Optimal Resource Allocation

Preface xi

1 BASIC MATHEMATICAL REDUNDANCY MODELS 1

1.1 Types of Models 2

1.2 Non-repairable Redundant Group with Active Redundant Units 3

1.3 Non-repairable Redundant Group with Standby Redundant Units 7

1.4 Repairable Redundant Group with Active Redundant Units 10

1.5 Repairable Redundant Group with Standby Redundant Units 13

1.6 Multi-level Systems and System Performance Estimation 15

1.7 Brief Review of Other Types of Redundancy 16

1.8 Time Redundancy 24

1.9 Some Additional Optimization Problems 27

Chronological Bibliography of Main Monographs on Reliability Theory (with topics on Optimization) 30

2 FORMULATION OF OPTIMAL REDUNDANCY PROBLEMS 33

2.1 Problem Description 33

2.2 Formulation of the Optimal Redundancy Problem with a Single Restriction 35

2.3 Formulation of Optimal Redundancy Problems with Multiple Constraints 39

2.4 Formulation of Multi-Criteria Optimal Redundancy Problems 43

Chronological Bibliography 45

3 METHOD OF LAGRANGE MULTIPLIERS 48

Chronological Bibliography 55

4 STEEPEST DESCENT METHOD 56

4.1 The Main Idea of SDM 56

4.2 Description of the Algorithm 57

4.3 The Stopping Rule 60

4.5 Approximate Solution 66

Chronological Bibliography 68

5 DYNAMIC PROGRAMMING 69

5.1 Bellman?s Algorithm 69

5.2 Kettelle?s Algorithm 73

Chronological Bibliography 84

6 UNIVERSAL GENERATING FUNCTIONS 85

6.1 Generating Function 85

6.2 Universal GF (U-function) 87

Chronological Bibliography 94

7 GENETIC ALGORITHMS 96

7.1 Introduction 96

7.2 Structure of Steady-State Genetic Algorithms 100

7.3 Related Techniques 102

Chronological Bibliography 104

8 MONTE CARLO SIMULATION 107

8.1 Introductory Remarks 107

8.2 Formulation of Optimal Redundancy Problems in Statistical Terms 108

8.3 Algorithm for Trajectory Generation 108

8.4 Description of the Idea of the Solution 111

8.5 Inverse Optimization Problem 114

8.6 Direct Optimization Problem 124

Chronological Bibliography 129

9 COMMENTS ON CALCULATION METHODS 130

9.1 Comparison of Methods 130

9.2 Sensitivity Analysis of Optimal Redundancy Solutions 135

10 OPTIMAL REDUNDANCY WITH SEVERAL LIMITING FACTORS 142

10.1 Method of ?Weighing Costs? 142

10.2 Method of Generalized Generating Functions 146

Chronological Bibliography 149

11 OPTIMAL REDUNDANCY IN MULTISTATE SYSTEMS 150

Chronological Bibliography 170

12 CASE STUDIES 172

12.1 Spare Supply System for Worldwide Telecommunication System Globalstar 172

12.2 Optimal Capacity Distribution of Telecommunication Backbone Network Resources 179

12.3 Optimal Spare Allocation for Mobile Repair Station 183

Chronological Bibliography 190

13 COUNTER-TERRORISM: PROTECTION RESOURCES ALLOCATION 191

13.1 Introduction 191

13.2 Written Description of the Problem 192

13.3 Evaluation of Expected Loss 195

13.4 Algorithm of Resource Allocation 197

13.5 Branching System Protection 201

13.6 Fictional Case Study 210

13.7 Measures of Defense, Their Effectiveness, and Related Expenses 217

13.8 Antiterrorism Resource Allocation under Fuzzy Subjective Estimates 223

13.9 Conclusion 232

Chronological Bibliography 232

About the author 235