The first part of the book defines the concept of uncertainties and the mathematical frameworks that will be used for uncertainty modeling. The application to system reliability assessment illustrates the concept. In the second part, evidential networks as a new tool to model uncertainty in reliability and risk analysis is proposed and described. Then it is applied on SIS performance assessment and in risk analysis of a heat sink. In the third part, Bayesian and evidential networks are used to deal with important measures evaluation in the context of uncertainties.--

Christophe Simon, Université de Lorraine, Centre de Recherche en Automatique de Nancy, France.

Philippe Weber, Université de Lorraine, Centre de Recherche en Automatique de Nancy, France.

Mohamed Sallak, PhD, Associate Professor University of Technologies of Compiègne, France.

Contents xi

Foreword xiii

Acknowledgments xiii

Chapter 1 Why and Where Uncertainties 1

1.1 Sources and forms of uncertainty 1

1.2 Types of uncertainty 3

1.3 Sources of uncertainty 3

1.4 Conclusion 6

Chapter 2 Models and Language of Uncertainty 9

2.1 Introduction 9

2.2 Probability theory 11

2.2.1 Interpretations 11

2.2.2 Fundamental notions 13

2.2.3 Discussion 15

2.3 Belief functions theory 15

2.3.1 Representation of beliefs 16

2.3.2 Combination rules 18

2.3.3 Extension and marginalization 20

2.3.4 Pignistic transformation 20

2.3.5 Discussion 21

2.4 Fuzzy set theory 21

2.4.1 Basic definitions 22

2.4.2 Operations on fuzzy sets 22

2.4.3 Fuzzy relations 23

2.5 Fuzzy arithmetic 25

2.5.1 Fuzzy numbers 26

2.5.2 Fuzzy probabilities 28

2.5.3 Discussion 29

2.6 Possibility theory 29

2.6.1 Definitions 30

2.6.2 Possibility and necessity measures 30

2.6.3 Operations on possibility and necessity measures 32

2.7 Random set theory 32

2.7.1 Basic definitions 33

2.7.2 Expectation of random sets 34

2.7.3 Random intervals 35

2.7.4 Confidence interval 35

2.7.5 Discussion 36

2.8 Confidence structures or c-boxes 36

2.8.1 Basic notions 36

2.8.2 Confidence distributions 37

2.8.3 P-boxes and C-boxes 38

2.8.4 Discussion 40

2.9 Imprecise probability theory 40

2.9.1 Definitions 41

2.9.2 Basic properties 42

2.9.3 Discussion 44

2.10 Conclusion 44

Chapter 3 Risk Graphs and Risk Matrices: Application of Fuzzy Sets and Belief Reasoning 47

3.1 SIL allocation scheme 48

3.1.1 Safety instrumented systems (SIS) 48

3.1.2 Conformity to standards ANSI/ISA S84.01-1996 and IEC 61508 49

3.1.3 Taxonomy of risk/SIL assessment methods 50

3.1.4 Risk assessment 50

3.1.5 SIL allocation process 52

3.1.6 The use of experts' opinions 53

3.2 SIL allocation based on possibility theory 54

3.2.1 Eliciting the experts' opinions 54

3.2.2 Rating scales for parameters 55

3.2.3 Subjective elicitation of the risk parameters 56

3.2.4 Calibration of experts' opinions 59

3.2.5 Aggregation of the opinions 61

3.3 Fuzzy risk graph 65

3.3.1 Input fuzzy partition and fuzzification 65

3.3.2 Risk/SIL graph logic by fuzzy inference system 66

3.3.3 Output fuzzy partition and defuzzification 67

3.3.4 Illustration case 69

3.4 Risk/SIL graph: belief functions reasoning 72

3.4.1 Elicitation of expert opinions in the belief functions theory 72

3.4.2 Aggregation of expert opinions 73

3.5 Evidential risk graph 75

3.6 Numerical illustration 77

3.6.1 Clustering of experts' opinions 77

3.6.2 Aggregation of preferences 78

3.6.3 Evidential risk/SIL graph 79

3.7 Conclusion 81

Chapter 4 Dependability Assessment Considering Interval-valued Probabilities 83

4.1 Interval arithmetic 84

4.1.1 Interval-valued parameters 84

4.1.2 Interval-valued reliability 85

4.1.3 Assessing the imprecise average probability of failure on demand 86

4.2 Constraint arithmetic 90

4.3 Fuzzy arithmetic 93

4.3.1 Application example 95

4.3.2 Monte Carlo sampling approach 97

4.4 Discussion 99

4.4.1 Markov chains 100

4.4.2 Multiphase Markov chains 101

4.4.3 Markov chains with fuzzy numbers 102

4.4.4 Fuzzy modeling of SIS characteristic parameters 104

4.5 Illustration 105

4.5.1 Epistemic approach 106

4.5.2 Enhanced Markov analysis 113

4.6 Decision-making under uncertainty 115

4.7 Conclusion 117

Chapter 5 Evidential Networks 119

5.1 Main concepts 119

5.1.1 Temporal dimension 121

5.1.2 Computing believe and plausibility measures as bounds 123

5.1.3 Inference 124

5.1.4 Modeling imprecision and ignorance in nodes 126

5.1.5 Conclusion 128

5.2 Evidential Network to model and compute Fuzzy probabilities 128

5.2.1 Fuzzy probability and basic probability assignment 128

5.2.2 Nested interval-valued probabilities to fuzzy probability 129

5.2.3 Computation mechanism 130

5.3 Evidential Networks to compute p-box 131

5.3.1 Connection between p-boxes and BPA 132

5.3.2 P-boxes and interval-valued probabilities 133

5.3.3 P-boxes and precise probabilities 133

5.3.4 Time-dependent p-boxes 134

5.3.5 Computation mechanism 134

5.4 Modeling some reliability problems 136

5.4.1 BPA for reliability problems 136

5.4.2 Building Boolean CMT (AND, OR) 137

5.4.3 Conditional mass table for more than two inputs (k-out-of-n:G gate) 138

5.4.4 Nodes for Pls and Bel in the binary case 140

5.4.5 Modeling reliability with p-boxes 140

5.5 Illustration by application of Evidential Networks 145

5.5.1 Reliability assessment of system 145

5.5.2 Inference for failure isolation 153

5.5.3 Assessing the fuzzy reliability of systems 155

5.5.4 Assessing the p-box reliability by EN 162

5.6 Conclusion 169

Chapter 6 Reliability Uncertainty and Importance Factors 171

6.1 Introduction 171

6.2 Hypothesis and notation 173

6.3 Probabilistic importance measures of components 174

6.3.1 Birnbaum importance measure 175

6.3.2 Component criticality measure 176

6.3.3 Diagnostic importance measure 176

6.3.4 Reliability achievement worth (RAW) 177

6.3.5 Reliability reduction worth (RRW) 177

6.3.6 Observations and limitations 178

6.3.7 Importance measures computation 179

6.4 Probabilistic importance measures of pairs and groups of components 179

6.4.1 Measures on minimum cutsets/pathsets/groups 181

6.4.2 Extension of RAW and RRW to pairs 182

6.4.3 Joint reliability importance factor (JR) 182

6.5 Uncertainty importance measures 184

6.5.1 Uncertainty probabilistic importance measures 184

6.5.2 Importance factors with imprecision 186

6.6 Importance measures with fuzzy probabilities 188

6.6.1 Fuzzy importance measures 189

6.6.2 Fuzzy uncertainty measures 190

6.7 Illustration 191

6.7.1 Importance factors on a simple system 192

6.7.2 Importance factors in a complex case 195

6.7.3 Illustration of group importance measures 197

6.7.4 Uncertainty importance factors 200

6.7.5 Fuzzy importance measures 203

6.8 Conclusion 206

Conclusion 207

Bibliography 211

Index 225