Equitable Resource Allocation: models, algorithms and applications / Edition 1

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Overview

A unique book that specifically addresses equitable resource allocation problems with applications in communication networks, manufacturing, emergency services, and more

Resource allocation problems focus on assigning limited resources in an economically beneficial way among competing activities. Solutions to such problems affect people and everyday activities with significant impact on the private and public sectors and on society at large.

Using diverse application areas as examples, Equitable Resource Allocation: Models, Algorithms, and Applications provides readers with great insight into a topic that is not widely known in the field. Starting with an overview of the topics covered, the book presents a large variety of resource allocation models with special mathematical structures and provides elegant, efficient algorithms that compute optimal solutions to these models.

Authored by one of the leading researchers in the field, Equitable Resource Allocation:

  • Is the only book that provides a comprehensive exposition of equitable resource allocation problems
  • Presents a collection of resource allocation models with applications in communication networks, transportation, content distribution, manufacturing, emergency services, and more
  • Exhibits practical algorithms for solving a variety of resource allocation models
  • Uses real-world applications and examples to explain important concepts
  • Includes end-of-chapter exercises

Bringing together much of the equitable resource allocation research from the past thirty years, this book is a valuable reference for anyone interested in solving diverse optimization problems.

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Editorial Reviews

From the Publisher
I am very pleased to see this book available. Formercomprehensive book on mathematical methods for resource allocationby Ibaraki and Katoh is excellent but it is already almost 25 yearsold. Meantime, a lot of new methods has been developed. Models andalgorithms for equitable resource allocation are likely the mostimportant advancements among them. They are extremely useful in avariety of practical application areas, but are not widely known.They had been scattered among specific research and applicationareas.

The book fills out the gap by presenting the equitabletechniques in a coherent and convenient form to readers from wideareas of engineering and operations management. It is indeed aunique book that specifically addresses equitable resourceallocation problems with applications in many areas, not restrictedto the information and communication technologies. Actually, it isan excellent book. Various models are widely motivated while thealgorithms are clearly presented in details as ready to implement.Each chapter is also accompanied by a set of interestingexercises.

I strongly recommend this book to professionals in OperationsManagement, Industrial Engineering, Computer Science andTelecommunications as well as a textbook for graduate students.

                                                                                                           - Wlodzimierz Ogryczak

I am very pleased to have this book available.  Algorithmsfor equitable resource allocation are extremely useful in a varietyof practical application areas, but are not as widely known as theyshould be among engineering and operations researchprofessionals. 

Much of the research has taken place in the last 20 years or so,and had been scattered among various journals.  It has nowbeen brought together into one coherent and convenientvolume.  Dr. Luss does an excellent job of motivating thevarious models and of describing the algorithms in a logicalstep-by-step fashion.

The set of problems that can be solved using these lexicographicmin-max algorithms is quite broad.  Initially, they weredeveloped to solve resource allocation problems in themanufacturing area.  Specifically, they addressed the questionof how to allocate electronic components to various product lines,when there was a shortage of components.  This can benaturally extended to allocating other sorts of scarce resources(e.g. manpower, computing resources, funding).

But what I find exciting is that these very same mathematicalprogramming techniques can be directly applied to problems thatseem totally unrelated.  For example, they can be used toimpute a traffic matrix for a packet communications network (suchas the network operated by an Internet Service Provider).

I wholeheartedly recommend this book to professionals –both in academia and in industry – in Operations Research,Management Science, Industrial Engineering, Telecommunications andComputerScience.                                                                                                         -John G. Klincewicz

Mathematical models and methods for optimization enable resourcesof various kinds to be used ‘as best as possible’ undergiven constraints, and have been responsible for major advances invarious fields, including control systems, operations research, andtelecommunication networks. When there are multiple and competingobjectives to be considered for optimization, the trade-off amongthe competing objectives introduces the new dimension of‘fairness’ into the optimization. In such cases, theuse of a single criterion for optimization is often inadequate andartificial. A particular form of posing multiple optimizationcriteria that captures a notion of fairness among competingobjectives gives rise to the class of problems known as‘lexicographic’ optimization, which goes beyond theusual minimax or maximin criterion to define the concept of‘equitable’ optimization. Such equitable optimizationis the subject of the book “Equitable Resource Allocation:Models, Algorithms, and Applications” by Dr. HananLuss.

The book is a clear and systematic exposition of lexicographicoptimization. After introductory chapters on single-criterionoptimization, the book discusses algorithms for the usual minimax(or maximin) criterion for dealing with multi-objective problems,and shows how algorithms for lexicographic optimization can bebuilt up from those for the minimax (or maximin) criterion. Thelater chapters consider various extensions of the basic model totake account of substitutable resources and multi-periodoptimization. The book considers theory and algorithms for bothcontinuous and discrete decision variables.

The book contains a variety of illustrative applications of theoptimization models, drawn from the author’s long anddistinguished research career at AT&T Labs andBellcore/Telcordia. The material is organized in a clear andhelpful manner among the chapters and within each chapter, and thewriting is crisp and precise. A notable feature of the book is theneat classification of the various algorithms that are presented,making it a valuable compendium of optimization models andalgorithms. The book will be a valuable text-book for an advancedcourse in optimization and a comprehensive reference for scientistsand practitioners in operations research, engineering,telecommunications, and economics.
                                                                                   -K.R.Krishnan (Bellcore/Telcordia - Retired)

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Product Details

Meet the Author

HANAN LUSS, PhD, serves as an Adjunct Professor, teaching operations research courses at Columbia University. Dr. Luss was at AT&T Bell Laboratories/AT&T Labs for twenty-five years, serving as technical manager of the Operations Research Studies Group, and at Telcordia Technologies for twelve years, serving as senior scientist. He led research activities and applied work with an emphasis on operations research methodologies for resource allocation, communication network design, capacity expansion, manufacturing, and related topics. A Fellow of the Institute for Operations Research and the Management Sciences (INFORMS), Dr. Luss has published over seventy papers in major refereed journals and books and has been granted more than ten patents.

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Table of Contents

Preface xi

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: LexicographicMinimax and Maximin Optimization 30

1.5.3 Chapter 4: Equitable Resource Allocation withSubstitutable Resources 31

1.5.4 Chapter 5: Multiperiod Equitable Resource Allocation32

1.5.5 Chapter 6: Equitable Network Resource Allocation 33

1.5.6 Chapter 7: Equitable Resource Allocation with IntegerDecisions 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 ComputerNetworks 38

1.6.3 Point-to-Point Throughput Estimation in Networks 39

1.6.4 Equitable Bandwidth Allocation for Content Distribution39

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 andMaximin 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 Resources123

4.1 Representations of Substitutable Resources 124

4.1.1 Transitive Substitutable Resources Represented by Trees124

4.1.2 Transitive Substitutable Resources Represented by AcyclicGraphs 125

4.1.3 Nontransitive Substitutable Resources Represented byBipartite Graphs 127

4.1.4 Activity-Dependent Substitutable Resources Represented byBipartite Graphs 128

4.1.5 Solution Approach 129

4.2 Transitive Substitutable Resources Represented by Trees131

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 AcyclicGraphs 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 byBipartite 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 SpecialCase 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 Path223

6.2 Multicommodity Network Flows with Multiple Paths 227

6.3 Bandwidth Allocation for Content Distribution 237

6.4 Content Distribution with Node-Dependent PerformanceFunctions 248

6.5 Concluding Remarks and Literature Review 254

Exercises 257

7 Equitable Resource Allocation with Integer Decisions259

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 OneResource 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 Values294

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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