Uncertainty in Risk Assessment – The Representation and Treatment of Uncertainties by Probabilistic and Non–Probabilistic Methods

Terje Aven, University of Stavanger, Norway Piero Baraldi, Politecnico di Milano, Italy Roger Flage, University of Stavanger, Norway Enrico Zio, Politecnico di Milano, Italy

Preface ix PART I INTRODUCTION 1 1 Introduction 3 1.1 Risk 4 1.1.1 The concept of risk 4 1.1.2 Describing/measuring risk 6 1.1.3 Examples 6 1.2 Probabilistic risk assessment 8 1.3 Use of risk assessment: The risk management and decision-making context 11 1.4 Treatment of uncertainties in risk assessments 13 1.5 Challenges: Discussion 15 1.5.1 Examples 16 1.5.2 Alternatives to the probability-based approaches to risk and uncertainty assessment 17 1.5.3 The way ahead 19 References Part I 21 PART II METHODS 27 2 Probabilistic approaches for treating uncertainty 29 2.1 Classical probabilities 30 2.2 Frequentist probabilities 31 2.3 Subjective probabilities 35 2.3.1 Betting interpretation 36 2.3.2 Reference to a standard for uncertainty 36 2.4 The Bayesian subjective probability framework 37 2.5 Logical probabilities 39 3 Imprecise probabilities for treating uncertainty 41 4 Possibility theory for treating uncertainty 45 4.1 Basics of possibility theory 45 4.2 Approaches for constructing possibility distributions 49 4.2.1 Building possibility distributions from nested probability intervals 49 4.2.2 Justification for using the triangular possibility distribution 51 4.2.3 Building possibility distributions using Chebyshev s inequality 52 5 Evidence theory for treating uncertainty 53 6 Methods of uncertainty propagation 59 6.1 Level 1 uncertainty propagation setting 61 6.1.1 Level 1 purely probabilistic framework 62 6.1.2 Level 1 purely possibilistic framework 64 6.1.3 Level 1 hybrid probabilistic possibilistic framework 67 6.2 Level 2 uncertainty propagation setting 71 6.2.1 Level 2 purely probabilistic framework 73 6.2.2 Level 2 hybrid probabilistic evidence theory framework 75 7 Discussion 79 7.1 Probabilistic analysis 80 7.2 Lower and upper probabilities 82 7.3 Non-probabilistic representations with interpretations other than lower and upper probabilities 84 7.4 Hybrid representations of uncertainty 85 7.5 Semi-quantitative approaches 87 References Part II 93 PART III PRACTICAL APPLICATIONS 99 8 Uncertainty representation and propagation in structural reliability analysis 101 8.1 Structural reliability analysis 101 8.1.1 A model of crack propagation under cyclic fatigue 101 8.2 Case study 102 8.3 Uncertainty representation 104 8.4 Uncertainty propagation 105 8.5 Results 107 8.6 Comparison to a purely probabilistic method 107 9 Uncertainty representation and propagation in maintenance performance assessment 111 9.1 Maintenance performance assessment 111 9.2 Case study 113 9.3 Uncertainty representation 116 9.4 Uncertainty propagation 118 9.4.1 Maintenance performance assessment in the case of no epistemic uncertainty on the parameters 118 9.4.2 Application of the hybrid probabilistic theory of evidence uncertainty propagation method 122 9.5 Results 123 10 Uncertainty representation and propagation in event tree analysis 127 10.1 Event tree analysis 127 10.2 Case study 128 10.3 Uncertainty representation 134 10.4 Uncertainty propagation 135 10.5 Results 137 10.6 Comparison of the results to those obtained by using other uncertainty representation and propagation methods 138 10.6.1 Purely probabilistic representation and propagation of the uncertainty 138 10.6.2 Purely possibilistic representation and propagation of the uncertainty 138 10.7 Result comparison 141 10.7.1 Comparison of results 141 10.7.2 Comparison of the results for the probability of occurrence of a severe consequence accident 145 11 Uncertainty representation and propagation in the evaluation of the consequences of industrial activity 147 11.1 Evaluation of the consequences of undesirable events 147 11.2 Case study 148 11.3 Uncertainty representation 150 11.4 Uncertainty propagation 152 11.5 Results 152 11.6 Comparison of the results to those obtained using a purely probabilistic approach 153 12 Uncertainty representation and propagation in the risk assessment of a process plant 155 12.1 Introduction 155 12.2 Case description 155 12.3 The textbook Bayesian approach (level 2 analysis) 156 12.4 An alternative approach based on subjective probabilities (level 1 analysis) 159 References Part III 163 PART IV CONCLUSIONS 167 13 Conclusions 169 References Part IV 173 Appendix A Operative procedures for the methods of uncertainty propagation 175 A.1 Level 1 hybrid probabilistic possibilistic framework 175 A.2 Level 2 purely probabilistic framework 176 Appendix B Possibility probability transformation 179 Reference 181 Index 183