**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 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