Importance Measures in Reliability, Risk and Optimization - Principles and Applications

Preface xv 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 , - B-importance 258 15.2.2 The , - 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