Yan-Fu Li is Full Professor at the Department of Industrial Engineering and the Director of the Institute for Quality & Reliability at Tsinghua University, China. He received his Ph.D in Industrial Engineering from National University of Singapore in 2010Enrico Zio is Full Professor at Mines-Paris, PSL University, and at the Energy Department of Politecnico di Milano, Italy. He received his Ph.D in nuclear engineering from Politecnico di Milano and in Probabilistic Risk Assessment from MIT in 1996 and 1998, respectively.

• Series Editor’s Foreword by Dr. Andre V. Kleyner xvPreface xviiAcknowledgments xixList of Abbreviations xxNotations xxiiPart I The Fundamentals 11 Reliability Assessment 31.1 Definitions of Reliability 31.1.1 Probability of Survival 31.2 Component Reliability Modeling 61.2.1 Discrete Probability Distributions 61.2.2 Continuous Probability Distributions 81.2.3 Physics-of-Failure Equations 131.3 System Reliability Modeling 151.3.1 Series System 151.3.2 Parallel System 161.3.3 Series-parallel System 161.3.4 K-out-of-n System 171.3.5 Network System 181.4 System Reliability Assessment Methods 181.4.1 Path-set and Cut-set Method 181.4.2 Decomposition and Factorization 191.4.3 Binary Decision Diagram 191.5 Exercises 20References 222 Optimization 232.1 Optimization Problems 232.1.1 Component Reliability Enhancement 232.1.2 Redundancy Allocation 242.1.3 Component Assignment 252.1.4 Maintenance and Testing 262.2 Optimization Methods 302.2.1 Mathematical Programming 302.2.2 Meta-heuristics 342.3 Exercises 36References 37Part II Reliability Techniques 413 Multi-State Systems (MSSs) 433.1 Classical Multi-state Models 433.2 Generalized Multi-state Models 453.3 Time-dependent Multi-State Models 463.4 Methods to Evaluate Multi-state System Reliability 483.4.1 Methods Based on MPVs or MCVs 483.4.2 Methods Derived from Binary State Reliability Assessment 483.4.3 Universal Generating Function Approach 493.4.4 Monte Carlo Simulation 503.5 Exercises 51References 514 Markov Processes 554.1 Continuous Time Markov Chain (CMTC) 554.2 In homogeneous Continuous Time Markov Chain 614.3 Semi-Markov Process (SMP) 664.4 Piecewise Deterministic Markov Process (PDMP) 744.5 Exercises 82References 845 Monte Carlo Simulation (MCS) for Reliability and Availability Assessment 875.1 Introduction 875.2 Random Variable Generation 875.2.1 Random Number Generation 875.2.2 Random Variable Generation 895.3 Random Process Generation 935.3.1 Markov Chains 935.3.2 Markov Jump Processes 945.4 Markov Chain Monte Carlo (MCMC) 975.4.1 Metropolis-Hastings (M-H) Algorithm 975.4.2 Gibbs Sampler 985.4.3 Multiple-try Metropolis-Hastings (M-H) Method 995.5 Rare-Event Simulation 1015.5.1 Importance Sampling 1015.5.2 Repetitive Simulation Trials after Reaching Thresholds (RESTART) 1025.6 Exercises 103Appendix 104References 1156 Uncertainty Treatment under Imprecise or Incomplete Knowledge 1176.1 Interval Number and Interval of Confidence 1176.1.1 Definition and Basic Arithmetic Operations 1176.1.2 Algebraic Properties 1186.1.3 Order Relations 1196.1.4 Interval Functions 1206.1.5 Interval of Confidence 1216.2 Fuzzy Number 1216.3 Possibility Theory 1236.3.1 Possibility Propagation 1246.4 Evidence Theory 1256.4.1 Data Fusion 1286.5 Random-fuzzy Numbers (RFNs) 1286.5.1 Universal Generating Function (UGF) Representation of Random-fuzzy Numbers 1296.5.2 Hybrid UGF (HUGF) Composition Operator 1306.6 Exercises 132References 1337 Applications 1357.1 Distributed Power Generation System Reliability Assessment 1357.1.1 Reliability of Power Distributed Generation (DG) System 1357.1.2 Energy Source Models and Uncertainties 1367.1.3 Algorithm for the Joint Propagation of Probabilistic and Possibilistic Uncertainties 1387.1.4 Case Study 1407.2 Nuclear Power Plant Components Degradation 1407.2.1 Dissimilar Metal Weld Degradation 1407.2.2 MCS Method 1457.2.3 Numerical Results 147References 149Part III Optimization Methods and Applications 1518 Mathematical Programming 1538.1 Linear Programming (LP) 1538.1.1 Standard Form and Duality 1558.2 Integer Programming (IP) 1598.3 Exercises 164References 1659 Evolutionary Algorithms (EAs) 1679.1 Evolutionary Search 1689.2 Genetic Algorithm (GA) 1709.2.1 Encoding and Initialization 1719.2.2 Evaluation 1729.2.3 Selection 1739.2.4 Mutation 1749.2.5 Crossover 1759.2.6 Elitism 1789.2.7 Termination Condition and Convergence 1789.3 Other Popular EAs 1799.4 Exercises 181References 18210 Multi-Objective Optimization (MOO) 18510.1 Multi-objective Problem Formulation 18510.2 MOO-to-SOO Problem Conversion Methods 18710.2.1 Weighted-sum Approach 18810.2.2 ε-constraint Approach 18910.3 Multi-objective Evolutionary Algorithms 19010.3.1 Fast Non-dominated Sorting Genetic Algorithm (NSGA-II) 19010.3.2 Improved Strength Pareto Evolutionary Algorithm (SPEA 2) 19310.4 Performance Measures 19710.5 Selection of Preferred Solutions 20010.5.1 “Min-Max” Method 20010.5.2 Compromise Programming Approach 20110.6 Guidelines for Solving RAMS+C Optimization Problems 20110.7 Exercises 203References 20411 Optimization under Uncertainty 20711.1 Stochastic Programming (SP) 20711.1.1 Two-stage Stochastic Linear Programs with Fixed Recourse 20911.1.2 Multi-stage Stochastic Programs with Recourse 21711.2 Chance-Constrained Programming 21811.2.1 Model and Properties 21911.2.2 Example 22111.3 Robust Optimization (RO) 22211.3.1 Uncertain Linear Optimization (LO) and its Robust Counterparts 22311.3.2 Tractability of Robust Counterparts 22411.3.3 Robust Optimization (RO) with Cardinality Constrained Uncertainty Set 22511.3.4 Example 22611.4 Exercises 228References 22912 Applications 23112.1 Multi-objective Optimization (MOO) Framework for the Integration of Distributed Renewable Generation and Storage 23112.1.1 Description of Distributed Generation (DG) System 23212.1.2 Optimal Power Flow (OPF) 23412.1.3 Performance Indicators 23512.1.4 MOO Problem Formulation 23712.1.5 Solution Approach and Case Study Results 23812.2 Redundancy Allocation for Binary-State Series-Parallel Systems (BSSPSs) under Epistemic Uncertainty 24012.2.1 Problem Description 24012.2.2 Robust Model 24112.2.3 Experiment 243References 244Index 245

Yan-Fu Li is Full Professor at the Department of Industrial Engineering and the Director of the Institute for Quality & Reliability at Tsinghua University, China. He received his Ph.D in Industrial Engineering from National University of Singapore in 2010Enrico Zio is Full Professor at Mines-Paris, PSL University, and at the Energy Department of Politecnico di Milano, Italy. He received his Ph.D in nuclear engineering from Politecnico di Milano and in Probabilistic Risk Assessment from MIT in 1996 and 1998, respectively.

• Series Editor’s Foreword by Dr. Andre V. Kleyner xvPreface xviiAcknowledgments xixList of Abbreviations xxNotations xxiiPart I The Fundamentals 11 Reliability Assessment 31.1 Definitions of Reliability 31.1.1 Probability of Survival 31.2 Component Reliability Modeling 61.2.1 Discrete Probability Distributions 61.2.2 Continuous Probability Distributions 81.2.3 Physics-of-Failure Equations 131.3 System Reliability Modeling 151.3.1 Series System 151.3.2 Parallel System 161.3.3 Series-parallel System 161.3.4 K-out-of-n System 171.3.5 Network System 181.4 System Reliability Assessment Methods 181.4.1 Path-set and Cut-set Method 181.4.2 Decomposition and Factorization 191.4.3 Binary Decision Diagram 191.5 Exercises 20References 222 Optimization 232.1 Optimization Problems 232.1.1 Component Reliability Enhancement 232.1.2 Redundancy Allocation 242.1.3 Component Assignment 252.1.4 Maintenance and Testing 262.2 Optimization Methods 302.2.1 Mathematical Programming 302.2.2 Meta-heuristics 342.3 Exercises 36References 37Part II Reliability Techniques 413 Multi-State Systems (MSSs) 433.1 Classical Multi-state Models 433.2 Generalized Multi-state Models 453.3 Time-dependent Multi-State Models 463.4 Methods to Evaluate Multi-state System Reliability 483.4.1 Methods Based on MPVs or MCVs 483.4.2 Methods Derived from Binary State Reliability Assessment 483.4.3 Universal Generating Function Approach 493.4.4 Monte Carlo Simulation 503.5 Exercises 51References 514 Markov Processes 554.1 Continuous Time Markov Chain (CMTC) 554.2 In homogeneous Continuous Time Markov Chain 614.3 Semi-Markov Process (SMP) 664.4 Piecewise Deterministic Markov Process (PDMP) 744.5 Exercises 82References 845 Monte Carlo Simulation (MCS) for Reliability and Availability Assessment 875.1 Introduction 875.2 Random Variable Generation 875.2.1 Random Number Generation 875.2.2 Random Variable Generation 895.3 Random Process Generation 935.3.1 Markov Chains 935.3.2 Markov Jump Processes 945.4 Markov Chain Monte Carlo (MCMC) 975.4.1 Metropolis-Hastings (M-H) Algorithm 975.4.2 Gibbs Sampler 985.4.3 Multiple-try Metropolis-Hastings (M-H) Method 995.5 Rare-Event Simulation 1015.5.1 Importance Sampling 1015.5.2 Repetitive Simulation Trials after Reaching Thresholds (RESTART) 1025.6 Exercises 103Appendix 104References 1156 Uncertainty Treatment under Imprecise or Incomplete Knowledge 1176.1 Interval Number and Interval of Confidence 1176.1.1 Definition and Basic Arithmetic Operations 1176.1.2 Algebraic Properties 1186.1.3 Order Relations 1196.1.4 Interval Functions 1206.1.5 Interval of Confidence 1216.2 Fuzzy Number 1216.3 Possibility Theory 1236.3.1 Possibility Propagation 1246.4 Evidence Theory 1256.4.1 Data Fusion 1286.5 Random-fuzzy Numbers (RFNs) 1286.5.1 Universal Generating Function (UGF) Representation of Random-fuzzy Numbers 1296.5.2 Hybrid UGF (HUGF) Composition Operator 1306.6 Exercises 132References 1337 Applications 1357.1 Distributed Power Generation System Reliability Assessment 1357.1.1 Reliability of Power Distributed Generation (DG) System 1357.1.2 Energy Source Models and Uncertainties 1367.1.3 Algorithm for the Joint Propagation of Probabilistic and Possibilistic Uncertainties 1387.1.4 Case Study 1407.2 Nuclear Power Plant Components Degradation 1407.2.1 Dissimilar Metal Weld Degradation 1407.2.2 MCS Method 1457.2.3 Numerical Results 147References 149Part III Optimization Methods and Applications 1518 Mathematical Programming 1538.1 Linear Programming (LP) 1538.1.1 Standard Form and Duality 1558.2 Integer Programming (IP) 1598.3 Exercises 164References 1659 Evolutionary Algorithms (EAs) 1679.1 Evolutionary Search 1689.2 Genetic Algorithm (GA) 1709.2.1 Encoding and Initialization 1719.2.2 Evaluation 1729.2.3 Selection 1739.2.4 Mutation 1749.2.5 Crossover 1759.2.6 Elitism 1789.2.7 Termination Condition and Convergence 1789.3 Other Popular EAs 1799.4 Exercises 181References 18210 Multi-Objective Optimization (MOO) 18510.1 Multi-objective Problem Formulation 18510.2 MOO-to-SOO Problem Conversion Methods 18710.2.1 Weighted-sum Approach 18810.2.2 ε-constraint Approach 18910.3 Multi-objective Evolutionary Algorithms 19010.3.1 Fast Non-dominated Sorting Genetic Algorithm (NSGA-II) 19010.3.2 Improved Strength Pareto Evolutionary Algorithm (SPEA 2) 19310.4 Performance Measures 19710.5 Selection of Preferred Solutions 20010.5.1 “Min-Max” Method 20010.5.2 Compromise Programming Approach 20110.6 Guidelines for Solving RAMS+C Optimization Problems 20110.7 Exercises 203References 20411 Optimization under Uncertainty 20711.1 Stochastic Programming (SP) 20711.1.1 Two-stage Stochastic Linear Programs with Fixed Recourse 20911.1.2 Multi-stage Stochastic Programs with Recourse 21711.2 Chance-Constrained Programming 21811.2.1 Model and Properties 21911.2.2 Example 22111.3 Robust Optimization (RO) 22211.3.1 Uncertain Linear Optimization (LO) and its Robust Counterparts 22311.3.2 Tractability of Robust Counterparts 22411.3.3 Robust Optimization (RO) with Cardinality Constrained Uncertainty Set 22511.3.4 Example 22611.4 Exercises 228References 22912 Applications 23112.1 Multi-objective Optimization (MOO) Framework for the Integration of Distributed Renewable Generation and Storage 23112.1.1 Description of Distributed Generation (DG) System 23212.1.2 Optimal Power Flow (OPF) 23412.1.3 Performance Indicators 23512.1.4 MOO Problem Formulation 23712.1.5 Solution Approach and Case Study Results 23812.2 Redundancy Allocation for Binary-State Series-Parallel Systems (BSSPSs) under Epistemic Uncertainty 24012.2.1 Problem Description 24012.2.2 Robust Model 24112.2.3 Experiment 243References 244Index 245

Yan-Fu Li is Full Professor at the Department of Industrial Engineering and the Director of the Institute for Quality & Reliability at Tsinghua University, China. He received his Ph.D in Industrial Engineering from National University of Singapore in 2010Enrico Zio is Full Professor at Mines-Paris, PSL University, and at the Energy Department of Politecnico di Milano, Italy. He received his Ph.D in nuclear engineering from Politecnico di Milano and in Probabilistic Risk Assessment from MIT in 1996 and 1998, respectively.

• Series Editor’s Foreword by Dr. Andre V. Kleyner xvPreface xviiAcknowledgments xixList of Abbreviations xxNotations xxiiPart I The Fundamentals 11 Reliability Assessment 31.1 Definitions of Reliability 31.1.1 Probability of Survival 31.2 Component Reliability Modeling 61.2.1 Discrete Probability Distributions 61.2.2 Continuous Probability Distributions 81.2.3 Physics-of-Failure Equations 131.3 System Reliability Modeling 151.3.1 Series System 151.3.2 Parallel System 161.3.3 Series-parallel System 161.3.4 K-out-of-n System 171.3.5 Network System 181.4 System Reliability Assessment Methods 181.4.1 Path-set and Cut-set Method 181.4.2 Decomposition and Factorization 191.4.3 Binary Decision Diagram 191.5 Exercises 20References 222 Optimization 232.1 Optimization Problems 232.1.1 Component Reliability Enhancement 232.1.2 Redundancy Allocation 242.1.3 Component Assignment 252.1.4 Maintenance and Testing 262.2 Optimization Methods 302.2.1 Mathematical Programming 302.2.2 Meta-heuristics 342.3 Exercises 36References 37Part II Reliability Techniques 413 Multi-State Systems (MSSs) 433.1 Classical Multi-state Models 433.2 Generalized Multi-state Models 453.3 Time-dependent Multi-State Models 463.4 Methods to Evaluate Multi-state System Reliability 483.4.1 Methods Based on MPVs or MCVs 483.4.2 Methods Derived from Binary State Reliability Assessment 483.4.3 Universal Generating Function Approach 493.4.4 Monte Carlo Simulation 503.5 Exercises 51References 514 Markov Processes 554.1 Continuous Time Markov Chain (CMTC) 554.2 In homogeneous Continuous Time Markov Chain 614.3 Semi-Markov Process (SMP) 664.4 Piecewise Deterministic Markov Process (PDMP) 744.5 Exercises 82References 845 Monte Carlo Simulation (MCS) for Reliability and Availability Assessment 875.1 Introduction 875.2 Random Variable Generation 875.2.1 Random Number Generation 875.2.2 Random Variable Generation 895.3 Random Process Generation 935.3.1 Markov Chains 935.3.2 Markov Jump Processes 945.4 Markov Chain Monte Carlo (MCMC) 975.4.1 Metropolis-Hastings (M-H) Algorithm 975.4.2 Gibbs Sampler 985.4.3 Multiple-try Metropolis-Hastings (M-H) Method 995.5 Rare-Event Simulation 1015.5.1 Importance Sampling 1015.5.2 Repetitive Simulation Trials after Reaching Thresholds (RESTART) 1025.6 Exercises 103Appendix 104References 1156 Uncertainty Treatment under Imprecise or Incomplete Knowledge 1176.1 Interval Number and Interval of Confidence 1176.1.1 Definition and Basic Arithmetic Operations 1176.1.2 Algebraic Properties 1186.1.3 Order Relations 1196.1.4 Interval Functions 1206.1.5 Interval of Confidence 1216.2 Fuzzy Number 1216.3 Possibility Theory 1236.3.1 Possibility Propagation 1246.4 Evidence Theory 1256.4.1 Data Fusion 1286.5 Random-fuzzy Numbers (RFNs) 1286.5.1 Universal Generating Function (UGF) Representation of Random-fuzzy Numbers 1296.5.2 Hybrid UGF (HUGF) Composition Operator 1306.6 Exercises 132References 1337 Applications 1357.1 Distributed Power Generation System Reliability Assessment 1357.1.1 Reliability of Power Distributed Generation (DG) System 1357.1.2 Energy Source Models and Uncertainties 1367.1.3 Algorithm for the Joint Propagation of Probabilistic and Possibilistic Uncertainties 1387.1.4 Case Study 1407.2 Nuclear Power Plant Components Degradation 1407.2.1 Dissimilar Metal Weld Degradation 1407.2.2 MCS Method 1457.2.3 Numerical Results 147References 149Part III Optimization Methods and Applications 1518 Mathematical Programming 1538.1 Linear Programming (LP) 1538.1.1 Standard Form and Duality 1558.2 Integer Programming (IP) 1598.3 Exercises 164References 1659 Evolutionary Algorithms (EAs) 1679.1 Evolutionary Search 1689.2 Genetic Algorithm (GA) 1709.2.1 Encoding and Initialization 1719.2.2 Evaluation 1729.2.3 Selection 1739.2.4 Mutation 1749.2.5 Crossover 1759.2.6 Elitism 1789.2.7 Termination Condition and Convergence 1789.3 Other Popular EAs 1799.4 Exercises 181References 18210 Multi-Objective Optimization (MOO) 18510.1 Multi-objective Problem Formulation 18510.2 MOO-to-SOO Problem Conversion Methods 18710.2.1 Weighted-sum Approach 18810.2.2 ε-constraint Approach 18910.3 Multi-objective Evolutionary Algorithms 19010.3.1 Fast Non-dominated Sorting Genetic Algorithm (NSGA-II) 19010.3.2 Improved Strength Pareto Evolutionary Algorithm (SPEA 2) 19310.4 Performance Measures 19710.5 Selection of Preferred Solutions 20010.5.1 “Min-Max” Method 20010.5.2 Compromise Programming Approach 20110.6 Guidelines for Solving RAMS+C Optimization Problems 20110.7 Exercises 203References 20411 Optimization under Uncertainty 20711.1 Stochastic Programming (SP) 20711.1.1 Two-stage Stochastic Linear Programs with Fixed Recourse 20911.1.2 Multi-stage Stochastic Programs with Recourse 21711.2 Chance-Constrained Programming 21811.2.1 Model and Properties 21911.2.2 Example 22111.3 Robust Optimization (RO) 22211.3.1 Uncertain Linear Optimization (LO) and its Robust Counterparts 22311.3.2 Tractability of Robust Counterparts 22411.3.3 Robust Optimization (RO) with Cardinality Constrained Uncertainty Set 22511.3.4 Example 22611.4 Exercises 228References 22912 Applications 23112.1 Multi-objective Optimization (MOO) Framework for the Integration of Distributed Renewable Generation and Storage 23112.1.1 Description of Distributed Generation (DG) System 23212.1.2 Optimal Power Flow (OPF) 23412.1.3 Performance Indicators 23512.1.4 MOO Problem Formulation 23712.1.5 Solution Approach and Case Study Results 23812.2 Redundancy Allocation for Binary-State Series-Parallel Systems (BSSPSs) under Epistemic Uncertainty 24012.2.1 Problem Description 24012.2.2 Robust Model 24112.2.3 Experiment 243References 244Index 245