basic models represent the main and first reference for the development of several and more complex models and methods dealing with a preventive maintenance strategy (Huang et al. 1995; Jiang et al. 2006). In particular, the preventive replacement should take place after the component/system has been significantly used and before it has aged for too long. As a consequence, a too early or too late scheduling of a preventive replacement action is not a good decision. The numerical example illustrated in Sect. 9.4.2 clarifies this important rule.

9.4.1 Time-Related Terms and Life Cycle Management The European standard EN 13306 gives a set of useful definitions related to maintenance strategies and rules. A few of them are reported as follows: • operating time ... time interval during which an item is performing its required function; • required time ... time interval during which the user requires the item to be in a condition to perform a required function; • standby time ... time interval during which an item is in a standby state; • idle time ... time interval during which an item is in an idle time;

320

9 Basic Models and Methods for Maintenance of Production Systems

• maintenance time ... time interval during which a maintenance is carried out an item either manually or automatically, including technical and logistic delays; • active maintenance time ... part of maintenance time during which active maintenance is carried out on an item, excluding logistic delays; • repair time ... part of active corrective maintenance time during which repair is carried out on an item; • logistic delay ... accumulated time during which maintenance cannot be carried out due to the necessity to acquire maintenance resources, excluding any administrative delay; • life cycle ... time interval that commences with the initiation of the concept and terminates with the disposal of the item. In particular, the logistic delay time can have a very significant contribution to maintenance time because of traveling to an unattended installation, pending arrival of spare parts (see Chap. 11), specialists, test equipment and information, and unsuitable environmental conditions. There are a lot of literature studies regarding life cycle management (LCM) and product lifecycle management (PLM). Life cycle management and product life cycle management can be especially defined as an integrated concept to assist in businesses managing the total life cycle of products and services towards more sustainable consumption and production patterns. Product life cycle goes through many phases, involves many professional disciplines, and requires many skills, tools, and processes; this is not the subject of this book, but reliability engineering and the optimization of maintenance management represent an effective set of quantitative and practical tools to support the optimization of life cycle management and product life cycle management.

9.4.2 Numerical Example. Preventive Replacement and Cost Minimization Consider a component whose failure behavior is well known, and in particular the probability distribution of the random variable ttf is a Weibull distribution (shape parameter ˇ D 2:1 and scale parameter ˛ D 1;531:4 h). Figure 9.3 reports the trend of the failure

Table 9.1 Numerical example. Corrective maintenance (CM) compared with preventative maintenance (PM) actions Performance of action

CM

PM

Spare part cost (¤/unit) Call cost (¤/replacement) Crew cost (¤/h) Nonproduction cost (¤/h) TTR (h)

400 200 100 600 18

350 100 100 600 8

TTR time to repair

probability function F .t/, reliability function R.t/, density function f .t/, and failure rate function .t/. The value of the mean time to repair (MTTR) is about 1;356 h and reliability referred to a period of time T D 1;000 h is about 0.665. The component is assumed to be repairable, and in particular to be as good as new after a maintenance action consisting of a replacement. The duration of the generic replacement action is supposed to be constant and equal to 18 h [i. e., time to repair (ttr) equals MTTR D 18) in the case of a corrective replacement and 8 h in the case of a preventive replacement. Table 9.1 summarizes the assumed variable and fixed costs of maintenance actions, distinguishing the following contributions: • Spare part cost, i. e., the cost of acquiring and storing the replacing new part C. • Call cost, i. e., the fixed cost of calling and organizing the maintenance crew activity. • Crew cost, i. e., the direct cost of the crew for unit time. • Nonproduction cost, i. e., the direct cost of nonproduction for unit time. This is generally called “lost production cost”. In particular, the second column of Table 9.1 refers to the corrective maintenance cost contributions, i. e., the cost performance in the case of a corrective replacement action. Similarly the third column reports the costs in case of a preventive maintenance (i. e., a preventive replacement action). Table 9.2 reports the results obtained in terms of system costs and reliability performance by the application of dynamic simulation to the component/system, assuming a period of time T equal to 32;200 h and 500 repetitions (simulation runs). In particular, configuration A refers to the component when the hypothesis of corrective replacement is adopted and no preventive maintenance rules are applied.

9.4 Component Replacement

321 Reliability vs Time 1.000

0.800

0.800

Reliability, R(t)

Unreliability, F(t)=1-R(t)

Unreliability vs Time 1.000

0.600

0.400

0.200

0.000 0.000

0.600

0.400

0.200

800.000

1600.000

2400.000

3200.000

0.000 0.000

4000.000

800.000

1600.000

Time, (t)

3200.000

4000.000

Failure Rate vs Time

Probability Density Function 0.004

4.800E-4

0.003

Failure Rate, f(t)/R(t)

6.000E-4

f(t)

3.600E-4

2.400E-4

0.002

0.002

8.000E-4

1.200E-4

4.000E-116 0.000

2400.000

Time, (t)

800.000

1600.000

2400.000

3200.000

4000.000

4.000E-116 0.000

800.000

Time, (t)

1600.000

2400.000

3200.000

4000.000

Time, (t)

Fig. 9.3 F .t /, R.t /, f .t /, and .t /, numerical example. ReliaSoft® software

Table 9.2 Different maintenance strategies and parameterizations tp (h)

Configuration A –

Configuration B 1,356

Configuration C 600

Mean availability CM downtime (h) PM downtime (h) Total downtime (h) W .T / (failures) Number of PR Maintenance cost (¤) Total cost (¤)

0.9871 415.71 0 415.71 23.1 0 55,192 304,618

0.9878 288.2 104.45 392.65 13.06 16.01 54,749 290,333

0.9842 128.61 380.56 509.17 7.15 47.58 76,614 382,116

T (h) Simulation repetitions (runs)

Configuration D 4,000 0.9871 416.62 0.03 416.65 23.15 0.004 55,557 305,547

32,200 500

PR preventative replacements

Configurations B, C, and D refer to the hypothesis that preventive replacement is also adopted and the component is preventively replaced when the number of hours from the last replacement (preventive or corrective) reaches the tp value. Configuration B adopts

1;356 h as the value for tp , configuration C adopts 600 h, and configuration D adopts 4;000 h. Figure 9.4 compares the values of the downtimes obtained for the set of simulated system configurations, distinguishing the contribution of

322

9 Basic Models and Methods for Maintenance of Production Systems

500

Downme [h]

400 300 CM downme [h]

200

PM downme [h] Total downme [h]

100 0 Config. A 415.71

CM downme [h] PM downme [h] Total downme [h]

Config. B 288.2

Config. C 128.61

Config. D 416.62

0

104.45

380.56

0.03

415.71

392.65

509.17

416.65

System Configuraon

Fig. 9.4 Downtime analysis in different system configurations. CM corrective maintenance, PM preventive maintenance

350000 300000

Costs [€]

250000 200000 Maintenance cost [

150000

Total cost [ €]

100000 50000 0 Maintenance cost [ €]

Config. A 55192

Config. B 54749

Config. C 76614

Config. D 55557

Total cost [ €]

304618

290333

382116

305547

System Configuraon

Fig. 9.5 Maintenance cost analysis in different system configurations

corrective maintenance from that of preventive maintenance. Finally, Fig. 9.5 presents the results obtained in terms of system costs distinguishing maintenance cost due to corrective maintenance and preventive maintenance actions from total cost, including the significant nonproduction cost contribution. These results clearly demonstrate how much the downtimes and system costs differ for the adoption of different parameterizations of a maintenance action,

and in particular for different values of the time tp . In general, it is possible to obtain advantages from the introduction of a preventive maintenance, e. g., a preventive replacement, but it is also possible to obtain disadvantages as demonstrated by a bad parameterization of the preventive maintenance action in configuration C (C25:4% total cost and C38:8% downtime) if compared with the absence of preventive maintenance. The following sections present and apply basic models for the control and minimization of these costs.

9.5 Time-Based Preventive Replacement – Type I Replacement Model

9.5 Time-Based Preventive Replacement – Type I Replacement Model This strategy refers to the practice of periodically replacing the deteriorating units and components of a production system. This practice is particularly effective for parts and components whose failure behaviors are closely correlated with the time or age of the unit in service. The so-called single unit model can be applied to systems with one unit, but also to each unit in a complex system where the economic dependency among components is weak. In this strategy, maintenance of the system means replacing the component/unit. Cf is the cost due to a replacement after failure;Cp is the unit cost due to a preventive replacement (assuming Cf > Cp ). The object of the problem is to determine the optimal preventive replacement age tp such that the expected system maintenance unit cost (i. e., the cost per unit of operation time, i. e., the total expected replacement cost per unit time) is minimized. Considering Fig. 9.6, when failures occur and failure replacements are performed, the time clock is reset to zero and the planning preventive replacement occurs when the component has been in use for a specified period tp . The following analytical model proposed by Barlow and Hunter (1960) quantifies the UEC, i. e., as a ratio of two expectations: the total expected replacement cost per cycle and the expected cycle length, defined as follows:

Corrective action

tp

0

time

Fig. 9.6 Time-based preventive replacement. Type I

the differentiation of products of differentiable functions (the so-called Leibniz law), and Eq. 9.1 can be explained as follows: UEC.tp / D

Cp R.tp / C Cf Œ1  R.tp / : R tp 0 R.t/ dt

(9.3)

The minimum UEC(tp / given by Eqs. 9.1 and 9.3 is as follows (Jiang et al. 2006): ˇ dUEC.t/ ˇˇ (9.4) ˇ  D 0; dt t Dt or r.t/G.t/ D

c  R.t/; c1

(9.5)

where c D Cf =Cp > 1; Zt G.t/ D

R.x/ dx;

(9.6)

0

r.t/ D

f .t/ 1  F .t/

failure rate function.

In particular, assuming a Weibull distribution of ttf for a generic component/system,

where R tp

tf .t/ dt ; 1  R.tp /

Preventive action

tp

expected total replacement cost per cycle expected cycle lenght Cp R.tp / C Cf Œ1  R.tp / D tp R.tp / C m.tp /Œ1  R.tp / Cp R.tp / C Cf Œ1  R.tp / D ; (9.1) Rt tp R.tp / C 0 p tf .t/ dt

1

Corrective action Preventive action

UEC.tp / D

m.tp / D

323

(9.2)

where tp is the age of the component/system and m.tp / is the mean time to failure (MTTF) if a corrective replacement occurs before tp (since the last preventive or corrective replacement). It is the mean of the truncated distribution at tp : Rt In particular, the term tp R.tp /C 0 p tf .t/ dt is equal Rt to 0 p R.t/ dt by applying the integration by parts, i. e.,

r.t/ D 8 ˆ ˆ < ˆ ˆ :

˛ G.t/ D ˇ

Zz

ˇ  t ˇ 1 : ˛ ˛

z 1=ˇ 1 ez dz D

(9.7)

˛  .1=ˇ; z/ ˇ (9.8)

0

z D .t=˛/ˇ ;

where ˇ is a shape parameter of the Weibull distribution, ˛ is scale parameter of the Weibull distribution,

324

9 Basic Models and Methods for Maintenance of Production Systems UEC(tp) Type I model

9.165

9.16

UEC [€/h]

9.155

9.15

9.145

9.14

9.135 1300

1320

1340

1360

1380

1400

1420

1440

1460

1480

1500

tp [h]

Fig. 9.7 Unit expected cost of replacement (UEC) minimization, type I model numerical example

and Zz  .k; z/ D

x k1 ex dx

(9.9)

0

 .k; z/ is the lower incomplete gamma function whose properties are illustrated by Weisstein (2008). In particular, Table 5.5 reports the values of the gamma function for different values of k, and assuming z equal to C1.

9.5.1 Numerical Example. Type I Replacement Model Consider the numerical example illustrated in Sect. 9.4.2 where the values of maintenance cost per action, including nonproduction costs, are: • corrective maintenance, Cf D ¤ 13;200 per action; • preventive maintenance, Cp D ¤ 6;050 per action. These values refer to the hypothesis of a fixed ttr in both the preventive maintenance and the corrective maintenance, and are equal to 8 and 18 h, respectively. The analytical model introduced above for the type I replacement model does not consider the existence of a repair duration: it is assumed to be equal to zero, i. e., the replacement is instantaneous. As a consequence, to properly apply this model it is necessary to quantify the cost of replacement due to the repair duration

and neglect the repair duration.2 The next model proposed faces this problem explicitly by introducing the duration of replacements, as discussed in Sect. 9.6. Figure 9.7 presents the results obtained by the application of the analytical model in terms of UEC(tp /. The best value of tp is 1;429 h, while the minimum value obtained for UEC is about ¤ 9.14 per hour. By the application of Monte Carlo simulation (assuming T D 32;200 h and 2,000 simulation repetitions3 ) the following results can be obtained and compared with those illustrated in Sect. 9.4.2: • • • • • • • •

mean availability 0.988; corrective maintenance downtime 293:38 h; preventive maintenance downtime 94:09 h; total downtime 387:47 h; W .T / 16.3 failures; number of preventive replacement 11.76; maintenance cost ¤ 53,823; total cost ¤ 286,305.

The total cost of ¤ 286,305 is about 25:07% if compared with previously defined configuration C (see Table 9.2) and 1:39% if compared with configuration B (see Table 9.2). Figure 9.8 summarizes the results obtained in terms of costs, comparing configurations A–D with the best configuration, E, and corresponding to tp D 1;429 h.

2

See Eq. 9.3 The number of repetitions is 2,000 in order to obtain more precise values of system performance. 3

9.5 Time-Based Preventive Replacement – Type I Replacement Model

325

Type I model - tp* determinaon 350000

Costs [€]

300000 250000 200000 150000

Maintenance cost [

100000

Total cost [ €]

50000 0

Maintenance cost [ €]

Config. A 55192

Config. B 54749

Config. C 76614

Config. D 55557

Config. E 53823

Total cost [ €]

304618

290333

382116

305547

286305

System Configuraon

Fig. 9.8 Maintenance costs minimization, type I model

9.5.2 Numerical Example. Type I Model and Exponential Distribution of ttf This numerical example relates to a component/system whose probability distribution of ttf is exponential (i. e., the density function is a Weibull distribution with shape parameter ˇ D 1) and consequently it differs from the distribution of the application illustrated in Sect. 9.5.1. The failure event is random because the failure rate is constant. Table 9.3 presents the results of the performance evaluation and comparison carried on the component/system for different values of time tp by the application of the Monte Carlo simulation.

In particular, the total downtime cumulated on a period of time T equal to 32;200 h (about 5 years) increases when a preventive maintenance replacement strategy is adopted. Consequently, it decreases when the adopted tp interval of time increases. A similar conclusion can be drawn from the analysis of both the maintenance cost and the total cost (see also Fig. 9.9). These results support the rule that it is not convenient to apply preventive maintenance actions of replacement on a component/system whose ttf is subject to an exponential distribution. This thesis is further supported by the following section, which presents and demonstrates universal results.

Table 9.3 Type I model and exponential distribution of time to failure (ttf) for ˇ D 1 tp (h)

Configuration A –

Configuration B 1,356

Configuration C 600

Configuration D 4,000

Mean availability CM downtime (h) PM downtime (h) Total downtime (h) W .T / (failures) Number of PR Maintenance cost (¤) Total cost (¤)

0.988 379.65 0 379.65 21.09 0 50,624 278,414

0.985 375.53 113.55 489.09 20.87 14.2 67,817 361,265

0.978 377.09 338.31 715.39 20.95 42.29 102,641 531,875

0.988 375.95 12.1 388.05 20.9 1.51 52,025 284,855

T (h) Simulation repetitions (runs)

32,200 500

326

9 Basic Models and Methods for Maintenance of Production Systems

Weibull β=1, α=1531.4 350000

Costs [€]

300000 250000 200000 Maintenance cost [

150000

Total cost [ €]

100000 50000 0 Maintenance cost [ €]

Config. A 50624

Config. B 67817

Config. C 102641

Config. D 52025

Total cost [ €]

278414

361265

531875

284855

System Configuraon

Fig. 9.9 Maintenance costs minimization, type I model and ˇ D 1

9.5.3 Type I Replacement Model for Weibull distribution of ttf Figure 9.10 presents the UEC for different Weibull distributions of ttf. These probability distributions differ for different values of shape parameter ˇ (called b in the figure). Cp and Cf values are assumed to be equal to 100 units of cost (e. g., dollars or euros) and 1,000 units of cost, respectively. In particular, for values of ˇ greater than 1 it is possible to identify an optimal value of tp in terms of units of time (e. g., hours or days). Values of the shape parameter lower than 1 are not supported by the determination of the best tp value, as clearly demonstrated by Fig. 9.11. Figure 9.12 presents the expected total cost and the expected cycle length for different values of shape parameter ˇ. Finally, Fig. 9.13 presents the UEC values for different shape parameters of the Weibull distribution, with Cp passing from a value equal to 100 units of cost to a new value equal to 10 units of cost.

9.5.4 The Golden Section Search Method This is a method to find a minimum of a unimodal continuous function over an interval without using derivatives. It can therefore be applied for the minimization

of an objective function similar to Eq. 9.10. Consider a function g over an interval Œa; b; g.t/ is continuous and unimodal (i. e., it has only one minimum) over Œa; b. This method applies as well to finding the maximum of g.t/. The basic idea is to narrow the interval that contains the minimum value, comparing different function values: min fg.t/g:

at b

(9.10)

A method based on five steps for the determination of the minimum (maximum) follows. This algorithm is based on an allowable final tolerance level, ı: Step 1.

Let Œa1 ; b1  D Œa; b; 1 D a1 C .1  ˛/.b1  a1 /; (9.11) 1 D a1 C ˛.b1  a1 /; p 1 C 5 ˛D D 0:6180: 2

(9.12)

˛ is a constant reduction factor for the determination of the size of the interval. Set k D 1. Evaluate g.1 / and g.1 /.

9.5 Time-Based Preventive Replacement – Type I Replacement Model

327

Type I Replacement Model - UEC for Weibull distributions of ttf 100 Weibull (a=50, b=1), Cp=100,Cf=1000 Weibull (a=50, b=2), Cp=100,Cf=1000 Weibull (a=50, b=3), Cp=100,Cf=1000 Weibull (a=50, b=4), Cp=100,Cf=1000 Weibull (a=50, b=0.5), Cp=100,Cf=1000 Weibull (a=50, b=0,2), Cp=100,Cf=1000

90

80

70

UEC(tp)

60

50

40

30

20

10

0

0

10

20

30

40

50

60

70

80

90

100

110

120

130

140

150

160

170

180

190

200

tp [unit of time]

Fig. 9.10 Weibull distribution of time to failure (ttf ). UEC.tp / for different values of shape parameter ˇ (i. e., b), UEC D Œ0; 100

Type I Replacement Model - UEC for Weibull distributions of ttf 30 Weibull (a=50, b=1), Cp=100,Cf=1000 Weibull (a=50, b=2), Cp=100,Cf=1000 Weibull (a=50, b=3), Cp=100,Cf=1000 Weibull (a=50, b=4), Cp=100,Cf=1000 Weibull (a=50, b=0.5), Cp=100,Cf=1000 Weibull (a=50, b=0,2), Cp=100,Cf=1000

25

UEC(tp)

20

15

10

5

0

0

10

20

30

40

50

60

70

80

90

100

110

120

130

140

150

160

170

180

190

200

tp [unit of time]

Fig. 9.11 Weibull distribution of ttf. UEC.tp / for different values of shape parameter ˇ (i. e., b), UEC D Œ0; 30

Step 2.

If bk  ak < ı, the optimal solution t  is defined as ak C bk : t D 2 Otherwise if g.k / > g.k /, go to step 3 and if g.k /  g.k /, go to step 4.

(9.13)

Step 3.

Let akC1 D k ; bkC1 D bk ; kC1 D k ;

(9.14)

kC1 D akC1 C ˛.bkC1  akC1 /: (9.15) Evaluate g.kC1 / and go to step 5.

328

9 Basic Models and Methods for Maintenance of Production Systems Type I Replacement Model - Total Expected Cost for Weibull distributions of ttf

1100

1000

900

Total Expected Cost

800

700

600

500 Weibull (a=50, b=1), Cp=100, Cf=1000 Weibull (a=50, b=2), Cp=100, Cf=1000 Weibull (a=50, b=3), Cp=100, Cf=1000 Weibull (a=50, b=4), Cp=100, Cf=1000 Weibull (a=50, b=0.5), Cp=100, Cf=1000 Weibull (a=50, b=0.2), Cp=100, Cf=1000

400

300

200

100

0

10

20

30

40

50

60

70

80

90

100

110

120

130

140

150

160

170

180

190

200

tp [units of time]

Type I Replacement Model - Expected Cycle Length for Weibull distributions of ttf 70

Expected cycle lenght [units of time]

60

50

40

30 Weibull (a=50, b=1), Cp=100,Cf=1000 Weibull (a=50, b=2), Cp=100,Cf=1000 Weibull (a=50, b=3), Cp=100,Cf=1000 Weibull (a=50, b=4), Cp=100,Cf=1000 Weibull (a=50, b=0.5), Cp=100,Cf=1000 Weibull (a=50, b=0,2), Cp=100,Cf=1000

20

10

0

0

10

20

30

40

50

60

70

80

90

100

110

120

130

140

150

160

170

180

190

200

tp [unit of time]

Fig. 9.12 Weibull distribution of ttf. Expected total cost and expected cycle length

Step 4.

9.5.5 Numerical Example. Type I Model and the Golden Section Method

Let akC1 D ak ; bkC1 D k ; kC1 D k ;

(9.16)

kC1 D akC1 C .1  ˛/.bkC1  akC1 /: (9.17) Evaluate g.kC1 / and go to step 5. Step 5.

Set k D k C 1 and go to step 2.

Consider a component whose ttf probability density function f .t/ between Œ0; 7 weeks is defined as follows: 8 1 ˆ ˆ ; 0t t/ D 1  P .Sr < t/ D 1  Fr .t/; (9.26) P ŒN.t/ > r D P .SrC1 < t/ D FrC1 .t/;

(9.27)

where Fr .t/ is the cumulative distribution function of variable Sr . Equation 9.26 measures the probability of cumulating fewer than r failures in a period of time t. The complementary equation (Eq. 9.27) obviously measures the probability of cumulating fewer than r failures in t, as illustrated in Fig. 9.28.

9.7 Block Replacement Strategy – Type II

341

N(t)

Sr = t1 + t2 +…+ tr S3 = t1 + t2 +t3 S2 = t1 + t2 S1 = t1

t t1

t2

1

t3

2

t4

3

tr

4

r-1

time tr+1

r

r+1

0

number of failures

Fig. 9.28 Renewal process. Variable t and Sr

As a consequence, it is possible to accept the following identical equations: ( P ŒN.t/ < r C P ŒN.t/ D r C P ŒN.t/ > r D 1 P ŒN.t/ D r D Fr .t/  FrC1 .t/: (9.28) Now the expected value W .t/ of N.t/ can be quantified by the following equation: EŒN.t/ D D

1 X rD0 1 X

rŒFr .t/  FrC1 .t/ D

0

Then,

rP ŒN.t/ D r

rD0

In particular, if ttf is distributed in accordance with a negative exponential function, from Eq. 9.29, 8  f  .s/ ˆ  ˆ W D 2 .s/ D ˆ ˆ  sŒ1  f .s/ s ˆ ˆ ˆ <  f .s/ D LŒf .t/ D  et  (9.31) ˆ Z1 ˆ ˆ  ˆ ˆ D : est f .t/ dt D ˆ ˆ : Cs

1 X

Fr .t/:

rD1

(9.29)

   W .t/ D L1 W  .s/ D 2 D t: s

(9.32)

As a consequence, the number of expected failures increases as a linear function of time t.

9.7.2 Laplace Transformation: W(t) and w(t)9 Determination

9.7.3 Renewal Process and W(t) Determination, Numerical Example

Applying Laplace integral transforms to both sides of Eq. 9.29, we have (Jardine and Tsang 2006) 8 f  .s/ ˆ  ˆ W .s/ D ˆ ˆ ˆ sŒ1  f  .s/ < (9.30) Z1 ˆ ˆ st  ˆ f .s/ D LŒf .t/ D e f .t/ dt ; ˆ ˆ :

In order to exemplify the application of the Laplace transform consider the following probability distribution of the random variable ttf:

0

where f .t/ is the probability density function of the random variable ttf. 9 This is m.x/, the renewal density function introduced in Sect. 5.5 in accordance with the basic hypotheses of the renewal process as illustrated in Sect. 9.7.1.

f .t/ D

1 ; 10

0  t  10:

Applying Laplace transforms, f  .s/ D

1 ; 10s

1 ; s.10s  1/      1 1 W .t/ D 2 exp t sinh t : 20 20

W  .s/ D

342

9 Basic Models and Methods for Maintenance of Production Systems

Figure 9.29 presents the values assumed by W .t/ in the case of immediate replacement of failed components and t D Œ0; 10. Similarly, Fig. 9.30 presents the trend of W .t/ for the period of time t D Œ0; 100. How is possible to determine the renewal density w.t/ for an item subject to a renewal process? By Eq. 9.22, w.t/ D

In particular, considering the example illustrated in this section,     1 1 1 dW .t/ D exp t sinh t w.t/ D dt 10 20 20     1 1 1 C exp t cosh t : 10 20 20 It is important to remember that this w.t/ is not the generic failure rate function defined for a repairable

dW .t/ : dt

1.8

W(t)=2*exp(1/20*t)*sinh(1/20*t)

1.6 1.4 1.2 1 0.8 0.6 0.4 0.2

Fig. 9.29 Renewal process. Transforms of Laplace W .t /, t D Œ0; 10

0

0

1

2

3

4

5 t

6

7

8

9

10 -3 0

10

20

30

40

50

60

70

80

90

10

10 5

10 4

10 3

W(t)

10 2

10 1

10 0

10 -1

10 -2

Fig. 9.30 Renewal process. Transforms of Laplace W .t /, t D Œ0; 100

t

100

9.7 Block Replacement Strategy – Type II

343 2 1.8 1.6

w(t). λ(t). f(t)

1.4 1.2

f(t)

1

λ(t) w(t)

0.8 0.6 0.4 0.2 0

Fig. 9.31 Renewal process .t /, f .t /, and w.t /, numerical example

0

1

f .t/ f .t/ D R1 R.t/ t f .x/ dx D

3

4

5

6

7

8

9

10

t

component subjected to a sequence of operative cycles (FFR), i. e., a sequence of failures and repairs (see footnote 9). Figure 9.31 presents the trend assumed by renewal density w.t/, .t/, and f .t/. In particular, for a repairable component the rate function .t/ represents the failure rate at point in time t measured from the last replacement: .t/ D

2

1 1=10 D : 1  .1=10/t 10  t

As a consequence, it is not correct to strictly compare these functions which are defined for different ranges of values: Œ0; 10 for .t/ and f .t/, Œ0; 1/ for w.t/.

9.7.4 Numerical Example, Type II Model This example relates to the application introduced in Sect. 9.4.2. The component is subject to preventive maintenance and possibly corrective maintenance actions in accordance with the model of Eq. 9.20. In particular, Monte Carlo analysis has been applied to different operating scenarios, from configuration A, corresponding to the absence of preventive maintenance actions, to configuration F identified in Sects. 9.4.2,

9.5.1, and 9.6.1. Configuration G will be properly justified in Sect. 9.9.1.1 (the application of the so-called Type I – Minimum Downtime model will justify a replacement time equal to 1,392 h). The proposed scenarios differ from the value of tp adopted in Eq. 9.20. Both preventive and corrective actions perform replacements in accordance with the “as good as new” hypothesis. Figure 9.32 shows that corrective maintenance downtime increases when the value of tp increases too, while preventive maintenance downtime decreases. In terms of maintenance and total costs the first scenario, configuration A, turns out to be the best one. In order to identify the best value of tp , in accordance with Eq. 9.20, it is necessary to quantify the renewal function W .t/, i. e., the expected number of failures. For the two-parameter Weibull distribution, W .t/ is not computable in explicit form, and for its guesstimate several functions, lower and upper bounds not the subject of this book (e. g., Soland 1969; Bilgen and Deligönül 1987; Constantine and Robinson 1997; Yannaros 1994; Jiang et al. 2006; Politis and Koutras 2006), are outlined in the literature. In particular, Soland (1969) and Constantine and Robinson (1997) presented useful tables for computing the renewal function W .t/. In order to evaluate W .t/ as an approximation, two alternative and practical methods are proposed:

344

9 Basic Models and Methods for Maintenance of Production Systems 600

500

downtime [h]

400

300

CM downtime [h] PM downtime [h] Total downtime [h]

200

100

0

Config. A

Config. C

Config. B

Config. G

CM downtime [h]

416

124

253

259

265

263

PM downtime [h]

0

421

182

182

174

174

63

416

545

435

441

438

436

427

Total downtime [h]

Config. E

Config. F

Config. D 364

Fig. 9.32 Downtime contributions, type II model

1. assuming w.t/ D .t/; 2. applying Monte Carlo simulation analysis.

9.7.4.1 Approximation Method 1 for W(t) In this method Eq. 5.75 is applied as follows: Zt W .t/ D 0

Z t   b1   x b b x dx D .x/ dx D : a a a 0

The trend of the approximated renewal function W .t/ is illustrated in Fig. 9.33. Figure 9.34 reports the estimated values of UEC(tp ) as a result of the application of the analytical model (Eq. 9.20), and assuming the W .t/ trend in Fig. 9.33 and the following unit costs10 : Cf D ¤ 13,200 per action and Cp D ¤ 6,050 per action. The minimum value of UEC(tp / obtained is about ¤ 11.45 per hour, for tp equal to 1;043 h. For t equal to 1,043 time units the number of expected failures obtained by the assumption w.t/ D .t/ is about 0.445. This value can now be compared with the values obtained by the application of the Monte Carlo simulation, see Table 9.8. In particular, in configuration B the expected number of failures W .t D 32;200 h/ was about 14.09, 10

In coherence with the hypotheses summarized in Table 9.1

assuming the hypothesis of random failure events but also constant repair times (see ttr values in Table 9.1). It is worth observing that this value obtained by the Monte Carlo simulation refers to 32;200 h, while the number of failures, 0.445, obtained by the assumption w.t/ D .t/ refers to a period of time of 1;043 h. This is a good result as it can be checked by a simple proportion: 14:09  1043 Š 0:456; 32;200 which is a value very close to 0.445. UEC evaluation In order to complete the comparative what-if analysis conducted on different values of tp (see Table 9.8 Conf. A–G), we also present the results obtained by the application of Monte Carlo simulation assuming tp D 1;043 h, T D 32;200 h, and 2,000 repetitions/runs (we call this configuration H). These results are obtained for configuration H: mean availability 0.9861, corrective maintenance downtime 209.87, preventive maintenance downtime 237.22, total downtime 447.10, W .T / D 11:67, number of preventive maintenance actions 29.65, maintenance cost ¤ 65,051, and total cost ¤ 333,311. These values further suggest it would not be useful

9.7 Block Replacement Strategy – Type II

345

Table 9.8 Performance evaluation and comparison, type II model Type II tp (h)

Configuration A –

Configuration C 600

Configuration B 1,356

Configuration G 1,392

Configuration E 1,429

Configuration F 1,445

Configuration D 4,000

Mean availability CM downtime (h) PM downtime (h) Total downtime (h) W .T / (failures) Number of PR

0.9871 416 0 416 23.1 0

0.9831 124 421 545 6.9 52.58

0.9865 253 182 435 14.09 22.72

0.9863 259 182 441 14.4 22.71

0.9864 265 174 438 14.71 21.7

0.9864 263 174 436 14.6 21.71

0.9867 364 63 427 20.23 7.91

Maintenance cost (¤) Total cost (¤)

55,192 304,618

82,285 409,189

62,210 323,390

62,935 327,409

T (h) Simulation repetitions (runs)

62,415 325,377

62,164 323,986

58,429 314,833

32,200 2,000

W(t)

W(t) determination by failure rate

Fig. 9.33 W .t / determination by the failure rate .t /

130 120 110 100 90 80 70 60 50 40 30 20 10 0 0

2000

to apply a preventive maintenance strategy based on the type II replacement rule to the current case study. How is it possible that Fig. 9.34 clearly identifies an optimal value of tp and what-if analysis demonstrates that it is not economic to apply a type II based preventive replacement? First of all, the analytical model illustrated by Eq. 9.20 does not consider the replacement times (Tp and Tf introduced in Sect. 9.6) which influence the alternating renewal process as a sequence of ttf and ttr values (see the definition introduced in Sect. 9.7) and the number of replacement cycles in the simulated period of time T , e. g., 32;200 h. In general, during a cycle of preventive replacement, therefore, some corrective actions take place, i. e., replacements based on the as good as new hypothesis. This is in contrast with the assumption w.t/ D .t/, because the density function w.t/ assumed to quantify the expected number of failures W .t/ is de-

4000

6000

8000

10000

12000

14000

t

fined for the whole preventive cycle that can include several corrective replacements. Consequently, by the assumption w.t/ D .t/, the function w.t/ increases during a preventive cycle as exemplified in Fig. 9.35 and corrective actions cannot be based on the as good as new hypothesis. The method illustrated in next section tries to bypass the limit of adopting w.t/ D .t/.

9.7.4.2 Approximation Method 2 for W(t) Evaluation By the application of the simulation analysis to the case study introduced for the first time in Sect. 9.4.2, it is possible to quantify the expected number of failures of the component in a period of time T . In particular, assuming T D 2;000 h the following what-if scenarios, based on different values of the corrective replace-

346

9 Basic Models and Methods for Maintenance of Production Systems UEC(tp) - Type II, W(t) approx. 11.85 11.8 11.75

UEC(tp)

11.7 11.65 11.6 11.55 11.5 11.45 11.4

Fig. 9.34 UEC(tp ) type II, numerical example

800

850

900

950

1000

1050

Corrective actions

0

Preventive action Corrective actions

tp

W1(t)

130

120

120

110

110

100

100

90

90

80

80

70

70

60

60

50

50

40

40

30

30 20

10 0

time

W(t) determination by failure rate

130

20

10 0

1000

2000

3000

4000

5000

6000

7000

8000

9000

0 10000 11000 12000 13000 14000 1500 0

1000

t

Fig. 9.35 Preventive maintenance and preventive replacement (PR) cycles when w.t / D .t /

2000

3000

4000

5000

6000

7000

8000

9000

10000 11000 12000 13000 14000 1500

t

PR cycle 1 w1(t)=λ(t) t0=0 t=time- t0

PR cycle 2 w2(t)=λ(t) t0=tp t=time- tp

t

t

W1 (t ) = ∫ w1 ( x)dx

W2 (t ) = ∫ w2 ( x)dx

W (t ) = W1 (t )

W (t ) = W1 (t ) + W2 (t )

0

ment time Tf and the restoration factor q 11 , have been simulated and compared, as illustrated in Fig. 9.36:

11

1200

Preventive action

2tp

W(t) determination by failure rate

Tf Tf Tf Tf

1150

tp

tp

1. 2. 3. 4.

1100

tp

D 18 h, q D 1 (as good as new hypothesis); D 18 h, q D 0 (minimal repair hypothesis); D 0 h, q D 1 (as good as new hypothesis); D 18 h, q D 0 (minimal repair hypothesis).

The percentage to which a component is restored after the execution of the maintenance action. In particular q D 1 corresponds to the well-known “as good as new hypothesis,” while q D 0 corresponds to the minimal repair hypothesis properly defined in Sect. 9.11.

0

UEC evaluation It is possible to quantify UEC(tp /, as illustrated in Fig. 9.37, by entering the values of W .t/ obtained in Eq. 9.20. Table 9.9 summarizes in detail the minimum values of UEC(tp / for scenarios A–D. From the values of UEC(tp / obtained, we see that the preventive maintenance replacement strategy based on the type II policy when q=1, i. e., in the presence of the “as good as new” hypothesis12, is not so attractive. This con clusion is coherent with the simu12

See footnote 11.

9.7 Block Replacement Strategy – Type II

347 1.8 1.6 1.4

W(T)

1.2 1.0

W(Tf=18,q=1) W(Tf=18,q=0)

0.8

W(Tf=0,q=1) W(Tf=0,q=0)

0.6 0.4 0.2

Fig. 9.36 Evaluation of the expected number of failures by simulation

0 0

200

400

600

800

1000

1200

1400

1600

1800

2000

me-t

100 90 80 70

UEC(tp)

60

UEC(A)

50

UEC(B) UEC(C)

40

UEC(D) 30 20 10 0 0

Fig. 9.37 UEC values, scenarios A–D

200

400

600

800

1000

1200

1400

1600

1800

2000

me replacement-tp

Table 9.9 UEC(tp ) values, scenarios A–D Scenario

Tf (h)

q

tp (h)

UEC(tp ) (¤=h)

A B C D

18 18 0 0

1 0 1 0

> 2;000 900 1,960 880

– 11.09 10.28 11.42

lated results obtained by the application of the Monte Carlo analysis and reported in Table 9.8 (see also the previously introduced configuration H, Sect. 9.7.4.1).

Otherwise, assuming the restoration factor q equal to 0, i. e., in the presence of “minimal repair” maintenance actions, the best value of tp exists in coherence with the previously illustrated analysis, conducted with basic and simplifying hypothesis w.t/ D .t/, whose results are illustrated in Fig. 9.34. This numerical example demonstrates how important it is to quantify the expected number of failures W .t/.

348

9 Basic Models and Methods for Maintenance of Production Systems

9.7.5 Discrete Approach to W(t) In the following a discrete analytical method to predict the expected number of failures in a period of time T , made up of several units or intervals, is presented. The basic assumption is that no more than one failure can occur in any unit of the period. As a consequence, this hypothesis is not so restrictive because it is possible to define the desired number of intervals for a given period of time T . The number of expected failures W .T / occurring in the interval .0; t D T / can be considered as the sum of the following contributions (as illustrated in Fig. 9.38): 1. Number of expected failures occurring in .0; T / when the first failure occurs in the first period .0; 1/, multiplied by the probability of the first failure occurringR in the interval .0; 1/, i. e., P .0  1 ttf < 1/ D 0 f .t/ dt . The number of expected failures that occurs in the interval .0; T / when the first failure occurs in the first period is 1 (e. g., the failure occurred in the first week) plus the expected number of failures in the remaining T  1 periods W .T  1/. 2. Number of expected failures that occur in interval .0; T / when the first failure occurs in the second period multiplied by the probability of the first failure occurring in the interval .1; 2/. 3. ...

T . Number of expected failures that occur in interval .0; T / when the first failure occurs in the T th period multiplied by the probability of the first failure occurring in the interval .T  1; T /. The events .1; : : : ; T / described above are disjunctive. As a consequence, W .T / D

T 1 X

Zi C1 Œ1 C W .T  i  1/ f .t/ dt :

i D0

i

(9.33) For example, the number of expected failures in the interval (0,5) is Z1   W .T D 5/ D Œ1 C W .4/ f .t/ dt 0

Z2   C Œ1 C W .3/ f .t/ dt 1

 C Œ1 C W .2/

Z3

 C Œ1 C W .1/

Z4

2

3 T ZD5   f .t/ dt : C Œ1 C W .0/

N(T)

0

2

3

1

T-1

1+W(T-1)

∫ f (t )dt 0

2

1+W(T-2)

∫ f (t )dt 1

3

∫ f (t )dt 2

1+W(T-3)

…

Fig. 9.38 W .T / determination. Discrete approach

T

 f .t/ dt

4

1

 f .t/ dt

time period

(9.34)

9.7 Block Replacement Strategy – Type II

349

Z1   D Œ1 C W .2/ f .t/ dt

9.7.6 Numerical Examples

0

The following two examples illustrate the application of the discrete approach for the determination of the expected number of failures W .t/ during a renewal process and in accordance with the hypotheses previously introduced. Example 2 also exemplifies the determination of the best tp value by the application of preventive replacement model type II, i. e., the minimization of UEC(tp /.

Z2   C Œ1 C W .1/ f .t/ dt 1

C Œ1 C W .0/

D Œ1 C W .0/ŒF .3/  F .2/ Š 0:022;

W .T D 4/ D

Zi C1 Œ1 C W .T  i  1/ f .t/ dt i

 D Œ1 C W .3/

Z1

 f .t/ dt

0

Z2   C Œ1 C W .2/ f .t/ dt Z1

W .T D 1/ D Œ1 C W .0/

1

Z3   C Œ1 C W .1/ f .t/ dt

f .t/ dt 0

2

D F .t D 1/  F .t D 0/ D

3 X i D0

W .0/ D 0;

zD tMTTF .TTF/

o f .t/ dt

2

9.7.6.1 Numerical Example 1 The variable ttf is assumed to be distributed in accordance with a normal distribution with a mean of 6 weeks and a standard deviation of 1.5 weeks. By the application of Eq. 9.33,

Z3

n

 C Œ1 C W .0/

˚.z D 3:34/  ˚.z D 4/ Š 0;

Z4

 f .t/ dt

3

W .T D 2/ D

1 X

Zi C1 Œ1 C W .T  i  1/

i D0

 D Œ1 C W .1/

f .t/ dt

Z3 2



f .t/ dt C

W .T D 5/ D

4 X

Zi C1 Œ1 C W .T  i  1/ f .t/ dt

i D0

 D Œ1 C W .4/

1

Z2

0

3

f .t/ dt

Z2   C Œ1 C W .0/ f .t/ dt

D

f .t/ dt

D F .4/  F .2/ D 0:087;

0

Z1

f .t/ dt C

Š

i

Z1

Z4

f .t/ dt

i

Z1

 f .t/ dt

0

Z2   C Œ1 C W .3/ f .t/ dt

1

Š F .2/  F .1/ Š 0:0034;

1

Zi C1 2 X W .T D 3/ D Œ1 C W .T  i  1/ f .t/ dt i D0

i

 C Œ1 C W .2/

Z3

 f .t/ dt

2

350

9 Basic Models and Methods for Maintenance of Production Systems

Z4   C Œ1 C W .1/ f .t/ dt  C Œ1 C W .0/

Z3   C Œ1 C W .4/ f .t/ dt

3

2

Z5

Z4

 C Œ1 C W .3/

 f .t/ dt

3

4

Z5 Š .1:0034/ŒF .3/  F .2/ C

Z5   C Œ1 C W .2/ f .t/ dt

f .t/ dt 3

4

D 0:249; W .T D 6/ D

5 X

Zi C1 Œ1 C W .T  i  1/ f .t/ dt

i D0

Z1

D Œ1 C W .5/



C 1:022ŒF .4/  F .3/

Z2   C Œ1 C W .4/ f .t/ dt

 C Œ1 C W .2/  C Œ1 C W .1/

Z5

 C Œ1 C W .0/

Z6

C 1:003ŒF .5/  F .4/ C ŒF .7/  F .5/ D 0:747:

1

Z4

 f .t/ dt

Š 1:087ŒF .3/  F .2/

0

 C Œ1 C W .3/

 C Œ1 C W .0/

Z7

 f .t/ dt

6

f .t/ dt

Z3

 C Œ1 C W .1/

Z6 5

i



 f .t/ dt

 f .t/ dt

2



9.7.6.2 Numerical Example 2

f .t/ dt 3

 f .t/ dt

4

 f .t/ dt

Consider the component introduced in the example illustrated in Sect. 9.5.5 and related to the application of the type I replacement model. The value of the expected number of failures can be quantified by the application of the discrete approach: W .0/ D 0;

5

Š .1:022/ŒF .3/  F .2/

Z1

C 1:003ŒF .4/  F .3/ C ŒF .6/  F .4/

W .T D 1/ D Œ1 C W .0/

D 0:496; W .T D 7/ D

6 X

0

Zi C1 Œ1 C W .T  i  1/ f .t/ dt

i D0



f .t/ dt

Z1 D

1 1 dt D D 0:125; 8 8

0

i

Z1

D Œ1 C W .6/



W .T D 2/ D

f .t/ dt

Zi C1 Œ1 C W .T  i  1/ f .t/ dt

i D0

0

 C Œ1 C W .5/

1 X

Z2

 f .t/ dt

1

 D Œ1 C W .1/

i

Z1

 f .t/ dt

0

9.7 Block Replacement Strategy – Type II

351

Z2   C Œ1 C W .0/ f .t/ dt

Z5 C

1

4

 Z1  Z2 1 1 1 dt C dt D 1C 8 8 8 0

22;185 1 D C Š 0:844; 32;768 6

1

  1 1 1 17 D 1C C D Š 0:266; 8 8 8 64 Zi C1 2 X W .T D 3/ D Œ1 C W .T  i  1/ f .t/ dt i D0

i

 D Œ1 C W .2/

Z1

 f .t/ dt

0

Z2   C Œ1 C W .1/ f .t/ dt 1

 C Œ1 C W .0/

Z3

 f .t/ dt

W .T D 6/ D

W .T D 4/ D

i D0

Zi C1 Œ1 C W .T  i  1/ f .t/ dt i

1 Œ1 C W .3/ C 1 C W .2/ 8 C 1 C W .1/ C 1 C W .0/   217 17 1 1 4C C C D 8 512 64 8 2;465 D Š 0:602; 4;096 D

W .T D 5/ D

4 X i D0

Zi C1 Œ1 C W .T  i  1/ f .t/ dt i

1 Œ1 C W .4/ C 1 C W .3/ C 1 C W .2/ 8 1 C 1 C W .1/ C Œ1 C W .0/ 6   1 2;465 217 17 1 D 4C C C C 8 4;096 512 64 8

5 X

Zi C1 Œ1 C W .T  i  1/ f .t/ dt

i D0

i

1 D Œ1 C W .5/ C 1 C W .4/ 8 C 1 C W .3/ C 1 C W .2/ 1 C Œ1 C W .1/ C 1 C W .0/ 6  1 22;185 1 2;465 D 4C C C 8 32;768 6 4;096    17 1 1 217 C C 2C C 512 64 6 8 Š 1:121;

2

1 D Œ1 C W .2/ C 1 C W .1/ C 1 C W .0/ 8  1 1 17 217 C Š 0:424; D 3C D 8 64 8 512 3 X

1 dt.1 C 0/ 6

W .T D 7/ D

6 X

Zi C1 Œ1 C W .T  i  1/ f .t/ dt

i D0

i

1 D Œ1 C W .6/ C 1 C W .5/ 8 C 1 C W .4/ C 1 C W .3/ 1 C Œ1 C W .2/ C 1 C W .1/ 6 C 1 C W .0/  22;185 1 4 C 1:121 C D 8 32;768  2;465 217 1 C C C 6 4;096 512   17 1 1 3C C C 6 64 8 Š 1:439:

D

Figure 9.39 illustrates the trend of W .T / values for the range Œ0; 7. Figure 9.40 illustrates the trend of UEC(tp / values for the range considered. The best value of tp is equal to 3 weeks, UEC(tp / D 8:73.

352

9 Basic Models and Methods for Maintenance of Production Systems Expected Number of Failures 1.6 1.4 1.2

W(T)

1 0.8 0.6 0.4 0.2 0

Fig. 9.39 Expected number of failures W .T /

0

1

2

3

4

5

6

7

T

UEC - Type II 12 11.5 11

UEC

10.5 10 9.5 9 8.5 8 7.5 7

1

Fig. 9.40 Type II model. Tp determination

2

3

4

5

6

7

Tp

Now, introducing the approximation w.t/ D .t/, Ztp EŒN.tp / D W .tp /approx. D

Ztp w.t/ dt Š

0

8 t p   ˆ ˆZ 8 1 ˆ ˆ dt D ln ; ˆ ˆ ˆ 8t 8  tp ˆ ˆ ˆ 0 ˆ ˆ ˆ ˆ Z4 Ztp ˆ < 1 1 dt C dt D 8t 7t ˆ ˆ 0 ˆ 4  ˆ ˆ 3 ˆ ˆ D ln 2 C ln ˆ ˆ ˆ ˆ 7  tp  ˆ ˆ 6 ˆ ˆ :D ln ; 7  tp

.t/ dt 0

0  t < 4;

9.7.7 Practical Shortcut to W(t) and tp Determination Similarly to Sect. 9.6.4, which relates to a practical shortcut for the optimal age replacement interval, this section presents a quick way to determine the renewal function W .t/. Smith (1954) proposed the following asymptotic approximation of W .t/, which is effective for large values of t: W .t/ Š

4  t < 7:

The values obtained, W .tp /approx. , approximate in a satisfactory way the expected number of failures W .tp / determined with the discrete approach, as illustrated in Fig. 9.41. Finally, Fig. 9.42 presents the values of UEC(tp / obtained by the application of both the approximation approach and the discrete approach.

t  2  2 C ;  22

(9.35)

where  is the mean of an arbitrary lifetime density function f .t/ and  2 is the variance of f .t/. In particular, for the Weibull distribution Eq. 9.35 with ˇ < 4 gives good numerical accuracy for t  3 and reasonable relative accuracy for t 2 Œ1; 3, where t is the variable time for the scaled Weibull distribution.13 For larger ˇ the accuracy is not very good 13

Because ˛ is the “scale” parameter of the generic Weibull density function. The condition t D 3 for the scaled function is equal to the condition t =˛ D 3 for the generic Weibull function.

9.8 Maintenance Performance Measurement in Preventive Maintenance

353 W(T) W(T) approx.

Expected Number of Failures

2 1.8 1.6 1.4 W(T)

1.2 1 0.8 0.6 0.4 0.2 0

Fig. 9.41 Expected number of failures, numerical example

0

1

2

3

4

5

6

7

T

UEC(Tp)

UEC - Type II

12

UEC(Tp) approx.

11.5 11 10.5 UEC

10 9.5 9 8.5 8 7.5

Fig. 9.42 UEC(Tp /. “Discrete” calculus compared with “approximation” calculus

7 1

2

for moderate values of t (Constantine and Robinson 1997). To exemplify this, consider the numerical example introduced in Sect. 9.4.2 and illustrated in Sect. 9.7.4 with regards to the type II preventive replacement model: Cf D ¤ 13,200 per action and Cp D ¤ 6,050 per action, ˇ D 2:1, and ˛ D 1;531:4 h. The MTTR is 1;356 h by Eq. 5.69 and the variance can be quantified by the following general equation (Abernethy 2007):       1 2 2   1C : (9.36)  D˛  1C ˇ ˇ 2

2

As a consequence (see also Table 5.5),  2 .˛ D 1531:4; ˇ D 2:1/    2   1 2   1C D 1531:42  1 C 2:1 2:1 Š 460;119:8:

3

4

5

6

7

Tp

Table 9.10 presents the results obtained by the application of Eq. 9.35. These values can be compared with those reported in Fig. 9.36. In particular, in accordance with the results and conclusions of Sect. 9.7.4, it seems there is not an optimal replacement time period tp .

9.8 Maintenance Performance Measurement in Preventive Maintenance Several authors have proposed some measures of effectiveness of preventive maintenance using the relative amount of preventive maintenance actions, such as the ratio of preventive maintenance hours and the total maintenance hours (Arts et al. 1998). They affirm that the benchmark data for appropriate preventive maintenance are about 75–97%. As a consequence, an effective preventive maintenance is supported by as few

354

9 Basic Models and Methods for Maintenance of Production Systems

Table 9.10 Asymptotic approximation of W .t / and UEC(tp /, numerical example t

1,000 1,500 2,000 2,500

3,000

3,500

4,000

4,500

5,000

5,500

6,000

6,500

7,000

7,500

W .t / 0.36 0.73 1.10 1.47 UEC(tp / 10.84 10.47 10.29 10.18

1.84 10.10

2.21 10.05

2.57 10.01

2.94 9.98

3.31 9.95

3.68 9.93

4.05 9.92

4.42 9.90

4.79 9.89

5.16 9.88

t

8,000 8,500 9,000 9,500 10,000 10,500 11,000 11,500 12,000 12,500 13,000 13,500 14,000 14,500

W .t / UEC.tp /

5.52 9.87

5.89 9.86

6.26 9.86

6.63 9.85

7.00 9.84

7.37 9.84

7.74 9.83

corrective maintenance (i. e., high occurrence rate for preventive maintenance) and as few preventive maintenance occurrences as possible. The first condition can be explained as follows:

8.11 9.83

8.47 9.83

8.84 9.82

number of corrective events after t ; (9.37) total number of events number of preventive events after t : Sm .t/ D total number of events (9.38) Sc .t/ D

Jiang et al. (2006) proposed preventive effect measures, which are not based on historic data (i. e., model-free) but are based on the previously introduced preventive maintenance replacement analytical models of type I and type II. For the age replacement policy (type I), they introduced the measure Pe .t/, called “preventive effect indicator”:

1 ; W .t/ C 1

10.32 9.81

Pe .0/ D 1;

Finally, Pe .t/ is decreasing. The authors demonstrated that a poor preventive effect implies a poor cost saving. In particular, for the age replacement policy the following measure of cost saving can be introduced: Scost D

(9.41)

Consider the numerical example illustrated in Sect. 9.5.5. The cost saving obtained by the introduction of the preventive maintenance type I replacement rule is14 Cf =MTTF  UEC.t  / Cf =MTTF 50=.15=4/  8:975 Š 32:7%; D 50=.15=4/

Scost .t  D 2:95/ D

where Z1 MTTF D

R.t/ dt 0

  Z4  Z7  Z1 1 7t dt C 0 dt D 1  t dt C 8 6

(9.40)

where W .t/ is the renewal function, i. e., the expected number of corrective maintenance actions.

Cf =MTTF  UEC.t  / : Cf =MTTF

9.8.1 Numerical Example

(9.39)

whereF .t/ measures the fraction of corrective replacement andR.t/ measures the fraction of preventive replacement. Similarly, for the block replacement policy (type II), where the number of preventive maintenance actions is 1 in the interval Œ0; t, PeII .t/ D

9.95 9.81

Pe .1/ D 0:

where

R.t/ D R.t/; F .t/ C R.t/

9.58 9.82

The generic preventive effect indicator Pe .t/ has the following properties:

Sm .t/  Sc .t/;

PeI .t/ D

9.21 9.82

0

15 D 4 14

4

7

.weeks/:

tp D 2:95 weeks and UEC.tp / D ¤ 8,975 per week.

9.9 Minimum Total Downtime

The expected maintenance cost per unit time decreases from ¤ 13,333 per week to ¤ 8,975 per week. Other significant performance indexes were introduced in some previously illustrated numerical examples and applications, e. g., Table 9.3: mean availability, corrective maintenance downtime, preventive maintenance downtime, total downtime, W .T /, number of preventive replacements, maintenance cost, and total cost. All these performance indexes are defined for a period of time15 T .

9.9 Minimum Total Downtime If the aim of the optimal replacement strategy is to maximize the throughput or the utilization of the equipment , the objective function of the supporting decision-making model can be the total downtime per unit time (due to both preventive and failure replacement actions and frequencies). The proposed model sets this function to a minimum. The type I and type II models previously described have been modified in accordance with this new objective function, and are described and exemplified separately in next sections.

9.9.1 Type I – Minimum Downtime This model supports the determination of the optimal age tp at which preventive replacements should occur in order to minimize the total downtime per unit time: total expected downtime cycle length Tp R.tp / C Tf Œ1  R.tp / : D .tp C Tp /R.tp / C Œm.tp / C Tf Œ1  R.tp / (9.42)

DT.tp / D

The cycle length is calculated in accordance with Eq. 9.18. 9.9.1.1 Type I – Minimum Downtime, Numerical Example 1 Consider the numerical example introduced in Sect. 9.4.2. The application of the analytical model 15

Mission time, also known as time of analysis – observing time.

355

(Eq. 9.42) generates the trend of the downtime DT.tp / illustrated in Fig. 9.43. The minimum value of the downtime is about 0.0122 for tp D 1;392 h. The results obtained by the application of the Monte Carlo simulation, in accordance with a preventive replacement executed after 1;392 h from the last preventive or corrective replacement, are reported in Table 9.11 (configuration G). These values are very similar to those related to the application of the original type I replacement model (configuration E, see Sect. 9.5.1) and to the application of the type I model with Tp and Tf (configuration F, see Sect. 9.6.1). All the results are obtained by a number of simulation runs, called “repetitions,” equal to 2,000. They are not deterministic values and this is why the total downtime does not seem to be at its minimum in configuration G.

9.9.1.2 Type I – Minimum Downtime, Numerical Example 2

Considering the numerical example introduced in Sect. 9.5.5, the total downtime per unit time is Tp R.tp / C Tf Œ1  R.tp / .tp C Tp /R.tp / C Œm.tp / C Tf Œ1  R.tp /



 8 Tp 1  18 tp C Tf 18 tp ˆ ˆ ˆ  t   ;

ˆ ˆ ˆ .tp C Tp / 1  18 tp C 2p C Tf 18 tp ˆ ˆ ˆ ˆ ˆ 0  tp < 4; ˆ ˆ ˆ ˆ ˆ

7tp 

tp 1  < Tp 6 C Tf 6 D  t 1  ;

7t  t 2 4 ˆ ˆ ˆ .tp C Tp / 6 p C 2.tp 1/ C Tf p 6 ˆ ˆ ˆ ˆ ˆ 4  tp < 7; ˆ ˆ ˆ ˆ ˆ ˆ Tf ˆ ˆ ; tp  7: : 45 12 C Tf

DT.tp / D

Table 9.12 summarizes the values of DT(tp / obtained for different operating scenarios and couplets of Tp and Tf values (see the scenarios introduced in Sect. 9.6.3). For example, in scenario C the best value of tp is 5 weeks, corresponding to a unit DT equal to 0.196. Figure 9.44 presents the graphic trend of the downtime values obtained.

356

9 Basic Models and Methods for Maintenance of Production Systems

Table 9.11 Monte Carlo analysis and type I model with downtime minimization Configuration A –

tp (h)

Configuration B 1,356

Configuration C 600

Configuration D 4,000

Configuration E 1,429

Configuration F 1,445

Configuration G 1,392

Mean availability CM downtime (h) PM downtime (h) Total downtime (h) W .T / (failures) Number of PR Maintenance cost (¤) Total cost (¤)

0.9871 0.9878 0.9842 0.9871 415.71 288.2 128.61 416.62 0 104.45 380.56 0.03 415.71 392.65 509.17 416.65 23.1 13.06 7.15 23.15 0 16.01 47.58 0.004 55,192 54,749 76,614 55,557 304,618 290,333 382,116 305,547

0.988 0.988 0.988 293.38 294.42 288.15 94.09 91.97 100 387.47 386.39 388.15 16.3 16.36 16.01 11.76 11.49 12.51 53,823 53,631 54,059 286,305 285,465 286,949

T (h) Simulation repetitions (runs)

32,200 500

32,200 2,000

Table 9.12 Analysis multiscenario. Downtime minimization, type I Type I – downtime minimization Id scenario Tp Tf A B C D

0.5 1 0.5 0.25

0.5 1 1 1

0

1

2

3

tp 4

5

6

7

8

1.00 1.00 1.00 1.00

0.35 0.52 0.38 0.27

0.22 0.36 0.26 0.20

0.17 0.29 0.22 0.18

0.14 0.25 0.20 0.17

0.13 0.23 0.20 0.18

0.12 0.21 0.20 0.19

0.12 0.21 0.21 0.21

11.76 11.76 11.76 11.76

Type I repl. model DT minimization

0.01226 0.01225

DT(tp)

0.01224 0.01223 0.01222 0.01221

Fig. 9.43 Type I model and downtime (DT) minimization, numerical example

0.01220 1300 1310 1320 1330 1340 1350 1360 1370 1380 1390 1400 1410 1420 1430 1440 1450 tp

1 0.9 0.8

DT(Tp)

0.7 0.6

A

0.5

B

0.4

C

0.3

D

0.2 0.1

Fig. 9.44 Downtime minimization, type I. Numerical example

0 0

1

2

3

4

tp

5

6

7

9.9 Minimum Total Downtime

357

9.9.1.3 Type I Replacement for Minimum Downtime. Weibull Distribution of ttf

Finally, Fig. 9.46 presents the expected total downtime per unit time when Tp passes from 0.5 to 0.1 units of time.

Figure 9.45 presents the expected total downtime per unit time for distributions of ttf which differ for the value of shape parameter ˇ, assuming Tp and Tf are equal to 0.5 and 1 unit of time (e. g., hour or day), respectively. For values of ˇ greater than 1 it is possible to identify an optimal value of tp in terms of units of time. Values of the shape parameter lower than 1 are not supported by a best tp value.

9.9.2 Type II – Downtime Minimization The following model supports the determination of the optimal replacement interval tp between preventive replacements adopting the block replacement strategy (type II) and setting the total downtime per unit time

Type I Replacement Model - DownTime minimization (Tp =0.5, Tf =1)

1

Weibull (a=50, b=1) 0.9

Weibull (a=50, b=2) Weibull (a=50, b=3)

0.8

Weibull (a=50, b=4) Weibull (a=50, b=0.5)

0.7

Weibull (a=50, b=0.2)

DT(tp)

0.6

0.5

0.4

0.3

0.2

0.1

0 0

10

20

30

40

50

60

70

80

90

100

110

tp [unit of time] Type I Replacement Model - DownTime minimization (Tp = 0.5, Tf = 1) 0.1 Weibull (a=50, b=1) 0.09

Weibull (a=50, b=2) Weibull (a=50, b=3)

0.08

Weibull (a=50, b=4) Weibull (a=50, b=0.5)

0.07

Weibull (a=50, b=0.2)

DT(tp)

0.06

0.05

0.04

0.03

0.02

0.01

0

0

10

20

30

40

50

60

70

80

90

100

110

120

tp [unit of time]

Fig. 9.45 Weibull distribution of ttf. Type I replacement model based on downtime minimization. Variable ˇ

130

140

150

160

170

180

190

200

358

9 Basic Models and Methods for Maintenance of Production Systems Type I Replacement Model - DownTime Minimization (Tf = 1) Weibull (a=50, b=1), Tp=0.5 Weibull (a=50, b=1), Tp=0.1 Weibull (a=50, b=3), Tp=0.5 Weibull (a=50, b=3), Tp=0.1 Weibull (a=50, b=0.2), Tp=0.5 Weibull (a=50, b=0.2), Tp=0.1

0.25

DT(tp)

0.2

0.15

0.1

0.05

0

0

10

20

30

40

50

60

70

80

90

100

110

120

130

140

150

160

170

180

190

200

t p [unit of time]

Fig. 9.46 Weibull distribution of ttf. Type I replacement model based on downtime minimization. Variables ˇ and Tp Preventive replacement Tp

Failure replacement

Failure replacement

Tf

Tf

Preventive replacement Tp

tp

time

Cycle length

Fig. 9.47 Cycle length. Downtime minimization, type II with Tp and Tf

DT(tp / to its minimum, as illustrated in Fig. 9.47. In particular, Fig. 9.47 illustrates the cycle length determination in the presence of fixed times of replacement Tp and Tf . failure replacement downtime C preventive replacement downtime DT.tp / D cycle length W .tp /Tf C Tp D : (9.43) tp C Tp 9.9.2.1 Type I – Minimum Downtime, Numerical Example Consider the numerical example introduced in Sect. 9.5.5. Table 9.13 summarizes the unit downtime values obtained by the application of Eq. 9.43, in accordance with the expected number of failures quantified by Eq. 9.33, the discrete approach.

Finally, Fig. 9.48 illustrates the trend of DT(Tp / for scenarios A, B, C, and D.

9.10 Group Replacement: The Lamp Replacement Problem Sometimes groups of similar items subject to failure (valves or filters in a piping system, lamps in a building or in a street, racks in a warehousing systems, etc.) are managed simultaneously in order to accomplish economies of scale. In such a situation, it could be useful to replace a generic item under group replacement conditions rather than replace only a single unit/entity. For example, it could be justifiable to replace all valves and filters of a piping system rather than only the failed ones. Replacing an item under group replacement, at the end of a fixed cycle length tp , is assumed to be less expensive than every failure replacement performed in

9.11 Preventive Maintenance Policies for Repairable Systems

359

Table 9.13 Analysis multiscenario. Downtime minimization, type II Type II – downtime minimization Id scenario Tp Tf 0.5 1 0.5 0.25

0.5 1 1 1

1

2

3

4

5

6

7

1.00 1.00 1.00 1.00

0.38 0.56 0.42 0.30

0.25 0.42 0.31 0.23

0.20 0.36 0.26 0.21

0.18 0.32 0.24 0.20

0.17 0.31 0.24 0.21

0.16 0.30 0.25 0.22

0.16 0.30 0.26 0.23

DT(Tp )

A B C D

tp 0

Fig. 9.48 Downtime minimization, type II

DT(Tp ) - Type II

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

A B C D 0

1

the course of tp ; in other words, the replacing cost per item, during a group replacement at tp , Cg is lower than the cost of failure replacement, during tp , Cf . Moreover, it is assumed that when an item fails in the course of tp that item is replaced by a new one before tp expires. The aim of the proposed model is to minimize the total expected cost of replacement per unit time UEC, defined as total expected cost in .0; tp / cycle length NCg C Cf ŒN W .tp / D ; (9.44) tp

UEC.tp / D

where N is the number of items in a group and W .t/ is the expected number of failures for one item. We now present a numerical example. Consider the application introduced in Sect. 9.5.5 and the definition of the probability distribution of the ttf f .t/ for the item subject to replacement. In particular, there is a group of 70 similar items subject to f .t/: It is possible to apply Eq. 9.44 assuming Cg and Cf are equal to ¤ 3,000 and ¤ 50,000 per replacement, respectively. Cg differs from Cp (equal to ¤ 5,000 per replacement) because the group replacement is performed in the presence of economies of scale. Figure 9.49 illustrates the values of the expected cost per

2

3

4

5

6

7

Tp

unit time obtained for different values of the period tp as follows: NCg C Cf ŒN W .tp / tp 70  3 C 50Œ70W .tp / D : tp

UEC.tp / D

Figure 9.49 shows the minimum of UEC for tp equal to 3 weeks, i. e., executing a block replacement on 70 items after three periods of time.

9.11 Preventive Maintenance Policies for Repairable Systems The analytical models proposed in this section assume, as a basic hypothesis, that the equipment, i. e., the production systems and components, is repairable and not as good as new immediately after the completion of the generic maintenance action (preventive or corrective). This is the reason why these models are not preventive replacement models, but are based on repair activity and/or replacement of a part of the whole system. In other words, the production system is subject to a continuous process of degradation and ageing.

360

9 Basic Models and Methods for Maintenance of Production Systems

Group replacement - UEC(tp)

800

UEC(t p) [k€/week]

750 700 650 600 550 500 1

Fig. 9.49 Group replacement. UEC(tp /

2

lim i .t/ D 1;

t !1

(9.45)

where i .t/ is the failure rate at time t (time from the last repair action) of the repairable component i in a system subjected to (i  1) repairs; •

i C1 .t/  i .t/; i C1 .0/  i .0/:

4

5

6

7

t p [week]

In particular, if the generic failure rate is not influenced and disturbed by any minimal repair of failures, we are in the presence of the so-called minimal repair action. It is also assumed that the state of the item is always known with certainty, in accordance with the adopted framework for the classification of maintenance strategies and the definition of preventive maintenance (see Sect. 9.2). It is therefore assumed that repair and/or replacement activities start immediately as soon as a failure occurs. In general (i. e., in the absence of the “as good as new hypothesis” and “minimal repair” actions), the life distribution of the equipment is assumed to change after each repair, i. e., the failure rate function increases after a generic maintenance action. We call this kind of repair activity a “not perfect or imperfect repair” action. As a consequence, the following properties follow: • i .t/ strictly increases,



3

t > 0;

(9.46)

From these assumptions and the property that the generic component degrades after the not perfect repair action, the following set of equations can be properly demonstrated:  MTTFi  MTTFi C1 ; (9.47) FNi .t/  FNi C1 .t/;

where MTTFi is the MTTF of the component subjected to (i  1) repairs and FNi .t/ is the survival function of the component at time t after the last (i  1) repair. The following sections present two analytical models for the determination of the best preventive policy for repairable systems subjected to replacement cycles (Nguyen and Murthy 1981) in accordance with the following considerations: 1. The replacement and repair costs of a failed component/system are generally greater than the replacement and repair costs of an entity that has not failed. 2. Continuing to repair a system is often costly compared with replacing it after a certain number of repairs.

9.11.1 Type I Policy for Repairable Systems The basic rule of this preventive maintenance model is to replace the component/system after (k  1) repairs. Considering an entity subjected to (i  1) repairs, that entity is repaired, or replaced if i D k, at the time of failure (breakdown action) or at the age Ti (preventive action) from the last repair or replacement. Figure 9.50 illustrates the replacement cycle with related costs, assumed to be constant. The notation adopted follows: Cr

Cp

is the replacement cost at the kth maintenance action. The replacement activity is coherent with the as good as new hypothesis. is the repair cost.

9.11 Preventive Maintenance Policies for Repairable Systems

361

Cp

Cp

Cp

Cp

Cr

Fi(Ti)Cf

Fi(Ti)Cf

Fi(Ti)Cf

Fi(Ti)Cf

Fi(Ti)Cf

i=1

i=2

i=3

T2

i = k-1

T3

Replacement with a new component

i=k

Tk

The same component “gets old”

Fig. 9.50 Type I policy, repairable systems. Replacement cycle and costs

The component is repaired after a failure Cp

Cp

Fi(Ti)Cf

Fi(Ti)Cf

i=1

i=2

C p + Cf

Cp

Cr + Cf

Replacement with a new component

Fi(Ti)Cf i=3

i = k-1

i=k

Tk

T2 T3 The same component “gets old”

Fig. 9.51 Type I policy, repairable systems. Example

Cf

is the breakdown cost; it is a cost additional to Cp . As a consequence, the generic corrective action, which follows a breakdown event, costs Cp C Cf :

This policy is characterized by k and fTi g variables, whose values have to be properly identified, and where fTi g denotes the set of maintenance ages T1 ; T2 ; : : : ; Tk . Figure 9.51 exemplifies a replacement cycle for a component which fails after a second preventive repair (i D 2) before waiting T3 and the third planned preventive action. The expected costs of a repair Cp .Ti / and of a replacement Cr .Ti / are, respectively, Cp .Ti / D .Cp C Cf /Fi .Ti / C Cp Œ1  Fi .Ti / D Cp C Cf Fi .Ti /

Applying this rule, UEC is UECŒk; T1 ; : : : ; Tk  D

P .k  1/Cp C Cr C Cf kiD1 Fi .Ti / ; D Pk R Ti FNi .t/ dt

where ECŒk; fTi g is the expected cost for a replacement cycle and LŒk; fTi g is the expected length of a replacement cycle. The optimal policy is to select k and maintenance ages fTi g so as to minimize Eq. 9.50. Differentiating Eq. 9.47 with respect to Ti and equating to zero, ri .Ti / D

Cr .Ti / D .Cr C Cf /Fi .Ti / C Cr Œ1  Fi .Ti / D Cr C Cf Fi .Ti /:

(9.49)

(9.50)

i D1 0

(9.48)

and

ECŒk; fTi g LŒk; fTi g

UECŒk; fTi g ; Cf

(9.51)

where ri .Ti / is failure rate of the component at time Ti after the last repair, called the “(i  1)th repair,”

362

and

9 Basic Models and Methods for Maintenance of Production Systems

8 ˆ r .T / D r1 .T1 /; 1 < i  k ˆ ˆ i i ˆ ˆ ˆ ˆ k  ZTi <  X r1 .T1 / FNi .t/ dt  Fi .Ti / ˆ ˆ ˆ i D1 ˆ 0 ˆ ˆ ˆ : Œ.k1/Cp CCr  D : Cf

and Zz

eu ux1 du:

.x; z/ D 0

(9.52)

Now we demonstrate that ZTi

The generic i th cycle is based on the set of functions fi .t/, ri .t/, etc. defined for the random variable ttfi . Consequently, the failure rate ri .t/ can be assumed to be equal to i .t/, where t is the point in time from the last maintenance action. The availability of the system is Pk R T i N i D1 0 Fi .t/ dt AŒk; fTi g D : P .k  1/Cp C Cr C Cf kiD1 Fi .Ti / RT P C kiD1 0 i FNi .t/ dt (9.53)

    1 Ti b N C Ti FN .Ti /; F .t/ dt D a 1 C ; b a

0

(9.57) where     1 Ti b a 1 C ; b a

T b i ! Za 1=b y Da y e dy 0

T b i Za

yD

The problem of maximizing AŒk; fTi g is equivalent to the problem of minimizing UECŒk; T1 ; : : : ; Tk . The following algorithm can be applied to compute optimal policy I, as demonstrated by Nguyen and Murthy (1981):

5. k  D k  1. Compute UECŒk; fTi g: Now consider the case of a Weibull distribution,     t ˇi ; Fi .t/ D 1  exp  ˛i (9.54) ˇi  t ˇi 1 ri .t/ D i .t/ D : ˛i ˛i RT In Eq. 9.50, in order to compute 0 i FNi .t/ dt when f .t/ is represented by a Weibull density function, it is useful to quantify the lower incomplete function .x; z/. Given a Weibull distribution (a scale parameter and b shape parameter), we know that

0

    t b exp  dt a

!

xDT Z i

D

b1 ab

a dx

x

x . x /b x b1 e a b b dx a a

b  x b . x /b e a dx a a

!

!

0

4. Set k D k C 1 and go to step 2.

ZTi

x . x /b e a dy a

yD0

Da

3. If T1 .k/  T1 .k  1/, go to step 5.

0

a

ZTi

2. Solve Eq. 9.52 for fTi .k/g.

FNi .t/ dt D

D

b yD. x a/

x b dyD dŒ. x a / Db

1. Set k= 1.

ZTi

(9.56)

ZTi  b x b x e. a / dx D b a 0

D

x b

b x b1 . a / f .x/D a .a/ e

ZTi  b1 b x x b e. a / x dx a a 0

ZTi D

xf .x/ dx 0

D

Eq. 5.38

Ti FN .Ti / C

ZTi

FN .x/ dx:

0

9.11.1.1 Numerical Example (9.55)

Consider a component subject to preventive maintenance type I general actions, in accordance with

9.11 Preventive Maintenance Policies for Repairable Systems

363

Table 9.14 T1 determination, k D 1 and ˇ D 2 T1

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4

F1 .T1 /

r1 .T1 /

1 C 1=ˇ

.T1 =˛/ˇ

A1 D

Œ1 C 1=ˇ I .T1 =˛/ˇ 

˛A1

0.002 0.010 0.022 0.039 0.061 0.086 0.115 0.148 0.183 0.221 0.261 0.302 0.345 0.387 0.430 0.473 0.514 0.555 0.594 0.632 0.668 0.702 0.734 0.763 0.790 0.815 0.838 0.859

0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.800 0.900 1.000 1.100 1.200 1.300 1.400 1.500 1.600 1.700 1.800 1.900 2.000 2.100 2.200 2.300 2.400 2.500 2.600 2.700 2.800

1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500

0.003 0.010 0.023 0.040 0.063 0.090 0.123 0.160 0.203 0.250 0.303 0.360 0.423 0.490 0.563 0.640 0.723 0.810 0.903 1.000 1.103 1.210 1.323 1.440 1.563 1.690 1.823 1.960

0.000 0.001 0.002 0.005 0.010 0.017 0.027 0.039 0.054 0.072 0.093 0.117 0.143 0.172 0.203 0.236 0.270 0.306 0.342 0.379 0.416 0.452 0.488 0.522 0.556 0.588 0.618 0.647

0.000 0.001 0.002 0.005 0.010 0.017 0.027 0.039 0.054 0.072 0.093 0.117 0.143 0.172 0.203 0.236 0.270 0.306 0.342 0.379 0.416 0.452 0.488 0.522 0.556 0.588 0.618 0.647

the previously illustrated model. We assume Cp D ¤ 5,000 per action, Cr D ¤ 15,000 per action, and Cf D ¤15,000 per action. The failure probability function, the rate function, and the scale parameter of the generic Weibull density function are defined as follows:     t ˇ ; Fi .t/ D 1  exp  ˛i ri .t/ D i .t/ D

ˇ  t ˇ 1 ; ˛i ˛i

˛i D .1:5/1i : The algorithm illustrated above and introduced by Nguyen and Murthy (1981) is applied to find the best .T /i values. In the first iteration, when k D 1, the value of ˛ is 1. Table 9.14 presents the calculus to

A2 D ˛A1 C T1 Œ1  F .T1 /

B1 D r1 .T1 /A2  F1 .T1 /  Cr =Cf

0.050 0.100 0.149 0.197 0.245 0.291 0.336 0.380 0.421 0.461 0.499 0.535 0.569 0.601 0.630 0.658 0.683 0.706 0.727 0.747 0.764 0.780 0.794 0.807 0.818 0.828 0.836 0.844

0:998 0:990 0:978 0:960 0:938 0:911 0:880 0:844 0:804 0:760 0:712 0:660 0:605 0:546 0:485 0:420 0:353 0:284 0:212 0:138 0:063 0:014 0.093 0.173 0.254 0.337 0.420 0.504

quantify T1 in the case ˇ D 2: ZT1 1 .T1 /

Cr FN1 .t/ dt  F1 .T1 /  D 0; Cf

0

where ZT1

    1 T1 ˇ C T1 FN .T1 /: FN .T1 / dt D ˛ 1 C ; ˇ ˛

0

In particular, T1 .k D 1/ 2 .1:05; 1:1/; r1 .T1 / 2 .2:1; 2:2/: This calculus was implemented in a spreadsheet in order to demonstrate that no particular informatics skills are required, and practitioners or managers can apply

364

9 Basic Models and Methods for Maintenance of Production Systems

the proposed model, even if it can appear very complicated. The value of T1 , which sets the B1 values to zero in the last column of Table 9.14, is the best T1 assuming k D 1. Before quantifying UEC, it is useful to quantify A2 as (see Table 9.14) ZT1 A2 .k D 1; T1 D 1:05/ D 



FN .t/ dt

For this purpose we propose the use of a new spreadsheet, reported in Table 9.15, which refers to Table 9.16 for the explanation of the symbols. Then, ˇ  ˇ 1 ˇ ˇ ˇˇ t 2t ˇ D 2 2 .t/ D ˇˇ ˇ ˛2 ˛2 ˛2 ˇ D2 and

0

1 T1 D a 1 C ; ˇ ˛

ˇ  ZT1

A2 .k D 1; T1 D 1:1/ D

T2 D

C T1 FN .T1 / D 0:764; FN .t/ dt D 0:780:

2 .t/ 2 ˛ 2 2

D

1 .t /D2 .t /

The last column in Table 9.15 reports the values of the following equation, called “B2”: ZT1

0

B2.T1 ; T2 / D 1 .T1 /

The value of UEC obtained, assuming T1 D 1:05 as a lower bound of T1 , is UECŒk D 1; T1 D 1:05 D

ZT2 C 1 .T1 /

.k  1/Cp C Cr C Cf i D1 Fi .Ti / Pk R Ti FNi .t/ dt

Š

FN2 .t/ dt  F2 .T2 /

0

Cp C Cr :  Cf

i D1 0

Table 9.14

FN1 .t/ dt  F1 .T1 /

0

Pk

15 C 15  0:668 Cr C Cf F1 .T1 / D D RT 1 N 0:764 0 Fi .t/ dt

1 .t/ 2 ˛ : 2 2

Equation B2 is equal to 0, in accordance with Eq. 9.52, when (see also Fig. 9.52)

¤ 32;740 per unit of time.

T1 .k D 2/ 2 .1; 1:05/; T2 .k D 2/ 2 .0:444; 0:467/;

When T1 is adopted for its upper bound T1 D 1:1,

r1 .T1 / D r2 .T2 / 2 .2:0; 2:1/:

UECŒk D 1; T1 D 1:1

P .k  1/Cp C Cr C Cf kiD1 Fi .Ti / D Pk R Ti FNi .t/ dt

Now

15 C 15  0:702 Cr C Cf F1 .T1 / D D RT 1 N 0:780 0 Fi .t/ dt

Consequently, values of unit cost lower than UEC.k D 1/ previously quantified are expected. Before the illustration of the calculus of the UEC we explicitly quantify the following values in accordance with Table 9.15:

i D1 0

Š

Table 9.14

¤ 32;730 per unit of time.

For k D 2 it is necessary to solve the following set of equations: 8 ˆ 1 .T1 / D 2 .T2 / ˆ ˆ ˆ ˆ ˆ ˆ T1 .k D 2/:

ZT1 b2.k D 2; T1 D 1/ D

FN .t/ dt

0

    1 T1 ˇ D a 1 C ; C T1 FN .T1 / D 0:747; ˇ ˛ ZT1 b2.k D 2; T1 D 1:05/ D FN .t/ dt D 0:764; 0

0.022 0.044 0.067 0.089 0.111 0.133 0.156 0.178 0.200 0.222 0.244 0.267 0.289 0.311 0.333 0.356 0.378 0.400 0.422 0.444 0.467 0.489 0.511 0.533 0.556 0.578 0.600 0.622 0.644 0.667 0.689 0.711 0.733 0.756 0.778 0.800 0.822 0.844 0.867 0.889 0.911

0.002 0.010 0.022 0.039 0.061 0.086 0.115 0.148 0.183 0.221 0.261 0.302 0.345 0.387 0.430 0.473 0.514 0.555 0.594 0.632 0.668 0.702 0.734 0.763 0.790 0.815 0.838 0.859 0.878 0.895 0.910 0.923 0.934 0.944 0.953 0.961 0.967 0.973 0.978 0.982 0.985

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5 1.55 1.6 1.65 1.7 1.75 1.8 1.85 1.9 1.95 2 2.05

0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.800 0.900 1.000 1.100 1.200 1.300 1.400 1.500 1.600 1.700 1.800 1.900 2.000 2.100 2.200 2.300 2.400 2.500 2.600 2.700 2.800 2.900 3.000 3.100 3.200 3.300 3.400 3.500 3.600 3.700 3.800 3.900 4.000 4.100

F1 .T1 / r1 .T1 / D r2 .T2 / T2

T1 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500

0.003 0.010 0.023 0.040 0.063 0.090 0.123 0.160 0.203 0.250 0.303 0.360 0.423 0.490 0.563 0.640 0.723 0.810 0.903 1.000 1.103 1.210 1.323 1.440 1.563 1.690 1.823 1.960 2.103 2.250 2.403 2.560 2.723 2.890 3.063 3.240 3.423 3.610 3.803 4.000 4.203

0.000 0.001 0.002 0.005 0.010 0.017 0.027 0.039 0.054 0.072 0.093 0.117 0.143 0.172 0.203 0.236 0.270 0.306 0.342 0.379 0.416 0.452 0.488 0.522 0.556 0.588 0.618 0.647 0.673 0.698 0.721 0.742 0.760 0.777 0.793 0.806 0.818 0.828 0.838 0.845 0.852

0.000 0.001 0.002 0.005 0.010 0.017 0.027 0.039 0.054 0.072 0.093 0.117 0.143 0.172 0.203 0.236 0.270 0.306 0.342 0.379 0.416 0.452 0.488 0.522 0.556 0.588 0.618 0.647 0.673 0.698 0.721 0.742 0.760 0.777 0.793 0.806 0.818 0.828 0.838 0.845 0.852

1 C 1=ˇ .T1 =˛1 /ˇ Œ1 C 1=ˇ I b1 .T1 =˛1 /ˇ  0.050 0.100 0.149 0.197 0.245 0.291 0.336 0.380 0.421 0.461 0.499 0.535 0.569 0.601 0.630 0.658 0.683 0.706 0.727 0.747 0.764 0.780 0.794 0.807 0.818 0.828 0.836 0.844 0.851 0.856 0.861 0.865 0.869 0.872 0.874 0.877 0.878 0.880 0.881 0.882 0.883

b2 0:998 0:990 0:978 0:960 0:938 0:911 0:880 0:844 0:804 0:760 0:712 0:660 0:605 0:546 0:485 0:420 0:353 0:284 0:212 0:138 0:063 0.014 0.093 0.173 0.254 0.337 0.420 0.504 0.589 0.674 0.760 0.846 0.933 1.020 1.107 1.195 1.283 1.370 1.458 1.547 1.635

b3 0.001 0.004 0.010 0.018 0.028 0.040 0.054 0.071 0.090 0.111 0.134 0.160 0.188 0.218 0.250 0.284 0.321 0.360 0.401 0.444 0.490 0.538 0.588 0.640 0.694 0.751 0.810 0.871 0.934 1.000 1.068 1.138 1.210 1.284 1.361 1.440 1.521 1.604 1.690 1.778 1.868

0.000 0.000 0.001 0.002 0.003 0.005 0.008 0.012 0.017 0.023 0.030 0.039 0.049 0.060 0.072 0.086 0.100 0.117 0.134 0.152 0.172 0.192 0.214 0.236 0.259 0.282 0.306 0.330 0.354 0.379 0.403 0.428 0.452 0.476 0.499 0.522 0.545 0.567 0.588 0.608 0.628

0.000 0.000 0.000 0.001 0.002 0.003 0.005 0.008 0.011 0.015 0.020 0.026 0.032 0.040 0.048 0.057 0.067 0.078 0.089 0.102 0.115 0.128 0.142 0.157 0.172 0.188 0.204 0.220 0.236 0.253 0.269 0.285 0.301 0.317 0.333 0.348 0.363 0.378 0.392 0.406 0.419

.T2 =˛1 /ˇ Œ1 C 1=ˇ I b4 .T2 =˛2 /ˇ 

Table 9.15 T1 and T2 determination, k D 2 and ˇ D 2 (the explanation of the symbols is given in Table 9.16)

0.022 0.044 0.066 0.088 0.110 0.132 0.153 0.174 0.194 0.214 0.234 0.253 0.272 0.290 0.308 0.325 0.341 0.357 0.372 0.387 0.400 0.414 0.426 0.438 0.450 0.461 0.471 0.480 0.489 0.498 0.506 0.513 0.520 0.526 0.532 0.538 0.543 0.548 0.552 0.556 0.559

b5 0.001 0.004 0.010 0.018 0.027 0.039 0.053 0.069 0.086 0.105 0.126 0.148 0.171 0.196 0.221 0.248 0.275 0.302 0.330 0.359 0.387 0.416 0.444 0.473 0.501 0.528 0.555 0.582 0.607 0.632 0.656 0.679 0.702 0.723 0.744 0.763 0.782 0.799 0.815 0.831 0.846

F2 .T2 / 1:330 1:319 1:301 1:276 1:244 1:205 1:159 1:107 1:049 0:984 0:914 0:838 0:756 0:670 0:578 0:482 0:382 0:277 0:169 0:058 0:057 0.175 0.296 0.419 0.545 0.673 0.803 0.934 1.067 1.202 1.338 1.475 1.614 1.753 1.893 2.035 2.176 2.319 2.462 2.605 2.749

B2

9.11 Preventive Maintenance Policies for Repairable Systems 365

366

9 Basic Models and Methods for Maintenance of Production Systems

Table 9.16 Table 9.15

Explanation of symbols b1, b2, and b3 in

b1

˛1 Œ1 C 1=ˇ I .T1 =˛1 /ˇ 

b2

b1 C T1 Œ1  F .T1 /

b3

r1 .T1 /b2  F1 .T1 /  Cr =Cf

b4

˛2 Œ1 C 1=ˇ I .T2 =˛2 /ˇ 

b5

b4 C T2 Œ1  F .T2 /

D

5 C 15 C 15  0:632 C 15  0:359 0:747 C 0:387

D ¤ 30,745 per unit of time. If the upper bounds values of time T1 D 1:05 and T2 D 0:668 are assumed, UECŒk D 2; T1 D 1:05; T2 D 0:668 P .k  1/Cp C Cr C Cf kiD1 Fi .Ti / D Pk R T i FNi .t/ dt

B2(T1,T2)

i D1 0

5.000

D

4.000

Table 9.15 3.000

D B2

2.000

Cp C Cr C Cf F1 .T1 / C Cf F2 .T2 / R T1 RT FN1 .t/ dt C 2 FN2 .t/ dt 0

0

5 C 15 C 15  0:668  15  0:387 0:764 C 0:400

Š ¤ 30,777 per unit of time.

1.000

0.000 0

0.2

0.4

0.6 0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

-1.000

-2.000 T1

Fig. 9.52 B2.T1 ; T2 /, numerical example

and ZT2 b5.k D 2; T2 D 0:444/ D 



1 T2 D a 1 C ; ˇ ˛

FN .t/ dt

0

ˇ 

1

C T2 FN .T2 / D 0:387;

ZT2 b5.k D 2; T2 D 0:467/ D

As expected these values of unit cost are lower than UEC.k D 1/, and a further iteration of the proposed algorithm is performed as follows (see k D k C 1 D 3). For k D 3 it is necessary to solve the following: 8 ˆ ˆ ˆ1 .T1 / D 2 .T2 / D 3 .T3 / ˆ ˆ ˆ ˆ .T / R T1 FN .t/ dt  F .T / ˆ ˆ 1 1 0 1 1 1 ˆ ˆ < R T2 C1 .T1 / 0 FN2 .t/ dt  F2 .T2 / ˆ ˆ ˆ RT ˆ 2Cp C Cr ˆ ˆ C1 .T1 /  0 1 FN3 .t/ dt  F3 .T1 / D ˆ ˆ Cf ˆ ˆ ˆ :˛ D 1; ˛ D .1:5/1 ; ˛ D .1:5/2 :

FN .t/ dt D 0:400:

2

3

Then, ˇ  ˇ 1 ˇ ˇ ˇ ˇ T3 2T3 ˇ D 2 : 3 .T3 / D 1 .T1 / D ˇˇ ˇ ˛3 ˛3 ˛3 ˇ D2

0

The UEC assuming the lower bounds values of time T1 D 1 and T2 D 0:444 is therefore UECŒk D 2; T1 D 1; T2 D 0:444 P .k  1/Cp C Cr C Cf kiD1 Fi .Ti / D Pk R T i N i D1 0 Fi .t/ dt Š

Table 9.15

Cp C Cr C Cf F1 .T1 / C Cf F2 .T2 / R T1 RT FN1 .t/ dt C 2 FN2 .t/ dt 0

0

Table 9.17 presents the spreadsheet used to support the algorithm calculus for k D 3, adopting the explanation of symbols in Tables 9.16 and 9.18. The values of time obtained are T1 .k D 3/ 2 .1:05; 1:1/; T2 .k D 3/ 2 .0:467; 0:489/; T3 .k D 3/ 2 .0:207; 0:217/; r1 .T1 / D r2 .T2 / D r3 .T3 / 2 .2:1; 2:2/:

0.002 0.010 0.022 0.039 0.061 0.086 0.115 0.148 0.183 0.221 0.261 0.302 0.345 0.387 0.430 0.473 0.514 0.555 0.594 0.632 0.668 0.702 0.734 0.763 0.790 0.815 0.838 0.859 0.878 0.895 0.910 0.923 0.934 0.944 0.953 0.961 0.967 0.973 0.978 0.982 0.985

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5 1.55 1.6 1.65 1.7 1.75 1.8 1.85 1.9 1.95 2 2.05

0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.800 0.900 1.000 1.100 1.200 1.300 1.400 1.500 1.600 1.700 1.800 1.900 2.000 2.100 2.200 2.300 2.400 2.500 2.600 2.700 2.800 2.900 3.000 3.100 3.200 3.300 3.400 3.500 3.600 3.700 3.800 3.900 4.000 4.100

F1 .T1 / r1 .T1 /

T1

0.022 0.044 0.067 0.089 0.111 0.133 0.156 0.178 0.200 0.222 0.244 0.267 0.289 0.311 0.333 0.356 0.378 0.400 0.422 0.444 0.467 0.489 0.511 0.533 0.556 0.578 0.600 0.622 0.644 0.667 0.689 0.711 0.733 0.756 0.778 0.800 0.822 0.844 0.867 0.889 0.911

T2

0.010 0.020 0.030 0.040 0.049 0.059 0.069 0.079 0.089 0.099 0.109 0.119 0.128 0.138 0.148 0.158 0.168 0.178 0.188 0.198 0.207 0.217 0.227 0.237 0.247 0.257 0.267 0.277 0.286 0.296 0.306 0.316 0.326 0.336 0.346 0.356 0.365 0.375 0.385 0.395 0.405

T3 0.003 0.010 0.023 0.040 0.063 0.090 0.123 0.160 0.203 0.250 0.303 0.360 0.423 0.490 0.563 0.640 0.723 0.810 0.903 1.000 1.103 1.210 1.323 1.440 1.563 1.690 1.823 1.960 2.103 2.250 2.403 2.560 2.723 2.890 3.063 3.240 3.423 3.610 3.803 4.000 4.203

0.000 0.001 0.002 0.005 0.010 0.017 0.027 0.039 0.054 0.072 0.093 0.117 0.143 0.172 0.203 0.236 0.270 0.306 0.342 0.379 0.416 0.452 0.488 0.522 0.556 0.588 0.618 0.647 0.673 0.698 0.721 0.742 0.760 0.777 0.793 0.806 0.818 0.828 0.838 0.845 0.852

.T1 =˛1 /ˇ Œ1 C 1=ˇ I .T1 =˛1 /ˇ  0.000 0.001 0.002 0.005 0.010 0.017 0.027 0.039 0.054 0.072 0.093 0.117 0.143 0.172 0.203 0.236 0.270 0.306 0.342 0.379 0.416 0.452 0.488 0.522 0.556 0.588 0.618 0.647 0.673 0.698 0.721 0.742 0.760 0.777 0.793 0.806 0.818 0.828 0.838 0.845 0.852

b1 0.050 0.100 0.149 0.197 0.245 0.291 0.336 0.380 0.421 0.461 0.499 0.535 0.569 0.601 0.630 0.658 0.683 0.706 0.727 0.747 0.764 0.780 0.794 0.807 0.818 0.828 0.836 0.844 0.851 0.856 0.861 0.865 0.869 0.872 0.874 0.877 0.878 0.880 0.881 0.882 0.883

b2 0:998 0:990 0:978 0:960 0:938 0:911 0:880 0:844 0:804 0:760 0:712 0:660 0:605 0:546 0:485 0:420 0:353 0:284 0:212 0:138 0:063 0.014 0.093 0.173 0.254 0.337 0.420 0.504 0.589 0.674 0.760 0.846 0.933 1.020 1.107 1.195 1.283 1.370 1.458 1.547 1.635

b3 0.001 0.004 0.010 0.018 0.028 0.040 0.054 0.071 0.090 0.111 0.134 0.160 0.188 0.218 0.250 0.284 0.321 0.360 0.401 0.444 0.490 0.538 0.588 0.640 0.694 0.751 0.810 0.871 0.934 1.000 1.068 1.138 1.210 1.284 1.361 1.440 1.521 1.604 1.690 1.778 1.868

0.000 0.000 0.001 0.002 0.003 0.005 0.008 0.012 0.017 0.023 0.030 0.039 0.049 0.060 0.072 0.086 0.100 0.117 0.134 0.152 0.172 0.192 0.214 0.236 0.259 0.282 0.306 0.330 0.354 0.379 0.403 0.428 0.452 0.476 0.499 0.522 0.545 0.567 0.588 0.608 0.628

.T2 =˛1 /ˇ Œ1 C 1=ˇ I .T2 =˛2 /ˇ  0.000 0.000 0.000 0.001 0.002 0.003 0.005 0.008 0.011 0.015 0.020 0.026 0.032 0.040 0.048 0.057 0.067 0.078 0.089 0.102 0.115 0.128 0.142 0.157 0.172 0.188 0.204 0.220 0.236 0.253 0.269 0.285 0.301 0.317 0.333 0.348 0.363 0.378 0.392 0.406 0.419

b4 0.022 0.044 0.066 0.088 0.110 0.132 0.153 0.174 0.194 0.214 0.234 0.253 0.272 0.290 0.308 0.325 0.341 0.357 0.372 0.387 0.400 0.414 0.426 0.438 0.450 0.461 0.471 0.480 0.489 0.498 0.506 0.513 0.520 0.526 0.532 0.538 0.543 0.548 0.552 0.556 0.559

b5 0.001 0.004 0.010 0.018 0.027 0.039 0.053 0.069 0.086 0.105 0.126 0.148 0.171 0.196 0.221 0.248 0.275 0.302 0.330 0.359 0.387 0.416 0.444 0.473 0.501 0.528 0.555 0.582 0.607 0.632 0.656 0.679 0.702 0.723 0.744 0.763 0.782 0.799 0.815 0.831 0.846

0.000 0.002 0.004 0.008 0.012 0.018 0.024 0.032 0.040 0.049 0.060 0.071 0.083 0.097 0.111 0.126 0.143 0.160 0.178 0.198 0.218 0.239 0.261 0.284 0.309 0.334 0.360 0.387 0.415 0.444 0.475 0.506 0.538 0.571 0.605 0.640 0.676 0.713 0.751 0.790 0.830

0.000 0.000 0.000 0.000 0.001 0.002 0.002 0.004 0.005 0.007 0.009 0.012 0.015 0.019 0.023 0.028 0.033 0.039 0.045 0.052 0.060 0.068 0.076 0.086 0.095 0.106 0.117 0.128 0.140 0.152 0.165 0.179 0.192 0.206 0.221 0.236 0.251 0.266 0.282 0.298 0.314

0.000 0.000 0.000 0.000 0.000 0.001 0.001 0.002 0.002 0.003 0.004 0.005 0.007 0.008 0.010 0.012 0.015 0.017 0.020 0.023 0.026 0.030 0.034 0.038 0.042 0.047 0.052 0.057 0.062 0.068 0.073 0.079 0.085 0.092 0.098 0.105 0.112 0.118 0.125 0.132 0.140

F2 .T2 / .T3 =˛3 /ˇ Œ1 C 1=ˇ I b6 .T3 =˛3 /ˇ 

Table 9.17 T1 , T2 , and T3 determination, k D 3 and ˇ D 2 (the explanation of the symbols is given in Table 9.18)

0.010 0.020 0.030 0.039 0.049 0.059 0.069 0.078 0.088 0.097 0.107 0.116 0.125 0.134 0.143 0.152 0.160 0.169 0.177 0.185 0.193 0.201 0.209 0.216 0.224 0.231 0.238 0.245 0.251 0.258 0.264 0.270 0.276 0.282 0.287 0.292 0.297 0.302 0.307 0.312 0.316

b7 0.000 0.002 0.004 0.008 0.012 0.018 0.024 0.031 0.039 0.048 0.058 0.069 0.080 0.092 0.105 0.119 0.133 0.148 0.163 0.179 0.196 0.213 0.230 0.248 0.266 0.284 0.302 0.321 0.340 0.359 0.378 0.397 0.416 0.435 0.454 0.473 0.491 0.510 0.528 0.546 0.564

F3 .T3 /

1:663 1:650 1:630 1:601 1:565 1:521 1:469 1:409 1:342 1:269 1:188 1:101 1:007 0:908 0:802 0:692 0:576 0:455 0:329 0:200 0:066 0.072 0.213 0.358 0.505 0.656 0.809 0.965 1.123 1.283 1.445 1.609 1.775 1.942 2.111 2.281 2.452 2.624 2.798 2.972 3.148

B3

9.11 Preventive Maintenance Policies for Repairable Systems 367

368

9 Basic Models and Methods for Maintenance of Production Systems Table 9.18 Explanation of symbols b6 and b7 in Table 9.17

Now, T1 .k D 3/  T1 .k D 2/:

b3

˛3 Œ1 C 1=ˇ I .T3 =˛3 /ˇ 

b7

b6 C T3 Œ1  F .T3 /



Consequently, the best value of k, k , is 2 and for k D 3 values of unit cost UEC.k D 3/ greater than UEC.k D 2/ previously quantified are expected. Before the illustration of the calculus of UEC.k D 3/ we explicitly quantify the following values in accordance with Table 9.17: ZT1 b2.k D 3; T1 D 1:05/ D  D a 1 C



1 T1 ; ˇ ˛

2Cp C Cr C Cf F1 .T1 / C Cf F2 .T2 / C Cf F3 .T3 / R T1 R T2 R T3 Table 9.17 N N N 0 F1 .t/ dt C 0 F2 .t/ dt C 0 F3 .t/ dt Š

10 C 15 C 15  0:668 C 15  0:387 C 15  0:196 0:764 C 0:400 C 0:193 D ¤32,251 per unit of time.

D FN .t/ dt

0

ˇ  ZT1

b2.k D 3; T1 D 1:1/ D

C T1 FN .T1 / D 0:764;

FN .t/ dt D 0:780;

0

then ZT2 b5.k D 3; T2 D 0:444/ D

Now, assuming the upper bounds values, T1 D 1:1, T2 D 0:489, and T3 D 0:217, the unit cost is UECŒk D 3; T1 D 1:1; T2 D 0:489; T3 D 0:217 P .k  1/Cp C Cr C Cf kiD1 Fi .Ti / D Pk R T i N i D1 0 Fi .t/ dt Š

FN .t/ dt

Table 9.17

10 C 15 C 15  0:702 C 15  0:416 C 15  0:213 0:780 C 0:414 C 0:201 D ¤32,233 per unit of time.

D

0

    1 T2 ˇ C T1 FN .T1 / D 0:4; D a 1 C ; ˇ ˛ ZT2 b5.k D 3; T1 D 0:467/ D FN .t/ dt D 0:414; 0

and finally ZT3 b7.k D 3; T3 D 0:207/ D

FN .t/ dt

0

    1 T2 ˇ D a 1 C ; C T1 FN .T1 / D 0:193; ˇ ˛ ZT3 b7.k D 3; T3 D 0:217/ D FN .t/ dt D 0:201: 0

UEC.k D 3/ assuming T1 D 1:05, T2 D 0:467, and T3 D 0:207 is therefore UECŒk D 3; T1 D 1:05; T2 D 0:467; T3 D 0:207 P .k  1/Cp C Cr C Cf kiD1 Fi .Ti / D Pk R Ti N i D1 0 Fi .t/ dt

2Cp C Cr C Cf F1 .T1 / C Cf F2 .T2 / C Cf F3 .T3 / R T1 R T2 R T3 N N N 0 F1 .t/ dt C 0 F2 .t/ dt C 0 F3 .t/ dt

Figure 9.53 illustrates the trend of the expected lower and upper bounds of UEC for different values of k, further demonstrating that the best value is k  D 2. From Eqs. 9.46 and 9.52, and from the assumed values of the scale parameter of the Weibull function ai , the trend of the failure rate increases on passing from the generic period i to i +1, as properly illustrated in Fig. 9.54 when k= 3 and i assumes the values f1; 2; 3g. Figure 9.55 presents the sawtooth trend of the failure rate r.t/ [i. e., .t/] for the component/system during the time period made up of three identical replacement cycles16 , each made up of three periods of duration T1 , T2 , and T3 . The results obtained are in accordance with the as good as new hypothesis adopted and with Eq. 9.52, where 8 ˆ

G; D.t/  S P0 .t/ D P Y .t/ D W A C eBt # "    ˇt  t 1 1  uM 1 ˇt .a / D a uM t ˇ  e2 t

1 X .2 t/j

jŠ

j D0

.j /

FX .S /:

(10.20)

The probability for a catastrophic failure state F is given by  eBt PF .t/ D P Y .t/ D W  G; D.t/ > S A C eBt   ˇ    ˇ t 1 1  u1 t .a1 t / D 1 a u1 t ˇ   1 X .2 t/j .j / 2 t FX .S / (10.21)  1 e jŠ j D0

Hence, the reliability RM .t/ is given by RM .t/ D

M X

Pk .t/

kD1

"

#  ˇ   ˇ 1 1  uM t t .a1 t / D 1 a uM a t ˇ   1 X .2 t/j .j / 2 t FX .S / : (10.22)  e jŠ j D0

10.4 Modeling of Inspection-Maintenance Repairable Degraded Systems The system is assumed to be periodically inspected at times fI; 2I; : : : ; nI; : : : g and the state of the system can only be detected by inspection. After a preventive maintenance or corrective maintenance action the system will be restored to the as-good-as-new state. Assume that the degradation fY .t/gt 0 and random shock fD.t/gt 0 are independent, and a corrective maintenance action is more costly than a preventive maintenance and a preventive maintenance costs much more than an inspection. In other words, Cc > Cp > Ci . From Sect. 10.3, T is defined as the time to failure T D infft > 0 W Y .t/ > G or D.t/ > S g, where G is the critical value for fY .t/gt 0 and S is the threshold level for fD.t/gt 0 . The material in this section is mostly based on the study conducted by Li and Pham (2005). The two threshold values L and G (G is fixed) effectively divide the system state into three zones as shown in Fig. 10.3. They are as follows: doing nothing zone when Y .t/  L and D.t/  S ; preventive maintenance zone when L < Y .t/  G and D.t/  S ; and corrective maintenance zone when Y .t/ > G and D.t/ > S . The maintenance action will be performed when either of the following situations occurs: 1. The current inspection reveals that the system condition falls into the preventive maintenance zone; however, this state is not found at the previous inspection. At the inspection time iI , the system falls into the preventive maintenance zone, which means fY ..i 1/I /  L, D..i 1/I /  S g\fL < Y .iI /  G, D.iI /  S g. Then preventive maintenance action is performed and it will take a random time R1 .

10.4 Modeling of Inspection-Maintenance Repairable Degraded Systems

403

Y(t)

CM zone

G PM zone

L

Doing nothing zone

D(t)

S Fig. 10.3 The evolution of the system. CM corrective maintenance, PM preventative maintenance. (Li and Pham 2005)

I1 L

I 1 ... I i I i +1 R1 W1

2. When the system fails at T , a corrective maintenance action is taken immediately and would take a random time R2 .

I i T R2 W2

I1 L I i T W3

R2

10.4.1 Calculate EŒNI  Let EŒNI  denote the expected number of inspections during a cycle. Then,

Note that after a preventive maintenance or a corrective maintenance action has been performed, the system is renewed and the cycle ends. From a concept of renewal reward theory, the average long-run maintenance cost per unit time is given by EC.L; I / D

EŒC1  : EŒW1 

(10.23)

EŒNI  D

Note that there is a probability Pp that the cycle will end as a result of a preventive maintenance action and it will take on average EŒR1  amount of time to complete a preventive maintenance action with a corresponding cost Cp EŒR1 Pp . Similarly, if a cycle ends as a result of a corrective maintenance action with probability P , it will take on average cEŒR2  amount of time to complete a corrective maintenance action with corresponding cost Cc EŒR2 Pc . We next discuss the analytical analysis of EŒC1 .

.i /P fNI D i g;

(10.25)

i D1

where P fNI D i g is the probability that there are a total of i inspections in a renewal cycle. It can be shown that P fNI D i g D P fY Œ.i  1/I   L; DŒ.i  1/I   S g  P fL < Y .iI /  G; D.iI /  S g C P fY .iI /  L; D.iI /  S g

The expected total maintenance cost during a cycle EŒC1  is defined as EŒC1  D Ci EŒNI  C Cp EŒR1 Pp C Cc EŒR2 Pc : (10.24)

1 X

 P fiI < T  .i C 1/I g: (10.26) Hence, EŒNI  D

1 X i D1

i fP fY Œ.i  1/I   L; DŒ.i  1/I   S g  P fL < Y .iI /  G; D.iI /  S g

C P fY .iI /  L; D.iI /  S g  P fiI < T  .i C 1/I g:

(10.27)

Assume Y .t/ D A C Bg.t/, where A N.A ; A2 /, B N.B ; B2 /, and A and B are independent. We

404

10 Advanced Maintenance Modeling

now calculate the probabilities P fY Œ.i  1/I   L, DŒ.i  1/I   S g and P fL < Y .iI /  G, D.iI /  S g. PN.t / Given g.t/ D t, D.t/ D i D0 Xi , where the Xi are independent and identically distributed, and N.t/ Poisson./, then, P fY Œ.i  1/I   L; DŒ.i  1/I   S g D P fA C B.i  1/I  Lg N..i 1/I / n o X  P DŒ.i  1/I  D Xi  S i D0

L  .A C B .i  1/I / D˚ q A2 C B2 Œ.i  1/I 2  e.i 1/I

!

1 X Œ.i  1/I j .j / FX .S / jŠ

j D0

(10.28) and

creasing function of time t, assume that A fA .a/, B fB .b/. Let  y1 D a C bg.iI / (10.31) y2 D a C bgŒ.i C 1/I : After simultaneously solving the above equations in terms of y1 and y2 , we obtain y1 gŒ.i C 1/I   y2 g.iI / D h1 .y1 ; y2 /; (10.32) gŒ.i C 1/I   g.iI / y2  y1 D h2 .y1 ; y2 /: (10.33) bD gŒ.i C 1/I   g.iI /

aD

Then the random vector .Y .iI /; Y Œ.i C 1/I / has a joint continuous probability distribution function as follows: fY.iI /;Y Œ.i C1/I  .y1 ; y2 / D jJ jfA Œh1 .y1 ; y2 /fB Œh2 .y1 ; y2 /;

(10.34)

where the Jacobian J is given by P fL < Y .iI /  G; D.iI /  S g " ! G  .A C B iI / D ˚ q A2 C B2 .iI /2 L  .A C B iI / ˚ q A2 C B2 .iI /2  eiI

1 X j D0

ˇ @h ˇ 1 ˇ ˇ @y1 J D ˇˇ ˇ @h2 ˇ @y1

!#

j

.iI / .j / FX .S /: jŠ

(10.29)

Since T D infft > 0 W Y .t/ > G or D.t/ > S g, we have P fiI < T  .i C 1/I g D P fY .iI /  L; Y Œ.i C 1/I  > Gg  P fDŒ.i C 1/I   S g C P fY Œ.i C 1/I   Lg  P fD.iI /  S; DŒ.i C 1/I  > S g: (10.30) In Eq. 10.30, since Y .iI / and Y Œ.i C 1/I  are not independent, we need to obtain the joint probability distribution function fY.iI /;Y Œ.i C1/I  .y1 ; y2 / in order to compute P fY .iI /  L; Y Œ.i C 1/I  > Gg. In general, as for when A > 0 and B >0 are two independent random variables, and g.t/ is an in-

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ 1 ˇ: ˇDˇ ˇ ˇ g.iI /  gŒ.i C 1/I  ˇ @h2 ˇ ˇ @y2 (10.35) @h1 @y2

As for the term P fD.iI /  S;PDŒ.i C 1/I  > S g N.t / in Eq. 10.30, since D.t/ D i D0 Xi is a compound Poisson process, the compound Poisson process has a stationary independent increment property. Therefore, the random variables D.iI / and DŒ.i C 1/I   D.iI / are independent. Using the Jacobian transformation, random vector .D.iI /; DŒ.i C 1/I D.iI // is distributed the same as vector .D.iI /; DŒ.i C 1/I /. Note that D.iI / and D.Ii C1 / are independent; therefore, P fD.iI /  S; DŒ.i C 1/I  > S g D P fD.iI /  S gP fDŒ.i C 1/I  > S g: (10.36)

10.4.2 Calculate Pp Note that either a preventive maintenance or a corrective maintenance action will end a renewal cycle. In other words, preventive maintenance and corrective maintenance events are mutually exclusive at the re-

10.4 Modeling of Inspection-Maintenance Repairable Degraded Systems

newal time point. As a consequence, Pp C Pc D 1. The probability Pp can be obtained as follows: Pp D P fpreventative maintenance ending a cycleg D

1 X

P fY Œ.i  1/I   L; L < Y .iI /  Gg

i D1

 P fD.iI /  S g:

(10.37)

10.4.3 Expected Cycle Length Analysis Since the renewal cycle ends as a result of either a preventive maintenance action with probability Pp or a corrective maintenance action with probability Pc , the mean cycle length EŒW1  is calculated as follows:

The expression for EŒT  depends on the probability P fY .t/  Gg and sometimes it is not easy to obtain a closed form.

10.4.4 Optimization of Maintenance Cost Rate Policy We determine the optimal inspection time I and preventive maintenance threshold L such that the longrun average maintenance cost rate EC.L; I / is minimized. Mathematically, we wish to minimize the following objective function: EC.L; I / P1

i D1

EŒW1  D

1 X

EŒ.iI C R1 /IPM occurs in Œ.i 1/I;iI  

i D1

C EŒ.T C R2 /ICM occurs  X 1 iIP fY Œ.i  1/I   L; DŒ.i  1/I   S g D i D1

  P fL < Y .iI /  G; D.iI /  S g

C EŒR1 Pp C .EŒT  C EŒR2 /Pc ;

(10.38)

where IPM occurs in ..i 1/I;iI  and ICM occurs are the indicator functions. The mean time to failure, EŒT , is given by Z1 EŒT  D

P fT > tg dt 0

Z1 P fY .t/  G; D.t/  S g dt

D 0

Z1 D

P fY .t/  Gg

1 X .2 t/j e2 t .j / FX .S / dt jŠ

j D0

0

(10.39) or, equivalently, by Z 1 X FX.j / .S / P fY .t/  Gg.2 t/j e2 t dt: EŒT  D jŠ 1

j D0

0

(10.40)

405

iP fY .Ii 1 /  L; D.Ii 1 /  S g P fL < Y .Ii /  G; D.Ii /  S g

D P1 . i D1 Ii P fY .Ii 1 /  L; D.Ii 1 /  S g P fL < Y .Ii /  G; D.Ii /  S g/ CEŒR1 Pp C EŒR2 Pc P1 i D1 iVi fP fY .Ii /  L; Y .Ii C1 / > Gg P fD.Ii C1 /  S g CP fY .Ii C1 /  Lg P fD.Ii /  S; D.Ii C1 / > S gg C P1 . i D1 Ii P fY .Ii 1 /  L; D.Ii 1 /  S g P fL < Y .Ii /  G; D.Ii /  S g/ CEŒR1 Pp C EŒR2 Pc P Cp EŒR1  1 i D1 P fY .Ii 1 /  L; D.Ii 1 /  S g P fL < Y .Ii /  G; D.Ii /  S g C P . 1 I P fY .Ii 1 /  L; D.Ii 1 /  S g i D1 i P fL < Y .Ii /  G; D.Ii /  S g/ CEŒR1 Pp C EŒR2 Pc Cc EŒR2  P f1  1 i D1 P fY .Ii 1 /  L; D.Ii 1 /  S g P fL < Y .Ii /  G; D.Ii /  S gg C ; P1 . i D1 Ii P fY .Ii 1 /  L; D.Ii 1 /  S g P fL < Y .Ii /  G; D.Ii /  S g/ CEŒR1 Pp C EŒR2 Pc where Ii 1 D .i  1/I , Ii D iI , Ii C1 D .i C 1/I , and Vi D P fY .iI /  L; D.iI /  S g. The above complex objective function is a nonlinear optimization problem. Li and Pham (2005) discussed a step-by-step algorithm based on the Nelder– Mead downhill simplex method shown as follows:

406

• Step 1: Choose .n C 1/ distinct vertices as an initial set fZ .1/ ; : : : ; Z .nC1/ g, then calculate the value of the function f .Z/ for i D 1; 2; : : : ; .n C 1/, where f .Z/ D EC.I; L/. Put the values f .Z/ in increasing order, where f .Z .1/ / D minfEC.I; L/g and f .Z .nC1/ / D maxfEC.I; L/g and set k D 0. P • Step 2: Compute X .k/ D n1 niD1 Z .i / . • Step 3: Use the centroid X .k/ in step 2 to compute X .kC1/ D X .k/  Z .nC1/ . • Step 4: Set  D 1 and compute f .X .k/ C X .kC1/ /. If f .X .k/ C X .kC1/ /  f .Z .1/ /, go to step 5. If f .X .k/ C X .kC1/ /  f .Z .n/ /, go to step 6. Otherwise, fix  D 1 and go to step 8.

10 Advanced Maintenance Modeling

p spectively, and g.t/ D t e0:005t . Assume that the ranPN.t / dom shock damage is described by D.t/ D i D1 Xi , where Xi follows the exponential distribution, i. e., Xi exp.0:04t/ and N.t/ Poisson.0:1/. Given G D 50, S D 100, Ci D 900 per inspection Cc D 5600 per corrective maintenance, Cp D 3000 per preventative maintenance, R1 exp.0:1t/, and R2 exp.0:04t/, we now determine the values of both I and L so that the average total cost per unit time EC.I; L/ is minimized. The step-by-step procedure follows: Step 1:

I and L are two decision variables. We need .n C 1/ D 3 initial distinct vertices, which are Z .1/ D .25; 20/, Z .2/ D .20; 18/, and Z .3/ D .15; 10/. Set k D 0. Calculate the value of f .Z .  / / corresponding to each vertex and sort them in increasing order in terms of EC.I; L/.

Step 2:

Calculate: X .0/ D .Z .1/ C Z .2/ /=2 D .22:5; 19/.

Step 3:

Generate the searching direction: X D X .0/  Z .3/ D .7:5; 9/.

Step 4:

Set  D 1; it will produce a new minimum EC.30; 28/ D 501:76 that leads trying an expansion with  D 2, i. e., .37:5; 38/.

Step 5:

Set  D 2. Similarly, calculate f .Z/ that leads to EC.37:5; 38/ D 440:7: Go to step 8 in Sect. 10.4.4. This result turns out to be a better solution; hence .15; 10/ is replaced by .37:5; 38/.

• Step 5: Set  D 2 and compute f .X .k/ C 2X .kC1/ /: If f .X .k/ C 2X .kC1/ /  f .X .k/ C X .kC1/ /, set  D 2. Otherwise, set  D 1. Then go to step 8. • Step 6: If f .X .k/ C X .kC1/ /  f .Z .nC1/ /, set  D 1=2: Compute f .X .k/ C 12 X .kC1/ /. If f .X .k/ C 12 X .kC1/ /  f .Z .nC1/ /, set  D 1=2 and go to step 8. Otherwise, set  D 1=2 and if f .X .k/  12 X .kC1/ /  f .Z .nC1/ /, set  D 1=2 and go to step 8. Otherwise, go to step 7. • Step 7: Shrink the current solution set toward the best Z .1/ by Z .i / D 12 .Z .1/ C Z .i / /, i D 2; : : : ; n C 1. Compute the new f .Z .2/ /, : : : , f .Z .nC1/ /, let k D k C 1, and return to step 2. • Step 8: Replace worst Z .nC1/ by X .k/ C q the P 1 .kC1/ .i / N2 X . If nC1 nC1 i D1 Œf .Z /  f  < ", where fN is an average value, stop. Otherwise, let k D k C 1 and return to step 2. It should be noted that " denotes the difference between the maximum and the minimum values of f . In the following example, " D 0:5, which also indicates how soon we would like the algorithm to stop when the vertices function values are close.

10.4.5 Numerical Example Assume that the degradation process is described by Y .t/ D ACBg.t/, where A andB are independent and follow the uniform distribution with parameter interval Œ0; 4 and an exponential distribution with parameter 0.3, i. e., A U.0; 4/ and B exp.0:3t/, re-

The iteration continues q P and stops at k D 6 (see Table 10.1) since 13 3i D1 ŒEC.Z .i / /  EC.I; L/2 < 0:5, where EC.I; L/ is the average value. From Table 10.1, the optimal values are I  D 37:5 and L D 38 and the corresponding cost value is EC .I; L/ D 440:7. Figure 10.4 shows the relationship between L and Pc for different I values, i. e., 35, 37.5, and 40. We also observe that Pc is an increasing function on L. This means a higher preventive maintenance threshold is more likely to result in a failure.

10.5 Warranty Concepts A warranty is a contract under which the manufacturers of a product and/or service agree to repair or replace the product or provide a service when a product

10.5 Warranty Concepts

407

Table 10.1 Optimal values I and L (Li and Pham 2005) k

Z .1/

Z .2/

Z .3/

Search result

0

(25,20) EC.I; L/ D 564:3

(20,18) EC.I; L/ D 631:1

(15,10) EC.I; L/ D 773:6

(37.5,38) EC.I; L/ D 440:7

1

(37.5,38) EC.I; L/ D 440:7

(25,20) EC.I; L/ D 564:3

(20,18) EC.I; L/ D 631:1

(42.5,40) EC.I; L/ D 481:2

2

(37.5,38) EC.I; L/ D 440:7

(42.5,40) EC.I; L/ D 481:2

(25,20) EC.I; L/ D 564:3

(32.5,29) EC.I; L/ D 482:2

3

(37.5,38) EC.I; L/ D 440:7

(42.5,40) EC.I; L/ D 481:2

(32.5,29) EC.I; L/ D 482:2

(32.5,33.5) EC.I; L/ D 448:9

4

(37.5,38) EC.I; L/ D 440:7

(32.5,33.5) EC.I; L/ D 448:9

(42.5,40) EC.I; L/ D 481:2

(38.75,37.125) EC.I; L/ D 441:0

5

(37.5,38) EC.I; L/ D 440:7

(38.75,37.125) EC.I; L/ D 441:0

(32.5,33.5) EC.I; L/ D 448:9

(35.3125,35.25) EC.I; L/ D 441:1

6

(37.5,38) EC.I ; L / D 440:7

(38.75,37.125) EC.I; L/ D 441:0

(35.3125,35.25) EC.I; L/ D 441:4

Stop

for a variety of products (Bai and Pham 2004, 2005, 2006; Murthy and Blischke 2006). Warranty types are dependent on the kind of product that it protects. For larger or more expensive products with many components, it may be cheaper to repair the product rather than to replace it. These items are called “repairable products.” Other warranties simply result in replacement of an entire product because the cost to repair it is either close to or exceeds its original price. These products are considered nonrepairable. The following are the most common types used in warranties:

Fig. 10.4 Pc versus L

fails or the service does not meet the intended requirements. These agreements exist because of the uncertainty present in the supply of products or services, especially in a competitive environment. Warranties are important factors in both the consumers’ and the manufacturers’ decision making (Wang and Pham 2006b). A warranty can be the deciding factor for the purchase of a particular item when different products have similar functions and prices. The length and type of warranty is often thought of as a reflection of the reliability of a product as well as the company’s reputation. Many researchers have developed different models to provide guidance in selecting a successful warranty plan

• Ordinary free replacement. Under this policy, when an item fails before a warranty expires it is replaced at no cost to the consumer. The new item is then covered for the remainder of the warranty period. This is the most common type of a warranty and often applies to cars and kitchen appliances. • Unlimited free replacement. This policy is the same as the ordinary free replacement policy but each replacement item carries a new identical warranty. This type of warranty is often used for electronic appliances with high early failure rates and usually has a shorter length because of this. • Pro rata warranty. The third most common policy takes into account how much an item is used. If the product fails before the end of the warranty period, then it is replaced at a cost that is discounted proportional to its use. Items that experience wear or aging, such as tires, are often covered under these warranties.

408

10 Advanced Maintenance Modeling

Table 10.2 Maintenance and warranty modeling and analysis literature Group

References

General modeling

Amari and Pham (2007), Bai and Pham (2006), Beichelt and Fisher (1980), Brown and Proschan (1983), Esary et al. (1973), Kijimma (1989), Özekici (1996), Sheu (1998), Wang and Pham (1999)

Maintenance modeling

Barlow and Proschan (1965), Ben-Daya et al. (2000), Lie et al. (1995), Pham (2003a), Wang and Pham (2006b)

Age, block replacements

Ansell et al. (1984), Beichelt (1981), Berg (1995), Block et al. (1988), Bris et al. (2003), Fox (1966), Lam (1991), Nakagawa (1981a, b), Park and Yoo (1993), Savits (1988), Wang and Pham (1999)

Imperfect repairs

Bagai and Jain (1994), Hollander et al. (1992), Ebrahimi (1985), Nakagawa (1977), Park (1979), Wang and Pham (1996a–c)

Optimal policies

Chen and Feldman (1997), Feldman (1977), Lam and Yeh (1994), Makis and Jardine (1992), Nakagawa and Yasui (1987), Phelps (1983), Sheu (1994), Suresh and Chaudhuri (1994), Wang and Pham (1996a–c)

Inspection policies

Dieulle et al. (2003), Li and Pham (2005a, b), Zuo et al. (2000), Zuckerman (1989)

Warranty modeling

Bai and Pham (2005, 2006), Murthy and Blischke (2006)

Optimization

Canfield (1986), Inagaki et al. (1980), Lam and Yeh (1994), Pham and Wang (2000), Wang and Pham (1997, 2006a), Zheng (1995)

Different warranty models may include a combination of these three types as well as offering other incentives such as rebates, maintenance, or other services that can satisfy a customer and extend the life of the product. Table 10.2 presents a brief summary of references to research papers and books on maintenance and warranty modeling and analysis for quick reference.

10.6 Conclusions In this chapter, we presented reliability and maintenance models for systems with multiple competing failure processes such as degradation and random shock. The results of the maintenance models can be used as decision-tools to help practitioners and

inspectors as well as marketing managers to allocate the resources and also for the purposes of promotion strategies of the new products, including warranty policies. It should be noted that maintenance system costs associated with inspections, preventive maintenance, corrective maintenance, and downtime are often difficult to obtain, even though they are applicable in practice. For some critical systems, the overriding goal is to ensure that the system is available when needed; therefore, in many cases, the cost is, however, secondary. To achieve as high a level of availability as possible for a specified inspection rate, it is worth determining the optimal policies, including the number of inspections with respect to imperfect repairs (i. e., minimal and opportunistic schemes), that maximizes the degraded system availability.

11

Spare Parts Forecasting and Management

Contents 11.1 Spare Parts Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 409 11.2 Spare Parts Characterization . . . . . . . . . . . . . . . . . . 410 11.3 Forecasting Methods . . . . . . . . . . . . . . . . . . . . . . . . . 411 11.4 Croston Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 11.5 Poisson Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 11.6 Binomial Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 11.6.1 Numerical Example . . . . . . . . . . . . . . . . . . . . . 415 11.7 Spare Parts Forecasting Accuracy . . . . . . . . . . . . . 416 11.8 Spare Parts Forecasting Methods: Application and Case Studies . . . . . . . . . . . . . . . . . . 417 11.8.1 Case Study 1: Spare Parts Forecasting for an Aircraft . . . . . . . . . . . . . . . . . . . . . . . . . . 417 11.8.2 Case Study 2: Spare Parts Forecasting in a Steel Company . . . . . . . . . . . . . . . . . . . . . 418 11.9 Methods of Spare Parts Management . . . . . . . . . . . 422 11.9.1 Spare Parts Management: Qualitative Methods . . . . . . . . . . . . . . . . . . . . . 423 11.9.2 Spare Parts Management: Quantitative Methods . . . . . . . . . . . . . . . . . . . . 426

Even without considering crashes and other damage during its life, a car needs parts such as its engine oil, tires, and brake pads to be replaced or changed. Similarly, production systems have the same need for spare parts. Although a very relevant topic, spare parts management has rarely been studied. How many spare parts are there in the local warehouse of the company? How can future demand be forecast? This chapter deals with these questions, and presents several methodologies to support decision making on this theme.

11.1 Spare Parts Problem During its working life a production system needs spare parts to fix breakdowns and other reliability problems, while equipment also wears out with use. Spare parts management is therefore very important in economic terms and also technical terms. Figure 11.1 shows a typical sequence of activities performed during corrective maintenance requiring spare parts (e. g., electronic card, gearbox, chains, and other components) or expendable material (e. g., oil, glue). The procurement of spare parts is often included in the sequence. The duration of this activity is strongly related to the presence of spare parts in the local warehouse of the company. If the required spare part is available in the company’s warehouse, the procurement lead time is only a few minutes, but otherwise it is days or even weeks (e. g., when the supplier is located very far away or has to manufacture the items). The absence of a spare part can lead to production stopping or being curtailed, and so to a very significant increase in related costs. Furthermore, adapting not original spare parts that are not perfectly interchangeable with failed components leads to further damage to the equipment occurring rather than to its swift and effective repair. Spare parts are typically expensive and are at great risk of becoming obsolete (see Sect. 11.2). In addition, they may or may not be used and this uncertainty usually makes storing them expensive. In conclusion, spare parts management must consider two opposing factors: the lack of production cost and the procurement and storage cost. As shown in

R. Manzini, A. Regattieri, H. Pham, E. Ferrari, Maintenance for Industrial Systems © Springer 2010

409

410

11 Spare Parts Forecasting and Management stop equipment

restart

failure

time

uptime

downtime

uptime

intervention

ΔT

ΔT

ΔT

ΔT

ΔT

ΔT

ΔT

ΔT

ΔT

call

preparation

disassembly

procurement

repair

calibration

assembly

check

reporting

Fig. 11.1 Typical corrective maintenance activities

Lack of production cost

Total cost

Storage cost Costs

Spare parts in stock Optimal level

Fig. 11.2 The spare parts trade-off problem

Fig. 11.2, this is a trade-off problem in which the goal is to determine the optimal set (kind and quantity) of spare parts required at the company’s local warehouse. This set gives the minimum total cost. Two subproblems arise out of this optimal level: the forecasting of spare part consumption and the economic management of actual consumption. The important first step is to forecast the number of spare parts that the system will use in the future very carefully. Then these parts need to be procured and managed as efficiently as possible.

11.2 Spare Parts Characterization Compared with other materials flowing in a supply chain, the behavior of spare parts is very peculiar. The consumption of spare parts is basically intermittent and storage usually requires a wide variety of spare parts combined with few units per type. According to Williams (1984), Syntesos (2001), and Syntetos et al.

(2005), the parameters usually adopted to characterize spare part properties are: ADI Average interdemand interval: the average time interval between two successive consumptions of a spare part. It is usually expressed in time periods (e. g., months). CV 2 Squared coefficient of variation: standard deviation of consumption divided by the average value of consumption. It is adimensional. Figure 11.3 shows the typical consumption of a spare part in agreement with which the following can be defined: PN i (11.1) ADI D i D1 ; N 12 0qP N i D1

B CV D @ 2

."ri "a /2 N

"a

C A ;

(11.2)

PN "a D

i D1 "ri

N

;

(11.3)

where "ri is the spare part demand (units), i is the time interval between two successive spare part demands (periods), and N is the number of time intervals analyzed Several studies in the literature (Syntetos 2001; Syntetos and Boylan 2005, 2006; Syntetos et al. 2005; Ghobbar and Friend 2002, 2003; Boylan and Syntetos 2006; Boylan et al. 2008) introduce different patterns of spare parts according to ADI and CV2 values. In particular, they suggest different cutoff values for the classification depending on the context of the application. For example, Fig. 11.4 shows the different pattern discussed in Syntetos et al. (2005).

11.3 Forecasting Methods τ1

411

τ2

0

εr1

τi

εr2

εri

time

Fig. 11.3 Typical spare parts consumption

0

ADI = 1.32 periods

Smooth

Intermittent

Erratic

Lumpy

CV2 = 0.49

prevent problems or damage (i. e., thermal or hygrometric conditions). For the reasons mentioned above, spare part acquisition and storage can lead to a significant financial investment. Spare parts have tricky specific properties. They represent a particular category of materials in a production system that needs to be managed very carefully.

Fig. 11.4 Patterns of spare parts. ADI average interdemand interval, CV 2 squared coefficient of variation

11.3 Forecasting Methods Using Fig. 11.4, one can classify the patterns into four categories according to the state and size of the demand: 1. Intermittent demand is random, and a lot of time periods have no demand. 2. Erratic demand is (highly) variable and there is erratic behavior of the size of the demand rather than the demand per single time period. 3. Smooth demand, also occurs at random with a lot of time periods having no demand. When there is demand, it occurs in single or very few units. 4. Lumpy demand is similarly random with many time periods having no demand. Moreover, when the demand occurs, it is (highly) variable. The concept of lumpy corresponds to an extremely irregular demand, with great differences between each period’s requirements and with a large number of periods with zero requirements. Another fundamental peculiarity of spare parts is their specificity of use. In other words, a spare part is not usually of general purpose but only has its own use. Consequently, the risk of obsolescence is very high. For example, when a machine is superseded in a production system by a new one, most of the spare parts are not reusable on other equipment and so immediately become obsolete. Spare parts are usually expensive because their technological content is significant. Furthermore, specific storage devices are required in some situations to

The goal of an efficient spare parts management system is to minimize the total cost. This general observation is not true when safety or environmental questions impose specific constraints. Generally speaking, a trade-off between storage costs and production downtime costs needs to be found. This determination of the optimal level of spare parts requires two levels of analysis: the forecasting of future demand and the consequent optimal management of this demand. Several different approaches are available in order to determine the future requirement of spare parts in the real world of industry: • Experience and the know-how possessed by maintenance personnel. The experience of operators often represents a unique source of information. • Information from suppliers. Several suppliers develop lists of “suggested” spare parts for local stock. These lists are developed according to the work experience of the supplier or using suitably developed tests. • Forecasting models. Statistical models elaborate the consumption of spare parts registered in the past and estimate future demand. These categories of methods need different investments in terms of time and cost. The simultaneous use of all of these approaches can produce the best practice: the forecasting models provide good results that can then be fine-tuned using the know-how of maintenance operators and suppliers. The existence of a sig-

412

11 Spare Parts Forecasting and Management

Table 11.1 Spare parts forecasting methods Method

Abbreviation

Description

Moving averages Weighted moving averages

MA WMA

Exponential weighted moving averages

EWMA

Single exponential smoothing

SES

Croston method

Croston

Double exponential smoothing Additive Holt–Winter

DES AW

Multiplicative Holt–Winter

MW

Adaptive-response-rate single exponential smoothing Time series decomposition (seasonal regression model) Autoregressive integrated moving average Poisson model

ARRSES SRM

Rolling average value based on past demand data A simple variation on the moving average technique that weights the data in order to average them Applies weighting factors that decrease exponentially. The weighting for each older data point decreases exponentially, giving much more importance to recent observations while not discarding older observations entirely Similar to exponential weighted moving averages, weights decrease exponentially. It produces interesting results in the case of low and intermittent demand Adjustment of single exponential smoothing to consider series with zero value of demand occurring many times. Forecasting in the case of low and intermittent demand A factor considering trend effects is introduced into single exponential smoothing Extension of single exponential smoothing to linear exponential smoothing. Assumes that the seasonal effects are constant in size Assumes that the seasonal effects are proportional in size to the local deseasonalized mean level This is a variation of single exponential smoothing that continually adjusts the smoothing parameter to allow for changes in the trend Identifies different separate components of the basic pattern

ARIMA

Based on autocorrelation of residual (noise) in the data of the series

Poisson

Binomial model

BM

Models based on the Poisson distribution with the customer’s service level being defined Method based on a two-factor consumption model

nificant maintenance information system (see Chap. 7) containing information on the past consumption of spare parts is fundamental to the application of statistical methods. In the literature the forecasting of spare parts using a statistical approach is usually based on the general demand forecasting problem. This very broad approach can be focused by taking the specific peculiarities of spare parts into account, therefore avoiding the way a great many statistical methods underperform. Several studies on this topic are reported in the technical literature (Makridakis et al. 1998; Willemain et al. 2004; Ghobbar and Friend 2004; Regattieri et al. 2005; Ferrari et al. 2006), and their conclusions sometimes differ. Moreover, a group of interesting methods can be selected from the experimental evidence (Table 11.1). The following sections only deal with “nonconventional” approaches such as the Croston, Poisson, and binomial methods specifically devoted to the intermittent-demand case. Other models are very well known and very frequently used in the product demand forecasting problem (Madrikakis et al. 1998).

11.4 Croston Model Croston’s method is a widely used approach for intermittent-demand forecasting, and is based on exponential smoothing. In particular, it involves separate simple exponential smoothing forecasts of the demand size and the time period between demands. This approach is devoted to the situation where the time series has several zero values. Let Yt0 be the expected consumption of a spare part for the period (t C 1) defined at period t: Yt0 D

zt : pt

If yt D 0, then pt D pt 1 ; zt D zt 1 ; q D q C 1; otherwise pt D pt 1 C ˛.q  pt 1 /; zt D zt 1 C ˛.yt  zt 1 /; q D 1;

(11.4)

11.5 Poisson Model

413

where yt is the spare part consumption at period t, pt is the time interval between period t and the last period with a positive consumption of spare part(s), zt is the average consumption of a spare part upgraded at period t, q is the number of periods between period t and the last period with a positive consumption of spare part(s), and ˛ is a smoothing factor (optimized by a trial-and-error procedure). Some authors, including Johnston and Boylan (1996), Syntetos and Boylan (2001), Syntesos et al. (2005), and Boylan et al. (2008), have proposed modifications to Croston’s method for the purpose of improving the accuracy of the forecast. In particular, Syntetos and Boylan (2001) proposed a modification in the final calculus of the forecast: Yt0 D

zt ; pt c pt 1

d

d

d

Time (periods)

0

T x components

Fig. 11.5 Average rate of consumption and time horizon

(11.5)

where c is a constant optimized by a trial-and-error procedure (c usually ranges from 100 to 200). Fig. 11.6 Component XC100 Table 11.2 Probability of consumption for component XC100

11.5 Poisson Model The Poisson method is based on the Poisson distribution and forecasts the probability of a rare event. It is a direct consequence of the binomial distribution. When applied to the spare parts forecasting problem, it provides an estimate of the probability of consumption for a fixed value of spare parts. The starting point of this approach is the average consumption rate of a spare part (called d ). The probability that x spare parts will be used in a time horizon T at an average rate of consumption d is given by Pd;T;x D

.d T /x e .d T / ; xŠ

(11.6)

where d is the average rate of spare part consumption (pieces per period), x is the number of pieces consumed, and T is the time horizon (periods). The cumulative probability of the maximum consumption of x spare parts is given by P CUM d;T;x D

x X .d T /k e .d T / : kŠ

kD0

Figure 11.5 shows the situation.

(11.7)

T

dT

P .d; T; 1/

P .d; T; 2/

P .d; T; 3/

0 1 10 100 1,000 2,000 3,000 4,000 5,000 ... 10,000

0 0.00024 0.0024 0.024 0.24 0.48 0.72 0.96 1.2 ... 2.4

0 0.00024 0.002394 0.023431 0.188791 0.297016 0.350462 0.367577 0.361433 ... 0.217723

0 2.88E-08 2.87E-06 0.000281 0.022655 0.071284 0.126166 0.176437 0.21686 ... 0.261268

0 2.3E-12 2.3E-09 2.25E-06 0.001812 0.011405 0.03028 0.05646 0.086744 ... 0.209014

We now present an application. The average consumption for component XC100 (a secondary shaft for a conveyor system; Fig. 11.6) registered in the past and extracted from the database management system is d D 2:4  104 pieces=h, i. e., one replacement approximately every 4;200 h. The probability of consumption relating to a time horizon T (in hours) is given by Eq. 11.6 (see also Fig. 11.7). Table 11.2 shows results for consumption of one, two, and three spare parts [P .d; T; 1/, P .d; T; 2/, and P .d; T; 3/, respectively] according to different time horizons.

414

11 Spare Parts Forecasting and Management

11.6 Binomial Model

0.4 0.35

P(d,T,x)

0.3 0.25

P(d,T,1) P(d,T,2) P(d,T,3)

0.2 0.15 0.1 0.05 0 0

2000

4000

6000

8000

10000

12000

T(h)

Fig. 11.7 Probability plot of expected consumption for XC100 Table 11.3 Probability of consumption for component XC100 in T D 5;000 h X

dT

P .d; T; X/

0 1 2 3 4 5 6 7 8 9 10

1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20

0.301194 0.361433 0.21686 0.086744 0.026023 0.006246 0.001249 0.000214 3.21E-05 4.28E-06 5.14E-07

N D x1 C x2 :   T n; x1 D 1=d

0.4 0.3

P(d,t,X)

0.25 0.2 0.15 0.1 0.05 0 1

2

3

4

(11.8) (11.9)

where N is the spare part forecast (pieces), d is the average consumption of spare parts (pieces per period), and T is the forecasting time horizon. x2 is related to the probability of at maximum x2 failures occurring; hence, spare part consumption in the time interval Tresidual is defined as   1 T : (11.10) Tresidual D T  1=d d

0.35

0

This method was proposed by Regattieri (1996) and is based on the binomial distribution. When the lumpiness of demand is significant, the simple application of the Poisson formula can give an inconsistent forecast (usually overestimated). In this method the spare part forecast is composed of two terms: the first, x1 , considers the average consumption of the spare part for a fixed period T and the second, x2 , tries to link the consumption to a desired level of service using the binomial approach. This method also considers the multiple use of the same spare part on different items of equipment in the system by applying the number of installations parameter (n). The forecast is given by

5

6

7

8

9

10

11

X

Fig. 11.8 Probability trend for expected consumption of XC100 in T D 5;000 h

In real applications the time horizon T is fixed and usually represents the lead time of supply. The main problem is to define the expected probability of consumption for different values of spare parts. The approach used is the same as in Eq. 11.6, i. e., fixed T and variable x. Table 11.3 and Fig. 11.8 show the expected consumption for time horizon T D 5;000 h and number of spare parts from zero to ten.

Let p be the cumulative probability of the spare part being used (i. e., to have a failure) in the Tresidual period. Assuming an exponential distribution of time to failure (other competitive distributions are Weibull and normal ones), F .Tresidual / D 1  e

  1  1=d Tresidual

D p:

(11.11)

Let n be the number of examined components contemporaneously installed and LS the a priori fixed probability of satisfying the demand of spare parts forecast. Using the binomial formula, ! x2 X n (11.12) .1  p/ni p i  LS: P .x2 / D i i D0

The iterative application of Eq. 11.12 means the minimum value x2 satisfying the disequation can be defined.

11.6 Binomial Model

415

d =2×10-3 pieces/h

0

time

T = 640 h x components

Fig. 11.9 P000-303 valve

11.6.1 Numerical Example

F .Tresidual / D 1  e D 1e

On a power and free transportation system installed in a car production plant for chassis handling there are n D 10 elements of a solenoid valve named “P000303” (Fig. 11.9). The past consumption of this spare part according to the database management system is d D 2  103 pieces=h. The plant engineer must forecast the expected consumption of item P000-303 for a time horizon of 640 h, corresponding to the time interval between two consecutive procurements. The fixed service level is 90%. The starting point is Eq. 11.8:

 x1 D

T 1=d

$

 nD

1 2

640 h  103 pieces=h

 10

D 10 pieces: Tresidual and the corresponding cumulative probability of failure p are   1 T Tresidual D T  1=d d % $ 640 h D 640 h  1 3 pieces=h 2  10 1  103 pieces=h 2 D 140 h; 

D 0:244:

Let x2 be equal to one unit: ! 1 X n P .x2 D 1/ D .1  p/ni p i i i D0

D

%

1 1 10 2

Dp

 140 3

i D0

1 X i D0

Then,

 

The service level in Tresidual is 90%, and the value of x2 is obtained from the recursive application of ! x2 X n P .x2 / D .1  p/ni p i  0:90: i

D

N D x1 C x2 :

  1  1=d Tresidual

nŠ .1  p/ni p i i Š.n  i /Š

10Š .1  0:244/100  0:2440 0Š.10  0/Š

C

10Š  .1  0:244/101  0:2441 1Š.10  1/Š

D 0:061 C 0:197 Š 0:258: In conclusion, P .x2 D 1/ < LS and x2 D 1 is not the solution. The following attempt value must be x2 D 2: ! 2 X n P .x2 D 2/ D .1  p/ni p i i i D0

D

10Š .1  0:244/100  0:2440 0Š.10  0/Š 10Š .1  0:244/101  0:2441 C 1Š.10  1/Š

416

11 Spare Parts Forecasting and Management

10Š .1  0:244/102  0:2442 2Š.10  2/Š D 0:061 C 0:197 C 0:286 Š 0:544: C

Furthermore, in this case the disequation (Eq. 11.12) is not satisfied. In brief, if x2 D 4, P .x2 D4/

! 4 X n D .1  p/ni p i i i D0

D

10Š .1  0:244/10  0:2440 0Š  10Š 10Š .1  0:244/104 C ::: C 4Š.10  4/Š  0:2444

D 0:060 C 0:197 C 0:286 C 0:246 C 0:136 D 0:928 > 0:90; then the expected value of x2 is four pieces. In conclusion, the total forecast of spare parts demand in 640 h is N D x1 C x2 D 10 C 4 D 14 pieces.

the outcome: in this case the expected values are very uncertain. There are many parameters to evaluate the forecast error. Let At be the actual value at time t, Ft the forecast value at time t, and n the couples (At ; Ft ) considered: • Mean deviation (MD): Pn Pn .At  Ft / t D1 et MD D D t D1 I n n

(11.13)

• Mean square deviation (MSD): Pn Pn 2 .At  Ft /2 t D1 et MSD D D t D1 I (11.14) n n • Mean absolute deviation (MAD): Pn Pn jAt  Ft j t D1 jet j MAD D D t D1 I (11.15) n n • Mean absolute percentage error (MAPE): ˇ n ˇ 1 X ˇˇ At  Ft ˇˇ I MAPE D n t D1 ˇ At ˇ

(11.16)

• Standardized MAD (SMAD):

11.7 Spare Parts Forecasting Accuracy

SMAD D

MAD MAD D Pn :  A t D1 At

(11.17)

n

The forecast error is the difference between the actual/real and the predicted/forecast value of a time series or any other phenomenon of interest. In simple cases, a forecast is compared with an outcome at a single point in time and a summary of forecast errors is constructed over a collection of these samples. Here the forecast may be assessed using the difference or using a proportional error. By convention, the error is defined using the value of the outcome minus the value of the forecast. Obviously, the forecast accuracy is linked to the forecast error. In particular, if the error E is expressed as a percentage, the accuracy is equal to (1  E)%. The evaluation of the forecast error (or accuracy) is of critical importance in choosing the best method according to the real data in the analysis. Furthermore, the evaluation of forecasting error, and in particular its value compared with the real outcomes, means the robustness of the choice can be evaluated. At times the forecasting error is greater than (or comparable with)

MD is the basic error but suffers from a significant problem linked to a “compensation” of errors (minus sign and plus sign). Consequently, MSD and MAD are introduced. MSD and MAD can introduce relevant bias effects when couples with significant differences in terms of value (e. g., different orders of magnitude) are compared. MAPE skips this problem by introducing the concept of percentage error. MD, MSD, MAD, and MAPE are typical error measurements normally used for the demandforecasting problem in which there are no periods with null demand. But in spare parts forecasting a null demand for components in a period is very frequently observed: in this situation a MAPE is not available and other methods can introduce the above-mentioned bias. The SMAD, defined as the ratio between MAD and the average value of the actual time series, is an efficient parameter to evaluate the accuracy of a forecast for an intermittent demand.

11.8 Spare Parts Forecasting Methods: Application and Case Studies

11.8 Spare Parts Forecasting Methods: Application and Case Studies 11.8.1 Case Study 1: Spare Parts Forecasting for an Aircraft Accurate spare parts demand forecasting is a very critical issue in the management of an aircraft fleet. Airline operators often base their predictions on work experience and on information from aircraft manufacturers. Stocking costs, obsolescence risks, or costs incurred by the unavailability of the airplane can be very important. A large stock of spare parts is often required for many reasons, thus making the management of aircraft fleets very difficult. Safety issues and costs due to interruption of service by airplanes being out of service while undergoing maintenance require efficient maintenance policies in cooperation with continuous inspection and preventive maintenance. Airline companies must have a policy for coping with unanticipated mechanical problems when their aircraft are away from their base. The management of spare parts inventory becomes a significant issue in this context. In particular, accurate forecasts of consumption are important and influence both the performance of an airline fleet and economic returns on capital. As demonstrated by Ghobbar and Friend (2002, 2004) and others, lumpiness is a direct consequence of the inner structural features of the operations performed by an airline company, in particular the fierce competition between companies to meet performance targets expected by customers while still making a profit.

417

There are two broad approaches to spare parts selection: the first is based on the operational experience of an enterprise and the second on the application of forecasting techniques. Ghobbar and Friend (2004) found that only 9–10% of companies use forecasting models. Airline operators usually base predictions on their operational experience, on annual budgets, and on information from lists of spare parts recommended by the aircraft manufacturers. The application presented here is a comparison of different forecasting techniques applied to the spare parts of a fleet of Airbus A320 aircraft belonging to an important national airline company. The airline’s technical division collects daily records of the demand for each component. These records are aggregated to provide monthly data. This database covers the 6 years from 1998 to 2004, and more than 3,000 different items are affected with five different levels, or classes, of lumpiness. Each class contains a population of many items, but for the sake of brevity the following analysis refers only to one item per class as a sample. These five groups of lumpiness seem to be typical for aircraft spare parts. To maintain confidentiality, the items are referred to as a, x, y, z, and w. Figure 11.10 presents an illustrative time series of the demand for item z. Table 11.4 contains the values of CV2 on a monthly basis, and ADI for these five items, while positions inside the lumpy area are given in Fig. 11.11. The performance of forecasting methods is evaluated using the MAD as defined in Eq. 11.15. The comparison of the forecasting methods, in terms of evaluating forecast accuracy using MAD and

demand

20

15

10

5

0 1

4

7

10

13

16

19

22

25

28

31

month

Fig. 11.10 Time series of demand for item z

34

37

40

43

46

49

52

55

58

418

11 Spare Parts Forecasting and Management

Table 11.4 Values of the squared coefficient of variation (CV 2 ) and the average interdemand interval (ADI) for the five representative items Item

CV2

ADI

a x y z w

2.56 2.15 0.79 0.59 2.59

1.63 2.19 1.55 1.34 3.17

tio of maximum SMAD to the minimum SMAD for a single item found using different techniques is approximately 1.56, ranging from 1.45 to 1.71, while for any forecasting method the same average fluctuation for different items is approximately 2.16, ranging from 1.84 and 2.31. This clearly demonstrates the dominant influence of lumpiness. These empirical experiments are summarized in Tables 11.6 and 11.7 that show the effectiveness of each model, and that the weighted moving averages (WMA), Croston, exponential weighted moving averages (EWMA), and trend adjusted exponential smoothing models are the best performers. However, the seasonal regression model (SRM) does perform well, particularly for small values of ADI (less than 1.70). Its forecast error for items y and z is very close to that of the best methods. Small values of CV2 and ADI, as for items y and z, improve the performance of each method. In particular, the Holt–Winter method is very competitive in these conditions, with the additive version generally being more effective than the multiplicative one.

Fig. 11.11 Lumpy coordinates of the five representative items

11.8.2 Case Study 2: Spare Parts Forecasting in a Steel Company SMAD parameters, applied to the five selected items, is reported in Table 11.5. The value of SMAD for each item returned using each forecasting method is presented in Table 11.6 in descending order. The column “position” represents the relative weight of a forecast’s performance, using which a comprehensive comparison and evaluation of strengths and weaknesses for each method can be carried out. Table 11.7 shows the total and average scores based on the collected values and the relative weights in Table 11.6. SMAD makes comparison possible in terms of performance of the forecasting methods on different items, as well as their behavior in different conditions of lumpiness (Fig. 11.12). As clearly seen from Fig. 11.12, the dominant parameter is item lumpiness. The choice of the forecasting method is a side issue. All methods generally perform better when applied to items with shady lumpiness such as y and z compared with the best performer applied to items with glaring lumpiness, such as x and w. Moreover, lumpiness is an independent variable and is uncontrollable. The average fluctuation of the ra-

A European leader in the metallurgy sector and trading worldwide has to cope with forecasting the spare parts requirements in its manufacturing plants. This application is focused on comparing the performance of the “traditionally” good methods for spare parts (particularly Croston, WMA, and EWMA) with the family of autoregressive integrated moving average (ARIMA) methods. The available data set contains information covering 5 years for approximately 2,500 items (Fig. 11.13). The analysis of 12 items (A–N) is now reported with different patterns according to Fig. 11.4, while Table 11.8 summarizes the characteristics of the components presented. ARIMA methods are a populated family of forecasting methods whose object is to express the forecast as a function of the previous values of the series (autoregressive terms) and previous values of forecasting error (moving average terms). The model is generally referred to as an ARIMA(p; d; q) model, where p, d , and q are integers greater than or equal to zero linked,

9.26 1.70 0.57 4.54 5.08

1.00 1.11 0.71 0.65 1.31 0.65 1.31 2.02

10.31 1.35 0.60 3.71 5.10

1.12 0.88 0.74 0.53 1.31 0.53 1.31 2.48

1.22 1.42 0.96 0.82 1.56 0.82 1.56 1.90

11.24 2.18 0.77 5.75 6.04

SES DES

1.36 1.39 0.99 0.86 1.66 0.86 1.66 1.94

12.55 2.12 0.79 6.01 6.44 1.19 1.38 0.92 0.74 1.55 0.74 1.55 2.10

11.02 2.12 0.74 5.17 5.99 1.10 1.26 0.88 0.68 1.43 0.68 1.43 2.09

10.17 1.93 0.71 4.80 5.54 1.17 1.22 0.85 0.71 1.42 0.71 1.42 2.00

10.76 1.87 0.68 4.98 5.51 1.16 1.25 0.81 0.68 1.51 0.68 1.51 2.21

10.74 1.91 0.65 4.78 5.84 1.08 1.22 0.83 0.65 1.49 0.65 1.49 2.31

10.00 1.86 0.67 4.54 5.79 1.08 1.20 0.82 0.63 1.50 0.63 1.50 2.37

10.01 1.84 0.66 4.44 5.82 1.10 1.17 0.78 0.65 1.45 0.65 1.45 2.23

10.19 1.79 0.63 4.55 5.61 1.12 1.14 0.75 0.66 1.47 0.66 1.47 2.23

10.33 1.75 0.60 4.63 5.70 1.09 1.14 0.78 0.65 1.48 0.65 1.48 2.27

10.10 1.75 0.62 4.59 5.76 1.11 1.18 0.78 0.62 1.44 0.62 1.44 2.31

10.23 1.80 0.63 4.36 5.58

0.95 1.12 0.69 0.57 1.14 0.57 1.14 2.00

8.77 1.71 0.55 4.00 4.42

1.04 1.10 0.69 0.53 1.18 0.53 1.18 2.23

9.60 1.68 0.55 3.72 4.58

0.92 1.08 0.71 0.55 1.21 0.55 1.21 2.19

8.48 1.65 0.57 3.90 4.71

0.90 1.03 0.65 0.50 1.18 0.50 1.18 2.36

8.31 1.58 0.52 3.51 4.58

0.95 1.10 0.68 0.55 1.20 0.55 1.20 2.18

8.77 1.68 0.54 3.86 4.66

MA(2) MA(3) MA(4) MA(5) MA(6) MA(7) MA(8) MA(9) MA(10) MA(11) MA(12) SRM TAES EWMA WMA Croston

MAD mean absolute deviation, SMAD standardized mean absolute deviation, TAES trend-adjusted exponential smoothing

MAD a 9.79 x 1.38 y 0.62 z 4.05 w — SMAD a 1.06 x 0.90 y 0.77 z 0.58 w Min 0.58 Max 1.06 Max/ 1.84 min

MW AW

Table 11.5 Evaluation of the forecast accuracy for different methods (MAD and SMAD = MAD/A)

11.8 Spare Parts Forecasting Methods: Application and Case Studies 419

420

11 Spare Parts Forecasting and Management

Table 11.6 Classification of methods based on performance evaluation Item a

x

y

z

w

Position

WMA EWMA Croston SES TAES MW SRM MA(7) MA(8) MA(11) MA(4) MA(9) MA(12) AW MA(10) MA(6) MA(5) MA(3) DES MA(2)

AW MW WMA SRM EWMA Croston TAES SES MA(10) MA(11) MA(9) MA(12) MA(8) MA(7) MA(5) MA(6) MA(4) MA(3) MA(2) DES

WMA CROSTON TAES EWMA SES SRM AW MA(10) MW MA(11) MA(12) MA(9) MA(6) MA(8) MA(7) MA(5) MA(4) MA(3) DES MA(2)

WMA AW TAES Croston EWMA SRM MW MA(12) MA(8) MA(7) SES MA(9) MA(11) MA(10) MA(6) MA(4) MA(5) MA(3) DES MA(2)

SRM WMA TAES Croston EWMA SES AW MA(5) MA(4) MA(12) MA(9) MA(10) MA(11) MA(7) MA(8) MA(6) MA(3) DES MA(2)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

1.80 1.60

MAD/A

1.40 x y z w a

1.20 1.00 0.80 0.60

(9 )

(1 0 M ) A (1 1 M ) A (1 2) SR M TA ES EW M A W M A C R O S TO N

M A

M A

(7 )

(8 )

M A

(6 )

M A

(5 )

M A

(4 )

M A

M A

(2 )

(3 )

M A

M A

S

ES D

AW

SE

M W

0.40

Fig. 11.12 Accuracy evaluation of forecasting methods using SMAD = MAD/A. See Table 11.1 for an explanation of the methods item L demand (pieces )

60 50 40 30 20 10 0 1

3

5

7

9

11

13

15

17 19

21

23

25

27

29 31

33

time (months)

Fig. 11.13 Time series of the demand for item L

35

37

39

41 43

45

47

49

51

53 55

57

59

11.8 Spare Parts Forecasting Methods: Application and Case Studies Table 11.7 Total and average score for forecasting methods applied to different items Method

Total score

Average score

WMA Croston EWMA TAES SRM MW AW SES MA(12) MA(11) MA(9) MA(10) MA(8) MA(7) MA(4) MA(5) MA(6) MA(3) DES MA(2)

8 18 21 21 24 24 31 34 54 56 58 58 60 61 70 73 76 89 95 98

1.6 3.6 4.2 4.2 4.8 6.0 6.2 6.8 10.8 11.2 11.6 11.6 12.0 12.2 14.0 14.6 15.2 17.8 19.0 19.6

Table 11.8 Values of CV2 and ADI for the 12 representative items Item

CV2

ADI

Pattern

A B C D E F G H I L M N

0.80 1.03 0.30 0.30 0.69 0.58 0.18 0.00 0.13 0.95 2.26 0.54

1.94 3.28 1.59 1.48 1.28 1.20 2.14 1.23 1.11 2.57 1.43 3.00

Lumpy Lumpy Intermittent Intermittent Erratic Erratic Intermittent Smooth Smooth Lumpy Lumpy Lumpy

421

respectively, to the order of the autoregressive, integrated, and moving average parts of the model. Detailed information can be found in Makridakis et al. (1998). The choice of the ARIMA model and the optimization of its parameter is performed by an iterative process programmed in software for statistical analysis (e. g., Minitab Statistical Software). Figures 11.14 and 11.15, respectively, show the autocorrelation analysis, complete and partial, and the parameter estimations for item L calculated by Minitab Statistical Software. Table 11.9 summarizes the performance of different forecasting methods in terms of MAD and SMAD. The parameters input in several methods such as WMA, EWMA, and seasonal ARIMA (ARIMAs) are optimized by an iterative process programmed in Minitab Statistical Software. For item L, the ARIMA methods perform best, especially ARIMA(2,0,2)(1,0,0)12. The forecast error is still relevant: SMAD is approximately 0.850, as is typical for spare parts forecasting. The application of the same process to the whole set of 12 items is summarized in the classification of merit in Table 11.10. Several guidelines can be formulated in this heterogeneous situation. The Croston method performs best for erratic patterns, with a significant reduction in SMAD of 20–30%. This trend is also confirmed for slow-moving patterns, but the difference is considerably less at 5–8 %. For an intermittent pattern, ARIMA models perform best. Lumpy patterns are generally forecast well by ARIMA models, but the EWMA method is strong when values of ADI and CV2 are high, i. e., near the

Autocorrelation Function for item L (with 5% significance limits for the autocorrelations)

Partial Autocorrelation Function for item L (with 5% significance limits for the partial autocorrelations) 1.0

0.8

0.8 Partial Autocorrelation

1.0

Autocorrelation

0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0

0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0

1

2

3

4

5

6

7

8 9 Lag

10

11

12 13

14

15

1

Fig. 11.14 Autocorrelation analysis for item L. Minitab Statistical Software

2

3

4

5

6

7

8 9 Lag

10 11 12 13 14 15

422

11 Spare Parts Forecasting and Management 0.49 ADI

ARIMA model: item L Estimates at each iteration Iteration 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

SSE 2983.20 2978.28 2975.19 2972.31 2969.10 2965.18 2959.97 2951.94 2937.25 2908.05 2822.13 2797.74 2767.09 2764.72 2764.51 2764.14 2764.10

0.100 0.112 0.262 0.412 0.562 0.711 0.860 1.009 1.153 1.303 1.406 1.445 1.446 1.452 1.455 1.457 1.458

0.100 0.112 -0.002 -0.109 -0.216 -0.325 -0.445 -0.585 -0.733 -0.842 -0.857 -0.904 -0.886 -0.887 -0.888 -0.888 -0.887

Parameters 0.100 0.100 0.088 0.089 0.238 -0.021 0.388 -0.125 0.538 -0.228 0.688 -0.335 0.838 -0.453 0.988 -0.591 1.133 -0.741 1.280 -0.851 1.344 -0.868 1.374 -0.920 1.354 -0.914 1.354 -0.915 1.355 -0.917 1.355 -0.917 1.355 -0.918

EWMA 3.520 3.354 3.197 3.011 2.826 2.652 2.527 2.487 2.506 2.331 1.964 2.029 1.964 1.957 1.951 1.943 1.940

3.0

1.32 1.0

Croston 0

Relative change in each estimate less than 0.0010

Fig. 11.15 Estimation of autoregressive integrated moving average (ARIMA) parameters for item L. Minitab Statistical Software. SSE sum of squares due to error Table 11.9 Performance comparison of forecasting methods for item L

a b

Method

MAD

MAD/A

WMA(3 periods) WMA(5 periods) WMA(7 periods) EWMA ARIMA(1,0,0) ARIMA(0,0,1) ARIMA(1,0,1) ARIMA(2,0,0) ARIMA(0,0,2) ARIMA(2,0,2) ARIMA(2,0,1) ARIMA(1,0,2) ARIMA(2,1,2) ARIMA(1,1,1) a ARIMA(2,0,2)(0,1,0)12 ARIMA(2,0,2)(0,1,0)12 without constant ARIMA(1,0,2)(0,1,0)12 without constant ARIMA(2,0,2)(1,0,0)12 without constant ARIMA(2,0,2)(0,0,1)12 ARIMA(2,0,2)(1,0,0)12 Croston Croston modified b

6.316 6.291 6.617 4.750 5.317 5.317 5.310 5.319 5.322 5.001 5.315 5.072 5.217 5.249 4.854 4.880 5.525 4.432 4.473 4.327 7.055 5.053

1.237 1.232 1.296 0.930 1.041 1.041 1.040 1.042 1.042 0.979 1.041 0.993 1.021 1.028 0.950 0.956 1.082 0.868 0.876 0.847 1.381 0.989

Seasonal ARIMA (Makridakis et al. 1998) Modified according to Syntetos and Boylan (2001)

lower limit of the lumpy class. ARIMA methods are very interesting when “seasonality” is present, which is very difficult to discover in lumpy behavior (e. g., item B): ARIMAs, with SMAD of 50%, work significantly better than the others.

ARIMA

2.0

0

0.5

1.0

1.5

2.0

CV2

Fig. 11.16 The technical merit of forecasting methods. EWMA exponential weighted moving averages

It is important to note that the number of iterations required to optimize the parameters in the ARIMA approach can be high, especially in a lumpy pattern. The modified Croston method (Syntetos and Boylan 2001) mainly works better than the original Croston method for lumpy patterns with high lumpiness in terms of ADI. The generalization of the proposed approach, which was applied to almost 900 items, results in the development of a technical diagram of merit for the forecasting methods that presents the best method according to the area of the item in terms of CV2 and ADI values (Fig. 11.16).

11.9 Methods of Spare Parts Management The goal of an efficient spare parts management system is to minimize the total cost. The forecasting problem of spare parts has been investigated in the preceding pages. When the requirement of spare parts is based on an estimate, the challenge is to optimize the management of these items. The main questions for a manager forecasting the future demand for a set of spare parts are which items to stock at the maintenance division of the company, and how many? There are two fundamental strategies in tackling this problem: the first is based on qualitative methods and the second on quantitative methods.

11.9 Methods of Spare Parts Management

423

Table 11.10 Technical merit for different items Item

CV2

ADI

Pattern

Forecasting technical merit 1°(best)

2°

3°

4°

5°(worst)

Croston modified Croston WMA WMA EWMA EWMA Croston WMA WMA WMA ARIMA Croston

WMA

A

0.80

1.94

Lumpy

ARIMA

EWMA

Croston

B C D E F G H I L M N

1.03 0.30 0.30 0.69 0.58 0.18 0.00 0.13 0.95 2.26 0.54

3.28 1.59 1.48 1.28 1.20 2.14 1.23 1.11 2.57 1.43 3.00

Lumpy Intermittent Intermittent Erratic Erratic Intermittent Smooth Smooth Lumpy Lumpy Lumpy

ARIMAs ARIMA ARIMA Croston Croston ARIMA Croston Croston ARIMAs Croston Croston mod

Croston modified Croston Croston Croston modified Croston modified EWMA ARIMA ARIMA EWMA Croston modified ARIMA

EWMA EWMA EWMA ARIMAs ARIMAs Croston modified EWMA EWMA Croston modified EWMA EWMA

WMA Croston modified Croston modified WMA WMA WMA Croston modified Croston modified Croston WMA WMA

ARIMAs seasonal ARIMA

11.9.1 Spare Parts Management: Qualitative Methods The goal of these approaches is to determine which spare parts should be stocked in the local warehouse. A set of significant spare parts management parameters is evaluated qualitatively. Several authors (Botter and Fortuin 2004; Cobbaert and Van Oudheusden 1996; Braglia et al. 2004) support qualitative approaches to approximating low levels of demand, and reject sophisticated mathematical models with complex distribution functions on the grounds that all the work involved in applying and preserving them is not worth the result. Some suitable qualitative solutions are now reported.

11.9.1.1 The VED Approach The starting point of the VED approach (Botter and Fortuin 2000) is a qualitative classification of service parts into vital, essential, and desirable, which is carried out by analyzing a set of factors. Criticality is the main rule and it can relate closely to several parameters, as reported in Table 11.11. Table 11.12 shows an example of the decision levels for each factor. The three decision levels usually correspond to VED classification. The cut sets for these criteria are clearly related to the specific case. The authors suggest that the specific case be analyzed by focusing on a small number of factors, and

that the analysis of each factor concentrates on a small number of feasible values. Since the choice is absolutely arbitrary, an unfounded situation is very commonly generated. Figure 11.17 presents an example of a framework built on these three factors. Assuming a two-level value for these three factors (i. e., short–long, low–high) it is possible to identify eight areas, each corresponding to a particular management strategy for the spare parts: 1. Low price, short response time, high usage. These cheap, fast-moving spare parts have to be stocked in local warehouses in large quantities. 2. Low price, short response time, low usage. These cheap, slow-moving items also have to be stocked close to the market, but in lower quantities. 3. Low price, long response time, high usage. Inventory and transport costs for these items should be investigated in order to determine whether or not the local stock level of these items is the most economical. For instance, some or all of the local stock of fast-moving parts could be positively substituted and absorbed by transport costs through the shipping of larger quantities using more economical means of transport. 4. Low price, long response time, low usage. In this case no stock can be a good solution. 5. High price, short response time, high usage. Because of the short response time, the expensive stocking of these items, primarily in local warehouses, must be managed particularly carefully.

424

11 Spare Parts Forecasting and Management

Table 11.11 Criticality factors Factor

Comment

Response time Functionality

Maximum time between a call for help and restoration of the system’s functionality Effect the failure of an item has on the system’s availability: an item is functional if the system cannot function without it, or is merely cosmetic if the system can continue to run without it, possibly with minor restrictions Total demand for an item in a unit of time, expressed in units or in money Newly developed, established, continued, or soon to be phased out An item can be (relatively) cheap or expensive Time between placing an order with the supplier of an item and the moment it is ready for use The possibility of restoring an item’s functionality after failure

Consumption Stage of the life cycle Price Purchase lead time Repairability

1

Criteria

Options

Response time

2–4 h Next business day Later than next business day AMC < 5 5 < AMC < 100 AMC > 100 Introduction Maturity Decline P < 100 100 < P < 1;000 P > 1;000 LT < 1 1 < LT < 3 LT > 3

high

Demand (AMC) 8 low

high

short long Response time

Pr

low

Life cycle

ice

Consumption (pieces)

Table 11.12 Choice of criteria and options for spare parts criticality (example)

Fig. 11.17 Framework based on three factors: consumption, response time, and price (Botter and Fortuin 2000)

The quantities should be the minimum required to meet the desired level of service. 6. High price, short response time, low usage. In this case no stock can be the best solution. 7. High price, long response time, high usage. A trade-off analysis is required to choose between maintaining a local stock or no stock for these parts. 8. High price, long response time, low usage. The same observation as in area 7.

P (dollars)

Purchase LT (weeks)

AMC average monthly consumption, P price, LT lead time Table 11.13 Inventory policy matrix Stocking policy

No stock Single item inventory Just-in-time inventory Multi-item inventory

Spare parts classification A

B

C

 

  



D



11.9.1.2 Multiattribute Spare Tree Analysis As a result of taking a decision tree approach (Braglia et al. 2004), the authors define four stocking policies and four classes of spare parts. As before, the classification in this case is based on qualitative parameters. The method provides one or more eligible policies for each class. Table 11.13 shows the grid of spare part classes and stocking policies.

Fig. 11.18 The main choice for criticality (Braglia et al. 2004)

As in the VED approach, the method is centered on the concept of spare part plant criticality with its three levels: desirable, important, and critical (see

11.9 Methods of Spare Parts Management

425

Fig. 11.19 Subtree 1: spare part plant criticality as critical (Braglia et al. 2004)

Fig. 11.20 Subtree 2: spare part plant criticality as important (Braglia et al. 2004)

Fig. 11.18). Each level is given by the application of an analytic hierarchy process, as in Sharaf and Helmy (2001). This main choice is influenced by a great many attributes, such as quality, production loss, domino effect, safety, and spare part characteristics, making a qualitative evaluation necessary. For example, if the rating for quality is between 0 and 85%, the authors suggest an evaluation is critical, while between 85 and 95% it is important, and it is desirable between 95 and 100%. Dealing with qualitative evaluation, these cut sets are absolutely arbitrary. Following this, a tun-

ing phase is usually required, preferably carried out by maintenance experts according to the application examined. Three alternative decision-making subtrees are reported in Figs. 11.19–11.21, and are covered by the assigned level. The final result is the classification of the spare part; hence, the related best policy is that shown in Table 11.13. In conclusion, it is important to note that qualitative methods are simple, rapid, and usually cheap, very interesting features especially when the mainte-

426

11 Spare Parts Forecasting and Management

Fig. 11.21 Subtree 3: spare part plant criticality as desirable (Braglia et al. 2004)

nance database is poor. Furthermore, it is possible to consider several intangible additional factors such as the quality of suppliers, obsolescence, and transport. They can help to reduce the size of the problem, subsequently making it positively suitable for a quantitative approach.

N

Pd,T,1 N–1

Pd,T,2

N–2 Pd,T,N-1 1 t0

t0+T

time

Fig. 11.22 Trend of spare parts stock with time

11.9.2 Spare Parts Management: Quantitative Methods The aim of these procedures is very ambitious: the accurate prediction of the numbers of spare parts to stock in the company’s local warehouse. Several authors (Giri et al. 2005; Chang et al. 2005; Gutierrez et al. 2008) proposed different approaches based on linear programming, simulation, and many other methods, normally focusing on the optimization of the level of service or inventory costs. Unfortunately, the extreme degree of complexity or oversimplification sometimes tends to restrict their effectiveness in real applications. Several methodologies successfully tested in industrial applications are now presented.

11.9.2.1 Minimum Total Cost Method This basic method (Roversi and Turco 1974) considers two fundamental costs, the holding cost and the

shortage cost. These two result in a trade-off problem. Moreover, spare parts storage is an expensive activity in which the probability of the use of the components is low. On the other hand, should there be a shortage of spare parts, the related loss of production can be very expensive. (11.18) Ctot .N / D CH C CS ; where Ctot is the total cost, N is the number of spare parts to stock, CH is the holding cost, and CS is the shortage cost. The holding cost term CH can be estimated as follows. In considering a single item, N is the number of spare parts in the warehouse at the point in time t0 D 0, and T is the interval between two consecutive supplies. The number of items in stock remains N if there is no consumption in T , the corresponding probability being Pd;T;0. In the case of a single consumption, the stock decreases to (N  1) with probability Pd;T;1 , and so on. The situation concerning spare parts stock is explained in Fig. 11.22.

11.9 Methods of Spare Parts Management

427

In conclusion, the holding cost is estimated by

CH D R NPd:T;0 C .N  1/Pd:T;1

 C .N  2/Pd:T;2 C : : : C Pd:T;N 1 ; (11.19)

where R is the purchase cost of the item, ' is the annual stock percentage cost (on purchase cost), Pd;T;x is the probability of x items being consumed during supply time T , and d is the average consumption of the item. Production losses could occur if the number of failures during the time T exceeds the number N of parts initially supplied. The cumulative probability of this event is calculated by P D Pd:T:N C1 CPd:T:N C2 CPd:T:N C3 C: : : (11.20) The probability Pd;T;x is easily computed using the Poisson formula (see Sect. 11.5). Assuming Cm is the cost resulting from the shortage of a single spare part, the shortage cost is CS D Cm dP:

(11.21)

These two cost terms are functions of N with opposite trends, and the optimal value for N sets their sum to a minimum, i. e., the total cost Ctot is minimized.

11.9.2.2 Numerical Application The main motor of an automatic labeling machine has an average consumption d D 2 pieces/year and a purchase cost R D ¤ 5,000 per piece. The manufacturer purchases spare parts every 4 months (i. e., supply time T D 4 months). Should a failure occur, this

7000 6000

[€/year]

5000 CH

4000

motor causes expensive production losses and emergency procurements that from historic Cm amount to ¤ 6,700 per piece. The annual stock percentage value is D 22%. Using the Poisson formula to calculate the probability Pd;T;n , we get the values given in Table 11.14. According to Eqs. 11.19 and 11.21 and considering different values of N , the total cost Ctot can be estimated. For N D 2,

CH D R  NPd:T;0 C .N  1/Pd:T;1  C .N  2/Pd:T;2 C : : : C Pd:T;N 1 D 5000  0:22 Œ2  0:513 C .2  1/  0:342 D ¤ 1504 per year and CS D Cm dP D Cm d.Pd;T;3 C Pd;T;4 C : : :/

 D Cm d 1  .Pd;T;0 C Pd;T;1 C Pd;T;2 / D 6700  2  0:031 D ¤ 415 per year. Table 11.15 shows the costs as a function of N . The optimal solution provides two motors in stock as spare parts (Fig. 11.23). This minimal cost model is very easy to use and is widely diffused among engineers and practitioners. The literature presents several graphical abacuses based on this model in support of the rapid solution of the proposed problem. Figure 11.24 shows one of these abacuses: the entering side (input zone) requires the average item consumption d and the supply time T . Following the different zones and using the remain-

Table 11.14 Poisson probability distribution N

Pd;T;n

0 1 2 3 4 5

0.513 0.342 0.114 0.025 0.004 ...

CS

3000

Ctot (€/year)

Table 11.15 Cost plan according to N values

2000 1000 0 0

1

2

3

N

Fig. 11.23 Optimal number of spare parts

4

N

CH (¤/year)

CS (¤/year)

Ctot (¤/year)

0 1 2 3

0 564 1,504 2,571

6,525 1,943 415 80

6,525 2,507 1,919 2,651

428

11 Spare Parts Forecasting and Management I n p u t region

Output region

T constant 1 w 2 ee w ks ee ks 1 m on th 3 m on th 6 m s on t h 1 ye s a 2 r

N= 0

N= 1

N= 2

N= 3 N= 4 N= 5

s ar

ye

0.001

d (pieces/year)

0.01

0.1

N= 6

1

10

Cm ( €/ p i e c e )

5 50 5 00 5

5 000

50

5 0000

(€/ p i e c e × year) 5 00

5 00000

Rϕ

5 000

Fig. 11.24 Example of abacus for the optimal number N of spare parts

ing input (i. e., R, ', Cm /, the final result in the output zone is the optimal number of spare parts.

11.9.2.3 Stock Level Conditioned to Least-Availability Method This method (Regattieri 1999) is based on the definition of asymptotic availability: AD

MTTF ; MTTF C MTTR

The MTTR is derived from different factors, as shown in Fig. 11.25. Its value is strongly dependent on the number of spare parts stored in the particular case, as reported in Fig. 11.26. Assuming no spare parts are in stock, let tN be the point in time corresponding to a failure and f .tN / its corresponding probability distribution. Consequently, the expected waiting time is ta D T  tN , with an average value of M.ta / given by RT

(11.22)

where MTTF is the mean time to failure and MTTR is the mean time to repair.

M.ta / D

0

.T  tN /f .tN / dtN : RT f .t / dt N N 0

(11.23)

11.9 Methods of Spare Parts Management

429

MTTR

Predictive maintenance

call

set-up

disassembly

Breakdown maintenance

reparation

supply

from market

calibration

assembly

check

closure

from stock

Fig. 11.25 Structure of the mean time to repair (MTTR)

tN ; f(tN)

The average value of MTTR can be expressed by the sum of the average waiting time M.ta / and Treplace , i. e., the amount of time due to all the remaining factors (except time of supply, e. g., disassembling, repairs):

time 0

T

2T

Fig. 11.26 Spare parts waiting time evaluation

MTTR D Treplace C M.ta / RT .T  tN /f .tN / dtN : D Treplace C 0 R T 0 f .tN / dtN (11.24) It is worth noting that increasing N lowers MTTR, with corresponding improvement of availability A and collapse of downtime costs. Moreover, the method affords the quantitative definition of the holding cost CH as in Eq. 11.19. An iterative process can be used to evaluate CH in order to find the optimum value of N by setting CH to a minimum while allowing the minimum level of availability Amin .N / to guarantee ontime dispatch of several technical requests (e. g., safety questions or productivity level):

8 ˚

ˆ min CH D min R NPd;t;0 C .N  1/Pd;T;1 C    ˆ ˆ ˆ  < C Pd;T;N 1 ˆ ˆ ˆ MTTF ˆ :A.N / D  Amin : MTTF C MTTR.N / (11.25)

11.9.2.4 Case Study: Spare Parts Optimization for a Motorcycle Manufacturer A worldwide leader in motorcycle race championships was having problems with spare parts optimization in

Table 11.16 Classes and values for the factors in the criticality index Lead time of supply LTmin

S

EV

LTmax

PAC

¤/piece

XEV

(%)

XS

Days

XLTm

Days

XLTM

Pieces/year

XPAC

0–15 16–50 51–200 201–500 501–...

0.2 0.4 0.6 0.8 1.0

0–10 11–50 51–89 90–100

0.2 0.5 0.8 1.0

0–1 2–5 6–22 23–...

0.2 0.4 0.7 1.0

0–5 6–10 11–44 45–...

0.2 0.4 0.7 1.0

0–5101 5101 –1 1.1–5 5.1 –...

1.0 0.8 0.45 0.2

EV economic value of the item, S possibility of substituting the item with a similar item when the part is not in stock, LT lead time of supply, PAC past average consumption

430

11 Spare Parts Forecasting and Management 100

C

90

Cumulative percentage

80 70

B

60 50 40

A

30 20 10 0

Fig. 11.27 ABC analysis for the criticality index

192

Number of items

3,000

2

spare parts classificaon (CV and ADI) 1.00 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00

2

Errac

Lumpy

N L

U

M G

H

F

C

O

A D 0.4

0.6

0.8

1.0

V

E 1.2

Q 1.4

F - BC06SD3476 G - DF573 H - FG563ER52 I - FF456LK86 L - EW986GH527

A - FG56WT356 B - AA65GH354 C - QA3465 D - QA3489 E - RT678YH452

S

P

smooth

0.2

R

I

B

0.0

T

1.6 ADI

1.8

Intermient

2.0

2.2

M - XC190FG0236 N - WE345JK784 O - WE3729RF45 P - D4239 Q - FG56WT654

2.4

2.6

2.8

3.0

3.2

R - DC0946SS4572 S - WE34897KL213 T - GG54TY673 U - DF254 V - AA65DFGE523

Fig. 11.28 Example of spare parts classification (CV2 and ADI)

item BC06SD3476 250

forecast (pieces)

200

150

historical

100

binomial

Poisson

50

0 0

1000

2000

3000

4000

T (h)

Fig. 11.29 Forecasting performance for a smooth item

its main production plant. These spare parts were routinely managed using the experience of staff involved in manufacturing and assembly operations. Significant

losses in production due to shortage of spare parts occurred, with high costs being incurred due to large numbers of parts in stock and the risk of obsolescence. The high number of items to be considered is often a problem. In this case more than 3,000 spare parts are codified and used, and on which, in terms of time and cost, it is neither convenient nor possible to carry out a systematic analysis. Rather, it is important to “simplify” the data set by, e. g., introducing an “item criticality index” composed of several factors. In this case the economic value of the item, the possibility of substituting it with a similar item when the part is not in stock, the lead time of supply, and the past average consumption are related to each other in an average weighted value as follows: PF i D1 pi Xi ; Criticality index D 1  P F i D1 pi

(11.26)

11.9 Methods of Spare Parts Management

431

16

Table 11.17 DX345SS387

14

N (pieces)

CH (¤/year)

CS (¤/year)

Ctot (¤/year)

12

1 2 3 4 5 6 7 8 9 10

43 65 94 120 164 230 327 462 664 950

910 468 240 116 49 21 16 4 3 1

953 533 334 236 213 251 343 466 667 950

forecast (pieces)

item DC0946SS4572

10

historical

8

Poisson binomial

6 4 2 0 0

1000

2000

3000

4000

Minimum total

cost

method

for

item

T (h)

Fig. 11.30 Forecasting performance for an intermittent item item XC190FG0236 25

forecast (pieces)

20

15

historical binomial Poisson

10

5

0

0

1000

2000

3000

4000

T (h)

Fig. 11.31 Forecasting performance for an erratic (or lumpy) item

where F is the number of factors, pi is the relative weight assigned to the factor, and Xi is the value as-

signed to the item as a function of its range for each factor. The smaller the index, the greater the criticality. Table 11.16 shows these factors, their ranges, and the values Xi for each class and each factor. The relative weights pi used in this application for the economic value of the item, the possibility of substituting it with a similar item when the part is not in stock, the lead time of supply, and the past average consumption are 0.4 (40%), 0.2 (20%), 0.1 (10%), and 0.3 (30%), respectively. All the items are ranked in increasing order of the criticality index, and for every item the specific criticality index is expressed as a percentage of the total amount of the indexes in order to obtain a cumulative percentage order. Hence, the items are classified in three categories A, B, and C, as reported in Fig. 11.27. The first 30% of the cumulative percentage forms a reduced set composed of 192 critical items representing approximately 73% in value of the whole set of spare

costs 1000

CH [€/year] CS [€/year] Ctot [€/year]

900 800

[€/year]

700 600 500 400

Saving

300 200 100 0 0

Fig. 11.32 Costs and saving for item DX345SS387

1

2

3

4

Optimal value

5

6

7

8

9

10

Spare parts in stock

432

parts in stock. This set is analyzed according to the well-known lumpiness parameters CV2 and ADI (see Fig. 11.28), leading to some interesting observations about the consumption forecast and optimal management. Forecasting of each item in the set was performed using both Poisson and binomial distributions, and the results were compared with the historical consumption in order to specify the best alternative. In brief, smooth items are best forecast by the Poisson formula, while for intermittent items the results from Poisson and binomial distributions are substantially good, and the bi-

11 Spare Parts Forecasting and Management

nomial distribution is slightly better for lumpy and erratic patterns. Figures 11.29–11.31 present a collection of different forecasting results for smooth, intermittent, and erratic items, respectively. The optimal management of the consumption forecast for the 192 items in the set is performed by the minimum total cost method as reported for a specific item in Table 11.17 and Fig. 11.32. Regarding the whole set, the optimal quantity is often smaller than the average number of spare parts in stock, with a consequent saving of several hundred thousand euros per year (approximately 37% of total cost).

12

Applications and Case Studies

Contents 12.1 Preventive Maintenance Strategy Applied to a Waste to Energy Plant . . . . . . . . . . . . . . . . . . . . 12.1.1 Motor System Reliability Evaluation . . . . . . . 12.1.2 Bucket Reliability Evaluation . . . . . . . . . . . . . 12.1.3 Motor System. Determination of Maintenance Costs . . . . . . . . . . . . . . . . . . . . 12.1.4 Time-Based Preventive Replacement for the Motor System . . . . . . . . . . . . . . . . . . . . 12.1.5 Time-Based Preventive Replacement for the Bucket Component . . . . . . . . . . . . . . . . 12.1.6 Time-Based Preventive Replacement with Durations Tp and Tf . . . . . . . . . . . . . . . . . 12.1.7 Downtime Minimization . . . . . . . . . . . . . . . . . 12.1.8 Monte Carlo Dynamic Analysis . . . . . . . . . . . 12.1.9 Monte Carlo Analysis of the System . . . . . . . 12.2 Reliability, Availability, and Maintainability Analysis in a Plastic Closures Production System for Beverages . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.1 RBD construction . . . . . . . . . . . . . . . . . . . . . . . 12.2.2 Rotating Hydraulic Machine . . . . . . . . . . . . . . 12.2.3 Data Collection and Reliability Evaluation of Components . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.4 Reliability Evaluation, Nonrepairable Components/Systems . . . . . . . . . . . . . . . . . . . . 12.2.5 Data on Repairs and Maintenance Strategies 12.2.6 Monte Carlo Analysis of the Repairable System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.7 Alternative Scenarios and System Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . .

433 434 436 437 439 439 441 442 442 446

446 448 449 449 454 456 456 460

12.3 Conclusions and Call for New Contributions . . . . 462

Reliability, availability, maintainability, quality, safety, preventive maintenance, inspections, spare parts, failure modes and effects analysis (FMEA), failure mode, effects, and criticality analysis (FMECA), reliability block diagrams, fault tree analysis, Markov

chains, etc. are just a few measures, parameters, and tools introduced in the previous chapters. Several models and methods, together with supporting numerical examples and applications, have been illustrated. This chapter aims to present some applications for industrial case studies where the previously introduced models and methods have been effectively applied. We hope that readers will be able to contribute to a new edition of this book by proposing industrial case studies which we can analyze further.

12.1 Preventive Maintenance Strategy Applied to a Waste to Energy Plant With reference to the case study introduced in Sect. 8.8 dealing with the maintenance optimization of a waste to energy (WtE) plant, this application presents the results obtained by the introduction of the analytical models for planning of preventive maintenance actions, as discussed in Chap. 9. The production system is made up of two lines for the waste treatment, supplied by two bridge cranes acting as primary loaders of the furnace hoppers. Every bridge crane has a motor system and a bucket among its most important elements. Tables 12.1 and 12.2 collect the time records of failure for these elements during a period of time T and quantify the time to failure (ttf) values representing the input of a statistical analysis and reliability evaluation for the determination of the best preventive rule. In particular, considering the bridge crane 1, the motor system registered 15 failure events from January 2005 to March 2007, while the bucket had nine fail-

R. Manzini, A. Regattieri, H. Pham, E. Ferrari, Maintenance for Industrial Systems © Springer 2010

433

434

12 Applications and Case Studies

ures during the period from July 2005 to March 2007. Figure 12.1 presents the frequency distribution of the ttf for the motor system and the bucket of the bridge crane 1. The analytical models for the optimization of the replacements of components and systems are based on the determination of reliability measures such as the reliability function, hazard rate, and expected number of failures. For this purpose, it is possible to apply a parameter estimation of these measures using literature statistical distributions such as Weibull, exponential, and normal, or a nonparametric evaluation, e. g., based on the Kaplan–Meier method and on a confidence interval (see the theory and applications illustrated in Chaps. 5 and 6). The following section presents some of the most significative results achieved by the application of both evaluation approaches, for the motor system and the bucket components respectively.

12.1.1 Motor System Reliability Evaluation

t – TTF

R.t /

LCI limit

UCI limit

.t /

2 4 5 10 12 14 37 47 51 52 59 74 86 151 168

0.933 0.867 0.800 0.733 0.667 0.600 0.533 0.467 0.400 0.333 0.267 0.200 0.133 0.067 0.000

0.807 0.695 0.598 0.510 0.428 0.352 0.281 0.214 0.152 0.095 0.043 0.000 0.000 0.000 0.000

1.000 1.000 1.000 0.957 0.905 0.848 0.786 0.719 0.648 0.572 0.490 0.402 0.305 0.193 0.000

0.067 0.071 0.077 0.083 0.091 0.100 0.111 0.125 0.143 0.167 0.200 0.250 0.333 0.500 1.000

TTF time to failure, LCI lower confidence interval, UCI upper confidence interval

tion f .t/, survival function R.t/, and hazard rate .t/, assuming a Weibull distribution with shape parameter 0.9455 and scale parameter 50.21 days. Figure 12.4 presents the same reliability functions but assuming an exponential distribution for the historical observations, with a correspondent Anderson–Darling index equal to 1.020. The scale parameter is 51.467 days, i. e., the constant hazard rate is 0.194 day1 . Finally, Figs. 12.5 and 12.6 present the results obtained by a nonparametric reliability evaluation. In particular, the values of the survival function assuming a confidence interval of 95% (Fig. 12.5) and a hazard rate (Fig. 12.6) are reported in Table 12.1.

3

3

2

2

Frequency

Frequency

Figure 12.2 presents the well-known four-way probability plot for the ttf. This is a very useful tool to identify the best parametric statistical distribution function capable of fitting the historical observations of failure events. The values obtained for the Anderson–Darling index suggest the Weibull probability function is the best-fitting distribution for the available data. In particular, Fig. 12.3 presents the trend of the most important estimated functions, the probability density func-

Table 12.1 Motor system. Nonparametric estimation. Values of R.t / and .t /

1

0

1

0

0

20

40

60

80

100 120 140 160

ttf - motor system Fig. 12.1 Time to failure (ttf ) (days) distribution for two components of bridge crane 1

0

25

50

75

100

125

ttf - bucket

150

175

200

12.1 Preventive Maintenance Strategy Applied to a Waste to Energy Plant

435

Motor System. Four-way Probability Plot of ttf ML Estimates - Complete Data

Lognormal base e

99 95 90 80 70 60 50 40 30

99

Anderson–Darling

95

Percent

Percent

Weibull

20 10 5

Weibull 0.991

80 70 60 50 40 30 20

Lognormal base e 1.137 Exponential 1.020

10

3 2

5

1

Normal

1 1

10

1.484

100

1

10

Exponential

100

1000

Normal

99

99

98

95

97

Percent

Percent

95 90 80 70 60 50

80 70 60 50 40 30 20 10 5

30 10

1 0

100

200

0

100

200

Fig. 12.2 Motor system. Four-way probability plot of ttf. Minitab® Statistical Software. ML maximum likelihood

Motor System. Overview Plot of ttf. Weibull function ML Estimates - Complete Data

Probability Density Function

Weibull Probability

Percent

0.02

0.01

99 95 90 80 70 60 50 40 30

Shape Scale

0.9455 50.21

MTTF

51.503

Failure Censor

15 0

20

Goodness of Fit

10

AD*

5

0.991

1

0.00 0

50

100

150

200

1

250

10

Survival Function

100

Hazard Function

1.0

0.025

0.9

0.024

0.8

0.023 0.022

0.6 0.5

Rate

Probability

0.7

0.4

0.021 0.020

0.3 0.019

0.2

0.018

0.1 0.0

0.017 0

50

100

150

200

250

0

50

100

150

200

250

Fig. 12.3 Motor system. Parametric estimation. Weibull function. Minitab® Statistical Software. MTTF mean time to failure

436

12 Applications and Case Studies

Motor System. Overview Plot of ttf. Exponential function ML Estimates - Complete Data

Probability Density Function

Exponential Probability

0.02 99

Percent

95 0.01

100

MTTF

51.467

Failure Censor

15 0

Goodness of Fit AD*

70 60 50 40 30 20 10 5 1 0

1 51.467

90 80

0.00

Shape Scale

0

200

100

Survival Function

1.02

200

Hazard Function 0.0200

1.0 0.9 0.8

0.6 0.0195

0.5

Rate

Probability

0.7

0.4 0.3 0.2 0.1

0.0190

0.0 0

100

200

0

100

200

Fig. 12.4 Motor system. Parametric estimation. Exponential function. Minitab® Statistical Software

Nonparametric Survival Plot for ttf - motor system Kaplan-Meier Method - 95.0% CI Complete Data 1.0

MTTF

51.467

0.9

Median IQR

47.000 64.000

0.8

Probability

0.7 0.6 0.5 0.4 0.3

Fig. 12.5 Motor system. Nonparametric estimation. Survival function and confidence interval (CI). Minitab® Statistical Software. IQR interquartile range

0.2 0.1 0.0 0

12.1.2 Bucket Reliability Evaluation Figures 12.7–12.9 and Table 12.2 report the estimated values and trends obtained by the application of both

50

100

150

Time to Failure

parametric and nonparametric statistical evaluation analyses, as similarly performed for the most important reliability functions for the motor system reported in the previous section.

12.1 Preventive Maintenance Strategy Applied to a Waste to Energy Plant

437

Nonparametric Hazard Plot for ttf - motor system Empirical Hazard Function Complete Data 1.0

MTTF

51.467

0.9

Median IQR

47.000 64.000

0.8 0.7

Rate

0.6 0.5 0.4 0.3 0.2 0.1

Fig. 12.6 Motor system. Nonparametric estimation. Hazard function. Minitab® Statistical Software

0.0 0

50

100

150

Time to Failure

Bucket comp. Four-way Probability Plot of ttf ML Estimates - Complete Data

Lognormal base e

99 95 90 80 70 60 50 40 30 20

99

Anderson–Darling

95

Percent

Percent

Weibull

10 5

Weibull 1.313

80 70 60 50 40 30 20

Lognormal base e 1.379 Exponential 1.324

10

3 2

5

1

Normal

1 1

10

100

1.521 10

100

Exponential

1000

Normal

99

99

98

95

97

Percent

Percent

95 90 80 70 60 50

80 70 60 50 40 30 20 10 5

30 10

1 0

100

200

300

0

100

200

®

Fig. 12.7 Bucket component. Four-way probability plot of ttf. Minitab Statistical Software

12.1.3 Motor System. Determination of Maintenance Costs The motor system and the bucket of bridge crane 1 are both subject to corrective and preventive maintenance, mainly consisting of actions of replacement. The com-

ponent is assumed to be “as good as new” after the completion of the generic recovery action, whose duration is the time to repair (ttr). The mean value of ttr (MTTR) is about 8 h, equivalent to about 10 h=year of idle time for the WtE plant. The corresponding cost for the lost production of electricity is about US$ 2,103

438

12 Applications and Case Studies

Bucket comp. Overview Plot of ttf. Weibull function ML Estimates - Complete Data Weibull Probability

Probability Density Function

Percent

0.010

0.005

99 95 90 80 70 60 50 40 30 20

Shape Scale

1.0807 69.247

MTTF

67.218

Failure Censor

9 0

Goodness of Fit

10

AD*

5

1.313

1

0.000 0

100

200

1

300

Survival Function

10

100

Hazard Function 0.018

1.0 0.9

0.017

0.8 0.016

0.6

0.015

0.5

Rate

Probability

0.7

0.4 0.3

0.014 0.013

0.2 0.012

0.1 0.0

0.011 0

100

200

300

0

100

200

300

Bucket comp. Overview Plot of ttf. Esponential function ML Estimates - Complete Data

Probability Density Function

Exponential Probability

0.015

99

0.010

Percent

95

80

0.000

70 60 50 40 30 20 10 5 1 100

200

1 67.222

MTTF

67.222

Failure Censor

9 0

90

0.005

0

Shape Scale

Goodness of Fit AD*

0

300

Survival Function

100

200

1.324

300

Hazard Function 0.0154

1.0 0.9 0.8

0.6 0.0149

0.5

Rate

Fig. 12.8 Bucket component. Parametric estimation. Weibull compared with exponential. Minitab® Statistical Software

Probability

0.7

0.4 0.3 0.2 0.1

0.0144

0.0 0

100

per year, considering a cost per hour equal to US$ 0.21 per kilowatt-hour, while the cost for the not produced and distributed heat is about US$ 2,277 per year, considering a cost per hour equal to US$ 0.08 per kilowatthour. Consequently the related average cost of a failure is supposed to be about US$ 150.7 per failure. The average cost of a preventive action Cp on the components is about US$ 111.5 per action, made up of US$ 101.5 per action for man work and US$ 10

200

300

0

100

200

300

per action for materials and spare parts. The average global cost of a failure maintenance action Cf is about US$ 1,348.7 per action, made up of the following contributions: • • • •

Cost of failure: US$ 150.7 per failure; Cost of man work: US$ 548 per action; Cost of materials and spare parts: US$ 650 per year; Cost of emissions: US$ 0 per year.

12.1 Preventive Maintenance Strategy Applied to a Waste to Energy Plant

Nonparametric Analysis - Bucket component Kaplan-Meier Method - Complete Data

Kaplan-Meier Survival Function

Probability

1.0

0.5

0.0 0

100

200

Nonparametric Hazard Function

Rate

1.0

0.5

0.0 0

100

200

Fig. 12.9 Bucket component. Nonparametric estimation. Survival function and hazard function. Minitab® Statistical Software

Table 12.2 Bucket component. Nonparametric estimation. Values of R.t / and .t / t – TTF

R.t /

LCI limit

UCI limit

.t /

5 13 16 31 52 81 99 108 200

0.889 0.778 0.667 0.556 0.444 0.333 0.222 0.111 0.000

0.684 0.506 0.359 0.231 0.120 0.025 0.000 0.000 0.000

1.000 1.000 0.975 0.880 0.769 0.641 0.494 0.316 0.000

0.111 0.125 0.143 0.167 0.200 0.250 0.333 0.500 1.000

TTF time to failure, LCI lower confidence interval, UCI upper confidence interval

439

Figure 12.11 reports the values assumed by the expected unit cost of maintenance UEC.tp / by the application of Eq. 9.3 and the continuous dynamic analysis. In particular, the best value of interval tp is about 46.99 days. This value generates a UEC equal to US$ 22.464 per day, as the result of the expected total replacement cost per cycle of US$ 688.86 and the expected cycle length of about 30.7 days. The trends presented in Figs. 12.12 and 12.13 are similar to those illustrated and exemplified in Sect. 9.5 about the generic Weibull density function for the variable ttf (see, in particular, Fig. 9.10). Nevertheless, they are different in shape: in detail, their sawtooth shape is due to the introduction of historical observations of failure events, while the results of Sects. 12.1.1 and 12.1.2 are the output of a parametric evaluation by a failure probability function. The expected cost per unit time displayed in Fig. 12.13 was derived assuming a Weibull distribution for ttf. As previously demonstrated in Sect. 12.1.1, the distribution of ttf values is well modeled by a shape factor equal to 0.9455, corresponding to a nonaging component subject to early wear out, as discussed in Sect. 5.10.5. This value is very close to 1: as a consequence, an exponential function can justify the UEC values, reported in Fig. 12.13 and obtained by a best fitting Weibull analytical model assuming ˇ D 0:9455 and ˛ D 50:21 days. In this case, a preventive maintenance strategy is not suitable. These remarks seem to be in conflict with the previous best tp value (46.99 days), but Fig. 12.11 demonstrates that the minimal UEC is also obtained with a larger interval of time tp . In particular, the expected cost of replacement per unit time is about US$ 24.6 per day in tp D 160 days, while the Weibull assumption justifies a unit cost equal to US$ 26.4 per day in tp D 160 days.

12.1.4 Time-Based Preventive Replacement for the Motor System Figure 12.10 illustrates the logic diagram of the continuous dynamic model developed in Simulink® (MATLAB / to support the determination of the best tp value by the application of the time-based preventive replacement model illustrated in Sect. 9.5. This is a parametric model useful for identifying the best tp value for the generic distribution of ttf values and couples of (Cp , Cf ).

12.1.5 Time-Based Preventive Replacement for the Bucket Component This section provides the results obtained by the application of replacement model type I to the decisions regarding the maintenance activity for the bucket component, similarly to Sect. 12.1.4 for the motor system.

440

12 Applications and Case Studies 1 s

Int. R(t)

Clock

R(t) 1 Constant

Divide

Cp

Cf

F(t)

Expected cost & Expected length

111.5

1348.7

Expected cost

R(t) & F(t) UEC(tp)

Fig. 12.10 Motor system. Type I model. Simulink® , MATLAB®

simout

To Workspace

100 90

UEC(t p) [$/day]

80 70 60 50 40 30

Fig. 12.11 Motor system, nonparametric analysis. UEC.tp /

20

10

20

30

40

50

60

70

80 90 t p [day]

100

110

120

130

140

150

160

Fig. 12.12 Motor system, nonparametric analysis. Expected total replacement cost per cycle

Expected total replacement cost per cycle [$/cycle]

1400

1200

1000

800

600

400

200

0

0

10

20

30

40

50

60

70

80 90 t p [day]

100

110

120

130

140

150 160

12.1 Preventive Maintenance Strategy Applied to a Waste to Energy Plant

441

100 90

UEC(t p) [$/day]

80 70 60 50 40 30

Fig. 12.13 Motor system, parametric analysis. Weibull distribution, UEC.tp /

20 0

10

20

30

40

50

60

70

80 90 t p [day]

100

110

120

130

140

150

160

90 80

UEC(t p) [$/day]

70 60 50 40 30 20

Fig. 12.14 UEC.tp / for the bucket component, nonparametric analysis

10 0

10

20

30

40

In particular, Fig. 12.14 reports the trend of the UEC values assuming a nonparametric distribution of ttf, as reported in Fig. 12.9. The minimum value of UEC.tp /, for tp D 80:99 days, is US$ 16.3 per day. The expected total replacement cost per cycle is US$ 798.83 per cycle and the expected cycle length is about 48.99 days. Figure 12.15 shows the outcomes of the application of the type I model to the Weibull distribution function, having scale value ˛ D 69:247 days and shape factor ˇ D 1:0807, reported in Fig. 12.8. The best tp value is about 98 days and the minimum UEC.tp / is US$ 19.94 per day.

50

60

70

80

90

100 110 120 130 140 150 160 170 180 190 200 t p [day]

12.1.6 Time-Based Preventive Replacement with Durations Tp and Tf Figure 12.16 illustrates the results obtained by the application of the analytical model for the determination of the best interval of time tp , as illustrated in Sect. 9.6, for the preventive replacement of the bucket component in bridge crane 1. The duration of a preventive replacement is assumed to be equal to 0.5 days per replacement, while the duration of a replacement in the case of failure is 1 day per replacement. The min-

442

12 Applications and Case Studies 22

UEC(tp) [$/day]

21.5

21

20.5

20

Fig. 12.15 Bucket component, parametric analysis. Weibull distribution. UEC.tp /

19.5

30

40

50

60

70

80

90

100 110 120 130 140 150 160 170 180 190 200 t p [day]

Type I vs Type I with duration of replacement (Tp = 0.5, Tf = 1)

40

35 Type I with duration of replacement

UEC(t p) [$/day]

Type I

Fig. 12.16 Bucket component, nonparametric analysis. Type I model versus type I model with duration of replacement

30

25

20

15 10

20

30

40

imum UEC based on the duration of replacement is about US$ 16.05 per day, when tp D 80:99 days. Figure 12.16 compares in detail the UEC values obtained by applying the type I model and the type I model with the duration of replacement.

12.1.7 Downtime Minimization With reference to the bucket component of bridge crane 1, Fig. 12.17 reports the values of the downtime obtained by the application of the model illustrated in Sect. 9.9 (see Eq. 9.42). From 0 to 200 days, there is not a tp value corresponding to a minimum for the downtime.

50

60

70

80

90

100 110 120 130 140 150 160 170 180 190 200 t p [day]

12.1.8 Monte Carlo Dynamic Analysis This section deals with the most important results obtained by the application of the Monte Carlo dynamic simulation. First of all, each component has to be considered separately and subsequently, as discussed in Sect. 12.1.9, it is possible to analyze the failure and repair behavior for each component in detail. Table 12.3 and Figs. 12.18 and 12.19 refer to the motor system. The following scenarios have been simulated with a period of time of 3,650 days, i. e., 10 years: • Corrective maintenance strategy. The component is subject to corrective maintenance and not to preventative maintenance.

12.1 Preventive Maintenance Strategy Applied to a Waste to Energy Plant

443 Downtime Minimization

25

Dow DT [%]

20

15

10

5

Fig. 12.17 Bucket component, nonparametric analysis. Downtime (DT) minimization

0

0

10

20

30

40

50

60

70

80

90

100 110 120 130 140 150 160 170 180 190 200 t p [day]

Table 12.3 Motor system. Multiscenario analysis. q D 1 Motor system (1,000 rep. – 3,650 days)

CM

Mean availability (all events) Point availability (all events) at 3,650 MTTFF Uptime CM downtime PM downtime Total downtime Number of failures Number of CMs Number of PMs Total events CMs and PMs Total costs

0.981 0.986 54.062 3,580.510 69.491 0.000 69.491 69.498 69.498 0.000 69.498 93,731.953

CM and PM (tp D 46:99)

CM and PM (tp D 160)

0.974 0.974 48.927 3,555.128 71.884 22.988 94.872 71.892 71.892 45.980 117.872 102,087.510

0.980 0.970 48.840 3,578.573 69.600 1.827 71.427 69.612 69.612 3.654 73.266 94,293.125

CM corrective maintenance, PM preventative maintenance, rep. replacements, MTTFF mean time to first failure

Motor system - System Failures (t)

• Corrective maintenance and preventive maintenance (tp D 46:99 days) strategy. The component is subject to corrective maintenance upon failure, and to preventative maintenance adopting the time-based replacement policy with tp D 46:99 days, in accordance with the optimization analysis conducted in Sect. 12.1.4. • Corrective maintenance and preventive maintenance (tp D 160 days) strategy. The component is subject to corrective maintenance and to preventative maintenance adopting the time-based replacement policy with tp D 160 days, in accordance with the analysis conducted in Sect. 12.1.4.

80 70 60 50 40 30

CM CM & PM (tp=46.99)

20

CM & PM (tp=160)

10 0 0

500

1000

1500

2000

2500

3000

3500

Fig. 12.18 Motor system. Number of failures, period 10 years. q D 1. CM corrective maintenance, PM preventative maintenance

All these scenarios are based on the “as good as new” assumption at the end of the generic replacement action, i. e., the adopted restoration factor is q D 1 after

444

12 Applications and Case Studies

Motor system - Availability A (t) 1

0.99

0.98 CM

0.97

CM & PM (tp=46.99) CM & PM (tp=160) 0.96

0.95

Fig. 12.19 Motor system. Point availabilityA.t /, period 10 years. q D 1

0.94 0

500

1000

both corrective and preventive actions. The assumed repair/restoration times are Tf D MTTRCM D 1 day and Tp D MTTRPM D 0:5 day. The first approach based on the corrective maintenance strategy has the lowest maintenance costs. The item is not an aging component, and the preventative maintenance strategy is not convenient: it is more affordable to act in the case of failure, even if the cost of corrective replacement is 1,348.7 unit of cost (u. c.) per action compared with 111.5 u. c. per action in the case of a preventative maintenance action. With reference to a finite number of historical observations, the best preventive replacement time tp seems to be equal to 46.99 days. Moreover, the number of corrective maintenance actions, about 69.49 events, is the same moving from the first to the third scenario, while the number of the preventive maintenance actions ranges from 0 to about 4 events. The number of failures slightly decreases instead, as shown in Fig. 12.18. This is behavior typical of the components subject to random failures, i. e., ageless, or without memory, items. The trend of the point availability A.t/ in the simulated hypothesis is reported in Fig. 12.19, where the greatest values are referred to all the components subject to corrective maintenance, and not to preventative maintenance. Similar remarks are made for the bucket component in Table 12.4 and Figs. 12.20 and 12.21. Different scenarios have been simulated. The period T is equal to 10 years:1 1

The restoration factor q is adopted equal to 1. Other scenarios illustrated below are based on different assumptions.

1500

2000

2500

3000

3500

Bucket comp. System Failures (t) 60 55 50 45 40 35 30 25 20

CM

15

CM & PM (tp=98)

10

CM & PM (tp=30)

5 0 0

500

1000

1500

2000

2500

3000

3500

Fig. 12.20 Bucket component. Number of failures, period 10 years. q D 1

• Corrective maintenance strategy. The component is subject to corrective maintenance and not to preventative maintenance; • Corrective maintenance and preventative maintenance (tp D 98 days) strategy. The component is subject to corrective maintenance, upon failure, and to preventative maintenance adopting the timebased replacement policy with tp D 98 days, in accordance with the optimization analysis conducted in Sect. 12.1.5; • Corrective maintenance and preventative maintenance (tp D 30 days) strategy. The component is subject to corrective maintenance, upon failure, and to preventative maintenance adopting the timebased replacement policy with tp is assumed to be equal to 30 days.

12.1 Preventive Maintenance Strategy Applied to a Waste to Energy Plant

445

Table 12.4 Bucket component. Multiscenario analysis. q D 1 CM

CM and PM (tp D 98)

CM and PM (tp D 30)

0.985 0.980 64.167 3,596.523 53.477 0.000 53.477 53.488 53.488 0.000 53.488 72,139.266

0.984 0.983 70.182 3,590.543 51.624 7.833 59.457 51.631 51.631 15.666 67.297 71,381.489

0.974 0.967 73.475 3,555.128 47.338 47.534 94.872 47.346 47.346 95.075 142.421 74,456.413

Bucket comp. (1,000 rep. – 3,650 days) Mean availability (all events) Point availability (all events) at 3,650 MTTFF Uptime CM downtime PM downtime Total downtime Number of failures Number of CMs Number of PMs Total events CMs and PMs Total costs

Bucket comp. Availability A (t) 1

0.99

CM

0.98

CM & PM (tp=98) CM & PM (tp=30) 0.97

Fig. 12.21 Bucket component. Point availability A.t /, period 10 years. q D 1

0.96 0

500

In this case the best maintenance strategy corresponds to the second scenario, because the shape factor of the Weibull function is greater than 1. With reference to the number of failures, the third scenario has the best performance, as pointed out in Fig. 12.20. This strategy is also based on 95 preventative maintenance actions compared with 15.7 for the second scenario, and the total downtime is about 95 days compared with 59 days (37:3%) for the second scenario and 53 days (43:63%) for the first scenario. Figure 12.22 reports the comparison of the total maintenance cost for the three scenarios, assuming alternatively the restoration factor q is equal to 1 and q D 0:5 in corrective maintenance and preventative maintenance. The maintenance cost is always higher for q D 0:5 because the aging component is not completely restored: moving from the first to the third scenario, the percentage increment is C8:07, C4:49, and C2:77%, respectively.

1000

1500

2000

2500

3000

3500

Bucket comp. Total cost (€) 78000 77000 76000 75000 74000 73000 72000

CM - q=1

71000

CM - q=0.5

70000

CM & PM (tp=98) - q=1

69000

CM & PM (tp=98) - q=0.5

68000 1 CM - q=1

72139

CM - q=0.5

77964

CM & PM (tp=98) - q=1

71381

CM & PM (tp=98) - q=0.5

74584

CM & PM (tp=30) - q=1

74456

CM & PM (tp=30) - q=0.5

76522

CM & PM (tp=30) - q=1 CM & PM (tp=30) - q=0.5

Fig. 12.22 Bucket component. Total maintenance cost (period 10 years) q D 1 compared with q D 0:5

Figures 12.23 and 12.24 compare severally corrective maintenance downtimes (days) and preventative maintenance downtimes for these six simulated operating scenarios.

446

12 Applications and Case Studies

CM Downme [days] 60

Fig. 12.25 Reliability block diagram. System with two basic components

50 40

Motor system

Bucket

30 CM - q=1

20

CM - q=0.5

10

CM & PM (tp=98) - q=1 CM & PM (tp=98) - q=0.5

0 1 CM - q=1

53.477

CM - q=0.5

57.7989

CM & PM (tp=98) - q=1

51.6244

CM & PM (tp=98) - q=0.5

53.4166

CM & PM (tp=30) - q=1

47.3378

CM & PM (tp=30) - q=0.5

48.0118

CM & PM (tp=30) - q=1 CM & PM (tp=30) - q=0.5

Fig. 12.23 Bucket component. CM downtime (period 10 years) q D 1 versus q D 0:5

PM Downme days 60 50 40 30 CM - q=1 20

CM - q=0.5

10

CM & PM (tp=98) - q=1 CM & PM (tp=98) - q=0.5

0 1

CM & PM (tp=30) - q=1

CM - q=1

0

CM & PM (tp=30) - q=0.5

CM - q=0.5

0

CM & PM (tp=98) - q=1

7.8326

CM & PM (tp=98) - q=0.5

11.3323

CM & PM (tp=30) - q=1

47.5342

CM & PM (tp=30) - q=0.5

52.7175

Fig. 12.24 Bucket component. PM downtime (period 10 years) q D 1 compared with q D 0:5

12.1.9 Monte Carlo Analysis of the System This section deals with the Monte Carlo analysis of a system made up of two components as in Fig. 12.25. It is possible to design several combinations of maintenance strategies to apply to the two components. In detail, the following scenarios, or systems, are considered: 1. System 1. The motor system and the bucket are both subject to corrective maintenance and not to preventative maintenance, q D 1. 2. System 2. In addition to the scenario for system 1, the bucket is subject to preventative maintenance with tp D 98 days with a related restoration factor q(PM) equal to 1 (as good as new replacement).

3. System 3. This differs from system 2 because the restoration factor in the case of corrective maintenance, q(CM), is equal to 0.5. 4. System 4. This differs from system 3 because the restoration factor of the motor system is equal to 1 and not to 0.5. Table 12.5 presents the results obtained by the simulation analysis of these scenarios. System 1 is the best performer regarding the total events preventative maintenance actions + corrective maintenance actions and total downtime because the less onerous preventive actions are not admissible. Moving to system 2 where these actions are applied on the bucket, i. e., on an aging component, the total costs range from US$ 163,348 to US$ 162,841. In the case of improper application of corrective maintenance, i. e., restoration factor q(CM) < 1, the expected cost for system 3 is lower. This astonishing result must be compared with the results for system 4, where q(CM) D 1 for the motor system and q(CM) D 0:5 for the bucket, and the expected cost of about US$ 165,000 is the greatest. In fact, if it is possible to choose between q D 1 and q < 1 for the motor system, i. e., a nonaging item, the first option, q D 1, is not convenient. This is a case where the “as good as new” hypothesis does not perform better than an incomplete restoration action. In conclusion, the best combination can be summarized as follows: only corrective maintenance with q < 1 for the motor system; corrective maintenance with q D 1 and preventative maintenance with tp D 98 and q D 1 for the bucket.

12.2 Reliability, Availability, and Maintainability Analysis in a Plastic Closures Production System for Beverages The aim of this section is to illustrate how a reliability, availability, and maintainability (RAM) analysis can result in cost reduction and productivity improvement.

12.2 Reliability, Availability, and Maintainability Analysis in a Plastic Closures Production System for Beverages

447

Table 12.5 System performance. Multiscenario analysis, q D 1 versus q D 0:5 System configuration

System 1

System 2

System 3

System 4

CM on motor system (Y/N) q(CM) on motor system CM on bucket (Y/N) q(CM) on bucket PM on motor system (Y/N) PM_tp on motor system (day) q(CM) on motor system PM on bucket (Y/N) PM_tp on bucket (day) q(CM) on bucket

Y 1 Y 1 N – – N – –

Y 1 Y 1 N – – Y 98 1

Y 0.5 Y 0.5 N – – Y 98 1

Y 1 Y 0.5 N – – Y 98 1

Mean availability (all events) Expected number of failures MTTFF Uptime CM downtime PM downtime Total downtime Number of failures Number of CMs Number of PMs Total events PMs and CMs Total costs

0.967 121.115 30.175 3,528.905 121.095 0.000 121.095 121.115 121.115 0.000 121.115 163,348

0.965 119.474 26.697 3,522.890 119.457 7.653 127.110 119.474 119.474 15.306 134.780 162,841

0.965 116.832 29.799 3,522.105 116.816 11.079 127.895 116.832 116.832 22.159 138.991 160,042

0.964 120.630 30.127 3,518.283 120.616 11.102 131.717 120.630 120.630 22.203 142.833 165,169

This second case study applies the reliability and availability evaluation analysis and the maintenance planning models to a complex mechanical system, illustrated in Fig. 12.26. This system is manufactured by an Italian company operating in the beverage and packaging sector for the production of plastic closures, such as the bottle top in Fig. 12.27. This production line is essentially made up of three main pieces of equipment: the compression-molding group, the folding (processing the corrugated effect on the cap to allow a proper “capping”) and scoring group, and a lining group for the insertion of a polyethylene-based liner through the cap. The analysis is focused on the first functional group, made up of a rotating hydraulic machine for the molding of plastic closures. The caps are made of a plastic compound (highdensity polyethylene, polypropylene) and their manufacturing process schedules several tasks such as extrusion, metering, pelleting, insertion, and molding. The core of the compression-molding group is a rotating carousel, represented in Fig. 12.28, driving the whole manufacturing process; Fig. 12.29 details the molding task identified by number 3 in Fig. 12.28. The compression-molding process has a high level of quality and repeatability, i. e., a small deviation from the quality standards.

Folding and scoring group Lining group

Compression molding group

Fig. 12.26 Rotating and compression-molding system

Fig. 12.27 X-ray picture of a cap for water

448

12 Applications and Case Studies

icy rules, spare parts numbers, maintenance action costs, availability of maintenance crews, etc.), and system optimization.

1 2 3 4 5 6

Placement Cavity lifting Molding Cooling Mold opening Ejection

Melt Pellet Caps

Fig. 12.28 Rotating compression-molding manufacturing process

A Punch B Stripper C Melt feeding D Mold E Pellet portion

Fig. 12.29 Compression-molding task

The application of the RAM analysis follows the decisional steps theoretically illustrated and discussed in the previous chapters: • production system analysis and reliability block diagram (RBD) construction; • failure modes identification and analysis by the application of FMEA or FMECA techniques; • data collection: failure and repair times for all basic components or “blocks” and failure modes of the production system, maintenance action costs (corrective, preventive, inspection, etc.), spare parts availability (e. g., storage costs and fulfillment costs), availability of crews, etc.; • evaluation of system reliability parameters assuming the hypothesis of nonrepairable components/systems; • system availability evaluation by dynamic simulation assuming the hypothesis of repairable components/systems; • evaluation of maintenance costs, multiscenario comparison assuming different configurations and parameterizations of the system (maintenance pol-

Some of these steps and the related results are presented in the following sections.

12.2.1 RBD construction The RBD construction is obtained by the analysis of the production system, and/or a concurrent and advanced design of the system, which simultaneously involves the following group of technicians: • research and development department with responsibility for the development of new products; • product’s engineers and designers with responsibility for the products; • production managers with the responsibility for the production system in accordance with production plans; • quality managers with responsibility for the quality of products, processes, and the whole production system; • logistics manager with responsibility for the production system’s spare parts; • maintenance manager with responsibility for the availability of the equipment (components and production system). The construction of the RBD is influenced by the results of a FMEA and FMECA conducted on the basic components/parts of the production system. In particular, these analyses support the identification of the failure modes of the basic components that are critical for the system function. At the same time, it is possible to evaluate the whole system, both from a qualitative and from a quantitative point of view, in order to point out its most important failure modes by an ad hoc fault tree analysis. The RBD model is a powerful tool for the accurate evaluation of a set of reliability parameters involving the whole production system, and it is very useful for the planning and organizing of costsaving maintenance actions on components and subsystems. The following case study does not explicitly describe the identification of the failure modes affecting the RBD construction.

12.2 Reliability, Availability, and Maintainability Analysis in a Plastic Closures Production System for Beverages

12.2.2 Rotating Hydraulic Machine

449

strength and life of products that very quickly wear out over a certain age, is given by

The RBD concerning the whole production line of plastic closures, showed in Fig. 12.26, is made up of thousands of basic reliability blocks. The focus is on the rotating hydraulic machine, the subject of the application of the RAM analysis discussed in the current chapter, having the operational diagram shown in Fig. 12.30. This functional group can be modeled by 259 basic blocks, as properly illustrated in the RBD presented in Fig. 12.31, and is made up of several pieces of equipment, e. g., thrust, driving belts, cams, and O-rings, grouped in operational sets named A, B, C, D, and E. Set A is made up of two components, set C is made up of three components, set D is a 10=12 (k=n) redundant system, and, finally, set E is made up of 48 units, each of five items.

f .x/ D

1 x e x ˇ e ˇ e ; ˇ

(12.1)

where  is the location parameter and ˇ is the scale parameter. The Gumbel mean or MTTF is  C 0:5772ˇ, where 0.5772 is the Euler’s constant. The standard p . The Gumbel cumulative distribution deviation is ˇ 6 function is: x F .x/ D e e : (12.2) Figure 12.34 presents the histogram of the distribution of failures. The very little number of available data gives the statistical analysis a significant level of uncertainty. Obviously, the greater the number of available data on failures, the more accurate the evaluation of the estimated reliability parameters. For 200 available ttf values, Figs. 12.35–12.37 present the result of the parametric probability plot analysis conducted by Minitab® Statistical Software for different distributions of ttf. Some of these distributions were introduced in Chap. 5, while the others, such as loglogistic, three-parameter Weibull, and three-parameter exponential, are illustrated by several literature references on statistics science and applications. The result of the goodness-of-fit analysis by Minitab® Statistical Software, based both on the Anderson–Darling index and the correlation coefficient, is reported in Table 12.6. With reference to the correlation coefficient, in spite of the good performance of the three-parameter Weibull distribution, the adopted probability distribution of the ttf is the Gumbel distribution, whose probability plot is illustrated in Fig. 12.38, having a very good correlation coefficient too but a better Anderson– Darling index.

12.2.3 Data Collection and Reliability Evaluation of Components This section exemplifies the reliability evaluation analysis conducted for a specific component E.4 of the production system. In particular, Fig. 12.32 illustrates five values of ttf, 50, 4,000, 4,200, 4,950, and 5;060 h, collected on five different applications of this component. In the case of a few historical values of the random ttf, as in this situation, the best-fit parametric analysis and evaluation suggests adopting the so-called Gumbel distribution, whose related probability plot is illustrated in Fig. 12.33. The probability density function of the Gumbel distribution, also called “smallest extreme value distribution,” used in general to model the

D.1 D.2 A.1

A.2 A

B

C.1

C.2

C.3

E.1

E.2

C

E.3 E (1th)

D.1 D k/n parallel (10/12)

Fig. 12.30 Operational diagram of the rotating hydraulic machine

E.4

E.5

E.1

E.2

E.3 E (48th)

E.4

E.5

Driv.Belt

Cam

E.2

A.2

Distr.

E.3

Water o-ring

B

Oil o-ring

E.4

E.5

C.1

C.2

Extruder12

Extruder11

Extruder10

Extruder9

Extruder8

Extruder7

C.3

UP flange (stripper)

UP flange (stripper)

UP flange (stripper)

UP flange (stripper)

UP flange o-ring

UP flange o-ring

UP flange o-ring

UP flange o-ring

UP flange o-ring

UP flange o-ring

UP flange o-ring

PUNCH

PUNCH

PUNCH

PUNCH

PUNCH

PUNCH

PUNCH

PUNCH

DOWN DOWN flange flange (mould) o-ring

DOWN DOWN flange flange (mould) o-ring

DOWN DOWN flange flange (mould) o-ring

DOWN DOWN flange flange (mould) o-ring

DOWN DOWN flange flange (mould) o-ring

DOWN DOWN flange flange (mould) o-ring

DOWN DOWN flange flange (mould) o-ring

DOWN DOWN flange flange (mould) o-ring

UP flange (stripper)

UP flange (stripper)

UP flange (stripper)

UP flange (stripper)

UP flange (stripper)

UP flange (stripper)

UP flange (stripper)

UP flange o-ring

UP flange o-ring

UP flange o-ring

UP flange o-ring

PUNCH

Molding unit (1–48)

Pelleting and inserting system (1–12)

UP flange o-ring

UP flange o-ring

PUNCH

PUNCH

PUNCH

PUNCH

PUNCH

PUNCH

PUNCH

UP flange o-ring

UP flange o-ring

UP flange (stripper)

Transmission pallet placement

k/n, 10/12

UP flange (stripper)

UP flange (stripper)

UP flange (stripper)

UP flange o-ring

Fig. 12.31 Reliability block diagram of the rotating hydraulic machine, 259 blocks

E.1

D

A.1

Legend

Thrust

Extruder6

Extruder5

Extruder4

Extruder3

Extruder2

Extruder1

UP flange (stripper)

DOWN DOWN flange flange (mould) o-ring

DOWN DOWN flange flange (mould) o-ring

DOWN DOWN flange flange (mould) o-ring

DOWN DOWN flange flange (mould) o-ring

DOWN DOWN flange flange (mould) o-ring

DOWN DOWN flange flange (mould) o-ring

DOWN DOWN flange flange (mould) o-ring

DOWN DOWN flange flange (mould) o-ring

UP flange (stripper)

UP flange (stripper)

UP flange (stripper)

UP flange (stripper)

UP flange (stripper)

UP flange o-ring

UP flange o-ring

UP flange o-ring

UP flange o-ring

UP flange o-ring

UP flange o-ring

UP flange o-ring

UP flange o-ring

UP flange (stripper)

UP flange (stripper)

UP flange (stripper)

PUNCH

PUNCH

PUNCH

PUNCH

PUNCH

PUNCH

PUNCH

PUNCH

DOWN DOWN flange flange (mould) o-ring

DOWN DOWN flange flange (mould) o-ring

DOWN DOWN flange flange (mould) o-ring

DOWN DOWN flange flange (mould) o-ring

DOWN DOWN flange flange (mould) o-ring

DOWN DOWN flange flange (mould) o-ring

DOWN DOWN flange flange (mould) o-ring

DOWN DOWN flange flange (mould) o-ring

UP flange (stripper)

UP flange (stripper)

UP flange (stripper)

UP flange (stripper)

UP flange (stripper)

UP flange o-ring

UP flange o-ring

UP flange o-ring

UP flange o-ring

UP flange o-ring

UP flange o-ring

UP flange o-ring

UP flange o-ring

UP flange (stripper)

UP flange (stripper)

UP flange (stripper)

PUNCH

PUNCH

PUNCH

PUNCH

PUNCH

PUNCH

PUNCH

PUNCH

DOWN DOWN flange flange (mould) o-ring

DOWN DOWN flange flange (mould) o-ring

DOWN DOWN flange flange (mould) o-ring

DOWN DOWN flange flange (mould) o-ring

DOWN DOWN flange flange (mould) o-ring

DOWN DOWN flange flange (mould) o-ring

DOWN DOWN flange flange (mould) o-ring

DOWN DOWN flange flange (mould) o-ring

UP flange (stripper)

UP flange (stripper)

UP flange (stripper)

UP flange (stripper)

UP flange (stripper)

UP flange o-ring

UP flange o-ring

UP flange o-ring

UP flange o-ring

UP flange o-ring

UP flange o-ring

UP flange o-ring

UP flange o-ring

UP flange (stripper)

UP flange (stripper)

UP flange (stripper)

PUNCH

PUNCH

PUNCH

PUNCH

PUNCH

PUNCH

PUNCH

PUNCH

DOWN DOWN flange flange (mould) o-ring

DOWN DOWN flange flange (mould) o-ring

DOWN DOWN flange flange (mould) o-ring

DOWN DOWN flange flange (mould) o-ring

DOWN DOWN flange flange (mould) o-ring

DOWN DOWN flange flange (mould) o-ring

DOWN DOWN flange flange (mould) o-ring

DOWN DOWN flange flange (mould) o-ring

UP flange (stripper)

UP flange (stripper)

UP flange (stripper)

UP flange (stripper)

UP flange (stripper)

UP flange o-ring

UP flange o-ring

UP flange o-ring

UP flange o-ring

UP flange o-ring

UP flange o-ring

UP flange o-ring

UP flange o-ring

UP flange (stripper)

UP flange (stripper)

UP flange (stripper)

PUNCH

PUNCH

PUNCH

PUNCH

PUNCH

PUNCH

PUNCH

PUNCH

DOWN DOWN flange flange (mould) o-ring

DOWN DOWN flange flange (mould) o-ring

DOWN DOWN flange flange (mould) o-ring

DOWN DOWN flange flange (mould) o-ring

DOWN DOWN flange flange (mould) o-ring

DOWN DOWN flange flange (mould) o-ring

DOWN DOWN flange flange (mould) o-ring

DOWN DOWN flange flange (mould) o-ring

450 12 Applications and Case Studies

12.2 Reliability, Availability, and Maintainability Analysis in a Plastic Closures Production System for Beverages

ReliaSoft Weibull++ 7 - www.ReliaSoft.com

F/S Timeline FS Timeline Failure

0.000

1200.000

2400.000

3600.000

4800.000

6000.000

Time, (t) μ=4412,8325, σ=1069,6069

Fig. 12.32 Failure timeline analysis, five applications. ReliaSoft® software

ReliaSoft Weibull++ 7 - www.ReliaSoft.com

Probability - Gumbel

99.000

Probability-Gumbel

Unreliability, F(t)

Data 1 Gumbel-2P MLE SRM K-M FM F=5/S=0 Data Points Probability Line

50.000

10.000 30.000

1224.000

2418.000

3612.000

4806.000

6000.000

Time, (t) μ=4412,8325, σ=1069,6069

Fig. 12.33 Probability plot, Gumbel distribution five data points. ReliaSoft® software

451

452

12 Applications and Case Studies

ReliaSoft Weibull++ 7 - www.ReliaSoft.com

F/S Histogram

4.000E-4

FS Histogram Pdf Line Failures

3.200E-4

Value

2.400E-4

1.600E-4

8.000E-5

0.000

0-501

501-1002

1002-1503 1503-2004 2004-2505 2505-3006 3006-3507 3507-4008 4008-4509 4509-5010 5010-5511 5511-6012

Period μ=4412.8325, σ=1069.6069

Fig. 12.34 Failure histogram, Gumbel distribution five data points. ReliaSoft® software

Probability Plot for ttf - comp. E.4 LSXY Estimates-Complete Data Correlation Coefficient Weibull 0.945 Lognormal 0.857 Exponential * Loglogistic 0.872

Lognorm al

99.9

99.9

90

99

50

90 Percent

Percent

Weibull

10 1

50 10 1

0.1 100

1000 ttf

0.1 100

10000

E xponential

1000 ttf

10000

Loglogistic

99.9

99.9

90

99 Percent

Percent

50 10

90 50 10

1 1 0.1

1

10

100 ttf

1000

10000

0.1 100

1000

10000 ttf

Fig. 12.35 Probability plot for ttf, part 1. Minitab® Statistical Software

12.2 Reliability, Availability, and Maintainability Analysis in a Plastic Closures Production System for Beverages

Probability Plot for ttf - comp. E.4 LSXY Estimates-Complete Data 3-Param et er Weibull 99.9

90

99

50

90 Percent

Percent

Correlation Coefficient 3-Parameter Weibull 0.996 3-Parameter Lognormal 0.985 2-Parameter Exponential * 3-Parameter Loglogistic 0.983

3-Param et er Lognorm al

99.9

10

50 10

1

1 0.1

0.1 8000

10000 12000 ttf - Threshold

14000

188000

2-Param et er Exponent ial

190000 192000 194000 ttf - Threshold 3-Param et er Loglogis t ic

99.9

99.9 90

99 Percent

Percent

50 10 1

90 50 10 1

0.1

01

0.

10

0.

00

1.

.

10

00

00

0.

10

185000

0.

.

00

10

0.1

00

00 00

10

187500 190000 ttf - Threshold

192500

ttf - Threshold

Fig. 12.36 Probability plot for ttf, part 2. Minitab® Statistical Software

Probability Plot for ttf - comp. E.4 LSXY Estimates-Complete Data Sm alles t Ext rem e V alue 99.9

90

99

50

90 Percent

Percent

Correlation Coefficient Smallest Extreme Value 0.993 Normal 0.985 Logistic 0.983

Norm al

99.9

10

50 10

1

1 0.1

0.1 0

4000

8000

ttf

0

2500

5000 ttf

Logistic 99.9

Percent

99 90 50 10 1 0.1 0

2500

5000

7500

ttf

Fig. 12.37 Probability plot for ttf, part 3. Minitab® Statistical Software

7500

453

454

12 Applications and Case Studies

ReliaSoft Weibull++ 7 - www.ReliaSoft.com

Probability - Gumbel

99.900

Probability-Gumbel Data 1 Gumbel-2P RRX SRM MED FM F=200/S=0 Data Points Probability Line

Unreliability, F(t)

50.000

10.000

5.000

1.000

0.500

0.100 100.000

1480.000

2860.000

4240.000

5620.000

7000.000

Time, (t) μ=4391.6311, σ=987.9175, ρ=0.9927

Fig. 12.38 Probability plot for ttf, Gumbel distribution. ReliaSoft® software Table 12.6 Goodness-of-fit analysis conducted with Minitab® Statistical Software Distribution Weibull Lognormal Exponential Loglogistic 3-Parameter Weibull 3-Parameter lognormal 2-Parameter exponential 3-Parameter loglogistic Smallest extreme value Normal Logistic

Anderson–Darling index

Correlation coefficient

5.269 10.924 91.066 9.718 0.386 1.537 85.054 1.532 0.869 1.487 1.489

0.945 0.857

Figure 12.39 presents the trend of reliability, unreliability, failure distribution, and conditional failure rate assuming a Gumbel distribution ( D 4;391,  D 988). Figures 12.40 and 12.41 present the reliability, unreliability, and failure rate obtained by the application of a nonparametric distribution analysis deriving from the Kaplan–Meier method with confidence interval equal to 95%. Table 12.7 summarizes the parametric probability distributions of the random ttf for the components of the system. These values are very important for one to be able to conduct an evaluation analysis on the whole

0.872 0.996 0.985 0.983 0.993 0.985 0.983

production system and to plan and optimize the maintenance actions.

12.2.4 Reliability Evaluation, Nonrepairable Components/Systems Once data collection has been completed and the failure behaviors of the parts and components of the system have been modeled, it is possible to evaluate the reliability of the whole system as a combination

12.2 Reliability, Availability, and Maintainability Analysis in a Plastic Closures Production System for Beverages ReliaSoft Weibull++ 7 - www.ReliaSoft.com

455

ReliaSoft Weibull++ 7 - www.ReliaSoft.com

Reliability vs Time Plot

Unreliability vs Time Plot

1.000

1.000

Reliability

Unreliability

Data 1 Gumbel-2P RRX SRM MED FM F=200/S=0 Data Points Reliability Line

Data 1 Gumbel-2P RRX SRM MED FM F=200/S=0 Data Points Unreliability Line 0.800

Unreliability, F(t)=1-R(t)

Reliability, R(t)=1-F(t)

0.800

0.600

0.400

0.200

0.000 0.000

0.600

0.400

0.200

1400.000

2800.000

4200.000

5600.000

0.000 0.000

7000.000

1400.000

2800.000

Time, (t)

4200.000

5600.000

7000.000

Time, (t) μ=4391.6311, σ=987.9175, ρ=0.9927

μ=4391.6311, σ=987.9175, ρ=0.9927

ReliaSoft Weibull++ 7 - www.ReliaSoft.com

ReliaSoft Weibull++ 7 - www.ReliaSoft.com

Probability Density Function

Failure Rate vs Time Plot

4.000E-4

0.060

Pdf

Failure Rate

Data 1 Gumbel-2P RRX SRM MED FM F=200/S=0 Pdf Line

Data 1 Gumbel-2P RRX SRM MED FM F=200/S=0 Failure Rate Line 0.048

Failure Rate, f(t)/R(t)

3.200E-4

f(t)

2.400E-4

1.600E-4

8.000E-5

0.000 0.000

0.036

0.024

0.012

2000.000

4000.000

6000.000

8000.000

0.000 400.000

10000.000

2120.000

Time, (t)

3840.000

5560.000

7280.000

9000.000

Time, (t) μ=4391.6311, σ=987.9175, ρ=0.9927

μ=4391.6311, σ=987.9175, ρ=0.9927

Fig. 12.39 R.t /, F .t /, f .t /, and .t /, Gumbel distribution. ReliaSoft® software Table 12.7 Probability distribution of ttf. Parts system Part

Probability distribution

A.1 A.2 B C.1 C.2 C.3 D (1–48) E.1 E.2 E.3 E.4 E.5

normal normal normal normal 3-parameters Weibull normal normal normal lognormal normal normal Gumbel





4,000 1,650 4,000 1,436

1,600 660 1,600 574.4

4,000 2,810 4,000 7.877 2,650 4,600 4,391

1,600 1,124 1,600 2.452 1,060 1,840 988

of different parts and components, in accordance with the elementary configurations illustrated in Chap. 6. Figure 12.42 reports the main results of the reliability evaluation for the system, i. e., the determination of the failure probability function FS .t/ and the reliability function RS .t/, very significant for prediction of the first failure in the case of nonrepairable components/systems.

ˇ



0.7612

1,531.4

1,212.65

The values of the reliability of the system RS .t/, where t is in hours, are very low: e. g., RS .10/ D 0:1301, RS .50/ D 0:0151, RS .100/ D 0:0018. The MTTF is about 4:185 h. This is the mean time to the first failure and in the presence of repairable components it is not a useful measure of system performance.

12 Applications and Case Studies Survival Plot for ttf

Cumulative Failure Plot for ttf

Kaplan-Meier Method - 95% CI Complete Data

Kaplan-Meier Method - 95% CI Complete Data

100

100

80

80

60

Table of M ean M edian IQ R

40

Statistics 3826.23 3980.76 1468.34

20

Percent

Percent

456

60

Table of M ean M edian IQ R

40

Statistics 3826.23 3980.76 1468.34

20

0

0 0

1000

2000

3000

4000

5000

6000

7000

ttf

0

1000

2000

3000

4000

5000

6000

7000

ttf

Fig. 12.40 Nonparametric distribution analysis, R.t / and F .t /

Hazard Plot for ttf Empirical Hazard Function Complete Data 1.0

Table of Mean Median IQR

0.8

Statistics 3826.23 3980.76 1468.34

Rate

0.6

0.4

0.2

0.0 0

1000

2000

3000

4000

5000

6000

7000

ttf

Fig. 12.41 Nonparametric distribution analysis, .t /

12.2.5 Data on Repairs and Maintenance Strategies In order to evaluate the availability of the components/system it is necessary to know the repair stochastic behavior that can be modeled by evaluating the parametric distribution of the generic random variable ttr. In this case study, the repair process is characterized by the parameterization, reported in Table 12.8, depending on the maintenance strategy adopted (corrective, preventive, or inspection). Table 12.8 also reports a lot of data grouped in three different sections: 1. Corrective maintenance data and parameters: ttr, cost per action,2 and restoration factor. The ttr for the generic component is assumed to be constant, but in general it is randomly distributed in accor2

In this case study, all costs are reported in a generic unit of measure, unit of cost, because the real values cannot be revealed.

dance with a generic probability distribution, such as Weibull or lognormal. The restoration factor specifies the level to which the block is restored after the maintenance action. In particular, type I assumes that the repair removes only the damage since the last repair, while type II, represented as II.q/ in Table 12.8, assumes that the generic repair is capable of removing any damage accumulated up to failure. As is well known, q D 1 means adopting the “as good as new” hypothesis. 2. Preventive maintenance data and parameters: replacement time tp in accordance with the type I analytical model illustrated in Chap. 9, duration of the preventive action assumed to be constant, cost per action and restoration factor. 3. Inspection maintenance (data and parameters: fixed interval time based on item age, duration of the inspection action, restoration factor assumed to be type II and q D 0.

12.2.6 Monte Carlo Analysis of the Repairable System The outcomes of the evaluation analysis for the repairable system are illustrated in the following figures. Scenario 1 represents the as-is configuration of the production system and maintenance system that the company needed to optimize in order to minimize the global production cost. This cost includes

195.6400 0.7000 8.6800 182.9200 0.9900 0.4900 1.3800 1.7000 0.0360 5.8000 2.5400 0.0010

II(1) II(1) II(1) II(1) II(1) II(1) II(1) II(1) II(1) II(1) II(1) II(1)

Type I/II

– 2,160 2,160 – 4,320 4,320 360 – 12,960 – – 12,960

Replacement tp (h)

A.1 A.2 B C.1 C.2 C.3 D (1–48) E.1 E.2 E.3 E.4 E.5

Part

– 1,000 1,000 – – – – – – 1,440 – –

Insp. fixed time interval – item age (h)

4,000 2,160 2,160 1,436 4,320 4,320 360 – 12,960 – – 12,960

5 0.167 1 1 4 4 0.5 – 0.3 – – 0.3

195.6400 0.7000 8.6800 182.9200 0.9900 0.4900 1.3800 – 0.0360 – – 0.0010

II(1) II(1) II(1) II(1) II(1) II(1) II(1) – II(1) – – II(1)

4,000 2,160 2,160 1,436 4,320 4,320 360 – 12,960 – – 12,960

– 0.167 0.033 – – – – – – 0.167 – –

20 0.167 1 5 4 4 0.5 – 0.3 – – 0.3

– – – – – – – – –

–

Cost per action (u. c./action)

195.6400 0.7000 8.6800 182.9200 0.9900 0.4900 1.3800 – 0.0360 – – 0.0010

Cost per action (u. c./action)

Insp. time duration (constant) (h)

Inspection maintenance

Prev. time duration (constant) (h)

Replacement tp (h)

Type I/II

– II(1) II(1) – II(1) II(1) II(1) – II(1) – – II(1)

Type I/II

Replacement tp (h) Cost per action (u. c./action)

– 0.7000 8.6800 – 0.9900 0.4900 1.3800 – 0.0360 – – 0.0010

Cost per action (u. c./action)

Scenario 3 Preventive maintenance Prev. time duration (constant) (h)

– 0.167 1 – 16 16 1 – 2 – – 2

Prev. time duration (constant) (h)

Preventive maintenance

Scenario 2 Preventive maintenance

Table 12.9 Preventive maintenance parameterization, scenarios 2 and 3

20 2 2.5 5 8 8 1 0.67 0.67 2 0.67 0.67

Cost per action (u. c./action)

Corrective maintenance

ttr (constant) (h)

u. c. unit of cost

A.1 A.2 B C.1 C.2 C.3 D (1–48) E.1 E.2 E.3 E.4 E.5

Part

Scenario

Table 12.8 Repair process parameterization, ttr constant (hours). Scenario 1

II(1) II(1) II(1) II(1) II(1) II(1) II(1) – II(1) – – II(1)

Type I/II

– – – II(0) – –

– II(0) II(0) – –

Type I/II

12.2 Reliability, Availability, and Maintainability Analysis in a Plastic Closures Production System for Beverages 457

458

12 Applications and Case Studies Unreliability vs Time

Reliability vs Time

1.000

1.000

Unreliability

Reliability

CompMould Unreliability Line

CompMould Reliability Line

0.800

Reliability, R(t)=F(t)

Unreliability, F(t)=1-R(t)

0.800

0.600

0.400

0.200

0.000 0.000

0.600

0.400

0.200

12.000

24.000

36.000

48.000

0.000 0.000

60.000

12.000

Time, (t)

24.000

36.000

48.000

60.000

Time, (t)

Fig. 12.42 System unreliability FS .t / and reliability RS .t /. ReliaSoft® software

Availability and Reliability vs Time 1.000

CompMould Point Availability Line Point Reliability Line

0.800

A(t), R(t)

0.600

0.400

0.200

0.000 0.000

6440.000

12880.000

19320.000

25760.000

32200.000

Time, (t)

Fig. 12.43 Availability and reliability, scenario 1. ReliaSoft® software

costs for corrective, preventive, and inspection maintenance met for crew and materials, e. g., spare parts and replacement costs. When the system is down, i. e., in the presence of a downtime of the system, a cost for nonproduction3 equal to 11:8 u: c:=h is assumed. Figure 12.43 shows the trend of the point availability AS .t/ and the reliability RS .t/ of the system for the range t 2 Œ0; 32;200 h by the application of the Monte Carlo dynamic simulation. The period of 3

Lost production cost

time of 32;200 h for the system corresponds to about five operating years. The reliability of the repairable systems is referred to the first failure. The simulation analysis is carried on 500 repeated runs, corresponding to 500 virtual periods of work under the same environmental conditions for the production systems. Figures 12.44 and 12.45 present the analysis of the most critical components/blocks considering the so-called failure criticality index and the downing events criticality index, respectively.

12.2 Reliability, Availability, and Maintainability Analysis in a Plastic Closures Production System for Beverages

459

RS FCI 1.144

Availability 100%

50%

0.915 0% 5 Item(s)

0.687

0.458

0.229

Fig. 12.44 Failure criticality index (FCI) – scenario 1. ReliaSoft® software

0.000

C.1

A.2

E.3

E.3

E.3

RS DECI 1.356

Availability 100%

50%

1.085 0% 5 Item(s)

0.814

0.543

0.271

Fig. 12.45 Downing events criticality index (DECI) – scenario 1. ReliaSoft® software

0.000

B

As previously exemplified in Chaps. 5 and 6, for a generic block the failure criticality index is obtained from the number of system downing failures due to the generic block divided by the whole number of system failures. Similarly, for a generic block the downing events criticality index is obtained from the number of system downing events, different from system downing failures, due to the generic block divided by the whole number of system failures. This implies that

C.1

A.2

C.2

E.3

1.36% of the times that the system is down is due to the down condition of component B. Figure 12.46 reports the state diagram (up/down) for the system and the most critical components/blocks. The Pareto chart of the costs for the corrective, preventive, and inspection maintenance of the blocks in Fig. 12.47 highlights the most critical components in terms of annual costs.

460

12 Applications and Case Studies Block Up/Down State Operating Time Time Under Repair

C.2

C.3

A.2

C.1

B

System

Fig. 12.46 Critical blocks up/down analysis. ReliaSoft® software

0.000

2000.000

4000.000

6000.000

8000.000

10000.000

Time, (t)

12.2.7 Alternative Scenarios and System Optimization 2500000

100

2000000

80

1500000

60

1000000

40

500000

20

0 Block Name

Percent

Total cost

Pareto Chart of Blocks/Components costs

0 3 1 1 4 1 5 2 2 3 1 2 4 1 0 7 6 5 2 9 r E. C . E. E. A . E. E. C . C . D.1 D. D. D. D.1 D. D. D. D.1 D. the O

Fig. 12.47 Pareto chart of components costs. Scenario 1

Component A.2 maintenance optimization N(1650, 660)

0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 200

Starting from the previously illustrated scenario 1, some other scenarios concerning alternative operating conditions for the production system can be discussed. Every scenario involves different maintenance strategies and rules to be applied to the components/blocks. Scenarios 2–4 have been simulated in order to identify the best to-be configuration of the production system capable of minimizing the whole production and maintenance costs, in accordance with the adoption of different maintenance strategies and decisions. In particular, as an alternative to the as-is situation, the following hypotheses are adopted:

300

400

500

600

700

800

900

1000

1100

1200

Fig. 12.48 Component A.2 preventive maintenance optimization, scenario 4

• Scenario 2 (see the first section of Table 12.9). Maintenance actions are planned for the following very expensive components, not preventively replaced in the as-is configuration: component A.1 (cost per action equal to about 196 u: c: added to crew costs) and component C.1 (cost per action equal to 183 u: c: added to crew costs). The adopted replacement times tp , 4;000 h for component A.1 and 1;436 h for component C.1, respectively, are the mean values of the parametric probability distribution of the related ttf random variable. The values of the preventive time duration are assumed to be

12.2 Reliability, Availability, and Maintainability Analysis in a Plastic Closures Production System for Beverages

461

Table 12.10 Preventive maintenance parameterization, scenario 4 Part

x 10000

A.1 A.2 B C.1 C.2 C.3 D (1–48) E.1 E.2 E.3 E.4 E.5

Replacement tp (h)

Cf (u. c./action)

Cp (u. c./action)

3,433 778 3,878

471.6400 28.3000 43.1800

264.6400 3.0000 22.5800

1,456 2,810 3,921

110.8800 15.1800 10.9400

6.0000 8.2800 5.8400

2,116 4,727

33.4000 11.7800

12.7000 6.6800

UEC (u. c./h)

Prev. time duration (constant) (h)

0.0822 0.0069 0.0086 no prev. replacement no prev. replacement 0.0078 0.0049 0.0025 no prev. replacement 0.0098 0.0024 no prev. replacement

Cost per action (u. c./action)

Type I/II

195.6400 0.7000 8.6800

II(1) II(1) II(1)

4 0.5 0.3

0.4900 1.3800 –

II(1) II(1) –

0.5 0.3

5.8000 2.5400

– –

20 0.167 1

80 70

Costs [u.c.]

60 50 40 Scenario 1

30

Scenario 2 Scenario 3

20

Scenario 4 10 0 Costs for Parts (CM)

Costs for Crews (CM)

Costs for Parts (PM)

Costs for Crews (PM)

Costs for Crews (IN)

Cost for not producon [€/year x 5]

Total Cost [€/year x 5]

Scenario 1

94600

311615

14735

73657

13484

274915

783006

Scenario 2

70053

296349

41363

78024

13529

265165

764481

Scenario 3

70175

295472

41050

90987

13504

275613

786801

Scenario 4

62248

187599

53616

73404

18124

210058

605048

Fig. 12.49 Multiscenario analysis. Cost evaluation and comparison

constant, and equal to 5 and 1 h, respectively (see Table 12.9). • Scenario 3 (see the second section of Table 12.9). This configuration differs from scenario 2 in the time duration of the preventive action, here equal to 20 and 5 h, respectively, for components A.1 and C.1. These values are the same as the constant ttr in the case of corrective action. • Scenario 4 (see Table 12.10). This configuration differs from scenario 2 in the identification and the adoption of the best value of the replacement time tp for the generic component/block of the

system, in accordance with the replacement analytical model of type I and the application of the multiscenario parametric analysis by Simulink® MATLAB® 7.0. Figure 12.48 exemplifies the output of this dynamic analysis for component A.2 with a best replacement time tp equal to 778 h. It can be concluded that it is not convenient to plan a preventive replacement for four basic components C.1, C.2, E.2, and E.5. Tables 12.11 and 12.12 compare the results for the set of four scenarios previously illustrated in or-

462

12 Applications and Case Studies

Table 12.11 Multiscenario analysis and performance evaluation Scenario 1 Mean availability (all events) Mean availability (w/o PM and inspection) Point availability (all events) at 32,200 Reliability (32,200) Expected number of failures MTTFF Uptime (ref. 5 years) CM downtime (ref. 5 years) Inspection downtime (ref. 5 years) PM downtime (ref. 5 years) Total downtime (ref. 5 years) Number of failures (ref. 5 years) Number of CMs (ref. 5 years) Number of inspections (ref. 5 years) Number of PMs (ref. 5 years) Total events (ref. 5 years) Total maintenance costs (u. c./year x 5)

0.928 0.930 0.928 0 1,771.77 4.63 29,870.21 2,243.62 0.46 85.71 2,329.79 1,771.77 1,771.77 14.00 109.97 1,895.73 50,809

Scenario 2 0.930 0.934 0.946 0 1,764.42 3.24 29,952.84 2,132.72 . 0.46 113.98 2,247.16 1,764.42 1,764.42 14.00 122.90 1,901.32 49,932

Scenario 3 0.928 0.934 0.926 0 1,759.39 5.35 29,864.30 2,126.06 0.46 209.18 2,335.70 1,759.39 1,759.39 14.00 121.18 1,894.57 51,119

Scenario 4 0.945 0.960 0.940 0 1,175.37 4.97 30,419.85 1,283.35 0.27 496.53 1,780.15 1,175.37 1,175.37 8.09 786.04 1,969.49 39,499

Table 12.12 Multiscenario analysis and cost evaluation Costs (u. c.) Costs for parts (CM) (ref. 5 years) Costs for crews (CM) (ref. 5 years) Total CM costs (ref. 5 years) Costs for parts (PM) (ref. 5 years) Costs for crews (PM) (ref. 5 years) Total PM costs (ref. 5 years) Costs for crews (IN) (ref. 5 years) Total inspection costs (ref. 5 years) Total maintenenace costs (u. c./year  5) Total Downtime (h) (ref. 5 years) Cost for not production (u. c./year  5) Total cost (u. c./year  5) Annual total cost (u. c./year)

Scenario 1

Scenario 2

Scenario 3

Scenario 4

94,600 311,615 406,216 14,735 73,657 88,391 13,484 13,484 508,091 466 274,915 783,006 156,601

70,053 296,349 366,402 41,363 78,024 119,387 13,529 13,529 499,317 449 265,165 764,481 152,896

70,175 295,472 365,647 41,050 90,987 132,036 13,504 13,504 511,188 467 275,613 786,801 157,360

62,248 187,599 249,847 53,616 73,404 127,019 18,124 18,124 394,991 356 210,058 605,048 121,010

der to demonstrate the efficacy of the applied models and methods to support decisions regarding maintenance planning. In particular, scenario 4 is the best performer in terms of mean availability, equal to 0.945. Moreover, in comparison with the as-is configuration of scenario 1, both the total maintenance cost (about 39,450 u. c. for 5 years, 22%) and the total annual costs, including nonproduction costs (about 121,009 u. c., 29:5%), are significantly reduced. Tables 12.11 and 12.12 outline the whole situation, as illustrated in Fig. 12.49.4

4

All costs are in unit of cost, see footnote 3.

12.3 Conclusions and Call for New Contributions This chapter is thought to be continuously under construction, and therefore it is open to new case studies from industrial and service applications, as outlined at the beginning with an explicit invitation for readers to submit original contributions. For this purpose the reader is invited to submit original contributions by contacting one of the authors.

A

Appendix

A.1 Standardized Normal Distribution 8 Zz ˆ ˆ ˆ ˆ f .x/ dx F .z/ D ˆ < 1

ˆ ˆ ˆ 1 x2 ˆ ˆ : f .x/ D p e 2 2 z

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4

0.50000 0.53983 0.57926 0.61791 0.65542 0.69146 0.72575 0.75804 0.78814 0.81594 0.84134 0.86433 0.88493 0.90320 0.91924 0.93319 0.94520 0.95543 0.96407 0.97128 0.97725 0.98214 0.98610 0.98928 0.99180 0.99379 0.99534 0.99653 0.99744 0.99813 0.99865 0.99903 0.99931 0.99952 0.99966

0.50399 0.54380 0.58317 0.62172 0.65910 0.69497 0.72907 0.76115 0.79103 0.81859 0.84375 0.86650 0.88686 0.90490 0.92073 0.93448 0.94630 0.95637 0.96485 0.97193 0.97778 0.98257 0.98645 0.98956 0.99202 0.99396 0.99547 0.99664 0.99752 0.99819 0.99869 0.99906 0.99934 0.99953 0.99968

0.50798 0.54776 0.58706 0.62552 0.66276 0.69847 0.73237 0.76424 0.79389 0.82121 0.84614 0.86864 0.88877 0.90658 0.92220 0.93574 0.94738 0.95.728 0.96562 0.97257 0.97831 0.98300 0.98679 0.98983 0.99224 0.99413 0.99560 0.99674 0.99760 0.99825 0.99874 0.99910 0.99936 0.99957 0.99969

0.51197 0.55172 0.59095 0.62930 0.66640 0.70194 0.73565 0.76730 0.79673 0.82381 0.84850 0.87076 0.89065 0.90824 0.92364 0.93699 0.94845 0.95818 0.96638 0.97320 0.97882 0.98341 0.98713 0.99010 0.99245 0.99430 0.99573 0.99683 0.99767 0.99831 0.99878 0.99913 0.99938 0.99957 0.99970

0.51595 0.55567 0.59483 0.63307 0.67003 0.70540 0.73891 0.77035 0.79955 0.82639 0.85083 0.87286 0.89251 0.90988 0.92507 0.93822 0.94950 0.95907 0.96712 0.97381 0.97932 0.98382 0.98745 0.99036 0.99266 0.99446 0.99585 0.99693 0.99774 0.99836 0.99882 0.99916 0.99940 0.99958 0.99971

0.51994 0.55962 0.59871 0.63683 0.67364 0.70884 0.74215 0.77337 0.80234 0.82894 0.85.314 0.87493 0.89435 0.91149 0.92647 0.93943 0.95053 0.95994 0.96784 0.97441 0.97982 0.98422 0.98778 0.99061 0.99286 0.99461 0.99598 0.99702 0.99781 0.99841 0.99886 0.99918 0.99942 0.99960 0.99972

0.52392 0.56356 0.60257 0.64058 0.67724 0.71226 0.74537 0.77637 0.80511 0.83147 0.85543 0.87698 0.89617 0.91309 0.92786 0.94062 0.95154 0.96080 0.96856 0.97500 0.98030 0.98461 0.98809 0.99086 0.99305 0.99477 0.99609 0.99711 0.99788 0.99846 0.99889 0.99921 0.99944 0.99961 0.99973

0.52790 0.56749 0.60642 0.64431 0.68082 0.71566 0.74857 0.77935 0.80785 0.83398 0.85769 0.87900 0.89796 0.91466 0.92922 0.94179 0.95254 0.96164 0.96926 0.97558 0.98077 0.98500 0.98840 0.99111 0.99324 0.99492 0.99621 0.99720 0.99795 0.99851 0.99893 0.99924 0.99946 0.99962 0.99974

0.53188 0.57142 0.61026 0.64803 0.68439 0.71904 0.75175 0.78230 0.81057 0.83646 0.85993 0.88100 0.89973 0.91621 0.93056 0.94295 0.95352 0.96246 0.96995 0.97615 0.98124 0.98537 0.98870 0.99134 0.99343 0.99506 0.99632 0.99728 0.99801 0.99856 0.99897 0.99926 0.99948 0.99964 0.99975

0.53586 0.57535 0.61409 0.65173 0.68793 0.72240 0.75490 0.78524 0.81327 0.83891 0.86214 0.88298 0.90147 0.91774 0.93189 0.94408 0.95449 0.96327 0.97062 0.97670 0.98169 0.98574 0.98899 0.99158 0.99361 0.99520 0.99643 0.99736 0.99807 0.99861 0.99900 0.99929 0.99950 0.99965 0.99976

R. Manzini, A. Regattieri, H. Pham, E. Ferrari, Maintenance for Industrial Systems © Springer 2010

463

464

A Appendix

A.2 Control Chart Constants

n

D3

D4

B3

B4

A2

A3

d2

c4

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

0 0 0 0 0 0.076 0.136 0.184 0.223 0.256 0.283 0.307 0.328 0.347 0.363 0.378 0.391 0.403 0.415 0.425 0.434 0.443 0.451 0.459

3.267 2.574 2.282 2.114 2.004 1.924 1.864 1.816 1.777 1.744 1.717 1.693 1.672 1.653 1.637 1.622 1.608 1.579 1.585 1.575 1.566 1.557 1.548 1.541

0 0 0 0 0.030 0.118 0.185 0.239 0.284 0.321 0.354 0.382 0.406 0.428 0.448 0.466 0.482 0.497 0.510 0.523 0.534 0.545 0.555 0.565

3.267 2.568 2.266 2.089 1.970 1.882 1.815 1.761 1.716 1.679 1.646 1.618 1.594 1.572 1.552 1.534 1.518 1.503 1.490 1.477 1.466 1.455 1.445 1.435

1.880 1.023 0.729 0.577 0.483 0.419 0.373 0.337 0.308 0.285 0.266 0.249 0.235 0.223 0.212 0.203 0.194 0.187 0.180 0.173 0.167 0.162 0.157 0.153

2.659 1.954 1.628 1.427 1.287 1.182 1.099 1.032 0.975 0.927 0.886 0.850 0.817 0.789 0.763 0.739 0.718 0.698 0.680 0.663 0.647 0.633 0.619 0.606

1.128 1.693 2.059 2.326 2.534 2.704 2.847 2.970 3.078 3.173 3.258 3.336 3.407 3.472 3.532 3.588 3.640 3.689 3.735 3.778 3.819 3.858 3.895 3.931

0.7979 0.8862 0.9213 0.9400 0.9515 0.9594 0.9650 0.9693 0.9727 0.9754 0.9776 0.9794 0.9810 0.9823 0.9835 0.9845 0.9854 0.9862 0.9869 0.9876 0.9882 0.9887 0.9892 0.9896

A.3 Critical Values of Student’s Distribution with  Degree of Freedom

A.3 Critical Values of Student’s Distribution with  Degree of Freedom



˛ 0.2

0.1

0.05

0.01

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

1.376 1.061 0.978 0.941 0.920 0.906 0.896 0.889 0.883 0.879 0.876 0.873 0.870 0.868 0.866 0.865 0.863 0.862 0.861 0.860 0.859 0.858 0.858 0.857 0.856 0.856 0.855 0.855 0.854 0.854

3.078 1.886 1.638 1.533 1.476 1.440 1.415 1.397 1.383 1.372 1.363 1.356 1.350 1.345 1.341 1.337 1.333 1.330 1.328 1.325 1.323 1.321 1.319 1.318 1.316 1.315 1.314 1.313 1.311 1.310

6.314 2.920 2.353 2.132 2.015 1.943 1.895 1.860 1.833 1.812 1.796 1.782 1.771 1.761 1.753 1.746 1.740 1.734 1.729 1.725 1.721 1.717 1.714 1.711 1.708 1.706 1.703 1.701 1.699 1.697

31.821 6.965 4.541 3.747 3.365 3.143 2.998 2.896 2.821 2.764 2.718 2.681 2.650 2.624 2.602 2.583 2.567 2.552 2.539 2.528 2.518 2.508 2.500 2.492 2.485 2.479 2.473 2.467 2.462 2.457

465

Bibliography

Abernethy RB (2007) The new Weibull handbook. Abernethy, North Palm Beach Ait Kadi D, Cléroux R (1988) Optimal block replacement policies with multiple choice at failure. Nav Res Logist 35:99– 110 Akhmedjanov FM (2001) Reliability databases: state of the art and perspectives. Riso Natl Lab R1235:1–37 Amari SV, Pham H (2007) A novel approach for optimal costeffective design of complex repairable systems. IEEE Trans Syst Man Cybern Part A 37(3):406–415 American Petroleum Institute (2002) API risk based inspection. API recommended practice 580. American Petroleum Institute, Washington Ansell J, Bendell A, Humble S (1984) Age replacement under alternative cost criteria. Manag Sci 30:358–367 Arts RHPM, Knapp GM, Mann L Jr (1988) Some aspects of measuring maintenance performance in the process industry. J Qual Maint Eng 4(1):6–11 Aven T (1983) Optimal replacement under a minimal repair strategy – a general failure model. Adv Appl Probab 15:198– 211 Aven T (1990) Availability formulae for standby systems of similar units that are preventively maintained. IEEE Trans Reliab 39:603–606 Bagai I, Jain K (1994) Improvement, deterioration, and optimal replacement under age-replacement with minimal repair. IEEE Trans Reliab 43:156–162 Bai DS, Jang JS, Kwon YI (1983) Generalized preventive maintenance policies for a system subject to deterioration. IEEE Trans Reliab 32:512–514 Bai J, Pham H (2004) Discounted warranty cost of minimally repaired series systems. IEEE Trans Reliab 53(1):37–42 Bai J, Pham H (2005) Repair-limit risk-free warranty policies with imperfect repair. IEEE Trans Syst Man Cybern Part A 35(6):765–772 Bai J, Pham H (2006) Cost analysis on renewable full-service warranties for multi-component systems. Eur J Oper Res 168:492–508 Baldin A (1983) La manutenzione e il rinnovo. Angeli, Milan Baldin A, Furlanetto L, Roversi A, Turco F (1978) Manuale della manutenzione degli impianti industriali. Angeli, Milan

Banerjee PK, Chuiv N (1996) Inspection policies for repairable systems. IIE Trans 28(12):1003–1010 Barlow RE, Hunter LC (1960) Optimum preventive maintenance policies. Oper Res 8:90–100 Barlow RE, Proschan F (1965) Mathematical theory of reliability. Wiley, New York Beichelt F (1981) A generalized block-replacement policy. IEEE Trans Reliab 30:171–172 Beichelt F, Fisher K (1980) General failure model applied to preventive maintenance policies. IEEE Trans Reliab 29:39– 41 Ben Daya M, Duffuaa SO (1997) Coordination of inspections for a group of machines. J Qual Maint Eng 3(3)199–208 Ben-Daya M, Duffuaa SO, Raouf A (2000) Maintenance, modeling and optimization. Kluwer, Boston. Berg M (1978) General trigger-off replacement procedures for two-unit system. Nav Res Logist Q 25:15–29 Berg MP (1995) The marginal cost analysis and its application to repair and replacement policies. Eur J Oper Res 82:214– 224 Bilgen S, Deligönül ZS (1987) Weibull renewal equation solution. Computing 39(1):71–76 Birolini A (1994) Quality and reliability of technical systems. Springer, Berlin Black JT, Hunter SL (2003) Lean manufacturing systems and cell design. Society of Manufacturing Engineers, Dearborn Block HW, Borges WS, Savits TH (1985) Age-dependent minimal repair. J Appl Probab 22:370–385 Block HW, Borges WS, Savits TH (1998) A general age replacement model with minimal repair. Nav Res Logist 35(5):365– 372 Boland PJ (1982) Periodic replacement when minimal repair costs vary with time. Nav Res Logist Q 29:541–546 Boland PJ, Proschan F (1982), Periodic replacement with increasing minimal repair costs at failure. Oper Res 30:1183– 1189 Botter R, Fortuin L (2000) Stocking strategy for service parts – a case study. Int J Oper Prod Manag 20(6):656–674 Boylan JE, Syntetos AA (2006) Accuracy and accuracy implication metrics for intermittent demand. Int J Appl Forecast 4:39–42

467

468 Boylan JE, Syntetos AA, Karakostas GC (2008) Classification for forecasting and stock control: a case study. J Oper Res Soc 59:473–481 Braglia M, Grassi A, Montanari R (2004) Multi-attribute classification method for spare parts inventory management. J Qual Manag Eng 10(1):55–65 Bris R, Chatelet E, Yalaoui F (2003) New method to minimize the preventive maintenance cost of series-parallel systems. Reliab Eng Syst Saf 82:247–255 Brown M, Proschan F (1983) Imperfect repair. J Appl Probab 20:851–859 Burr JT (2005) Elementary statistical quality control, 2nd edn. Dekker, New York Canfield RV (1986) Cost optimization of periodic preventive maintenance. IEEE Trans Reliab 35:78–81 Cattaneo M (2004) La manutenzione in Italia nelle PMI. http://aiman.gs-m.eu/documenti/manutenzione_italia_pmi_ 2000%20.pdf Chan FTS, Lau HCW, Ip L, Chan HK, Kong S (2005) Implementation of total productive maintenance: a case study. Int J Prod Econ 95(1):71–94 Chan PKW, Downs T (1978) Two criteria for preventive maintenance. IEEE Trans Reliab 27:272–273 Chand G, Shirvani B (2000) Implementation of TPM in cellular manufacture. J Mater Process Technol 4(3):149–154 Chang PL, Chou YC, Huang MG (2005) A (r; r; Q) inventory model for spare parts involving equipment criticality. Int J Prod Econ 76:66–74 Chelbi S, Ait-Kadi D (1999) An optimal inspection strategy for randomly failing equipment, Reliab Eng Syst Saf 63:127– 131 Chen M, Feldman RM (1997) Optimal replacement policies with minimal repair and age-dependent costs. Eur J Oper Res 98:75–84 Cheng SC (2005) Fuzzy estimator of the total failure time of censored data. In: Proceedings of conference of the North American Fuzzy Information Processing Society – NAFIPS, pp 505–509 Cheng SC, Mordeson JN (1985) A note on the notion of adequacy. Chin J Math 13(1):13–41 Cho DI. Parlar M (1991) A survey of maintenance models for multi-unit systems. Eur J Oper Res 51:1–23 Chuenusa C, Ramnik B, Fiju A (2004) The status of maintenance management in UK manufacturing organizations: results from a pilot survey. J Qual Maint Eng 10(1):5–15 Clavareau J, Labeau P-E (2009) Maintenance and replacement policies under technological obsolescence. Reliab Eng Syst Saf 94(2):370–381 Cléroux R, Dubuc S, Tilquin C (1979) The age replacement problem with minimal repair and random repair costs. Oper Res 27:1158–1167 Cobbaert K, Van Oudheusden D (1996) Inventory models for fast moving spare parts subject to sudden death obsolescence. Int J Prod Econ 44:239–248 Constantine AG, Robinson NI (1997) The Weibull renewal function for moderate to large arguments. Comput Stat Data Anal 24:9–27 Contreras CA (2007) Using the EM algorithm for inference in a mixture of distributions with censored but partially identifiable data. Comput Stat Data Anal 51(5):2769–2781

Bibliography Corrigo (2007) Preventive maintenance benchmark study results. http://www.corrigo.com/uploadedFiles/Products/ Resources/Benchmark_Reports/2007_Corrigo_PM_ Benchmark_Report.pdf Crocker J, Kumar DU (2000) Age related maintenance versus reliability centered maintenance: a case study on aeroengines. Reliab Eng Syst Saf 67(2):113–118 Croston JD (1972) Forecasting and stock control for intermittent demands. Oper Res Q 23:289–304 CSERGE, Warren Spring Laboratory, EFTEL (1993) Externalities from landfill and incineration. Report to the Department of Environment. HMSO, London Dal B, Tugwell P, Greatbanks R (2000) Overall equipment effectiveness as a measure of operational improvement, a practical analysis. Int J Oper Prod Manag 20(12):1488–1502 D’Augostino RB, Stephens MA (1986) Goodness-of-fit techniques. Dekker, New York de Almeida JR, Camargo JB, Basseto BA, Paz SM (2003) Best practices in code inspection for safety-critical software, IEEE Software 20(3):56–63 Dekker R (1996) Applications of maintenance optimization models: a review and analysis. Reliab Eng Syst Saf 51(3):229–240 Dekker R, Roelvink IFK (1995) Marginal cost criteria for preventive replacement of a group of components. Eur J Oper Res 84:467–480 Devore JL, Farnum NR (1999) Applied statistics for engineers and scientists. Duxbury, Pacific Grove Dhillon BS (2005) Reliability, quality, and safety for engineers. CRC, New York Dieulle L, Berenguer C, Gralland G, Roussignol M (2003) Sequential condition-based maintenance scheduling for a deteriorating system. Eur J Oper Res 150:451–461 Dijkgraaf E, Vollebergh H (2003) Burn or bury? A social comparison of final waste disposal methods SIEV. Erasmus University Rotterdam and OCFEB, Rotterdam Dimitrakos TD, Kyriakidis EG (2007) An improved algorithm for the computation of the optimal repair/replacement policy under general repairs. Eur J Oper Res 182:775–782 Duffuaa SO, Raouf A, Campbell JD (1999) Planning and control of maintenance systems. Modelling and analysis. Wiley, New York Ebeling CE (2005) An introduction to reliability and maintainability engineering, McGraw-Hill, New York Ebrahimi N (1985) Mean time to achieve a failure-free requirement with imperfect repair. IEEE Trans Reliab 34:34–37 EC (1996) Cost–benefit analysis of the different municipal solid waste management systems: objectives and instruments for the year 2000. Final report to EC, by Coopers and Lybrand, CSERGE and EFTEC EC (2000a) A study on economic valuation of environment externalities from landfill disposal and incineration of waste. Final main report. European Commission, DG Environment. http://europa.eu.int/comm/ environment/waste/studies/econ eva landfill.htm EC (2000b) A study on economic valuation of environment externalities from landfill disposal and incineration of waste. European Commission, DG Environment ECON, Senter for Okonomosk Analyse (1995) Environmental costs of different types of waste

Bibliography Economou M (2004) The merit and limitations of reliability predictions. In: IEEE – RAMS 2004 conference, pp 352–357 EMC (1996) Waste disposal management, analyzing direct and indirect costs. Final report. EMC, Israel, p 70 Enosh (1996) Solid waste management externalities. Final report. Enosh, Israel, p 140 Esary JD, Marshall AW, Proschan F (1973) Shock models and wear processes. Ann Probab 1:627–649 Eshet T, Ayalon O, Shechter M (2006) Valuation of externalities of selected waste management alternatives: a comparative review and analysis. Resour Conserv Recycl 46:335–364 Eti MC, Ogaji SOT, Probert SD (2004) Implementing total productive maintenance in Nigerian manufacturing industries. Appl Energy 79(4):385–40 Eunomia (2002) Economic analysis of option for managing biodegradable municipal waste. Final report and appendices. http://europa.eu.int/comm/environment/waste/compost/ econanalysis finalreport.pdf Evans RD (2005) Too small for a CMMS, working paper. Compliance Technologies, Woodstock. http://www. maintenanceworld.com/articles/evansr/toosmall.htm ExternE (1998) Extern: externalities of energy. New results, to be published by Extern Program of European Commission. Directorate-General XII, Science Research and Development, p 67 Fedele L, Furlanetto L, Saccardi D (2005) Progettare e gestire la manutenzione. Mc-Graw-Hill, Milan Feldman RM (1976) Optimal replacement with semi-Markov shock models. J Appl Probab 13:108–117 Feldman RM (1977) Optimal replacement for systems governed by Markov additive shock processes. Ann Probab 5:413–429 Fernandes P (2001) A framework for a strategy driven manufacturing system design in an aerospace environment – design beyond factory floor. MSc thesis, Massachusetts Institute of Technology, Boston Ferrari E, Pareschi A, Persona A, Regattieri A (2001) TPM: a total approach for industrial maintenance question. In: Proceedings of 7th international conference on reliability and quality in design – ISSAT01, Washington DC, pp 262–268 Ferrari E, Pareschi A, Persona A, Regattieri A (2002) TPM: situation and procedure for a soft introduction in Italian factories. Tqm Mag 14:350–358 Ferrari E, Pareschi A, Persona A, Regattieri A (2003) Plant layout computerised design: logistic and relayout program (LRP). Int J Adv Manuf Technol 21(12):917–922 Ferrari E, Pareschi A, Persona A, Regattieri A (2006) Statistical management and modeling for demand spare parts. In: Springer handbook of engineering statistics. Springer, Berlin, pp 905–929 Ferrari G (1994) La gestione delle manutenzione. Angeli, Milan Fox B (1966) Age replacement with discounting. Oper Res 14:533–537 Fu HM (2007) Integral estimate method for interval censored data. J Aerosp Power 22(2):175–179 Gamberi M, Manzini R, Regattieri A (2009) A new approach for the automatic analysis and control of material handling systems: integrated layout flow analysis (ILFA). Int J Adv Manuf Technol 41(1):156–167 Geraghty T (1996) Beyond TPM. Manuf Eng 7(5):183–185

469 Ghobbar AA, Friend CH (2002) Sources of intermittent demand for aircraft spare parts within airline operations. J Air Transp Manag 8:221–231. Ghobbar AA, Friend CH (2003) Evaluation of forecasting methods for intermittent parts demand in the field of aviation: a predictive model. Comput Oper Res 30(14):2097–2114. Ghobbar A.A, Friend CH (2004) The material requirements planning system for aircraft maintenance and inventory control: a note. J Air Transp Manag 10:217–221 Giri BC, Dohi T, Kaio N (2005) A discrete time order replacement model with time discounting and spare parts provisioning. J Qual Maint Eng 11(3):190–205 Glasser GJ (1967) The age replacement problem. Technometrics 9:83–91 Gosavi A (2006) A risk-sensitive approach to total productive maintenance. Automatica 42(8):1321–1330 Grall C, Berenguer L, Dieulle C (2002) A condition-based maintenance policy for stochastically deteriorating systems. Reliab Eng Syst Saf 76:167–180 Grall L, Dieulle C, Berenguer L, Roussignol M (2002) Continuous-time predictive-maintenance scheduling for a deteriorating system. IEEE Trans Reliab 51(2):141–150 Gutierrez RS, Solis AO, Mukhopadhyay S (2008) Lumpy demand forecasting using neural networks. Int J Prod Econ 111(2):409–420 Haange R (1995) Overview of remote-maintenance scenarios for the ITER machine. Fusion Eng Des 27:69–82 Hariga M (1996) A maintenance inspection model for a single machine with general failure distribution. Microelectron Reliab 36(3):353–358 Hariga M, Azaiez N, Ben Daya M (2006) A discounted integrated inspection-maintenance model for a single deteriorating production facility. Int Trans Oper Res 13:353–364 Hartman E (1992) Successfully installing TPM in a nonJapanese plant. TPM, London Health and Safety Executive (2005) European comparisons – fatal injuries. http://www.hse.gov.uk/statistics/european/ fatal.htm Hocley CJ, Appleton DP (1997) Setting the requirements for the Royal Air Force’s next generation aircraft. Paper presented at the annual reliability and maintainability symposium, Philadelphia Hollander M, Presnell B, Sethuraman J (1992) Nonparametric methods for imperfect repair models. Ann Stat 20(2):879– 887 Hsieh KL (2007) Applying neural networks approach to achieve the parameter optimization for censored data. WSEAS Trans Comput 6(5):858–863 Huang J, Miller CR, Okogbaa OG (1995) Optimal preventivereplacement intervals for the Weibull life distribution: solutions and applications. In: Proceedings of the annual reliability and maintainability symposium, Washington Hutchins D (1998) Introducing TPM. Manuf Eng 4(2):34–36 Inagaki T, Inoue K, Akashi H (1980) Optimization of staggered inspection schedules for protective systems. IEEE Trans Reliab 29:170–173 International Labor Organization (2006) Facts on safety at work. http://www.ilo.org/public/english/bureau/inf/download/ factsheets/pdf/wdshw.pdf

470 Jardine AKS, Buzacott JA (1985) Equipment reliability and maintenance. Eur J Oper Res 19:285–296 Jardine AKS, Tsang AHC (2006) Maintenance, replacement and reliability. Theory and applications. Taylor & Francis, Oxford Jeong IJ, Leon VJ, Villalobos JR (2007) Integrated decisionsupport system for diagnosis. Maint Plann Sched Manuf Syst 45(2):267–285 Jiang R, Ji P, Tsang AHC (2006) Preventive effect of optimal replacement policies. J Qual Maint Eng 12(3):267–274 JIPM – Japan Institute of Plant Maintenance (1995) Applichiamo il TPM. Guida operativa alla realizzazione del total productive maintenance. Angeli, Milan Johnston FR, Boylan JE (1996) Forecasting for items with intermittent demand. J Oper Res Soc 47:113–121 Kaspi M, Shabtay D (2003) Optimization of the machining economics problem for a multistage transfer machine under failure, opportunistic and integrated replacement strategies. Int J Prod Res 41(10):2229–2247 Khan FI, Haddara M, Krishnasamy LA (2008) New methodology for risk-based availability analysis reliability. IEEE Trans Reliab 57(1):103–111 Kijima M, Nakagawa T (1991) Accumulative damage shock model with imperfect preventive maintenance. Nav Res Logist 38:145–156 Kijima M, Nakagawa T (1992) Replacement policies of a shock model with imperfect preventive maintenance. Eur J Oper Res 57:100–110 Kijima M, Morimura H, Suzuki Y (1988) Periodical replacement problem without assuming minimal repair. Eur J Oper Res 37(2):194–203 Kijimma M (1989) Some results for repairable systems with general repair. J Appl Probab 26:89–102 Klutke GA, Yang YJ (2002) The availability of inspected systems subjected to shocks and graceful degradation. IEEE Trans Reliab 44:371–374 Koelsch JR (1993) A dose of TPM: downtime needn’t be a better pill. Manuf Eng 110(3):6–66 Konopka J, Fowler J (1994) A TPM inspired factory throughput analysis tool. In: Proceedings of international symposium on semiconductor manufacturing, pp 64–67 Koudate A, Suzue T (1990) Variety reduction program: a production strategy for product diversification. Productivity, New York Kumar DU, Knezevic J, Crocker J (1999) Maintenance free operating period – an alternative measure to MTBF and failure rate for specifying reliability? Reliab Eng Syst Saf 64:127– 131 Lai KK, Leung FKN, Tao B, Wang SY (2000) Eur J Oper Res 124(2):294–306 Lam CT, Yeh RH (1994) Optimal maintenance-policies for deteriorating systems under various maintenance strategies. IEEE Trans Reliab 43(3):423–430 Lam YA (1991) Optimal repairable replacement model for deteriorating system. J Appl Probab 28:843–851 Lee YB, Kim HE, Yoo YC, Park JH (2006) Accelerated life test model for life prediction of piston assemblies in hydraulic pump and motor. Key Eng Mater 326–328:649–652 Legat V, Zaludova AH, Cervenka V, Jurca V (1996) Contribution to optimization of preventive replacement. Reliab Eng Syst Saf 51:259–266

Bibliography Lewis EE (1987) Introduction to reliability engineering. Wiley, New York Li W (2005) Reliability and maintenance modeling of multistate degraded systems with multiple competing failure processes. PhD dissertation, Department of Industrial and Systems Engineering, Rutgers University Li W, Pham H (2003) An inspection-maintenance model for systems with multiple competing processes. IEEE Trans Reliab 54:318–327 Li W, Pham H (2005a) Reliability modeling of multi-state degraded systems with multi-competing failures and random shocks. IEEE Trans Reliab 54(2):297–303 Li W, Pham H (2005b) An inspection-maintenance model for systems with multiple competing processes. IEEE Trans Reliab 54(2):318–327 Lie XG, Makis V, Jardine AKS (1995) A replacement model with overhauls and repairs. Nav Res Logist 42:1063–1079 Maggard B (1992) The theory and design of total productive maintenance. A guide for implementing TPM. TPM, London Makis V, Jardine AKS (1992) Optimal replacement policy for a general model with imperfect repair. J Oper Res Soc 43:111–120 Makridakis S, Wheelwright SC, Hyndman RJ (1998) Forecasting: methods and applications. Wiley, New York Manzini R, Gamberi M, Regattieri A, Persona A (2004) Framework for designing a flexible cellular assembly system. Int J Prod Res 42(17):3505–3528 Manzini R, Gamberi M, Regattieri A (2005) Design and control of a flexible order-picking system (FOPS). A new integrated approach to the implementation of an expert system. J Manuf Technol Manag 16(1):18–35 Manzini R, Persona A, Regattieri A (2006a) Framework for designing and controlling a multi-cellular flexible manufacturing system. Int J Serv Oper Manag 2:1–21 Manzini R, Gamberi M, Regattieri A (2006b) Design and control of an AS/RS. Int J Adv Manuf Technol 28(7/8):766–774 Manzini R, Ferrari E, Meccariello A, Regattieri A (2007) A cost based model for failure modes analysis, reliability prediction and magnitude evaluation of a waste to energy plant. In: Proceedings of the ISSAT international conference on modeling of complex systems and environments, Ho Chi Minh City, Vietnam Manzini R, Gamberi M, Persona A, Regattieri A (2007) Design of a class based storage picker to product order picking system. Int J Adv Manuf Technol 32:811–821 Marin JJ, Pollard RW (2005) Experience report on the FIDES reliability predictions. In: IEEE – RAMS 2005 conference, vol 1, pp 8–13 McCall JJ (1965) Maintenance policies for stochastically failing equipment: a survey. Manag Sci 11(5):493–524 Miranda ML, Hale B (1997) Waste not, want not: the private and social costs of waste-to-energy production. Energy Policy 25:587–600 Monesi F (2008) La qualità totale nelle aziende di servizi: implementazione di un approccio lean six Sigma finalizzato all’efficienza gestionale. PhD thesis, University of Bologna Murthy DNP, Blischke WR (2006) Warranty management and product manufacture. Springer, Berlin Murthy DNP, Nguyen DG (1981) Optimal age-policy with imperfect preventive maintenance. IEEE Trans Reliab 30:80– 81

Bibliography Nakagawa T (1977) Optimum preventive maintenance policies for repairable systems. IEEE Trans Reliab 26:168–173 Nakagawa T (1979) Imperfect preventive maintenance. IEEE Trans Reliab 28(5):402–408 Nagakawa T (1980) A summary of imperfect preventive maintenance policies with minimal repair. Res Oper 14(3):249–255 Nakagawa T (1981a) A summary of periodic replacement with minimal repair at failure. J Oper Res Soc Jpn 24:213–227 Nakagawa T (1981b) Modified periodic replacement with minimal repair at failure. IEEE Trans Reliab 30:165–168 Nakagawa T (1983) Optimal number of failures before replacement time. IEEE Trans Reliab 32:115–116 Nakagawa T, Osaki S (1974) Some aspects of damage models. Microelectron Reliab 13:253–257 Nakagawa T, Yasui K (1987) Optimum policies for a system with imperfect maintenance. IEEE Trans Reliab 36:631–633 Nagano S (2008) Space systems verification program and management process. Syst Eng 11(1):27–38 Nakajima S (1988) Introduction to TPM. Productivity, Cambridge Nakajima S (1992) TPM-total productive maintenance. Productivity, Isedi Natvig B (1990) On information based minimal repair and the reduction in remaining system lifetime due to the failure of a specific module. J Appl Probability 27:365–375 Nelson WB (2005) A bibliography of accelerated test plans. IEEE Trans Reliab 54(2):194–197 Nguyen D.G, Murthy DNP (1981) Optimal preventive maintenance policies for repairable systems. Oper Res 29(6):1181– 1194 O’Hanlon (2005) Computerized maintenance management and enterprise asset management best practices. http://www.cmmscity.com/articles/cmms_best_practices.pdf O’Connor PDT (1988). RADC reliability engineer’s toolkit, USAF Rome Air Development Center. Wiley, New York Ohnishi M, Kawai H, Mine H (1986) An optimal inspection and replacement policy for a deteriorating system. J Appl Probab 23:973–988 Oke SA, Charles-Owaba OE (2006) Application of fuzzy logic control model to Gantt charting preventive maintenance scheduling. Int J Qual Reliab Manag 23(4):441–459 Özekici S (1996) Reliability and maintenance of complex systems. Springer, Berlin Pareschi A, Regattieri A (2002) Remote maintenance, teleservice and internet. Topics, methods and applications. In: Proceedings of 8th international conference on reliability and quality in design, Anaheim, USA Park D, Willemain TR (1999) The threshold bootstrap and threshold jackknife. Comput Stat Data Anal 31:187–202 Park KS (1979) Optimal number of minimal repairs before replacement. IEEE Trans Reliab 28:137–140 Park KS, Yoo YK (1993) (,k) block replacement policy with idle count. IEEE Trans Reliab 42:561–565 Persona A, Regattieri A (1997) Methodology for maintenance system establishment and management in little and medium industry. In: Proceedings of SMI conference, Bologna Pham H (2001) Recent advances in reliability and quality engineering, World Scientific, New York Pham H (ed) (2003a) Handbook of reliability engineering. Springer, Berlin

471 Pham H (2003b) Commentary: steady-state series-system availability. IEEE Trans Reliab 52(2):146–147 Pham H (ed) (2006a) Springer handbook of engineering statistics. Springer, Berlin Pham H (2006b) Reliability modeling, analysis and optimization. World Scientific, New York Pham H (2008) Recent advances in reliability and quality in design. Springer, Berlin Pham H, Wang HZ (1996) Imperfect maintenance. Eur J Oper Res 94:425–438 Pham H, Wang H (2000) Optimal (t,T) opportunistic maintenance of a k-out-of-n system with imperfect PM and partial failure. Nav Res Logist 47:223–239 Pham H, Suprasad A, Misra RB (1996) Reliability analysis of k-out-of-n systems with partially repairable multi-state components. Microelectron Reliab J 36(10):1407–1415 Pham H, Suprasad A, Misra RB (1997) Availability and mean life time prediction of multistage degraded system with partial repairs. Reliab Eng Syst Saf J 56:169–173 Phelps RI (1983) Optimal policy for minimal repair. J Oper Res Soc 34:452–457 Pierskalla WP, Voelker JA (1976) A survey of maintenance models: the control and surveillance of deteriorating systems. Nav Res Logist Q 23:353–388 Pinto FW (2008) Mechanical integrity – stationary equipment reliability. Process Saf Prog 27(2):105–111 Plant Maintenance Resource Center (2004) CMMS implementation survey results. Plant Maintenance Resource Center, Booragoon Politis K, Koutras MV (2006) Some new bounds for the renewal function. Probab Eng Inf Sci 20:231–250 Ponte M (2003) L’informazione essenziale di tecnica e legislazione per costruire. Tipografia del Genio Civile, Rome Powell JC, Brisson I (1994) The assessment of social costs and benefits of waste disposal. Working paper 0967-8875 Rabl A, Spadaro JV, Desaigues B (1998) Usefulness of damage cost estimates despite uncertainties: the example of regulations for incinerators. Environmental risk final report 1 annex 7: COSTS Rampersad HK (1995) Concurrent design of product, process and robotic assembly system, Assem Autom 15:21–28 Regattieri A (1996) Maintenance engineering: methodologies and applied instruments. Graduate thesis, Bologna University Regattieri A (1999) Maintenance engineering in industrial systems. PhD thesis, Parma University Regattieri A, Gamberi M, Gamberini R, Manzini R (2005) Managing lumpy demand for aircraft spare parts. J Air Transp Manag 11:426–431 Rezg N, Chelbi A, Xie X (2005) Modeling and optimizing a joint inventory control and preventive maintenance strategy for a randomly failing production unit: analytical and simulation approaches. Int J Comput Integr Manuf 18(2/3):225– 235 Ritchken P, Wilson JG (1990) (m,T) group maintenance policies. Manag Sci 36:632–639 Rosi L, Salemme R (2001) Remote monitoring & diagnostic for rotating equipment. Working paper. General Electric Power Systems

472 Rotival L, Gai S, Canova M, Lancioni R, Rossi C (2001) Internet changes the rules of the game. Working paper. Nuovo Pignone, Florence Roversi A, Turco F (1974) Il problema dei ricambi negli impianti industriali: la valutazione dei fabbisogni. Angeli, Milan Ryan C, Joiner J, Cryer D (2005) MINITAB handbook. Thomson Brooks/Cole, Belmont Saranga H (2004) Opportunistic maintenance using genetic algorithms. J Qual Maint Eng 10(1):66–74 Savits TH (1988) A cost relationship between age and block replacement policies. J Appl Probab 25:789–796 Sharaf MA, Helmy HA (2001) A classification model for inventory management of spare parts. In: Proceedings of 7th international conference on production, Alexandria, Egypt, vol 1, pp 375–382 Sherif YS, Smith ML (1981) Optimal maintenance models for systems subject to failure – a review. Nav Res Logist Q 28:47–74 Sheu SH (1994) Extended block replacement policy with used item and general random minimal repair cost. Eur J Oper Res 79:405–416 Sheu SH (1998) A generalized age block replacement of a system subject to shocks. Eur J Oper Res 108:345–362 Sheu SH, Griffith WS (1996) Optimal number of minimal repairs before replacement of a system subject to shocks. Nav Res Logist 43:319–333 Smith DJ (2001) Reliability maintainability and risk. Newnes, London Smith R (2003) Benchmark maintenance organization effectiveness. Life Cycle Engineering, Charleston Smith WL (1954) Asymptotic renewal theorems. Proc R Soc Edinb A 64:9–48 Smith WL, Leadbetter MR (1963) On the renewal function for the Weibull distribution. Technometrics 5:393–396 Soland RM (1969) Availability of renewal functions for gamma and Weibull distributions with increasing hazard rate. Oper Res 17(3):536–543 Srinivasan VS (1966) The effect of standby redundancy in system’s failure with repair maintenance. Oper Res 14:1024– 1036 Suresh PV, Chaudhuri D (1994) Preventive maintenance scheduling for a system with assured reliability using fuzzy set theory. Int J Reliab Qual Saf Eng 1(4):497–513 Swanson L (2003) An information-processing model of maintenance management. Int J Prod Econ 83:45–64 Syntesos AA (2001) Forecasting of intermittent demand. PhD thesis, Buckinghamshire Business School, Brunel University Syntetos AA, Boylan JE (2001) On the bias of intermittent demand estimates. Int J Prod Econ 71:457–466 Syntetos AA, Boylan JE (2005) The accuracy of intermittent demand estimates. Int J Forecast 21:303–314 Syntetos AA, Boylan JE (2006) Comments on the attribution of an intermittent demand estimator. Int J Forecast 22:201–208 Syntetos AA, Boylan JE, Croston JD (2005) On the categorization of demand patterns. J Oper Res Soc 56:495–503 Tam ASB, Chan WM, Price JWH (2007) Maintenance scheduling to support the operation of manufacturing and production assets. Int J Adv Manuf Technol 34:399–405

Bibliography Tellus Institute (1992) Tellus packaging study – assessing the impacts of production and disposal of packaging and public policy measures, vol 1. Tellus Institute, Boston Thonemann U, Bradley J (2002) The effect of product variety on supply-chain performance. Eur J Oper Res 143(3):548–569 United States Securities and Exchange Commission (2000) Form 10-Q. http://www.continental.com/web/en-US/content/ company/investor/docs/continental_10q_2000_q3.pdf Usher JS, Kamal AH, Syed WH (1998) Cost optimal preventive maintenance and replacement scheduling. IIE Trans 30:1121–1128 Valdez-Flores C, Feldman RM (1989) A survey of preventive maintenance models for stochastically deteriorating singleunit systems. Nav Res Logist 36:419–446 Vaughn A, Fernandes P, Shields JT (2002) Manufacturing system design framework manual. Massachusetts Institute of Technology, Boston Vesely WE, Roberts NH, Haasl DF (1981) Fault tree handbook, system and reliability. Research Office of Nuclear Regulatory Research, US Nuclear Regulatory Commission, Washington Wang H (2002) A survey of maintenance policies of deteriorating systems. Eur J Oper Res 139:469–489 Wang H, Pham H (1996a) Optimal age-dependent preventive maintenance policies with imperfect maintenance. Int J Reliab Qual Saf Eng 3(2):119–135 Wang H, Pham H (1996b) A quasi renewal process and its applications in imperfect maintenance. Int J Syst Sci 27(10):1055–1062 Wang H, Pham H (1996c) Optimal maintenance policies for several imperfect repair models. Int J Syst Sci 27(6):543–549 Wang H, Pham H (1997) Optimal opportunistic maintenance of a k-out-of-n:G system. Int J Reliab Qual Saf Eng 4(4):369– 386 Wang H, Pham H (2006a) Reliability and maintenance of series systems subject to imperfect repair and correlated failure and repair. Eur J Oper Res 174:1706–1722 Wang H, Pham H (2006b) Reliability and optimal maintenance. Springer, Berlin Wang HZ, Pham H (1999) Some maintenance models and availability with imperfect maintenance in production systems. Ann Oper Res 91:305–318 Weisstein EW (2008) Incomplete gamma function at MathWorld. http://mathworld.wolfram.com/ IncompleteGammaFunction.html Willemain TR, Smart CN, Shockor JH, DeSautels PA (1994) Forecasting intermittent demand in manufacturing: a comparative evaluation of Croston’s method. Int J Forecast 10(4):529–538 Willemain TR, Smart CN, Schwarz HF (2004) A new approach to forecasting intermittent demand for service parts inventories. Int J Forecast 20(3):375–387 Williams TM (1984) Stock control with sporadic and slowmoving demand. J Oper Res Soc 35(10):939–948 Wintle JB, Kenzie BW, Amphlett GH, Smalley J (2001) Best practice for risk based inspection as a part of plant integrity management. Contract research report 363/2001. Prepared by TWA and Royal & Sun Alliance Engineering for the Health and Safety Executive

Bibliography Wortman MA, Klutke GA, Ayhan H (1994) A maintenance strategy for systems subjected to deterioration governed by random shocks. IEEE Trans Reliab 43:439–445 Yannaros N (1994) Weibull renewal processes. Ann Inst Stat Math 46(4):641–648 Zheng X (1995) All opportunity-triggered replacement policy for multiple-unit systems. IEEE Trans Reliab 44:648–652

473 Zhou X, Wang J (2005) A genetic method of LAD estimation for models with censored data. Comput Stat Data Anal 48(3):451–466 Zuckerman D (1989) Optimal inspection policy for a multi-unit machine. J Appl Probab 26:543–551 Zuo MJ, Liu B, Murthy DNP (2000) Replacement-repair policy for multi-state deteriorating products under warranty. Eur J Oper Res 123:519–530

Index

A accelerated life testing, 204 accident, 53, 54 active maintenance time, 320 adaptive-response-rate single exponential smoothing, 412 additive Holt–Winter, 412 ADI average interdemand interval, 410, 430 Aerospace, 21 age-based replacement policy, 319 AIAG FMEA-3, 221 airlines, 6, 417 alternating renewal process, 261, 345 analytic hierarchy process, 425 Anderson–Darling, 43 ANEC, 22 ARP5580, 221 Arrhenius, 205 as bad as first failure, 124 as good as new, 96 asset management, 196 asset register, 190 associative law, 243 attribute data, 33 automation, 8 automotive, 220, 221 autonomous maintenance, 74 autoregressive integrated moving average, 412 availability, 91, 113, 127 B basic event, 237 basic statistics, 89 bathtub curve, 94 Bellcore, 213 binomial distribution, 27, 48 binomial model, 412 Birnbaum, 294 block diagram, 156 block replacement, 399 block replacement policy, 319, 339 Boolean algebra, 239, 243

breakdown, 65, 67, 236, 316 British Standards Institution, 221 BS 5760, 221 C c-chart, 39 call cost, 320 capability analysis, 25, 40 capital equipment, 372 CAPP, 12 case studies, 117 catastrophic risks, 58 causes by occurrence analysis, 227 CEN standard, 19, 21, 60 censored data, 118, 135, 145 central limit theorem, 23 check lists, 59 closure production system, 446 CM downtime, 334 CMMS, 196 Coffin–Manson model, 206 cold standby, 180 comakership with suppliers, 13 combined parallel–series system, 170 combined series–parallel system, 168 common causes, 25, 309 commutative law, 243 complete failure data, 134 component, 88 computer-aided design, 10 computer-aided manufacturing, 10 computerized maintenance management system, ix, 189 condition based maintenance, 315, 454 conditional probability, 89 conditioning event, 237 confidence interval, 137 constant failure rate, 95, 97, 247 constant interval replacement policy, 319, 339 continuous dryer system, 187 continuous improvement, 18 control charts, 25, 464

475

476 conventional risks, 58 corrective, 67, 70 corrective actions, 227 corrective maintenance, 314 cost, 3, 203 cost control, 68 cost of emissions, 438 cost of failure, 438 cost of man work, 438 cost of materials and spare parts, 438 cost rate, 405 crew cost, 320 critical path method, 11 criticality, 294, 430 criticality matrix, 231, 234 Croston method, 412 cumulative distribution, 90 cumulative failure, 152 customer, 5, 18 CV2 squared coefficient of variation, 410, 430 cycle length, 333, 405 cycles of replacement, 369 D danger, 54 data collection, 83, 134, 191, 196 data mining, 12 data warehousing, 12 decision tree, 12 defect, 24, 50 defectives, 75 deferred maintenance, 315 degradation process, 400, 402 demand analysis, 10 density function, 90 dependent event, 311 design, 10 design FMEA (DFMEA), 220 design for assembly, 5 design for disassembly, 5 design for manufacturing, 5 design modification, 318 detection, 222, 225 DFA, 4 DFD, 5 DFM, 4 direct method, 136 discounted cash flow rate of return, 11 discrete random variable, 36 disjunction, 243 distinct causes, 240 distribution function, 36 distribution management, 13, 14 distributive law, 243 double exponential smoothing, 412 downing event criticality index, 159 downtime, 65, 115 drink vending machine, 221 duration of replacements, 336

Index E early wear out, 110 economic order quantity, 13 economic value added, 12 ECOS, 22 effects classification, 227 Efficiency, 183 EFTA, 19 elasticity, 3 electric power supplier, 252 electrical hazards, 55 electromigration model, 205 elementary inspection model, 376 emergency situation, 57 EN ISO 14121, 55 EN ISO 9000, 17, 19 enterprise resource program, 195 environment factor, 207 environmental standards, 21 equivalent fault tree (EFT), 244 equivalent reliability block diagram, 244 ergonomic hazards, 56 erratic demand, 411 expected cycle length, 323 expected number of failures (ENF), 113 expected overall performance, 43 expected within performance, 43 exponential distribution, 97 exponential smoothing, 10 exponential voltage model, 205 exponential weighted moving averages, 412 Eyring, 206 F failure event, 91 failure mode, 233 failure mode and effects analysis (FMEA), 222 failure mode, effects, and criticality analysis (FMECA), 220, 231 failure modes and effects analysis (FMEA), 220, 224 failure process, 90 failure rate databank (FARADA), 206 failure rate prediction, 97, 204, 211 failure replacement, 333 failure report, 191, 192 failure to danger, 57 father event, 236 fault finding, 317 fault tree analysis (FTA), 237, 239, 244, 263 FFR, 113 fire service, 60 first failure, 248 fit analysis, 118, 145 flexible automation, 9 flexible manufacturing system, 8 forecasting, 11, 410 forecasting accuracy, 416 functional scheme, 152

Index

477

functional unit, 133 Fussell–Vesely, 294

key characteristic, 24 KPI, 71, 353

G

L

gamma function, 110 Gantt, 11 golden section search method, 326 goodness of the fit, 106, 145 Government–Industry Data Exchange Program (GIDEP), 206 great risks, 58 group replacement, 339, 358

lamp replacement problem, 358 Laplace transform, 302 law of absorption, 243 lean manufacturing, 73 least-square, 136, 145 left censored data, 134 life cycle management, 5, 320 life data analysis, 133 life–stress relationships, 205 linear regression, 145 location allocation problem, 13 logistic delay, 320 loglogistic function, 454 lognormal distribution, 103, 104 lower control limit, 26 lower incomplete gamma function, 324 lower specification limit, 24 lumpy demand, 411

H harm, 54 hazard, 54, 57 hazard operability, 59 hazard rate, 92, 94 head protection, 60 health, 21, 51 hearing protectors, 60 heating system, 263 hospitals, 6 hot standby, 180 I idempotent law, 243 idle time, 319 IEC 812, 221 immediate maintenance, 315 imperfect maintenance, 388, 398 improved indirect method, 136 in control, 25 incinerator, 278 independent events, 90, 239 individual censored data, 134 industrial management, 5 infant mortality, 94, 110 information technology, 8 INHIBIT gate, 237 inspection maintenance, 317, 373, 381 inspection units, 37, 38 intermediate event, 237 intermittent demand, 410, 411 International Electrotechnical Commission, 221 interval censored data, 134 inventory control, 68, 196 inverse Laplace transform, 305 inverse power rule, 205 item criticality, 232 J J1739, 220 just in time, 13 K k-out-of-n parallel, 170 Kaplan–Meier, 120, 136

M M –P diagram, 58 magnitude, 54, 224 maintainability, 96 maintenance, 65, 71, 398 maintenance control, 66 maintenance cost, 334 maintenance global service, 83, 215 maintenance information system, 189, 196 maintenance management, 65, 77 maintenance planning, 66 maintenance status survey, 80 maintenance strategies, 66, 315, 398, 437 maintenance-free operating period, 390 manufacturing systems, 8 market investigation, 12 market uncertainty, 2 Markov analysis, 116, 301 Martin Titan Handbook, 206 material handling device design, 11 material/substance hazards, 56 maximum likelihood estimator, 136, 149 mean absolute deviation (MAD), 416 mean absolute percentage error (MAPE), 416 mean availability, 115 mean deviation (MD), 416 mean square deviation (MSD), 416 mean time to failure (MTTF), 95, 137 mean time to repair (MTTR), 96, 429 mechanical hazards, 55 median rank, 136 memoryless, 94 micro-stops, 74 MIL-STD-1629A, 220 MIL-STD-217, 206 minimal cut sets (MCS), 239

478 minimal repair, 371 minimum total cost method, 426 minimum total downtime, 355 mirrored blocks, 244 Monte Carlo simulation, 128, 157, 260, 275, 442 motorcycle manufacturer, 429 moving average, 10, 412 multiattribute spare tree analysis, 424 multiple censored data, 134 multiscenario analysis, 337 N net present value, 11 neural network, 145 noise hazards, 55 nonconformity, 24, 27 nonnormal probability, 46 nonparametric reliability evaluation, 101, 120 nonproduction cost, 320 nonrepairable component, 91 normal distribution, 41, 103 not conditional failure rate, 92 np-chart, 37 number of failures, 159 O occurrence–severity matrix, 227 on condition monitoring, 70 on-line counseling, 215 operating time, 319 opportunistic maintenance, 317, 393 ordinary free replacement, 407 OSHA, 53 out of control, 26 out of specification, 49 outsourcing, 83 overall equipment effectiveness OEE, 76 overhaul, 83, 316 P P-AND gate (priority AND gate), 237 p-chart, 35 parallel configuration, 161 Pareto chart, 227 part stress analysis, 207 payback analysis, 11 performance, 2 piping system, 236 planned replacement, 317 plant control, 68 plant layout, 12 PM downtime, 334 point availability, 115 Poisson distribution, 27, 38, 413 population, 23, 35 power rating factor, 207 PPM, 48 predetermined maintenance, 315

Index predictive maintenance, 72, 316, 439 prevention strategy, 60 preventive maintenance, 57, 314, 317, 333 pro rata warranty, 407 proactive, 72 probability distribution function, 90 probability of event, 238 probability plot, 101 process capability, 2 process design, 10 process FMEA (PFMEA), 220 product design, 10 product life cycle management, 5, 9, 320 product limit estimator method, 136 product mix, 2, 3, 5 production efficiency, 75 production planning, 14 production process, 66 production system, 2, 11, 13 production system design framework, 4 profit analysis, 12 profit per unit time maximization, 378 program evaluation and review technique, 11 project execution, 12 project planning, 11 protection, 54 protection strategy, 60 protective action, 57, 60, 63 purchase order, 196 Q quality audit, 19 quality control, 23, 68 quality factor, 207 quality management system, 18 R R-chart, 26 RADC, 212 radiation hazards, 56 radio-frequency identification, 9 RAMS, 72 random failures, 110 rank adjustment method, 136, 140 rapid wear out, 110 rate of quality, 75 RCM, 71 reactor explosion, 240 redundant system, 161, 171, 246, 302 refurbishment, 316 relevant accident, 58 reliability, 88 reliability based preventive maintenance, 316 reliability block diagram, 152 reliability database, 267 reliability function, 91 reliability libraries, 268 reliability of system, 153, 163, 434 reliability parameters evaluation, 133, 454

Index remote maintenance, 190, 214 renewal process, 113, 115, 340 repair process, 91, 95, 99, 248 repair time, 320 replacement, 317 replacement upon failure, 317 required time, 319 research for productivity, 2 residual risk, 59 restoration, 316, 346 right censored data, 134 risk, 53, 56 risk analysis, 54, 57, 222 risk priority number (RPN), 220 Rome Air Development Center (RADC), 206 running in period, 94 S s-chart, 30 SABE, 21 safety, 53 safety of machinery, 61 safety stock, 13 scheduled-basis preventive maintenance, 316 scheduling, 10 sequencing, 10 serial configuration, 153 service life period, 94 severity, 222, 232 shock damage, 400 simple standby system, 174 simulation, 11, 157 single exponential smoothing, 412 Six Sigma analysis, 48 six-pack capability analysis, 43 spare parts, 195, 295, 320, 409 spare parts forecasting, 411, 414 spare parts management, 7, 423, 426 specific/minor risks, 58 specification limit, 24 stakeholders, 4 standardized MAD (SMAD), 416 standardized normal distribution, 463 standby system, 180, 246, 319 state diagram, 157 static reliability importance analysis, 252 statistical quality control, 24 steady-state availability, 115 stochastic failure and repair process, 89, 95, 117 storage cost, 409 stress factor, 207 student distribution, 137, 465 successful configuration, 171 supply plant, 152 survival function, 92 switching device, 180

479 telemaintenance, 214 thermal hazards, 55 thermal water treatments, 51 three stress models, 206 time series, 10, 59 time series decomposition, 412 time to failure, 90 time to market, 3 time to repair, 90 time-based preventive maintenance, 316 time-dependent analysis, 180, 301 top event, 237, 239 top-down analysis, 233 total expected replacement cost per cycle, 323 TPM, 71, 73, 76 transfer out block, 265 transporation, 13 traveling scheduling procedures, 11 two temperature/voltage models, 205 two-state diagram, 91 type I model, 324, 328 type II model, 319, 343, 357 U unavailability, 247 UNI, 19 unlimited free replacement, 407 up/down analysis, 132, 157 uptime, 65 use-based preventive maintenance, 316 V variety reduction program, 7, 12 VED approach, 423 vehicle routing, 12, 13 Venn diagrams, 241 vibration hazards, 55 VRP, 7, 12 W warm standby, 307 warranty, 406, 407 waste to energy plant, 433 waste treatment, 277 water supplier system, 185 wear out, 94 Weibull distribution, 47, 110, 454 weighted moving averages, 412 what-if analysis, 12 wood panel manufacturing, 216 work order, 191

T

X

Telcordia, 213

x-chart, N 29