The discrete ordered median problem: Models and solution methods.
type: Item
instance of: scholarly article
title: The discrete ordered median problem: Models and solution methods.
zbMATH Open document ID: 1042.90026
DOI: 10.1007/978-1-4419-8511-8
author: Patricia Domínguez-Marín
published in: Combinatorial Optimization
publication date: 2003-11-25T00:00:00Z
review text: In contrast to classical location theory ordered median problems deal with objective functions which are pointwise defined. For a new location the costs for satisfying the demand that is given at existing customer locations is determined. Moreover, if several new locations have to be considered only the costs of the closest location are taken into account. Then the components of the cost vector are sorted in non-decreasing order and the objective function value is calculated via the scalar product of the resulting ordered cost vector and a vector \(\Lambda\). Thereby it can be shown that by choosing appropriate \(\Lambda\) vectors all classical objective functions from location theory have an ordered median representation. In addition new objective functions, which have various applications but have not yet been treated in literature, can be formulated. Considering the decision-space, ordered median problems can, like classical location problems, be distinguished in continuous, network and discrete location problems. In this regard especially the discrete case, which is the topic of this monograph, has been more or less neglected so far. The book is divided into eight chapters and an appendix. In chapter one and two a motivation for ordered median problems is given and a first mathematical formulation for the Discrete Ordered Median Problem (DOMP) is derived. This formulation, being a combination of a feasibility problem of a generalized matching formulation and a classical \(N\)-Median problem, is nonlinear which is due to the sorting of the cost vector in the objective function. The third chapter is devoted to three linearizations of the nonlinear (DOMP) whereas, in addition to their validity, some structural results and several valid inequalities are proven. In order to compare these linearizations detailed statistical tests are reported which are based on an extensive number of test problems. Thereby, an analysis of variance and nonparametric statistical methods…
reviewed by: S. Nickel
Mathematics Subject Classification ID: 90B80
Mathematics Subject Classification ID: 90-02
Mathematics Subject Classification ID: 90C59
Mathematics Subject Classification ID: 90C10
Mathematics Subject Classification ID: 90C57
zbMATH DE Number: 2008935
zbMATH Keywords: discrete location
zbMATH Keywords: integer programming
zbMATH Keywords: heuristics
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-1-4419-8511-8
OpenAlex ID: W2477302469
description: scientific article; zbMATH DE number 2008935