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Professor Way Kuo, City University of Hong Kong Professor 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, USA Professor 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.
References xvii
Acknowledgements xix
Part One INTRODUCTION and BACKGROUND 1
Introduction 2
1 Introduction to Importance Measures 5
References 11
2 Fundamentals of Systems Reliability 13
2.1 Block Diagrams 13
2.2 Structure Functions 14
2.3 Coherent Systems 17
2.4 Modules within a Coherent System 18
2.5 Cuts and Paths of a Coherent System 19
2.6 Critical Cuts and Critical Paths of a Coherent System 21
2.7 Measures of Performance 23
2.7.1 Reliability for a mission time 24
2.7.2 Reliability function (of time t) 25
2.7.3 Availability function 27
2.8 Stochastic Orderings 28
2.9 Signature of Coherent Systems 28
2.10 Multilinear Functions and Taylor (Maclaurin) Expansion 31
2.11 Redundancy 32
2.12 Reliability Optimization and Complexity 33
2.13 Consecutive-k-out-of-n Systems 34
2.14 Assumptions 35
References 36
Part Two PRINCIPLES of IMPORTANCE MEASURES 39
Introduction 40
3 The Essence of Importance Measures 43
3.1 ImportanceMeasures in Reliability 43
3.2 Classifications 44
3.3 c-type and p-type ImportanceMeasures 45
3.4 ImportanceMeasures of a Minimal Cut and a Minimal Path 45
3.5 Terminology 45
References 46
4 Reliability Importance Measures 47
4.1 The B-reliability Importance 47
4.1.1 The B-reliability importance for system functioning and for system failure 52
4.1.2 The criticality reliability importance 52
4.1.3 The Bayesian reliability importance 53
4.2 The FV Reliability Importance 53
4.2.1 The c-type FV (c-FV) reliability importance 54
4.2.2 The p-type FV (p-FV) reliability importance 54
4.2.3 Decomposition of state vectors 54
4.2.4 Properties 56
References 57
5 Lifetime Importance Measures 59
5.1 The B-time-dependent-lifetime Importance 59
5.1.1 The criticality time-dependent lifetime importance 61
5.2 The FV Time-dependent Lifetime Importance 61
5.2.1 The c-FV time-dependent lifetime importance 61
5.2.2 The p-FV time-dependent lifetime importance 63
5.2.3 Decomposition of state vectors 64
5.3 The BP Time-independent Lifetime Importance 64
5.4 The BP Time-dependent Lifetime Importance 69
5.5 Numerical Comparisons of Time-dependent Lifetime ImportanceMeasures 69
5.6 Summary 71
References 72
6 Structure Importance Measures 73
6.1 The B-i.i.d. Importance and B-structure Importance 73
6.2 The FV Structure Importance 76
6.3 The BP Structure Importance 76
6.4 Structure ImportanceMeasures Based on the B-i.i.d. importance 79
6.5 The Permutation Importance and Permutation Equivalence 80
6.5.1 Relations to minimal cuts and minimal paths 81
6.5.2 Relations to systems reliability 83
6.6 The Domination Importance 85
6.7 The Cut Importance and Path Importance 86
6.7.1 Relations to the B-i.i.d. importance 87
6.7.2 Computation 89
6.8 The Absoluteness Importance 91
6.9 The Cut-path Importance,Min-cut Importance, and Min-path Importance 92
6.10 The First-term Importance and Rare-event Importance 93
6.11 c-type and p-type of Structure ImportanceMeasures 93
6.12 Structure ImportanceMeasures for Dual Systems 94
6.13 Dominant Relations among ImportanceMeasures 96
6.13.1 The absoluteness importance with the domination importance 96
6.13.2 The domination importance with the permutation importance 96
6.13.3 The domination importance with the min-cut importance and min-path importance 96
6.13.4 The permutation importance with the FV importance 96
6.13.5 The permutation importance with the cut-path importance, min-cut importance,
and min-path importance 100
6.13.6 The cut-path importance with the cut importance and path importance 101
6.13.7 The cut-path importance with the B-i.i.d. importance 101
6.13.8 The B-i.i.d. importance with the BP importance 102
6.14 Summary 102
References 105
7 ImportanceMeasures of Pairs and Groups of Components 107
7.1 The Joint Reliability Importance and Joint Failure Importance 107
7.1.1 The joint reliability importance of dependent components 110
7.1.2 The joint reliability importance of two gate events 110
7.1.3 The joint reliability importance for k-out-of-n systems 111
7.1.4 The joint reliability importance of order k 111
7.2 The Differential ImportanceMeasure 112
7.2.1 The first-order differential importance measure 112
7.2.2 The second-order differential importance measure 113
7.2.3 The differential importance measure of order k 114
7.3 The Total Order Importance 114
7.4 The Reliability AchievementWorth and Reliability ReductionWorth 115
References 116
8 ImportanceMeasures for Consecutive-k-out-of-n Systems 119
8.1 Formulas for the B-importance 119
8.1.1 The B-reliability importance and B-i.i.d. importance 119
8.1.2 The B-structure importance 122
8.2 Patterns of the B-importance for Lin/Con/k/n Systems 123
8.2.1 The B-reliability importance 123
8.2.2 The uniform B-i.i.d. importance 124
8.2.3 The half-line B-i.i.d. importance 126
8.2.4 The nature of the B-i.i.d. importance patterns 126
8.2.5 Patterns with respect to p 128
8.2.6 Patterns with respect to n 129
8.2.7 Disproved patterns and conjectures 131
8.3 Structure ImportanceMeasures 135
8.3.1 The permutation importance 135
8.3.2 The cut-path importance 135
8.3.3 The BP structure importance 135
8.3.4 The first-term importance and rare-event importance 136
References 137
Part Three IMPORTANCE MEASURES for RELIABILITY DESIGN 139
Introduction 140
References 141
9 Redundancy Allocation 143
9.1 Redundancy ImportanceMeasures 144
9.2 A Common Spare 145
9.2.1 The redundancy importance measures 145
9.2.2 The permutation importance 147
9.2.3 The cut importance and path importance 147
9.3 Spare Identical to the Respective Component 148
9.3.1 The redundancy importance measures 148
9.3.2 The permutation importance 149
9.4 Several Spares in a k-out-of-n System 150
9.5 Several Spares in an Arbitrary Coherent System 150
9.6 Cold Standby Redundancy 152
References 152
10 Upgrading System Performance 155
10.1 Improving Systems Reliability 156
10.1.1 Same amount of improvement in component reliability 156
10.1.2 A fractional change in component reliability 157
10.1.3 Cold standby redundancy 158
10.1.4 Parallel redundancy 158
10.1.5 Example and discussion 158
10.2 Improving Expected System Lifetime 159
10.2.1 A shift in component lifetime distributions 160
10.2.2 Exactly one minimal repair 160
10.2.3 Reduction in the proportional hazards 167
10.2.4 Cold standby redundancy 168
10.2.5 A perfect component 170
10.2.6 An imperfect repair 170
10.2.7 A scale change in component lifetime distributions 171
10.2.8 Parallel redundancy 171
10.2.9 Comparisons and numerical evaluation 172
10.3 Improving Expected System Yield 174
10.3.1 A shift in component lifetime distributions 175
10.3.2 Exactly one minimal repair / cold standby redundancy / a perfect component /
parallel redundancy 180
10.4 Discussion 182
References 182
11 Component Assignment in Coherent Systems 185
11.1 Description of Component Assignment Problems 186
11.2 Enumeration and Randomization Methods 187
11.3 Optimal Design based on the Permutation Importance and Pairwise Exchange 188
11.4 Invariant Optimal and InvariantWorst Arrangements 189
11.5 Invariant Arrangements for Parallel-series and Series-parallel Systems 191
11.6 Consistent B-i.i.d. Importance Ordering and Invariant Arrangements 192
11.7 Optimal Design based on the B-reliability Importance 194
11.8 Optimal Assembly Problems 196
References 197
12 Component Assignment in Consecutive-k-out-of-n and Its Variant Systems 199
12.1 Invariant Arrangements for Con/k/n Systems 199
12.1.1 Invariant optimal arrangements for Lin/Con/k/n systems 200
12.1.2 Invariant optimal arrangements for Cir/Con/k/n systems 200
12.1.3 Consistent B-i.i.d. importance ordering and invariant arrangements 202
12.2 Necessary Conditions for Component Assignment in Con/k/n Systems 204
12.3 Sequential Component Assignment Problems in Con/2/n:F Systems 206
12.4 Consecutive-2 Failure Systems on Graphs 207
12.4.1 Consecutive-2 failure systems on trees 208
12.5 Series Con/k/n Systems 208
12.5.1 Series Con/2/n:F systems 209
12.5.2 Series Lin/Con/k/n:G systems 209
12.6 Consecutive-k-out-of-r-from-n Systems 211
12.7 Two-dimensional and Redundant Con/k/n Systems 213
12.7.1 Con/(r, k)/(r, n) systems 214
12.8 Miscellaneous 216
References 217
13 B-importance based Heuristics for Component Assignment 219
13.1 The Kontoleon Heuristic 219
13.2 The LK Type Heuristics 221
13.2.1 The LKA heuristic 221
13.2.2 Another three LK type heuristics 221
13.2.3 Relation to invariant optimal arrangements 221
13.2.4 Numerical comparisons of the LK type heuristics 224
13.3 The ZK Type Heuristics 225
13.3.1 Four ZK type heuristics 225
13.3.2 Relation to invariant optimal arrangements 227
13.3.3 Comparisons of initial arrangements 227
13.3.4 Numerical comparisons of the ZK type heuristics 229
13.4 The B-importance based Two-stage Approach 229
13.4.1 Numerical comparisons with the GAMS/CoinBomin solver and enumeration
method 230
13.4.2 Numerical comparisons with the randomization method 230
13.5 The B-importance based Genetic Local Search 231
13.5.1 The description of algorithm 232
13.5.2 Numerical comparisons with the B-importance based two-stage approach and a
genetic algorithm 235
13.6 Summary and Discussion 236
References 238
Part Four RELATIONS and GENERALIZATIONS 241
Introduction 242
14 Comparisons of Importance Measures 245
14.1 Relations to the B-importance 245
14.2 Rankings of Reliability ImportanceMeasures 247
14.2.1 Using the permutation importance 247
14.2.2 Using the permutation importance and joint reliability importance 249
14.2.3 Using the domination importance 250
14.2.4 Summary 250
14.3 ImportanceMeasures for Some Special Systems 250
14.4 Computation of ImportanceMeasures 251
References 253
15 Generalizations of Importance Measures 255
15.1 Noncoherent Systems 255
15.1.1 Binary monotone systems 256
15.2 Multistate Coherent Systems 257
15.2.1 The µ, _ B-importance 258
15.2.2 The µ, _ cut importance 259
15.3 Multistate Monotone Systems 261
15.3.1 The permutation importance 261
15.3.2 The utility B-reliability importance 262
15.3.3 The utility-decomposition reliability importance 262
15.3.4 The utility B-structure importance, joint structure importance, and joint reliability
importance 263
15.3.5 The B-importance, FV importance, reliability achievement worth, and reliability
reduction worth with respect to system mean unavailability and mean performance 265
15.4 Binary Type Multistate Monotone Systems 266
15.4.1 The B-t.d.l. importance, BP t.i.l. importance, and L1 t.i.l. importance 267
15.5 Summary of ImportanceMeasures for Multistate Systems 268
15.6 Continuum Systems 270
15.7 Repairable Systems 272
15.7.1 The B-availability importance 272
15.7.2 The c-FV unavailability importance 273
15.7.3 The BP availability importance 273
15.7.4 The L1 t.i.l. importance 274
15.7.5 Simulation-based importance measures 275
15.8 Applications in the Power Industry 276
References 277
Part Five BROAD IMPLICATIONS to RISK and MATHEMATICAL
PROGRAMMING 281
Introduction 282
References 283
16 Networks 285
16.1 Network Flow Systems 285
16.1.1 The edge importance measures in a network flow system 286
16.1.2 The edge importance measures for a binary monotone system 288
16.1.3 The B-reliability importance, FV reliability importance, reliability reduction
worth, and reliability achievement worth 289
16.1.4 The flow-based importance and impact-based importance 290
16.2 K-terminal Networks 291
16.2.1 Importance measures of an edge 293
16.2.2 A K-terminal optimization problem 295
References 295
17 Mathematical Programming 297
17.1 Linear Programming 297
17.1.1 Basic concepts 298
17.1.2 The simplex algorithm 300
17.1.3 Sensitivity analysis 301
17.2 Integer Programming 303
17.2.1 Basic concepts and branch-and-bound algorithm 303
17.2.2 Branch-and-bound using linear programming relaxations 306
17.2.3 Mixed integer nonlinear programming 309
References 309
18 Sensitivity Analysis 311
18.1 Local Sensitivity and Perturbation Analysis 311
18.1.1 The B-reliability importance 311
18.1.2 The multidirectional sensitivity measure 312
18.1.3 The multidirectional differential importance measure and total order importance 317
18.1.4 Perturbation analysis 318
18.2 Global Sensitivity Analysis 319
18.2.1 ANOVA-decomposition based global sensitivity measures 320
18.2.2 Elementary effect methods and derivative-based global sensitivity measures 323
18.2.3 Relationships between the ANOVA-decomposition-based and the derivativebased
sensitivity measures 326
18.2.4 The case of random input variables 327
18.2.5 Moment-independent sensitivity measures 328
18.3 Systems reliability subject to uncertain component reliability 330
18.3.1 Software Reliability 332
18.4 Broad applications 335
References 336
19 Risk and Safety in Nuclear Power Plants 339
19.1 Introduction to Probabilistic Risk Analysis and Probabilistic Safety Assessment 339
19.2 Probabilistic (Local) ImportanceMeasures 340
19.3 Uncertainty and Global Sensitivity Measures 342
19.4 A Case Study 343
19.5 Review of Applications 345
19.6 System Fault Diagnosis and Maintenance 347
References 348
Afterword 350
References 354
APPENDIX 355
A Proofs 357
A.1 Proof of Theorem 8.2.7 357
A.2 Proof of Theorem 10.2.10 358
A.3 Proof of Theorem 10.2.17 359
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