Importance Measures in Reliability, Risk, and Optimization E-Book

Way Kuo

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 897

Veröffentlichungsjahr: 2012

Beschreibung

This unique treatment systematically interprets a spectrum of importance measures to provide a comprehensive overview of their applications in the areas of reliability, network, risk, mathematical programming, and optimization. Investigating the precise relationships among various importance measures, it describes how they are modelled and combined with other design tools to allow users to solve readily many real-world, large-scale decision-making problems. Presenting the state-of-the-art in network analysis, multistate systems, and application in modern systems, this book offers a clear and complete introduction to the topic. Through describing the reliability importance and the fundamentals, it covers advanced topics such as signature of coherent systems, multi-linear functions, and new interpretation of the mathematical programming problems. Key highlights: * Generalizes the concepts behind importance measures (such as sensitivity and perturbation analysis, uncertainty analysis, mathematical programming, network designs), enabling readers to address large-scale problems within various fields effectively * Covers a large range of importance measures, including those in binary coherent systems, binary monotone systems, multistate systems, continuum systems, repairable systems, as well as importance measures of pairs and groups of components * Demonstrates numerical and practical applications of importance measures and the related methodologies, including risk analysis in nuclear power plants, cloud computing, software reliability and more * Provides thorough comparisons, examples and case studies on relations of different importance measures, with conclusive results based on the authors' own research * Describes reliability design such as redundancy allocation, system upgrading and component assignment. This book will benefit researchers and practitioners interested in systems design, reliability, risk and optimization, statistics, maintenance, prognostics and operations. Readers can develop feasible approaches to solving various open-ended problems in their research and practical work. Software developers, IT analysts and reliability and safety engineers in nuclear, telecommunications, offshore and civil industries will also find the book useful.

Leseprobe

Contents

Cover

Title Page

Copyright

List of Figures

List of Tables

Preface

Acknowledgments

Acronyms and Notation

Part One: Introduction and Background

Introduction

Chapter 1: Introduction to Importance Measures

Chapter 2: Fundamentals of Systems Reliability

2.1 Block Diagrams

2.2 Structure Functions

2.3 Coherent Systems

2.4 Modules within a Coherent System

2.5 Cuts and Paths of a Coherent System

2.6 Critical Cuts and Critical Paths of a Coherent System

2.7 Measures of Performance

2.8 Stochastic Orderings

2.9 Signature of Coherent Systems

2.10 Multilinear Functions and Taylor (Maclaurin) Expansion

2.11 Redundancy

2.12 Reliability Optimization and Complexity

2.13 Consecutive-k-out-of-n Systems

2.14 Assumptions

Part Two: Principles of Importance Measures

Introduction

Chapter 3: The Essence of Importance Measures

3.1 Importance Measures in Reliability

3.2 Classifications

3.3 c-Type and p-Type Importance Measures

3.4 Importance Measures of a Minimal Cut and a Minimal Path

3.5 Terminology

Chapter 4: Reliability Importance Measures

4.1 The B-reliability Importance

4.2 The FV Reliability Importance

Chapter 5: Lifetime Importance Measures

5.1 The B-time-dependent-lifetime Importance

5.2 The FV Time-dependent Lifetime Importance

5.3 The BP Time-independent Lifetime Importance

5.4 The BP Time-dependent Lifetime Importance

5.5 Numerical Comparisons of Time-dependent Lifetime Importance Measures

5.6 Summary

Chapter 6: Structure Importance Measures

6.1 The B-i.i.d. Importance and B-structure Importance

6.2 The FV Structure Importance

6.3 The BP Structure Importance

6.4 Structure Importance Measures Based on the B-i.i.d. Importance

6.5 The Permutation Importance and Permutation Equivalence

6.6 The Domination Importance

6.7 The Cut Importance and Path Importance

6.8 The Absoluteness Importance

6.9 The Cut-path Importance, Min-cut Importance, and Min-path Importance

6.10 The First-term Importance and Rare-event Importance

6.11 c-type and p-type of Structure Importance Measures

6.12 Structure Importance Measures for Dual Systems

6.13 Dominant Relations among Importance Measures

6.14 Summary

Chapter 7: Importance Measures of Pairs and Groups of Components

7.1 The Joint Reliability Importance and Joint Failure Importance

7.2 The Differential Importance Measure

7.3 The Total Order Importance

7.4 The Reliability Achievement Worth and Reliability Reduction Worth

Chapter 8: Importance Measures for Consecutive-k-out-of-n Systems

8.1 Formulas for the B-importance

8.2 Patterns of the B-importance for Lin/Con/k/n Systems

8.3 Structure Importance Measures

Part Three: Importance Measures for Reliability Design

Introduction

Chapter 9: Redundancy Allocation

9.1 Redundancy Importance Measures

9.2 A Common Spare

9.3 Spare Identical to the Respective Component

9.4 Several Spares in a k-out-of-n System

9.5 Several Spares in an Arbitrary Coherent System

9.6 Cold Standby Redundancy

Chapter 10: Upgrading System Performance

10.1 Improving Systems Reliability

10.2 Improving Expected System Lifetime

10.3 Improving Expected System Yield

10.4 Discussion

Chapter 11: Component Assignment in Coherent Systems

11.1 Description of Component Assignment Problems

11.2 Enumeration and Randomization Methods

11.3 Optimal Design Based on the Permutation Importance and Pairwise Exchange

11.4 Invariant Optimal and Invariant Worst Arrangements

11.5 Invariant Arrangements for Parallel–series and Series–parallel Systems

11.6 Consistent B-i.i.d. Importance Ordering and Invariant Arrangements

11.7 Optimal Design Based on the B-reliability Importance

11.8 Optimal Assembly Problems

Chapter 12: Component Assignment in Consecutive-k-out-of-n and Its Variant Systems

12.1 Invariant Arrangements for Systems

12.2 Necessary Conditions for Component Assignment in Systems

12.3 Sequential Component Assignment Problems in :F Systems

12.4 Consecutive-2 Failure Systems on Graphs

12.5 Series Systems

12.6 Consecutive-k-out-of-r-from-n Systems

12.7 Two-dimensional and Redundant Systems

12.8 Miscellaneous

Chapter 13: B-importance-based Heuristics for Component Assignment

13.1 The Kontoleon Heuristic

13.2 The LK-Type Heuristics

13.3 The ZK-Type Heuristics

13.4 The B-importance-based Two-stage Approach

13.5 The B-importance-based Genetic Local Search

13.6 Summary and Discussion

Part Four: Relations and Generalizations

Introduction

Chapter 14: Comparisons of Importance Measures

14.1 Relations to the B-importance

14.2 Rankings of Reliability Importance Measures

14.3 Importance Measures for Some Special Systems

14.4 Computation of Importance Measures

Chapter 15: Generalizations of Importance Measures

15.1 Noncoherent Systems

15.2 Multistate Coherent Systems

15.3 Multistate Monotone Systems

15.4 Binary-Type Multistate Monotone Systems

15.5 Summary of Importance Measures for Multistate Systems

15.6 Continuum Systems

15.7 Repairable Systems

15.8 Applications in the Power Industry

Part Five: Broad Implications to Risk and Mathematical Programming

Introduction

Chapter 16: Networks

16.1 Network Flow Systems

16.2 -terminal Networks

Chapter 17: Mathematical Programming

17.1 Linear Programming

17.2 Integer Programming

Chapter 18: Sensitivity Analysis

18.1 Local Sensitivity and Perturbation Analysis

18.2 Global Sensitivity and Uncertainty Analysis

18.3 Systems Reliability Subject to Uncertain Component Reliability

18.4 Broad Applications

Chapter 19: Risk and Safety in Nuclear Power Plants

19.1 Introduction to Probabilistic Risk Analysis and Probabilistic Safety Assessment

19.2 Probabilistic (Local) Importance Measures

19.3 Uncertainty and Global Sensitivity Measures

19.4 A Case Study

19.5 Review of Applications

19.6 System Fault Diagnosis and Maintenance

Afterword

Allaying concerns

Reliability awareness

Orderings and ranges of component reliability

Importance-measure-based B&B method

Multidimensional approach

Practical decision-making

Reliability, risk, and optimization

Appendix: Proofs

A.1 Proof of Theorem 8.2.7

A.2 Proof of Theorem 10.2.10

A.3 Proof of Theorem 10.2.17

A.4 Proof of Theorem 10.3.11

A.5 Proof of Theorem 10.3.15

Bibliography

Index

This edition first published 2012

© 2012 John Wiley & Sons, Ltd

Registered officeJohn Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom

For details of our global editorial offices, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at www.wiley.com.

The right of the author to be identified as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988.

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the publisher.

Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books.

Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The publisher is not associated with any product or vendor mentioned in this book. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold on the understanding that the publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional should be sought.

Library of Congress Cataloging-in-Publication Data

Kuo, Way, 1951– Importance measures in reliability, risk, and optimization : principles and applications / Way Kuo, Xiaoyan Zhu. p. cm. Includes bibliographical references and index. ISBN 978-1-119-99344-5 (hardback) 1. Reliability (Engineering) 2. Risk assessment. 3. Industrial priorities. I. Zhu, Xiaoyan. II. Title. TA169.K855 2012 620′.00452–dc23 2011052840

A catalogue record for this book is available from the British Library.

Print ISBN: 978-1-119-99344-5

List of Tables

6.1 Structure importance measures

9.1 Greedy method 1

9.2 Greedy method 2

9.3 Greedy method 3

10.1 Comparisons of the TIL importance measures

10.2 Comparisons of the BP TIL importance and L1 TIL importance for the systems in Figures 2.4 and 2.5

11.1 Elimination procedure

11.2 Invariant optimal and invariant worst arrangements

12.1 Complete invariant optimal arrangements for Lin/Con/k/n systems

12.2 Complete invariant optimal arrangements for Cir/Con/k/n systems

12.3 Relationships of consistent B-i.i.d. importance ordering to invariant arrangements for Lin/Con/k/n systems

12.4 The uniform B-i.i.d. importance and necessary conditions for the optimal arrangements for Lin/Con/k/n:F systems

12.5 The greedy algorithm for Lin/Con/2/n:F systems

12.6 The greedy algorithm for Cir/Con/2/n:F systems

12.7 Invariant optimal arrangements for Con/(r,k)/(r,n):F systems

13.1 The Kontoleon heuristic

13.2 The Kontoleon heuristic procedure for the CAP in the system in Figure 6.4

13.3 The LKA heuristic

13.4 The LKA heuristic procedure for the CAP in the system in Figure 6.4

13.5 The LK-type heuristics

13.6 The ZKA heuristic

13.7 The ZK-type heuristics

13.8 The ZKA heuristic procedure for the CAP in the system in Figure 6.4

13.9 Comparisons of the initial and final arrangements associated with the ZKA heuristic

13.10 The BITS

13.11 The BIGLS framework

13.12 The B-importance-based local search

14.1 Relations to the B-importance

14.2 Importance measures for parallel and series systems

15.1 States of the system

17.1 The simplex algorithm (minimization problem)

17.2 The procedure of the simplex algorithm for the problem in Example 17.1.3

17.3 The procedure of the simplex algorithm for the problem in Example 17.1.4

17.4 The general B&B algorithm

18.1 Results in Example 18.3.2

Preface

Appearing on the scene in the late 1940s and early 1950s, reliability was first defined and applied to communication and transportation, and it has become a major part of performance measure for military systems. Much of the early work was confined to reliability analysis, and many of the early theories were developed hypothetically without considering the real problems encountered.

Along with reliability design, the concept of importance measures of components in a coherent system was proposed in the 1960s (Birnbaum 1969). At that moment, the system of interest was confined to a binary system of two states, functioning or failed. Strictly, the early version of the importance measures was sensitivity analysis of a probabilistic system. Given the increasing trend of using complex and complicated systems, other types of importance measures have also been investigated, one of which is the Fussell–Vesely importance (Fussell 1975; Vesely 1970), which was used in the design for reliability and safety in nuclear power plants. The precise relationships among various importance measures and their applications in the areas of reliability, risk, mathematical programming, and even broader categories have not yet been investigated thoroughly. This book will be the first to meet the need of addressing these unresolved issues.

In the 1980s, consecutive-k-out-of-n (Con/k/n) systems were proposed as a type of complex systems (Chiang and Niu 1981). It was quickly studied that optimal design for such systems is nearly impossible without further investigating other types of importance measures. Similarly, applications such as design for reliable software systems were needed, and with the aid of importance measures, one is able to improve software reliability with limited resources (Fiondella and Gokhale 2008). Still, optimal design has been only remotely possible. In the 1990s, as with other optimization design tools, heuristics were proposed along with importance measures to facilitate optimal reliability design (Kuo and Prasad 2000; Kuo and Wan 2007).

In this book, we provide a comprehensive picture of importance measures in reliability. Furthermore, we generalize the ideas of importance measures to solve some of the problems in reliability design, risk, and mathematical programming. Appendix A provides the proofs of some theorems. The book is divided into five parts.

Part One, “Introduction and Background,” includes two chapters. The importance measures carry very wide range of applications including both deterministic and stochastic systems. Often it is most economic to quickly identify the improvement for the system performance such as reliability or other measures, instead of searching for a global solution in the crisis management. Chapter 1 gives examples of importance measures in various areas. Chapter 2 presents the fundamentals of systems reliability. In additional to defining reliability and availability, we present the classical definitions of coherent systems, cuts and paths, system signatures, various system configurations, redundancy, reliability optimization, and complexity.

Part Two, “Principles of Importance Measures,” includes six chapters, introducing different types of importance measures in aspects of mathematical expressions, physical meanings, probabilistic interpretations, computation, and so on. Chapter 3 presents the essence and classifications of importance measures of components in coherent systems defined in Chapter 2. Chapters 4, 5, and 6 present the major importance measures in three different classes—reliability, lifetime, and structure importance measures, respectively. Structure importance measures evaluate the relative importance of positions in the system; thus, the structure importance of components actually represents the importance of their positions in the system.

The importance measures that are discussed in Chapters 4–6 are the index of individual components and evaluate the importance of an individual component with respect to system performance. In contrast, Chapter 7 discusses importance measures of pairs or groups of components, which evaluate the interaction of a pair or a group of components on system performance. The importance measures of pairs or groups of components can provide additional information that the importance measures of individual components cannot. Chapter 8 presents the importance measures and their relationships for Con/k/n systems, including both the F and G systems. In these chapters, we make general assumptions about the system and its components as presented in Chapter 2. Examples of using these importance measures in dealing with network reliability problems and those beyond the context of coherent systems are also given.

Part Three, “Importance Measures for Reliability Design,” includes five chapters. Through the many importance measures introduced in Part Two and the importance measures specially designed for their applications in Part Three, we are able to solve problems of redundancy allocation, system upgrading, and component assignment problems (CAP1). Chapters 9, 10, and 11–13 address these three applications, respectively. For example, a reliability analyst would present the system structure and evaluate the system performance, while a systems analyst would like to identify the critical components so that the systems can be improved with minimum additional resources. Importance measures in upgrading systems, along with special examples, are addressed in Chapter 10.

The CAP itself is of great interest in the area of reliability optimization. The importance measures have the close relationships to CAP and can be used to find optimal or approximate solutions. Chapter 11 introduces a CAP, which is to assign components to different positions in a system with the objective of maximizing systems reliability. Chapter 12 focuses on the CAP in Con/k/n systems and some variations of Con/k/n systems, and Chapter 13 presents heuristics for the CAP using the B-importance. For systems that are large and as complex as the various versions of Con/k/n systems, heuristics and sometimes metaheuristics are of great use in the optimal design.

Part Four, “Relations and Generalizations,” includes two chapters. Chapter 14 investigates the relationships of importance measures presented in Parts Two and Three and compares them in various aspects. The summary and comparisons center on the essential role of the B-importance, and the issues addressed in Chapter 14 include the computation of importance measures. In Chapter 15, we release one or more assumptions made in Chapter 2 and investigate the corresponding importance measures. In particular, Chapter 15 discusses the importance measures for noncoherent systems, multistate systems, continuum systems, and repairable systems.

Part Five, “Broad Implications to Risk and Mathematical Programming,” includes four chapters. Importance measures have been adopted and applied in various disciplines, such as network flows, -terminal networks, mathematical programming, sensitivity and uncertainty analysis, perturbation analysis, software reliability, fault diagnosis, and probabilistic risk analysis (PRA) and probabilistic safety assessment (PSA). Chapters 16–19 address these applications, showing that importance measures could be used in optimization problems, for example, to illustrate a simplex algorithm for linear programming and to design a branch-and-bound method for integer programming, as in Chapter 17.

A sketch note

As we deal with many modern systems using the fastest computing facilities, we still find it a challenge to create an optimal design for a generic system. It is even more difficult when component reliability values cannot be accurately estimated. This book provides a comprehensive view on modeling and combining importance measures with other design tools. Included are some solution methods that require only orderings or ranges of component reliability without the need to know their exact values; this feature lessens, to some extent, the trouble of estimating component reliability, a task that is normally necessary but hard to perform in addressing various reliability problems. With respect to this feature, some importance measures are designed under the assumption of unknown component reliability; some others are designed for different ranges of component reliability. Theoretical justifications are made in the book by rigorous proofs whenever possible although throughout the writing of this book, we identified many difficult problems that will need to be further solved by readers. This book presents many conclusive results for the first time based on our studies.

In preparing for this book, we have thoroughly reviewed articles that discuss and analyze many importance measures. We found major and minor errors in these articles, and in attempting to streamline the proofs, sometimes we reached different conclusions. After updating the theories, we have adopted the correct versions in this book. As much as we would wish to present the materials in the book as complete, we may inadvertently make errors. In all cases, we acknowledge the contributions made by the authors of those articles, who provided us with many insightful viewpoints although some of them reached erroneous conclusions.

This book is an integral part of reliability modeling (Kuo and Zuo 2003) and reliability design (Kuo et al. 2006). In broader research, importance-measure-based methods for solving various hard problems in the fields of reliability, risk, and mathematical programming deserve investigation, since we believe that the importance-measure-based methods are among the most practical decision tools. The concept of importance measures finds broad implications to mathematical optimization, but the optimization community is unfamiliar of this concept and its further development. Inspired by this book, we hope that readers can develop feasible, effective methods for solving various open-ended problems in their research and practical work. Finally, we hope that you enjoy reading this book, which has been planned for many years over the course of our careers as reliability practitioners and researchers.

References

Birnbaum ZW. 1969. On the importance of different components in a multicomponent system. In Multivariate Analysis, Vol. 2 (ed. Krishnaiah PR). Academic Press, New York, pp. 581–592.

Chiang DT and Niu SC. 1981. Reliability of consecutive-k-out-of-n:F system. IEEE Transactions on ReliabilityR-30, 87–89.

Fiondella L and Gokhale SS. 2008. Importance measures for a modular software system. Proceedings of the 8th International Conference on Quality Software, pp. 338–343.

Fussell JB. 1975. How to hand-calculate system reliability and safety characteristics. IEEE Transactions on ReliabilityR-24, 169–174.

Kuo W and Prasad V. 2000. An annotated overview of system-reliability optimization. IEEE Transactions on Reliability49, 176–187.

Kuo W, Prasad VR, Tillman FA, and Hwang CL. 2006. Optimal Reliability Design: Fundamentals and Applications, 2nd edn. Cambridge University Press, Cambridge, UK.

Kuo W and Wan R. 2007. Recent advances in optimal reliability allocation. IEEE Transactions on Systems, Man, and Cybernetics, Series A37, 143–156.

Kuo W and Zuo MJ. 2003. Optimal Reliability Modeling: Principles and Applications. John Wiley & Sons, New York.

Vesely WE. 1970. A time dependent methodology for fault tree evaluation. Nuclear Engineering and Design13, 337–360.

1. The singular and plural of an acronym are always spelled the same.

Acknowledgments

We are appreciative of the research funds granted from National Science Foundation during the past 25 consecutive years. We would also like to acknowledge Army Research Office, National Research Council, Nuclear Regulatory Commission, Bell Labs, and IBM. This book grows from the research projects, supported in part by the above agencies and companies. The Weisenbaker Chair fund of Texas A&M University and the Distinguished Professorships of National Tsing Hua University and National Taiwan University are acknowledged for providing us with the platform in conducting the exploratory study that leads to this book.

The early draft of the book has been used in classes and seminars attended by the graduate students and professionals. We are grateful for the input to this manuscript from Larry Yu-Chi Ho of Harvard University, Cambridge, Massachusetts; Min Xie of National University of Singapore and now City University of Hong Kong, Hong Kong; Fan-Chin Meng of Academia Sinica, Taipei; Nozer D. Singpurwalla of George Washington University, Washington, DC; Shu-Cherng Fang of North Carolina State University, Raleigh, North Carolina; Markos V. Koutras of the University of Piraeus, Piraeus; Baoding Liu of Tsinghua University, Beijing; Haijun Li of Washington State University, Pullman, Washington; and Enrico Zio of Ecole Centrale Paris, Châtenay-Malabry.