Acknowledgments xvii

1 Introduction 1

1.1 Perspective 1

1.2 Equitable Resource Allocation: Lexicographic Minimax (Maximin) Optimization 3

1.3 Examples and Applications 14

1.3.1 Allocation of High-Tech Components 14

1.3.2 Throughput in Communication and Computer Networks 15

1.3.3 Point-to-Point Throughput Estimation in Networks 18

1.3.4 Bandwidth Allocation for Content Distribution 20

1.3.5 Location of Emergency Facilities 23

1.3.6 Other Applications 25

1.4 Related Fairness Criteria 26

1.5 Outline of the Book 30

1.5.1 Chapter 2: Nonlinear Resource Allocation 30

1.5.2 Chapter 3: Equitable Resource Allocation: Lexicographic Minimax and Maximin Optimization 30

1.5.3 Chapter 4: Equitable Resource Allocation with Substitutable Resources 31

1.5.4 Chapter 5: Multiperiod Equitable Resource Allocation 32

1.5.5 Chapter 6: Equitable Network Resource Allocation 33

1.5.6 Chapter 7: Equitable Resource Allocation with Integer Decisions 34

1.6 Concluding Remarks and Literature Review 35

1.6.1 Equitable Allocation of High-Tech Components 38

1.6.2 Equitable Throughput in Communication and Computer Networks 38

1.6.3 Point-to-Point Throughput Estimation in Networks 39

1.6.4 Equitable Bandwidth Allocation for Content Distribution 39

1.6.5 Equitable Location of Emergency Facilities 39

1.6.6 Other Applications 39

2 Nonlinear Resource Allocation 41

2.1 Formulation and Optimality Properties 42

2.2 Algorithms 48

2.2.1 The Activity Deletion Algorithm 48

2.2.2 The Activity Addition Algorithm 53

2.2.3 The Constraints Evaluation Algorithm 55

2.2.4 Lower and Upper Bounds 58

2.3 Nonlinear Resource-Usage Constraint 58

2.3.1 Formulation and Optimality Properties 59

2.3.2 Algorithms 62

2.4 Multiple Resource Constraints: A Special Case 66

2.5 Concluding Remarks and Literature Review 73

Exercises 75

3 Equitable Resource Allocation: Lexicographic Minimax and Maximin Optimization 77

3.1 Formulation and Optimality Properties 78

3.2 Minimax Algorithms 84

3.2.1 The Minimax Activity Deletion Algorithm 84

3.2.2 The Minimax Activity Addition Algorithm 90

3.2.3 The Minimax Constraints Evaluation Algorithm 94

3.2.4 Lower and Upper Bounds 97

3.3 The Lexicographic Minimax Algorithm 98

3.4 Extension to Nonseparable Objective Function 107

3.5 Concluding Remarks and Literature Review 116

Exercises 120

4 Equitable Resource Allocation with Substitutable Resources 123

4.1 Representations of Substitutable Resources 124

4.1.1 Transitive Substitutable Resources Represented by Trees 124

4.1.2 Transitive Substitutable Resources Represented by Acyclic Graphs 125

4.1.3 Nontransitive Substitutable Resources Represented by Bipartite Graphs 127

4.1.4 Activity-Dependent Substitutable Resources Represented by Bipartite Graphs 128

4.1.5 Solution Approach 129

4.2 Transitive Substitutable Resources Represented by Trees 131

4.2.1 Formulation 131

4.2.2 The Minimax Algorithm 134

4.2.3 The Lexicographic Minimax Algorithm 143

4.2.4 Lower and Upper Bounds 151

4.3 Transitive Substitutable Resources Represented by Acyclic Graphs 153

4.3.1 Formulation 154

4.3.2 The Feasibility Problem 155

4.3.3 The Minimax Algorithm 161

4.3.4 The Lexicographic Minimax Algorithm 165

4.4 Activity-Dependent Substitutable Resources Represented by Bipartite Graphs 172

4.4.1 Formulation 173

4.4.2 The Minimax Algorithm 175

4.4.3 The Lexicographic Minimax Algorithm 179

4.5 Concluding Remarks and Literature Review 180

Exercises 181

5 Multiperiod Equitable Resource Allocation 183

5.1 Formulation for Storable Resource Allocation 184

5.2 Minimax Algorithms for Storable Resources 187

5.2.1 The Search-Based Algorithm 188

5.2.2 The Transformation-Based Algorithm 192

5.2.3 The Multiperiod Activity Deletion Algorithm: A Special Case 200

5.3 The Lexicographic Minimax Algorithm 203

5.4 Allocation of Nonstorable Resources 210

5.5 Multiperiod Allocation of Substitutable Resources 213

5.6 Concluding Remarks and Literature Review 218

Exercises 219

6 Equitable Allocation of Network Resources 221

6.1 Multicommodity Network Flows with a Single Fixed Path 223

6.2 Multicommodity Network Flows with Multiple Paths 227

6.3 Bandwidth Allocation for Content Distribution 237

6.4 Content Distribution with Node-Dependent Performance Functions 248

6.5 Concluding Remarks and Literature Review 254

Exercises 257

7 Equitable Resource Allocation with Integer Decisions 259

7.1 Knapsack Resource Constraints with Integer Variables 261

7.1.1 Formulation and Challenges 261

7.1.2 The Integer Minimax Problem 264

7.1.3 The Integer Lexicographic Minimax Problem with One Resource Constraint 270

7.2 Problems with a Limited Number of Distinct Outcomes 273

7.2.1 The Equitable Facility Location Problem 273

7.2.2 The Equitable Sensor Location Problem 279

7.2.3 Lexicographic Minimization of Counting Functions 281

7.3 Problems with a Large Number of Distinct Outcomes 290

7.3.1 Examples 291

7.3.2 Lexicographic Maximization of Performance Function Values 294

7.3.3 The Conditional Maximin Approach 301

7.3.4 The Ordered Weighted Averaging Approach 302

7.3.5 The Convex Integer Optimization Approach 305

7.4 Concluding Remarks and Literature Review 307

Exercises 311

Appendices 313

Appendix A. Summary of Models and Algorithms / 315

Appendix B. The Kuhn–Tucker Conditions / 323

Appendix C. Duality in Linear Programming / 327

References 331

Author Index 343

Subject Index 347

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