Network and Discrete Location

MARK S. DASKIN, PhD, is Clyde W. Johnson Collegiate Professor of Industrial and Operations Engineering as well as Department Chair at the University of Michigan. Dr. Daskin is the former editor-in-chief of both Transportation Science and IIE Transactions and continues to serve on the editorial boards of both journals. He is also on the editorial board of the IIE Transactions on Healthcare Systems Engineering and is the author of the award-winning book Service Science, also published by Wiley.

Preface to the First and Second Editions xi

Acknowledgments xvii

1. Introduction to Location Theory and Models 1

1.1 Introduction 1

1.2 Key Questions Addressed by Location Models 3

1.3 Example Problem Descriptions 4

1.3.1 Ambulance Location 4

1.3.2 Siting Landfills for Hazardous Wastes 10

1.3.3 Summary 10

1.4 Key Dimensions of Location Problems and Models 11

1.4.1 Planar Versus Network Versus Discrete Location Models 11

1.4.2 Tree Problems Versus General Graph Problems 12

1.4.3 Distance Metrics 13

1.4.4 Number of Facilities to Locate 14

1.4.5 Static Versus Dynamic Location Problems 15

1.4.6 Deterministic Versus Probabilistic Models 16

1.4.7 Single- Versus Multiple-Product Models 16

1.4.8 Private Versus Public Sector Problems 17

1.4.9 Single- Versus Multiple-Objective Problems and Models 17

1.4.10 Elastic Versus Inelastic Demand 18

1.4.11 Capacitated Versus Uncapacitated Facilities 18

1.4.12 Nearest Facility Versus General Demand Allocation Models 18

1.4.13 Hierarchical Versus Single-Level Models 19

1.4.14 Desirable Versus Undesirable Facilities 19

1.5 A Taxonomy of Location Models 20

1.5.1 Typology of Location Models 20

1.5.2 A Simple Analytic Model 22

1.6 Summary 26

Exercises 27

2. Review of Linear Programming 29

2.1 Introduction 29

2.2 The Canonical Form of a Linear Programming Problem 31

2.3 Constructing the Dual of an LP Problem 34

2.4 Complementary Slackness and the Relationships Between the Primal and the Dual Linear Programming Problems 36

2.5 Solving a Linear Programming Problem in Excel 43

2.6 The Transportation Problem 47

2.7 The Shortest Path Problem 64

2.7.1 The Shortest Path Problem in Excel 78

2.7.2 The Shortest Path Problem in AMPL 80

2.8 The Out-of-Kilter Flow Algorithm 80

2.9 Integer Programming Problems 92

2.10 Summary 96

Exercises 97

3. An Overview of Complexity Analysis 111

3.1 Introduction 111

3.2 Basic Concepts and Notation 112

3.3 Example Computation of an Algorithm’s Complexity 115

3.4 The Classes P and NP (and NP-Hard and NP-Complete) 117

3.5 Summary 122

Exercises 123

4. Covering Problems 124

4.1 Introduction and the Notion of Coverage 124

4.2 The Set Covering Model 125

4.3 Applications of the Set Covering Model 137

4.4 Variants of the Set Covering Location Model 140

4.5 The Maximum Covering Location Model 143

4.5.1 The Greedy Adding Algorithm: A Heuristic Algorithm for Solving the Maximum Covering Location Model 146

4.5.2 Lagrangian Relaxation: An Optimization-Based Heuristic Algorithm for Solving the Maximum Covering Location Model 154

4.5.3 Other Solution Approaches and Example Results 163

4.6 An Interesting Model Property or It Ain’t Necessarily So 164

4.7 The Maximum Expected Covering Location Model 168

4.8 Summary 174

Exercises 175

5. Center Problems 193

5.1 Introduction 193

5.2 Vertex P-Center Formulation 198

5.3 The Absolute 1- and 2-Center Problems on a Tree 201

5.3.1 Absolute 1-Center on an Unweighted Tree 201

5.3.2 Absolute 2-Centers on an Unweighted Tree 205

5.3.3 Absolute 1-Center on a Weighted Tree 206

5.4 The Unweighted Vertex P-Center Problem on a General Graph 211

5.5 The Unweighted Absolute P-Center Problem on a General Graph 215

5.5.1 Characteristics of the Solution to the Absolute P-Center Problem 215

5.5.2 An Algorithm for the Unweighted Absolute P-Center on a General Graph 219

5.6 Summary 229

Exercises 230

6. Median Problems 235

6.1 Introduction 235

6.2 Formulation and Properties 237

6.3 1-Median Problem on a Tree 241

6.4 Heuristic Algorithms for the P-Median Problem 246

6.5 An Optimization-Based Lagrangian Algorithm for the P-Median Problem 260

6.5.1 Methodological Development 260

6.5.2 Numerical Example 265

6.5.3 Extensions and Enhancements to the Lagrangian Procedures 271

6.6 Computational Results Using the Heuristic Algorithms and the Lagrangian Relaxation Algorithm 271

6.7 Another Interesting Property or It Still Ain’t Necessarily So 277

6.8 Summary 283

Exercises 285

7. Fixed Charge Facility Location Problems 294

7.1 Introduction 294

7.2 Uncapacitated Fixed Charge Facility Location Problems 297

7.2.1 Heuristic Construction Algorithms 298

7.2.2 Heuristic Improvement Algorithms 305

7.2.3 A Lagrangian Relaxation Approach 311

7.2.4 A Dual-Based Approach 314

7.3 Capacitated Fixed Charge Facility Location Problems 325

7.3.1 Lagrangian Relaxation Approaches 328

7.3.2 Bender’s Decomposition 345

7.4 Summary 355

Exercises 356

8. Extensions of Location Models 362

8.1 Introduction 362

8.2 Multiobjective Problems 362

8.3 Hierarchical Facility Location Models 375

8.3.1 Basic Notions of Hierarchical Facilities 375

8.3.2 Basic Median-Based Hierarchical Location Formulations 379

8.3.3 Coverage-Based Hierarchical Location Formulations 383

8.3.4 Extensions of Hierarchical Location Formulations 385

8.4 Models of Interacting Facilities 387

8.4.1 Flows Between Facilities 387

8.4.2 Facilities with Proximity Constraints 390

8.5 Multiproduct Flows and Production/Distribution Systems 393

8.6 Location/Routing Problems 399

8.7 Hub Location Problems 410

8.8 Dispersion Models and Models for the Location of Undesirable Facilities 425

8.8.1 Dispersion Models 426

8.8.2 A Maxisum Model for the Location of Undesirable Facilities 429

8.9 An Integrated Location-Inventory Model 435

8.9.1 A Multiobjective Location-Inventory/Covering Model 448

8.9.2 A Look at Aggregation Effects 452

8.10 Reliability and Facility Location Modeling 455

8.10.1 The Expected Failure Case 458

8.10.2 Modeling a Malevolent Attacker 461

8.11 Summary 466

Exercises 468

9. Location Modeling in Perspective 480

9.1 Introduction 480

9.2 The Planning Process for Facility Location 481

9.2.1 Problem Definition 481

9.2.2 Analysis 483

9.2.3 Communication and Decision 489

9.2.4 Implementation 495

9.2.5 Caveats on the Planning Process 496

9.3 Summary 496

Exercises 497

References 499

Index 509