# Discrete and Fractional Programming Techniques for Location Models

Springer Science & Business Media, Apr 30, 1998 - Mathematics - 180 pages
At first sight discrete and fractional programming techniques appear to be two com pletely unrelated fields in operations research. We will show how techniques in both fields can be applied separately and in a combined form to particular models in location analysis. Location analysis deals with the problem of deciding where to locate facilities, con sidering the clients to be served, in such a way that a certain criterion is optimized. The term "facilities" immediately suggests factories, warehouses, schools, etc. , while the term "clients" refers to depots, retail units, students, etc. Three basic classes can be identified in location analysis: continuous location, network location and dis crete location. The differences between these fields arise from the structure of the set of possible locations for the facilities. Hence, locating facilities in the plane or in another continuous space corresponds to a continuous location model while finding optimal facility locations on the edges or vertices of a network corresponds to a net work location model. Finally, if the possible set of locations is a finite set of points we have a discrete location model. Each of these fields has been actively studied, arousing intense discussion on the advantages and disadvantages of each of them. The usual requirement that every point in the plane or on the network must be a candidate location point, is one of the mostly used arguments "against" continuous and network location models.

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### Contents

 Introduction xvii Discrete Location Models 3 22 Multilevel Uncapacitated Facility Location Problems 9 222 The 2echelon Uncapacitated Facility Location Problem 11 23 Submodularity 14 24 A General Uncapacitated Facility Depot Location Model 18 242 Formulations 20 243 Linear Relaxation 22
 321 1level Fractional Location Problems 79 322 2level Fractional Location Problems 89 33 Conclusions 93 Generalized Fractional Programming 95 41 A Primal Approach 97 411 The Parametric Approach 98 412 An Allocation Model 106 413 A Nonstandard Class of Generalized Fractional Programs 110

 244 Lagrangian Relaxation 25 245 Heuristics 41 246 Branch and Bound 43 247 Computational Results 47 25 Conclusions 56 Location Models and Fractional Programming 59 31 Fractional Programming 62 311 Continuous Fractional Programming 63 312 Integer Fractional Programming 76 32 Fractional Location Models 78
 42 A Dual Approach 114 421 Solving the Standard Dual 115 422 A New Duality Approach 135 423 Computational Results 146 43 Conclusions 152 Summary and Remarks 155 Bibliography 157 Index 169 Copyright