ISBN: 3540243216
TITLE: Location Theory
AUTHOR: Nickel/Puerto
TOC: 

Part I Location Theory and the Ordered Median Function
1 Mathematical Properties of the Ordered Median Function. 3
1.1 Introduction 3
1.2 Motivating Example 4
1.3 The Ordered Median Function 6
1.4 Towards Location Problems 15
Part II The Continuous Ordered Median Location Problem
2 The Continuous Ordered Median Problem 23
2.1 Problem Statement 23
2.2 Distance Functions 27
2.2.1 The Planar Case 31
2.3 Ordered Regions, Elementary Convex Sets and Bisectors 34
2.3.1 Ordered Regions 34
2.3.2 Ordered Elementary Convex Sets 35
2.3.3 Bisectors 39
3 Bisectors 43
3.1 Bisectors - the Classical Case 43
3.2 Possible Generalizations 44
Bisectors- the General Case 46
3.3.1 Negative Results 47
3.3.2 Structural Properties 48
3.3.3 Partitioning of Bisectors 59
3.4 Bisectors of Polyhedral Ganges 65
3.5 Bisectors of Elliptic Ganges 74
3.6 Bisectors of a Polyhedral Gauge and an Elliptic Gauge 83
3.7 Approximation of Bisectors 96
4 The Single Facility Ordered Median Problem 105
4.1 Solving the Single Facility OMP by Linear Programming, 105
4.2 Solving the Planar Ordered Median Problem Geometrically 110
4.3 Non Polyhedral Case 117.
4.4 Continuous OMPs with Positive and Negative Lambdas 123
4.4.1 The Algorithms 127
4.5 Finding the Ordered Median in the Rectilinear Space 134
5 Multicriteria Ordered Median Problems 137
5.1 Introduction 137
5.2 Multicriteria Problems and Level Sets 138
5.3 Bicriteria Ordered Median Problems 139
5.4 The 3-Criteria Case 151
5.5 The Case Q > 3 158
5.6 Concluding Remarks 162
6 Extensions of the Continuous Ordered Median Problem 165
6.1 Extensions of the Single Facility Ordered Median Problem 165,
6.1.1 Restricted Case 165
6.2 Extension to the Multifacility Case 169
6.2.1 The Non-Interchangeable Multifacility Model 170
6.2.2 The Indistinguishable Multifacility Model 172
6.3 The Single Facility OMP in Abstract Spaces 174
6.3.1 Preliminary Results 176
6.3.2 Analysis of the Optimal Solution Set 185
6.3.3 The Convex OMP and the Single Facility Location Problem in Normed Spaces, 189
6.4 Concluding Remarks 193
Part III Ordered Median Location Problems an Networks
7 The Ordered Median Problem an Networks 197
7.1 Problem Statement 197
7.2 Preliminary Results 200
7.3 General Properties 203
8 On Finite Dominating Sets for the Ordered Median Problem 209
8.1 Introduction 209
8.2 FDS for the Single Facility Ordered Median Problem 210
8.3 Polynomial Size FDS for the Multifacility Ordered Median Problem 214
8.3.1 An FDS for the Multifacility Ordered Median Problem when a = lambda_l=... = lambda_k ?lambda_k+1 = lambda_M 215
8.3.2 An FDS for the Ordered 2-Median Problem with General Nonnegative lambda-Weights  225
8.4 On the Exponential Cardinality of FDS for the Multifacility Facility Ordered Median Problem 236
8.4.1 Some Technical Results 239
8.4.2 Main Results 245
9 The Single Facility Ordered Median Problem an Networks 249
9.1 The Single Facility OMP an Networks: Illustrative Examples 250
9.2 The k-Centrum Single Facility Location Problem 254
9.3 The General Single Facility Ordered Median Problem an Networks 266
9.3.1 Finding the Single Facility Ordered Median of a General Network 267
9.3.2 Finding the Single Facility Ordered Median of a Tree 269
10 The Multifacility Ordered Median Problem an Networks 275
10.1 The Multifacility k-Centrum Problem 275
10.2 The Ordered p-Median Problem with lambda^s-Vector
lambda^s = (a,^M-s ...,a,b,^s...,b) 281
10.3 A Polynomial Algorithm for the Ordered p-Median Problem
an Tree Networks with lambda^s-Vector, lambda^s = (a, b,. 9 b) 283
11 Multicriteria Ordered Median Problems an Networks 289
11.1 Introduction 289
11.2 Examples and Remarks 291
11.3 The Algorithm 293
11.4 Point Comparison 295
11.5 Segment Comparison 296
11.6 Computing the Set of Efficient Points Using Linear
Programming 307
12 Extensions of the Ordered Median Problem an Networks 311
12.1 Notation and Model Definitions 312
12.2 Tactical Subtree with Convex Ordered Median Objective 314
12.2.1 Finding an Optimal Tactical Subedge 314
12.2.2 Finding an Optimal Tactical Continuous Subtree Containing a Given Node 315
12.3 Strategic Subtree with Convex Ordered Median Objective 317
12.3.1 Finding an Optimal Strategic Subedge 318
12.3.2 Finding an Optimal Strategic Continuous Subtree Containing a Given Node 318
12.3.3 Submodularity of Convex Ordered Median Functions 318
12.4 The Special Case of the Subtree k-Centrum Problem 320
12.4.1 Nestedness Property for the Strategic and Tactical Discrete k-Centrum Problems 321
12.4.2 A Dynamic Programming Algorithm for the Strategic Discrete Subtree k-Centrum Problem 322
12.5 Solving the Strategic Continuous Subtree k-Centrum Problem 325
12.6 Concluding Remarks 327
Part IV The Discrete Ordered Median Location Problem
13 Introduction and Problem Statement 331
13.1 Definition of the Problem 332
13.2 A Quadratic Formulation for the Discrete OMP 335
13.2.1 Sorting as an Integer Linear Program (ILP) 336
13.2.2 Formulation of the Location-Allocation Subproblem 337
13.2.3 Quadratic Integer Programming Formulation for OMP 339
14 Linearizations and Reformulations 341
14.1 Linearizations of (OMP) 341
14.1.1 A First Linearization: (OMP^1) 341
14.1.2 A Linearization Using Less Variables: (OMP^2) 346 14.1.3 Simplifying Further: (OMP^3) 349
14.1.4 Comparison Between (OMP^2*) and (OMP3^*) 352
14.2 Reformulations 354
14.2.1 Improvements for (OMP^1) 356
14.2.2 Improvements for (OMP^3) 362
14.2.3 Improvements for (OMP^2) 368
14.2.4 Comparison Between (OMP^2*) and (OMP^3*) 371
14.3 Computational Results 372
15 Solution Methods 381
15.1 A Branch and Bound Method 381
15.1.1 Combinatorial Lower Bounds 382
15.1.2 Branching 387
15.1.3 Numerical Comparison of the Branching Rules 389
15.1.4 Computational Results 390
15.2 Two Heuristic Approaches for the OMP 393
15.2.1 An Evolution Program for the OMP 393
15.2.2 A Variable Neighbourhood Search for the OMP 399
15.2.3 Computational Results 407
16 Related Problems and Outlook 419
16.1 The Discrete OMP with lambda Element logisches und kleinergleich M, 419 16.1.1 Problem Formulation 419
16.1.2 Computational Results 421
16.2 Conclusions and Further Research 422
References 423
Index 435
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