Professor Way Kuo, City University of Hong KongProfessor Kuo is President and Distinguished Professor of City University of Hong Kong. He served as Distinguished Professor and Dean of Engineering at the University of Tennessee between 2003 and 2008, and between 2000 and 2003 he held the Wisenbaker Chair of Engineering in Innovation and was the Executive Associate Dean of Engineering at Texas A&M University. Professor Kuo is recipient of the IEEE Reliability Society Lifetime Achievement Award, and now serves as the Editor-in-Chief of IEEE Transactions on Reliability. He has co-authored six textbooks and is a member of the U.S. National Academy of Engineering, Academia Sinica (Taiwan), ad International Academy for Quality. He is a fellow of ASQ, ASA, IEEE, INFORMS and IIE. Professor Xiaoyan Zhu, University of Tennessee, USAProfessor Zhu is an assistant professor in the Department of Industrial and Information Engineering at University of Tennessee, Knoxville. She has taught both undergraduate and graduate courses in mathematical programming and operations research, for example Operations Research I (Linear Programming), Inventory Control, Production Planning, and Advanced Nonlinear Programming. Professor Zhu has published several papers related to importance measures, and she is a member of INFORMS, IIE, and IEEE.

• Preface xv References xviiAcknowledgements xixPart One INTRODUCTION and BACKGROUND 1Introduction 21 Introduction to Importance Measures 5References 112 Fundamentals of Systems Reliability 132.1 Block Diagrams 132.2 Structure Functions 142.3 Coherent Systems 172.4 Modules within a Coherent System 182.5 Cuts and Paths of a Coherent System 192.6 Critical Cuts and Critical Paths of a Coherent System 212.7 Measures of Performance 232.7.1 Reliability for a mission time 242.7.2 Reliability function (of time t) 252.7.3 Availability function 272.8 Stochastic Orderings 282.9 Signature of Coherent Systems 282.10 Multilinear Functions and Taylor (Maclaurin) Expansion 312.11 Redundancy 322.12 Reliability Optimization and Complexity 332.13 Consecutive-k-out-of-n Systems 342.14 Assumptions 35References 36Part Two PRINCIPLES of IMPORTANCE MEASURES 39Introduction 403 The Essence of Importance Measures 433.1 ImportanceMeasures in Reliability 433.2 Classifications 443.3 c-type and p-type ImportanceMeasures 453.4 ImportanceMeasures of a Minimal Cut and a Minimal Path 453.5 Terminology 45References 464 Reliability Importance Measures 474.1 The B-reliability Importance 474.1.1 The B-reliability importance for system functioning and for system failure 524.1.2 The criticality reliability importance 524.1.3 The Bayesian reliability importance 534.2 The FV Reliability Importance 534.2.1 The c-type FV (c-FV) reliability importance 544.2.2 The p-type FV (p-FV) reliability importance 544.2.3 Decomposition of state vectors 544.2.4 Properties 56References 575 Lifetime Importance Measures 595.1 The B-time-dependent-lifetime Importance 595.1.1 The criticality time-dependent lifetime importance 615.2 The FV Time-dependent Lifetime Importance 615.2.1 The c-FV time-dependent lifetime importance 615.2.2 The p-FV time-dependent lifetime importance 635.2.3 Decomposition of state vectors 645.3 The BP Time-independent Lifetime Importance 645.4 The BP Time-dependent Lifetime Importance 695.5 Numerical Comparisons of Time-dependent Lifetime ImportanceMeasures 695.6 Summary 71References 726 Structure Importance Measures 736.1 The B-i.i.d. Importance and B-structure Importance 736.2 The FV Structure Importance 766.3 The BP Structure Importance 766.4 Structure ImportanceMeasures Based on the B-i.i.d. importance 796.5 The Permutation Importance and Permutation Equivalence 806.5.1 Relations to minimal cuts and minimal paths 816.5.2 Relations to systems reliability 836.6 The Domination Importance 856.7 The Cut Importance and Path Importance 866.7.1 Relations to the B-i.i.d. importance 876.7.2 Computation 896.8 The Absoluteness Importance 916.9 The Cut-path Importance,Min-cut Importance, and Min-path Importance 926.10 The First-term Importance and Rare-event Importance 936.11 c-type and p-type of Structure ImportanceMeasures 936.12 Structure ImportanceMeasures for Dual Systems 946.13 Dominant Relations among ImportanceMeasures 966.13.1 The absoluteness importance with the domination importance 966.13.2 The domination importance with the permutation importance 966.13.3 The domination importance with the min-cut importance and min-path importance 966.13.4 The permutation importance with the FV importance 966.13.5 The permutation importance with the cut-path importance, min-cut importance,and min-path importance 1006.13.6 The cut-path importance with the cut importance and path importance 1016.13.7 The cut-path importance with the B-i.i.d. importance 1016.13.8 The B-i.i.d. importance with the BP importance 1026.14 Summary 102References 1057 ImportanceMeasures of Pairs and Groups of Components 1077.1 The Joint Reliability Importance and Joint Failure Importance 1077.1.1 The joint reliability importance of dependent components 1107.1.2 The joint reliability importance of two gate events 1107.1.3 The joint reliability importance for k-out-of-n systems 1117.1.4 The joint reliability importance of order k 1117.2 The Differential ImportanceMeasure 1127.2.1 The first-order differential importance measure 1127.2.2 The second-order differential importance measure 1137.2.3 The differential importance measure of order k 1147.3 The Total Order Importance 1147.4 The Reliability AchievementWorth and Reliability ReductionWorth 115References 1168 ImportanceMeasures for Consecutive-k-out-of-n Systems 1198.1 Formulas for the B-importance 1198.1.1 The B-reliability importance and B-i.i.d. importance 1198.1.2 The B-structure importance 1228.2 Patterns of the B-importance for Lin/Con/k/n Systems 1238.2.1 The B-reliability importance 1238.2.2 The uniform B-i.i.d. importance 1248.2.3 The half-line B-i.i.d. importance 1268.2.4 The nature of the B-i.i.d. importance patterns 1268.2.5 Patterns with respect to p 1288.2.6 Patterns with respect to n 1298.2.7 Disproved patterns and conjectures 1318.3 Structure ImportanceMeasures 1358.3.1 The permutation importance 1358.3.2 The cut-path importance 1358.3.3 The BP structure importance 1358.3.4 The first-term importance and rare-event importance 136References 137Part Three IMPORTANCE MEASURES for RELIABILITY DESIGN 139Introduction 140References 1419 Redundancy Allocation 1439.1 Redundancy ImportanceMeasures 1449.2 A Common Spare 1459.2.1 The redundancy importance measures 1459.2.2 The permutation importance 1479.2.3 The cut importance and path importance 1479.3 Spare Identical to the Respective Component 1489.3.1 The redundancy importance measures 1489.3.2 The permutation importance 1499.4 Several Spares in a k-out-of-n System 1509.5 Several Spares in an Arbitrary Coherent System 1509.6 Cold Standby Redundancy 152References 15210 Upgrading System Performance 15510.1 Improving Systems Reliability 15610.1.1 Same amount of improvement in component reliability 15610.1.2 A fractional change in component reliability 15710.1.3 Cold standby redundancy 15810.1.4 Parallel redundancy 15810.1.5 Example and discussion 15810.2 Improving Expected System Lifetime 15910.2.1 A shift in component lifetime distributions 16010.2.2 Exactly one minimal repair 16010.2.3 Reduction in the proportional hazards 16710.2.4 Cold standby redundancy 16810.2.5 A perfect component 17010.2.6 An imperfect repair 17010.2.7 A scale change in component lifetime distributions 17110.2.8 Parallel redundancy 17110.2.9 Comparisons and numerical evaluation 17210.3 Improving Expected System Yield 17410.3.1 A shift in component lifetime distributions 17510.3.2 Exactly one minimal repair / cold standby redundancy / a perfect component /parallel redundancy 18010.4 Discussion 182References 18211 Component Assignment in Coherent Systems 18511.1 Description of Component Assignment Problems 18611.2 Enumeration and Randomization Methods 18711.3 Optimal Design based on the Permutation Importance and Pairwise Exchange 18811.4 Invariant Optimal and InvariantWorst Arrangements 18911.5 Invariant Arrangements for Parallel-series and Series-parallel Systems 19111.6 Consistent B-i.i.d. Importance Ordering and Invariant Arrangements 19211.7 Optimal Design based on the B-reliability Importance 19411.8 Optimal Assembly Problems 196References 19712 Component Assignment in Consecutive-k-out-of-n and Its Variant Systems 19912.1 Invariant Arrangements for Con/k/n Systems 19912.1.1 Invariant optimal arrangements for Lin/Con/k/n systems 20012.1.2 Invariant optimal arrangements for Cir/Con/k/n systems 20012.1.3 Consistent B-i.i.d. importance ordering and invariant arrangements 20212.2 Necessary Conditions for Component Assignment in Con/k/n Systems 20412.3 Sequential Component Assignment Problems in Con/2/n:F Systems 20612.4 Consecutive-2 Failure Systems on Graphs 20712.4.1 Consecutive-2 failure systems on trees 20812.5 Series Con/k/n Systems 20812.5.1 Series Con/2/n:F systems 20912.5.2 Series Lin/Con/k/n:G systems 20912.6 Consecutive-k-out-of-r-from-n Systems 21112.7 Two-dimensional and Redundant Con/k/n Systems 21312.7.1 Con/(r, k)/(r, n) systems 21412.8 Miscellaneous 216References 21713 B-importance based Heuristics for Component Assignment 21913.1 The Kontoleon Heuristic 21913.2 The LK Type Heuristics 22113.2.1 The LKA heuristic 22113.2.2 Another three LK type heuristics 22113.2.3 Relation to invariant optimal arrangements 22113.2.4 Numerical comparisons of the LK type heuristics 22413.3 The ZK Type Heuristics 22513.3.1 Four ZK type heuristics 22513.3.2 Relation to invariant optimal arrangements 22713.3.3 Comparisons of initial arrangements 22713.3.4 Numerical comparisons of the ZK type heuristics 22913.4 The B-importance based Two-stage Approach 22913.4.1 Numerical comparisons with the GAMS/CoinBomin solver and enumerationmethod 23013.4.2 Numerical comparisons with the randomization method 23013.5 The B-importance based Genetic Local Search 23113.5.1 The description of algorithm 23213.5.2 Numerical comparisons with the B-importance based two-stage approach and agenetic algorithm 23513.6 Summary and Discussion 236References 238Part Four RELATIONS and GENERALIZATIONS 241Introduction 24214 Comparisons of Importance Measures 24514.1 Relations to the B-importance 24514.2 Rankings of Reliability ImportanceMeasures 24714.2.1 Using the permutation importance 24714.2.2 Using the permutation importance and joint reliability importance 24914.2.3 Using the domination importance 25014.2.4 Summary 25014.3 ImportanceMeasures for Some Special Systems 25014.4 Computation of ImportanceMeasures 251References 25315 Generalizations of Importance Measures 25515.1 Noncoherent Systems 25515.1.1 Binary monotone systems 25615.2 Multistate Coherent Systems 25715.2.1 The μ, _ B-importance 25815.2.2 The μ, _ cut importance 25915.3 Multistate Monotone Systems 26115.3.1 The permutation importance 26115.3.2 The utility B-reliability importance 26215.3.3 The utility-decomposition reliability importance 26215.3.4 The utility B-structure importance, joint structure importance, and joint reliabilityimportance 26315.3.5 The B-importance, FV importance, reliability achievement worth, and reliabilityreduction worth with respect to system mean unavailability and mean performance 26515.4 Binary Type Multistate Monotone Systems 26615.4.1 The B-t.d.l. importance, BP t.i.l. importance, and L1 t.i.l. importance 26715.5 Summary of ImportanceMeasures for Multistate Systems 26815.6 Continuum Systems 27015.7 Repairable Systems 27215.7.1 The B-availability importance 27215.7.2 The c-FV unavailability importance 27315.7.3 The BP availability importance 27315.7.4 The L1 t.i.l. importance 27415.7.5 Simulation-based importance measures 27515.8 Applications in the Power Industry 276References 277Part Five BROAD IMPLICATIONS to RISK and MATHEMATICALPROGRAMMING 281Introduction 282References 28316 Networks 28516.1 Network Flow Systems 28516.1.1 The edge importance measures in a network flow system 28616.1.2 The edge importance measures for a binary monotone system 28816.1.3 The B-reliability importance, FV reliability importance, reliability reductionworth, and reliability achievement worth 28916.1.4 The flow-based importance and impact-based importance 29016.2 K-terminal Networks 29116.2.1 Importance measures of an edge 29316.2.2 A K-terminal optimization problem 295References 29517 Mathematical Programming 29717.1 Linear Programming 29717.1.1 Basic concepts 29817.1.2 The simplex algorithm 30017.1.3 Sensitivity analysis 30117.2 Integer Programming 30317.2.1 Basic concepts and branch-and-bound algorithm 30317.2.2 Branch-and-bound using linear programming relaxations 30617.2.3 Mixed integer nonlinear programming 309References 30918 Sensitivity Analysis 31118.1 Local Sensitivity and Perturbation Analysis 31118.1.1 The B-reliability importance 31118.1.2 The multidirectional sensitivity measure 31218.1.3 The multidirectional differential importance measure and total order importance 31718.1.4 Perturbation analysis 31818.2 Global Sensitivity Analysis 31918.2.1 ANOVA-decomposition based global sensitivity measures 32018.2.2 Elementary effect methods and derivative-based global sensitivity measures 32318.2.3 Relationships between the ANOVA-decomposition-based and the derivativebasedsensitivity measures 32618.2.4 The case of random input variables 32718.2.5 Moment-independent sensitivity measures 32818.3 Systems reliability subject to uncertain component reliability 33018.3.1 Software Reliability 33218.4 Broad applications 335References 33619 Risk and Safety in Nuclear Power Plants 33919.1 Introduction to Probabilistic Risk Analysis and Probabilistic Safety Assessment 33919.2 Probabilistic (Local) ImportanceMeasures 34019.3 Uncertainty and Global Sensitivity Measures 34219.4 A Case Study 34319.5 Review of Applications 34519.6 System Fault Diagnosis and Maintenance 347References 348Afterword 350References 354APPENDIX 355A Proofs 357A.1 Proof of Theorem 8.2.7 357A.2 Proof of Theorem 10.2.10 358A.3 Proof of Theorem 10.2.17 359

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