2,393 526 21MB
Pages 490 Page size 827 x 1111 pts Year 2009
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 088548018 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 PierreJacques Courtois
Offshore Risk Assessment (2nd Edition) Jan Erik Vinnem
Reliability and Risk Issues in Large Scale Safetycritical 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
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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 088548018 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 16147839 ISBN 9781848825741 eISBN 9781848825758 DOI 10.1007/9781848825758 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 © SpringerVerlag 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 017602098, 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 857106703, 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 reproduced, 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 absence 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 information 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: letex publishing services GmbH, Leipzig, Germany Printed on acidfree paper Springer is part of Springer Science+Business Media (www.springer.com)
to Sara and Marta
Preface
Billions of dollars are currently spent producing hightechnology 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 illfamed company. The acquisition of a safer, or highquality, 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 socalled reliability and maintenance engineering
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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., reliabilitycentered 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.
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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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1.2.2
Product Quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.3
Production System Design Framework . . . . . . . . . . . . . . . . . . . . . .
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1.4
Models, Methods, and Technologies for Industrial Management
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1.4.1
The Product and Its Main Features . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.4.2
Reduction of Unremunerated Complexity: The Case of Southwest Airlines . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1.4.3
The Production Process and Its Main Features . . . . . . . . . . . . . . .
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1.4.4
The Choice of Production Plant . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1.5
Design, Management, and Control of Production Systems . . . . .
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1.5.1
Demand Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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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 . . . .
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1.5.5
Material Handling Device Design . . . . . . . . . . . . . . . . . . . . . . . . . .
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1.5.6
System Validation and Profit Evaluation . . . . . . . . . . . . . . . . . . . .
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1.5.7
Project Planning and Scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1.5.8
New Versus Existing Production Systems . . . . . . . . . . . . . . . . . . .
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1.6
Production System Management Processes for Productivity . . . .
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1.6.1
Inventory and Purchasing Management . . . . . . . . . . . . . . . . . . . . .
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1.6.2
Production Planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1.6.3
Distribution Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1.7
Research into Productivity and Maintenance Systems . . . . . . . . .
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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 RChart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.2 Numerical Example, RChart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.3 The xChart N ............................................ 2.7.4 Numerical Example, xChart N .............................. 2.7.5 The sChart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.6 Numerical Example, sChart and xChart N ................... 2.8 Control Charts for Attribute Data . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.1 The pChart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.2 Numerical Example, pChart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.3 The npChart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.4 Numerical Example, npChart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.5 The cChart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.6 Numerical Example, cChart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.7 The uChart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.8 Numerical Example, uChart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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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 ReliabilityCentered 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 . . . . . . . . . . . . . . . . . . . .
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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 . . . . . . . . . . . . . . . . . . . . . . . . .
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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
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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 . . . . . . . . . . . . . . . . . . . . . . . . . koutofn Redundancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Examples, koutofn Redundancy . . . . . . . . . . . . . . . . Simple Standby System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Example – TimeDependent Analysis: Standby System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.10 Production System Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.10.1 Water Supplier System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.10.2 Continuous Dryer System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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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
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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 StandardsBased 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, WhatIf Analysis . . . . . . . . . . . . . . . . . . . 8.8.10 System Modifications for ENF Reduction and Effects Analysis . 8.9 Markov Analysis and TimeDependent 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 TimeRelated Terms and Life Cycle Management . . . . . . . . . . . . 9.4.2 Numerical Example. Preventive Replacement and Cost Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 TimeBased 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
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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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TimeBased 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
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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 MaintenanceFree 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 InspectionMaintenance 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
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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 . . . . . . . . . . TimeBased Preventive Replacement for the Motor System . . . . TimeBased Preventive Replacement for the Bucket Component TimeBased 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
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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 metaframework 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 decisionmaking 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 longterm 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 highlevel 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 timeconsuming 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 standalone 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 decisionmaking process including: • a great degree of flexibility and elasticity in the production system; • short lead times; • highquality products and production processes; • short time to market; • control of production costs.
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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 reliabilitybased 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 metaframework 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 longterm objectives and programs, and is focused on creating operating capabilities. The corporatelevel 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 corporatelevel 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 metaframework gives the concurrent engineering approach a great and strategic importance and provides enlightenment on the validation analysis, and the continuous improvement (see the socalled 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 socalled 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 wellknown 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 wellknown 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 socalled 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 radiofrequency 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 wellmotivated 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 startup 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 abovementioned 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 startup 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 decisionmaking techniques and methods
1.5.1 Demand Analysis The starting point of the proposed method is the product (or service) market analysis, based on uptodate 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 decisionmaking 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 wellknown design automation tools, particularly computeraided design and computeraided 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 finetuning 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, whatif analysis). This ex ante evaluation checks the formal congruity of the whole design process, supporting the final choice of system configuration and the finetuning 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 wellknown net present value, payback analysis, and discounted cash flow rate of return are very frequently used. The best solution results from this doublecheck, 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 socalled 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 wellknown 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 QUANTITIESVARIANCE ANALYSIS
PRODUCTPROCESSMHD 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 WHATIF 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
decisionmaking 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 onthejob. 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 justintime strategy. Other eligible methodologies, such as consignment stock, electronic data interchange, comakership, business to business, and emarketplaces, 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 wellknown material requirement planning and manufacturing resource planning are usually suitable for this purpose, but the literature also contains several models and methods for socalled 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 decisionmaking 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 statisticsbased models and methods for the evaluation and prediction of reliability. This chapter also discusses the elementary reliability configurations of a production system, the socalled 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 RChart . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.2 Numerical Example, RChart . . . . . . . . . . . . . 2.7.3 The xChart N ........................... 2.7.4 Numerical Example, xChart N ............. 2.7.5 The sChart . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.6 Numerical Example, sChart and xChart N ...
26 26 29 29 30 30 33
2.8
Control Charts for Attribute Data . . . . . . . . . . . . . . 2.8.1 The pChart . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.2 Numerical Example, pChart . . . . . . . . . . . . . . 2.8.3 The npChart . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.4 Numerical Example, npChart . . . . . . . . . . . . 2.8.5 The cChart . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.6 Numerical Example, cChart . . . . . . . . . . . . . . 2.8.7 The uChart . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.8 Numerical Example, uChart . . . . . . . . . . . . . .
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 socalled 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
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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 processbased quality management system, as proposed by the ISO standards. Continual improvement of the quality management system
Management responsability
Interested parties
Requirements
Fig. 2.1 Processbased 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, thirdparty 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
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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 170501:2004) Conformity assessment – Supplier’s declaration of conformity – Part 2: Supporting documentation (ISO/IEC 170502:2004)
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 170501:2004 EN ISO/IEC 170502:2004
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 9104002:2008
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 endoflife. 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
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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 – Selfdeclared 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.
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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) • Onesided tolerance. It relates to characteristics with only one specification limit. • Twosided 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 socalled 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 RChart 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
Rchart
schart
xchart N
pchart
npchart
cchart
uchart
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, RChart
2.7.3 The xChart 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 xchart, N also called “xchart,” 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 Rchart 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 Rchart 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
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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 Rchart, numerical example. Minitab® Statistical Software. UCL upper control limit, LCL lower control limit
2.7.4 Numerical Example, xChart 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 sChart This chart for subgroup standard deviation can be used to support the construction of the xchart 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 schart. The average of standard deviation of subgroups, sO , is the centerline of the schart: 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
XbarR 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 xchart from R. Numerical example (25 samples). Minitab® Statistical Software
XbarR 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 xchart from R. Numerical example (24 samples). Minitab® Statistical Software
20 8 13
22 8 15
24 8 17
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2 Quality Management Systems and Statistical Quality Control
XbarR 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 xchart from R. Numerical example (23 samples). Minitab® Statistical Software
XbarR 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 xchart 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 xchart is now created assuming the centerline of the Rchart 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 xchart 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 xchart 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, sChart and xChart 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 Rchart, as constructed by the tool Minitab® Statistical Software (Fig. 2.12). From the Rchart 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
XbarR 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 Rchart and xchart 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 xchart from R, test results. Numerical example. Minitab® Statistical Software
XbarS 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 schart and xchart from s. Numerical example. Minitab® Statistical Software
2.8.1 The pChart The pchart 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 pchart 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, pChart
LCLp;i
Figure 2.15 presents the pchart 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 pchart with unequal sample sizes. Numerical example. Minitab® Statistical Software
Table 2.12 Rejected items. Numerical example
2.8.3 The npChart
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, npChart 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 npchart 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 cChart The cchart 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 npchart, 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 npchart, equal sample sizes. Numerical example. Minitab® Statistical Software
duction process. The number of nonconformities per inspection unit is called c. The centerline of the cchart 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, cChart 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 cchart 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 cchart 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 pchart, generated by Minitab® Statistical Software and shown in Fig. 2.19, is very similar to the cchart 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 cchart. 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 pchart. Inspection unit equal to 6,000 digits. Numerical example. Minitab® Statistical Software
2.8.7 The uChart
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 socalled uchart 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, uChart 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 uchart 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 uchart, 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 withinsubgroup variation, while the second relates to the betweensubgroup 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 indepth 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 uchart, 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 singlesided specification spread: LSL O or USL, O respectively. A conventionally accepted minimum value for these indexes is 1.33, corresponding to the socalled four sigma production process, as defined in Sect. 2.9. The performance of an incontrol process is predictable. Therefore, the capability analysis following the “incontrol analysis” can assess the ability of the production process to produce units that are “in spec” and predict the number of parts “outofspec.”
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 xchart based on the schart. 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 bestfitting statistical evaluation conducted with the within standard deviation. • “Expected overall performance.” They relate to the parametric distribution obtained by a bestfitting evaluation conducted with the overall standard deviation. In particular, the maximum expected value of PPM is about 96,620. The socalled sixpack 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 schart reported in Fig. 2.22, the Rchart is proposed to support the generation of the xchart. The standard deviations and capability indexes are hence quantified both in “overall” and “within” hypotheses. Finally, the socalled 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
XbarS 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 xchart and schart – 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 sixpack 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 Sixpack 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 Sixpack 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 Sixpack 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 141211:2007)
gonomics aspects), availability and suitability of protective measures, information for use, etc.
3.4 Classification of Risks MILSTD882 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 onfield 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, fireextinguisher 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 timeconsuming 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 10051:2001
Safety of machinery  Human physical performance  Part 1: Terms and deﬁnions
EN ISO 134062:2001
EN 10052:2003
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 134062:2001)
EN ISO 13407:1999
Humancentred 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 137321:2008
Erg.s of the thermal environment Methods for the assessment of human responses to contact with surfaces  Part 1: Hot surfaces (ISO 137321:2006)
EN ISO 137323:2008
Erg.s of the thermal environment Methods for the assessment of human responses to contact with surfaces  Part 3: Cold surfaces (ISO 137323:2005)
EN ISO 145052:2006
Erg.s of the thermal environment Evaluaon of thermal environments in vehicles  Part 2: Determinaon of equivalent temperature (ISO 145052:2006)
EN ISO 145053:2006
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 149151:2002
Soware Erg.s for mulmedia user interfaces  Part 1: Design principles and framework (ISO 149151:2002)
EN 10053:2002
Safety of machinery  Human physical performance  Part 3: Recommended force limits for machinery operaon
EN 10054:2005
Safety of machinery  Human physical performance  Part 4: Evaluaon of working postures and movements in relaon to machinery
EN 10055:2007
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 5471:1996+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 100751:2000
Erg. principles related to mental workload Part 1: General terms and deﬁnions (ISO 10075:1991)
EN ISO 20685:2005
3D scanning methodologies for internaonally compable anthropometric databases (ISO 20685:2005)
EN ISO 100752:2000
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 100753:2004
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 9241110:2006
Erg.s of humansystem 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 110641:2000
Erg. design of control centres  Part 1: Principles for the design of control centres (ISO 110641:2000)
EN ISO 110642:2000
Erg. design of control centres  Part 2: Principles for the arrangement of control suites (ISO 110642:2000)
EN ISO 110643:1999
Erg. design of control centres  Part 3: Control room layout (ISO 110643:1999)
EN ISO 110643:1999/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 5472:1996+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 149152:2003
Soware Erg.s for mulmedia user interfaces  Part 2: Mulmedia navigaon and control (ISO 149152:2003)
EN ISO 149153:2002
Soware Erg.s for mulmedia user interfaces  Part 3: Media selecon and combinaon (ISO 149153:2002) Erg.s of the thermal environment  Risk assessment strategy for the prevenon of stress or discomfort in thermal working condions (ISO 15265:2004)
EN 5473:1996+A1:2008
EN 6141:2006
EN ISO 15265:2004
EN 6142:2000+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 155361:2008
Erg.s  Computer manikins and body templates  Part 1: General requirements (ISO 155361:2005)
EN 8941:1997
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 155362:2007
Erg.s  Computer manikins and body templates  Part 2: Veriﬁcaon of funcons and validaon of dimensions for computer manikin systems (ISO 155362:2007)
EN 8942:1997
Safety of machinery  Erg.s requirements for the design of displays and control actuators  Part 2: Displays
EN ISO 15537:2004
EN 8943:2000
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 110644:2004
Erg. design of control centres  Part 4: Layout and dimensions of workstaons (ISO 110644:2004)
EN ISO 9241151:2008
Erg.s of humansystem interacon  Part 151: Guidance on World Wide Web user interfaces (ISO 9241151:2008)
EN ISO 110645:2008
Erg. design of control centres  Part 5: Displays and controls (ISO 110645:2008)
EN ISO 9241171:2008
EN ISO 110646:2005
Erg. design of control centres  Part 6: Environmental requirements for control centres (ISO 110646:2005)
EN ISO 9241400:2007
Erg.s of humansystem interacon  Part 171: Guidance on soware accessibility (ISO 9241171:2008) Erg.s of humansystem interacon  Part 400: Principles and requirements for physical input devices (ISO 9241400:2007)
EN ISO 110647:2006
Erg. design of control centres  Part 7: Principles for the evaluaon of control centres (ISO 110647:2006)
EN ISO 9241410:2008
Erg.s of humansystem interacon  Part 410: Design criteria for physical input devices (ISO 9241410:2008)
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 134061:1999
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 134061:1999)
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 selfexplanatory 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
ReliabilityCentered 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 timedependent 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, conditionbased 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: onsite, i. e., at the location where the item is used, online, 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 knowhow, 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 onfield 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 welldesigned 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 decisionsupporting 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 tradeoff 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 realworld instance examined. Before evaluating this tradeoff, 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 decisionmaking 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 manhours 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 “204040” 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 reliabilitycentered
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
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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 ReliabilityCentered Maintenance The RCM process is known as a “reliability by design” based approach and is reliabilitycentered 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 stepbystep 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 oncondition 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., failurefinding, redesign, and runtofailure, 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 crossfunctional team constituted of maintenance, operations, and
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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 peoplecentered methodology, generally considered as a critical addon 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 costeffectiveness 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:
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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 productchange adjustments until the production of the new item is completely satisfactory. 2. Speed losses due to: (a) Microstops: 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 startup phases (e. g., after periodic repairs, longtime 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 warmup 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:
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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 smallgroup activities where all members, usually including managers, participate and work jointly on a selfdiscipline basis.
4.7.3 TPM Operating Instruments In addition to the wellknown 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
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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 finetuning process of OEE can vary across different business sectors and industries. Generally speaking, a worldclass 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, “topdown” 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 onfield 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 warmup 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.
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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
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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 startup 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 startup activities (in hours) – mac_2
Fig. 4.12 Maintenance activities (in hours) – mac_2
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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
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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.
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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
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Table 4.6 Overall selfrating 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 selfrating 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 worldclass 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 macroactivities: 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 inhouse. This is typical, e. g., when the external contractors support the inhouse workforce during workintensive 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.
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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 incentivebased 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 inhouse).
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 hightechnology 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 selfimprovement. 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 timebased 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 socalled 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 socalled 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 twostate 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 Twostate 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 socalled 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 ncomponentsystem 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 wellknown bathtubcurve hazard rate. It is a parametric rate function that identifies the failure behaviors of components/systems subject to a runningin 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 “runin” 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 socalled 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
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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 8.000E6
FS Histogram Pdf Line Failures
6.400E6
Value
4.800E6
3.200E6
1.600E6
0.000
029453.... 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
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Probability  Exponential
99.900
ProbabilityExponential Folio2\Data 1 Exponential1P 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 socalled 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.
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Unreliability vs Time Plot
1.000
Unreliability Folio2\Data 1 Exponential1P RRX SRM MED FM F=100/S=0 Data Points Unreliability Line
Unreliability, F(t)=1R(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 Exponential1P RRX SRM MED FM F=100/S=0 Data Points Reliability Line
Reliability, R(t)=1F(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
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Probability Density Function Pdf
1.600E5
Folio2\Data 1 Exponential1P RRX SRM MED FM F=100/S=0 Pdf Line
f(t)
1.200E5
8.000E6
2.000E5
Failure Rate
1.600E5
Folio2\Data 1 Exponential1P RRX SRM MED FM F=100/S=0 Failure Rate Line
1.200E5
8.000E6
4.000E6
4.000E6
0.000 0.000
Failure Rate, f(t)/R(t)
Failure Rate vs Time Plot
2.000E5
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
KaplanMeier 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 timedependent 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.
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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)/[1F(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
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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)/[1G(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. Indepth 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 00.2 0.20.4 0.40.6 0.60.8 0.81.0 1.01.2 1.21.4 1.41.6 1.61.8 1.82.0 2.02.2 2.22.4 2.42.6 2.62.8 2.83.0 3.03.2 3.23.4 3.43.6 3.63.8 3.84.0 4.04.2 4.24.4 4.44.6 4.64.8 4.85.0
Period β = 8.8740, η = 3.2201, ρ = 0.9885
Fig. 5.28 ttf histogram. ReliaSoft® software
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Probability  Normal distribution
99.900
ProbabilityNormal (Normal2P)
Unreliability, F(t)
Folio1\Data 1 Normal2P 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 (Normal2P) Folio1\Data 1 Normal2P RRX SRM MED FM F=100/S=0 Data Points Unreliability Line
0.600
0.400
Reliability (Normal2P) Folio1\Data 1 Normal2P RRX SRM MED FM F=100/S=0 Data Points Reliability Line
0.800
Reliability, R(t)=1F(t)
0.800
Unreliability, F(t)=1R(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 (Normal2P)
Failure Rate (Normal2P)
Folio1\Data 1 Normal2P RRX SRM MED FM F=100/S=0 Pdf Line
Folio1\Data 1 Normal2P 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
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Probability  Lognormal distribution
99.900
ProbabilityLognormal (Lognormal2P)
Unreliability, F(t)
Folio1\Data 1 Lognormal2P 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 (Lognormal2P)
Failure Rate (Lognormal2P)
Folio1\Data 1 Lognormal2P RRX SRM MED FM F=100/S=0 Pdf Line
Folio1\Data 1 Lognormal2P 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 (Lognormal2P)
0.800
Folio1\Data 1 Lognormal2P RRX SRM MED FM F=100/S=0 Data Points Unreliability Line
Reliability, R(t)=1F(t)
Unreliability, F(t)=1R(t)
Time, (t)
ReliaSoft Weibull++ 7  www.ReliaSoft.com
ReliaSoft Weibull++ 7  www.ReliaSoft.com
0.600
0.400
1.000
Reliability (Lognormal2P)
0.800
Folio1\Data 1 Lognormal2P 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 timedependent 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 “burnin” 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)/[1F(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 socalled 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
ProbabilityWeibull Folio1\Data 1 Weibull2P 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 Weibull2P RRX SRM MED FM F=100/S=0 Data Points Unreliability Line
Reliability, R(t)=1F(t)
Unreliability, F(t)=1R(t)
10.000
Time, (t)
0.600
0.400
1.000
Reliability
0.800
Folio1\Data 1 Weibull2P 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 Weibull2P RRX SRM MED FM F=100/S=0 Pdf Line
Folio1\Data 1 Weibull2P 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 socalled mean availability. The steadystate 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 socalled 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 twostate 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 Œ A.t/ . C / C ln./ D t: C C (5.89) Applying the properties of logarithms, one gets
Œ 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 inhouse, 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. Fiftyseven 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 (MILHDBK
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 MILHDBK217F2 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 MILHDBK217F2 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 (MILHDBK217F2) Name
MILSTD217 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 pp2460 ght2456
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 wellknown 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 RADCTR85194 distributed in 1985. The abovementioned approaches, i. e., MILSTD217, 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, MILHDBK217F2 is conservative and the actual value is several
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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 MILSTD217 several other sources of reliability information have been developed, such as FIDES 2004, Telcordia SR332, 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 MILHDBK217F2 (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) SR332 Telcordia is the new name of Bellcore Company (Bell Communications Research, a spinoff of AT&T Bell Labs). Bellcore previously referred to MILHDBK217 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 MILHDBK217, mainly therefore electronic devices. As this standard becomes more widely used, it could become the international successor to the US MILHDBK217. 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; weatherprotected (equipment for stationary use on the ground in weatherprotected locations, operating permanently or otherwise); • ground; stationary; nonweatherprotected (equipment for stationary use on the ground in nonweatherprotected 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 MILSTD217 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 userfriendly.
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 highrisk 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 “capitalintensive” 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 mediumsized 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:
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• 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 online 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 offline, 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 realtime 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 online 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 highlevel competences developed by
Fleet of machines/equipment
Database Algorithms Support ….
Control parameter adjustment Performance feedback Maintenance history Telemaintenance and diagnosis Spare parts Training
Realtime controller Production management
Remote site
Internet, LAN, WAN
Production/service site
Fig. 7.20 Remote maintenance system structure
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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 startup 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 $1,500. Telecontrol required
Fig. 7.22 Product exiting from peeler. (Courtesy of Reni Ettore Spa)
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Table 7.28 Comparison between traditional maintenance and remote maintenance Traditional maintenance (2005)
Remote maintenance (2006)
Total hours available per year Production losses (h)a Production losses ($)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 lowcost 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 StandardsBased 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, WhatIf Analysis . . 8.8.10 System Modifications for ENF Reduction and Effects Analysis . . . . . . . . . . . . . . . . . . . . . 8.9
Markov Analysis and TimeDependent 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 healthcare 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
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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 timeconsuming 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: • MILSTD1629A; • J1739 from the Society of Automotive Engineers for the automotive industry;
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221
• AIAG FMEA3 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 (MILSTD1629A 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 topdown 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
………..
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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 nexthigherlevel effect (FMEA worksheet, column 4), i. e., involving items set on the nexthigher 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., MILSTD1629A) 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 lifecycle 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 MILSTD1629A 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 MILSTD1629A 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 magnitudeeffect 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 MILSTD1629A 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 (MILSTD1629A) 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 (MILSTD1629A) 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 (MILSTD1629A) 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 abovementioned 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 (MILSTD1629A 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 socalled 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 twodimensional space severity (on xaxes) and occurrence (on yaxes) 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
Highpriority 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
Mediumpriority 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
Lowpriority 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 (socalled 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 wellknown 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 followup 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 timeconsuming 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 MILSTD1629A 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
20081101
20081001
20081101
20081101
20081001
Planned start date
20090401
20090201
20090401
20090601
20090201
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 timedependent failure distributions through wellknown 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 topdown 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 crosscheck 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 decisionmaking 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 decisionmaking 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 bottomup 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 quasistatic, 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 nosignal 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 PAND 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
PAND 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)
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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 twoinput 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 topdown 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 socalled quantitative FTA, properly illustrated and exemplified in Sect. 8.6. The next section presents the “qualitative” FTA, whose aim is the identification of the socalled 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
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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
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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
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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 firstorder 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 .
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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 wellknown 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:
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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
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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 socalled 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)=1R(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
4.000E4
Failure Rate vs Time
5.000E4
Pdf
Failure Rate
Fault Tree1 Pdf Line
Fault Tree1 Failure Rate Line
4.400E4
Failure Rate, f(t)/R(t)
3.200E4
f(t)
2.400E4
1.600E4
8.000E5
0.000 0.000
20000.000
Time, (t)
3.800E4
3.200E4
2.600E4
4000.000
8000.000
12000.000
16000.000
2.000E4 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
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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)=1R(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
6.000E4
6.000E4
Pdf
Failure Rate
Fault Tree1 Pdf Line
Fault Tree1 System no electric power P1 broken P2 broken PR broken V1 broken V2 broken
4.800E4
Failure Rate, f(t)/R(t)
4.804E4
f(t)
3.608E4
2.412E4
1.216E4
2.000E6 0.000
9000.000
Time, (t)
3.600E4
2.400E4
1.200E4
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)=1R(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
8.000E4
0.003
Pdf
Failure Rate
Fault Tree1 Pdf Line
Fault Tree1 Failure Rate Line
0.002
Failure Rate, f(t)/R(t)
6.400E4
f(t)
4.800E4
3.200E4
1.600E4
0.000 0.000
0.002
0.001
7.600E4
600.000
1200.000
1800.000
2400.000
3000.000
2.000E4 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
2.000E4 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.
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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 socalled 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 fancoils 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 threeway valve) for each exchanger and for each loop. As a consequence, in the case of two fancoils 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 160m2 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 fancoils are fed in a redundant configuration, i. e., the heating system is supposed to be capable when at least one fancoil is operating.
8 O100 mm
subloop 2.1
subloop 2.2
5 subloop 2.1
subloop 1.1
10 6 9 1
subloop 1.1 subloop 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 3way zone valve 6 Fancoil 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 fancoils; • 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 startup, 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
3way 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
3way 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 fancoil redundancy and fill water (Fig. 8.64.) There is only a single boiler and two redundant fancoils, 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 fancoil redundancy (Fig. 8.65): There is only a single boiler and two redundant fancoils, 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 fancoil 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
3way 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 fancoil redundancy and fill water (Figs. 8.67 and 8.68). There are two alternative boilers (i. e., one is redundant) and two redundant fancoils. 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 fancoil redundancy (Fig. 8.69). There are two alternative boilers (i. e., one is redundant) and two redundant fancoils. 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 StandardsBased 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 socalled 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
3way 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 standardsbased reliability database of predefined components MIL217, 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 fancoil redundancy and fill water. Configuration A
Table 8.14 Failure rates from standardsbased 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 3way 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 fancoil 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 FanCoil 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 fancoils. 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 timedependent 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 ycoordinate). 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 FanCoil 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 fancoils, 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 timedependent 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 fancoil 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 FanCoil Redundancy
8.7.3.4 Configuration D – Two boilers and FanCoil 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 fancoils, 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 fancoils. 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
3way 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
3way zone valve closed 1.2
rupture hydronic pipe adduction
no supply H2O
Fig. 8.68 Thermic system, two boilers and fancoil redundancy and fill water. Configuration D
3way 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
3way 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 fancoil 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 FanCoil 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 fancoils, 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)=1R(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
3.000E4
Failure Rate vs Time
4.000E4
Pdf
Failure Rate
Fault Tree1 Pdf Line
Fault Tree1 Failure Rate Line
3.400E4
Failure Rate, f(t)/R(t)
2.401E4
f(t)
1.802E4
1.204E4
6.048E5
2.800E4
2.200E4
1.600E4
6.000E7 0.000
4000.000
8000.000
12000.000
16000.000
1.000E4 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
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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 Timedependent 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 costbased 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 hightemperature 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
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Block Failure Rate vs Time 4.000E4
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)
3.200E4
2.400E4
1.600E4
8.000E5
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 highefficiency furnaces and devices for continuous monitoring of emissions and air pollution control. There are various types of incinerator plants: • Simple incinerator made of a bricklined 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
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1.000
Reliability vs Time
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Unreliability
Reliability Fault Tree1 Reliability Line
Fault Tree1 Unreliability Line
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Reliability, R(t)=F(t)
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Probability Density Function
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Failure Rate vs Time
3.000E4
Pdf
Failure Rate
Fault Tree1 Pdf Line
Fault Tree1 Failure Rate Line
1.600E4
Failure Rate, f(t)/R(t)
2.400E4
f(t)
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8.012E5
4.016E5
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6.000E5
2.000E7 0.000
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0.000 0.000
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12000.000
18000.000
24000.000
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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,
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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 Timedependent 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 socalled 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
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Block Failure Rate vs Time 3.000E4
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)
2.400E4
1.800E4
1.200E4
6.000E5
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 halfhour observations. A halfhourly observation gives the average value of 30 values registered each minute. Figure 8.95 reports the trend of halfhour values during the 20month observation period. By an indepth 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
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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)=1R(t)
0.800
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0.600
0.400
0.200
0.000 0.000
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16000.000
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Time, (t) ReliaSoft BlockSim 7  www.ReliaSoft.com
Probability Density Function
4.000E4
Failure Rate vs Time
4.000E4
Pdf
Failure Rate
Fault Tree1 Pdf Line
Fault Tree1 Failure Rate Line
3.600E4
Failure Rate, f(t)/R(t)
3.201E4
f(t)
2.402E4
1.603E4
8.040E5
3.200E4
2.800E4
2.400E4
5.000E7 0.000
4000.000
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2.000E4 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
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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 Timedependent 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;
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Block Failure Rate vs Time 4.000E4
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)
3.220E4
2.440E4
1.660E4
8.800E5
1.000E5 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
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1.000
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1.000
Unreliability
Reliability
Fault Tree1 Unreliability Line
Fault Tree1 Reliability Line
0.800
Reliability, R(t)=F(t)
Unreliability, F(t)=1R(t)
0.800
0.600
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0.400
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0.000 0.000
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Time, (t)
12000.000
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20000.000
Time, (t)
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2.000E4
Failure Rate vs Time
4.000E4
Pdf
Failure Rate
Fault Tree1 Pdf Line
Fault Tree1 Failure Rate Line
3.200E4
Failure Rate, f(t)/R(t)
1.600E4
f(t)
1.200E4
8.000E5
4.000E5
2.400E4
1.600E4
8.000E5
0.000 0.000
4000.000
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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
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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;
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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 Timedependent 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
4.000E4
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)
3.200E4
2.400E4
1.600E4
8.000E5
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
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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)=1R(t)
0.800
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Probability Density Function
1.000E4
Failure Rate vs Time
3.000E4
Pdf
Failure Rate Fault Tree1 Failure Rate Line
Fault Tree1 Pdf Line
2.500E4
Failure Rate, f(t)/R(t)
8.010E5
f(t)
6.020E5
4.030E5
2.040E5
2.000E4
1.500E4
1.000E4
5.000E7 0.000
6000.000
12000.000
18000.000
24000.000
5.000E5 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 Timedependent 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 2.500E4
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)
2.000E4
1.500E4
1.000E4
5.000E5
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 unitvalues (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 halfhour observations)
Fig. 8.95 Halfhour 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, WhatIf 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 socalled 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 whatif 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 $6,000. Similar considerations can be made, considering different system alternative and/or simultaneous modifications, with reference to other externality costs, such as the emissions of CO2 , CH4 , PM10 , SO2 , CO, and N2 O.
Table 8.26 Valve redundancy introduction, whatif analysis 1 vs. 2 valves T (h) Mean availability (all events) Point availability (all events) at 8;760 h Reliability (8;760 h) Expected number of failures (failures) MTTFF (h) System uptime (h) System downtime (h) NOx (kg) NOx externality costs (2003 US$=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 TimeDependent Components/Systems
301
Fig. 8.100 System modification: valves VR1101 and VR1102
8.9 Markov Analysis and TimeDependent 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 socalled 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 TimeDependent 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 wellknown 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.
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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 TimeDependent 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)
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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 socalled 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 TimeDependent 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)
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8 Effects Analysis and Reliability Modeling of Complex Production Systems
State II  PII(t) 5.00E03 4.50E03 4.00E03
Probability
3.50E03 3.00E03 2.50E03 PII(t) 2.00E03 1.50E03 1.00E03 5.00E04 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”)?
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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 2dim 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 TimeRelated Terms and Life Cycle Management . . . . . . . . . . . . . . 319 9.4.2 Numerical Example. Preventive Replacement and Cost Minimization . . . . . . . 320
9.5
9.6
9.7
TimeBased 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 . . . . . . . . . . . TimeBased 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 MaintenanceFree 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 armchair. (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 wellknown toothpaste, is “prevention is better than cure,” known as “prevenire è meglio che curare” by the Italian authors of this book. This is the Hamletlike
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?, online or offline maintenance?, onsite or offsite 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 wellknown 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: maintenancefree 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 decisionmaking 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
Conditionbased 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 usebased preventive maintenance actions, performed on an hours run of the component/system basis, and the timebased preventive maintenance actions, performed on a calendar basis. These are also known as scheduledbasis preventive maintenance strategies. Another special class of preventive maintenance is the conditionbased preventive maintenance (the socalled 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 online or offline, 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 socalled 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 conditionbased 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 conditionbased preventive maintenance and conditionbased 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 conditionbased inspection maintenance schedules faultfinding 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. • Conditionbased 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 conditionbased 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 conditionbased 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 conditionbased 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 timebased preventive maintenance, or to inspection maintenance (i. e., conditionbased 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 CharlesOwaba 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 conditionbased (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)
Conditionbased 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. agebased 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 TimeRelated 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)=1R(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
4.800E4
0.003
Failure Rate, f(t)/R(t)
6.000E4
f(t)
3.600E4
2.400E4
0.002
0.002
8.000E4
1.200E4
4.000E116 0.000
2400.000
Time, (t)
800.000
1600.000
2400.000
3200.000
4000.000
4.000E116 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 TimeBased Preventive Replacement – Type I Replacement Model
9.5 TimeBased 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 socalled 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 Timebased preventive replacement. Type I
the differentiation of products of differentiable functions (the socalled 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 TimeBased 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 TimeBased 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
r1
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 socalled 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 twoparameter 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 whatif 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 whatif 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 whatif 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 wellknown “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
met
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 replacementtp
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
T1
1+W(T1)
∫ f (t )dt 0
2
1+W(T2)
∫ f (t )dt 1
3
∫ f (t )dt 2
1+W(T3)
…
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., modelfree) 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 decisionmaking 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 socalled 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 = k1
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 = k1
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 InspectionMaintenance 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 asgoodasnew 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 InspectionMaintenance 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 longrun 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 InspectionMaintenance 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 stepbystep 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 stepbystep 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), BenDaya 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 decisiontools 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 tradeoff problem
Fig. 11.2, this is a tradeoff 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 tradeoff 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 knowhow 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 finetuned using the knowhow of maintenance operators and suppliers. The existence of a sig
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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
Adaptiveresponserate 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 twofactor 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 intermittentdemand 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 intermittentdemand 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 trialanderror 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 trialanderror 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 2.88E08 2.87E06 0.000281 0.022655 0.071284 0.126166 0.176437 0.21686 ... 0.261268
0 2.3E12 2.3E09 2.25E06 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.
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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 3.21E05 4.28E06 5.14E07
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 =2×103 pieces/h
0
time
T = 640 h x components
Fig. 11.9 P000303 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 P000303 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
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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 abovementioned 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
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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 trendadjusted 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 slowmoving 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 twolevel 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, fastmoving spare parts have to be stocked in local warehouses in large quantities. 2. Low price, short response time, low usage. These cheap, slowmoving 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 fastmoving 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 tradeoff 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 Justintime inventory Multiitem 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 decisionmaking 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,N1 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 tradeoff 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 LeastAvailability 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
setup
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 wellknown 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 TimeBased Preventive Replacement for the Motor System . . . . . . . . . . . . . . . . . . . . 12.1.5 TimeBased Preventive Replacement for the Bucket Component . . . . . . . . . . . . . . . . 12.1.6 TimeBased 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
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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 wellknown fourway 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 bestfitting 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. Fourway 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. Fourway 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
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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 KaplanMeier 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. Fourway 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. Fourway 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
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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 kilowatthour, 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 KaplanMeier Method  Complete Data
KaplanMeier 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 TimeBased 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 timebased 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 TimeBased 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 TimeBased 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
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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 timebased 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 timebased 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
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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.
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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
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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 compressionmolding 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 polyethylenebased 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 compressionmolding 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 compressionmolding 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 compressionmolding system
Fig. 12.27 Xray picture of a cap for water
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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 compressionmolding manufacturing process
A Punch B Stripper C Melt feeding D Mold E Pellet portion
Fig. 12.29 Compressionmolding 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 Orings, 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, threeparameter Weibull, and threeparameter exponential, are illustrated by several literature references on statistics science and applications. The result of the goodnessoffit 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 threeparameter 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 bestfit parametric analysis and evaluation suggests adopting the socalled 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 oring
B
Oil oring
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 oring
UP flange oring
UP flange oring
UP flange oring
UP flange oring
UP flange oring
UP flange oring
PUNCH
PUNCH
PUNCH
PUNCH
PUNCH
PUNCH
PUNCH
PUNCH
DOWN DOWN flange flange (mould) oring
DOWN DOWN flange flange (mould) oring
DOWN DOWN flange flange (mould) oring
DOWN DOWN flange flange (mould) oring
DOWN DOWN flange flange (mould) oring
DOWN DOWN flange flange (mould) oring
DOWN DOWN flange flange (mould) oring
DOWN DOWN flange flange (mould) oring
UP flange (stripper)
UP flange (stripper)
UP flange (stripper)
UP flange (stripper)
UP flange (stripper)
UP flange (stripper)
UP flange (stripper)
UP flange oring
UP flange oring
UP flange oring
UP flange oring
PUNCH
Molding unit (1–48)
Pelleting and inserting system (1–12)
UP flange oring
UP flange oring
PUNCH
PUNCH
PUNCH
PUNCH
PUNCH
PUNCH
PUNCH
UP flange oring
UP flange oring
UP flange (stripper)
Transmission pallet placement
k/n, 10/12
UP flange (stripper)
UP flange (stripper)
UP flange (stripper)
UP flange oring
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) oring
DOWN DOWN flange flange (mould) oring
DOWN DOWN flange flange (mould) oring
DOWN DOWN flange flange (mould) oring
DOWN DOWN flange flange (mould) oring
DOWN DOWN flange flange (mould) oring
DOWN DOWN flange flange (mould) oring
DOWN DOWN flange flange (mould) oring
UP flange (stripper)
UP flange (stripper)
UP flange (stripper)
UP flange (stripper)
UP flange (stripper)
UP flange oring
UP flange oring
UP flange oring
UP flange oring
UP flange oring
UP flange oring
UP flange oring
UP flange oring
UP flange (stripper)
UP flange (stripper)
UP flange (stripper)
PUNCH
PUNCH
PUNCH
PUNCH
PUNCH
PUNCH
PUNCH
PUNCH
DOWN DOWN flange flange (mould) oring
DOWN DOWN flange flange (mould) oring
DOWN DOWN flange flange (mould) oring
DOWN DOWN flange flange (mould) oring
DOWN DOWN flange flange (mould) oring
DOWN DOWN flange flange (mould) oring
DOWN DOWN flange flange (mould) oring
DOWN DOWN flange flange (mould) oring
UP flange (stripper)
UP flange (stripper)
UP flange (stripper)
UP flange (stripper)
UP flange (stripper)
UP flange oring
UP flange oring
UP flange oring
UP flange oring
UP flange oring
UP flange oring
UP flange oring
UP flange oring
UP flange (stripper)
UP flange (stripper)
UP flange (stripper)
PUNCH
PUNCH
PUNCH
PUNCH
PUNCH
PUNCH
PUNCH
PUNCH
DOWN DOWN flange flange (mould) oring
DOWN DOWN flange flange (mould) oring
DOWN DOWN flange flange (mould) oring
DOWN DOWN flange flange (mould) oring
DOWN DOWN flange flange (mould) oring
DOWN DOWN flange flange (mould) oring
DOWN DOWN flange flange (mould) oring
DOWN DOWN flange flange (mould) oring
UP flange (stripper)
UP flange (stripper)
UP flange (stripper)
UP flange (stripper)
UP flange (stripper)
UP flange oring
UP flange oring
UP flange oring
UP flange oring
UP flange oring
UP flange oring
UP flange oring
UP flange oring
UP flange (stripper)
UP flange (stripper)
UP flange (stripper)
PUNCH
PUNCH
PUNCH
PUNCH
PUNCH
PUNCH
PUNCH
PUNCH
DOWN DOWN flange flange (mould) oring
DOWN DOWN flange flange (mould) oring
DOWN DOWN flange flange (mould) oring
DOWN DOWN flange flange (mould) oring
DOWN DOWN flange flange (mould) oring
DOWN DOWN flange flange (mould) oring
DOWN DOWN flange flange (mould) oring
DOWN DOWN flange flange (mould) oring
UP flange (stripper)
UP flange (stripper)
UP flange (stripper)
UP flange (stripper)
UP flange (stripper)
UP flange oring
UP flange oring
UP flange oring
UP flange oring
UP flange oring
UP flange oring
UP flange oring
UP flange oring
UP flange (stripper)
UP flange (stripper)
UP flange (stripper)
PUNCH
PUNCH
PUNCH
PUNCH
PUNCH
PUNCH
PUNCH
PUNCH
DOWN DOWN flange flange (mould) oring
DOWN DOWN flange flange (mould) oring
DOWN DOWN flange flange (mould) oring
DOWN DOWN flange flange (mould) oring
DOWN DOWN flange flange (mould) oring
DOWN DOWN flange flange (mould) oring
DOWN DOWN flange flange (mould) oring
DOWN DOWN flange flange (mould) oring
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
ProbabilityGumbel
Unreliability, F(t)
Data 1 Gumbel2P MLE SRM KM 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
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12 Applications and Case Studies
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F/S Histogram
4.000E4
FS Histogram Pdf Line Failures
3.200E4
Value
2.400E4
1.600E4
8.000E5
0.000
0501
5011002
10021503 15032004 20042505 25053006 30063507 35074008 40084509 45095010 50105511 55116012
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 EstimatesComplete 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 EstimatesComplete Data 3Param et er Weibull 99.9
90
99
50
90 Percent
Percent
Correlation Coefficient 3Parameter Weibull 0.996 3Parameter Lognormal 0.985 2Parameter Exponential * 3Parameter Loglogistic 0.983
3Param et er Lognorm al
99.9
10
50 10
1
1 0.1
0.1 8000
10000 12000 ttf  Threshold
14000
188000
2Param et er Exponent ial
190000 192000 194000 ttf  Threshold 3Param 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 EstimatesComplete 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
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12 Applications and Case Studies
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Probability  Gumbel
99.900
ProbabilityGumbel Data 1 Gumbel2P 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 Goodnessoffit analysis conducted with Minitab® Statistical Software Distribution Weibull Lognormal Exponential Loglogistic 3Parameter Weibull 3Parameter lognormal 2Parameter exponential 3Parameter 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
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ReliaSoft Weibull++ 7  www.ReliaSoft.com
Reliability vs Time Plot
Unreliability vs Time Plot
1.000
1.000
Reliability
Unreliability
Data 1 Gumbel2P RRX SRM MED FM F=200/S=0 Data Points Reliability Line
Data 1 Gumbel2P RRX SRM MED FM F=200/S=0 Data Points Unreliability Line 0.800
Unreliability, F(t)=1R(t)
Reliability, R(t)=1F(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
4.000E4
0.060
Pdf
Failure Rate
Data 1 Gumbel2P RRX SRM MED FM F=200/S=0 Pdf Line
Data 1 Gumbel2P RRX SRM MED FM F=200/S=0 Failure Rate Line 0.048
Failure Rate, f(t)/R(t)
3.200E4
f(t)
2.400E4
1.600E4
8.000E5
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 3parameters 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
KaplanMeier Method  95% CI Complete Data
KaplanMeier 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 asis 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)=1R(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 socalled 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 tobe 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 asis 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 asis 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 asis 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
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Index
A accelerated life testing, 204 accident, 53, 54 active maintenance time, 320 adaptiveresponserate single exponential smoothing, 412 additive Holt–Winter, 412 ADI average interdemand interval, 410, 430 Aerospace, 21 agebased replacement policy, 319 AIAG FMEA3, 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 cchart, 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 computeraided design, 10 computeraided 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 leastsquare, 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 koutofn 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 maintenancefree 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 microstops, 74 MILSTD1629A, 220 MILSTD217, 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 npchart, 37 number of failures, 159 O occurrence–severity matrix, 227 on condition monitoring, 70 online 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 PAND gate (priority AND gate), 237 pchart, 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 Rchart, 26 RADC, 212 radiation hazards, 56 radiofrequency 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 schart, 30 SABE, 21 safety, 53 safety of machinery, 61 safety stock, 13 scheduledbasis 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 sixpack 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 steadystate 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 timebased preventive maintenance, 316 timedependent analysis, 180, 301 top event, 237, 239 topdown 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 twostate 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 usebased 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 whatif analysis, 12 wood panel manufacturing, 216 work order, 191
T
X
Telcordia, 213
xchart, N 29