A flexible approach to location problems

Abstract

In continuous location problems we are given a set of existing facilities and we are looking for the location of one or several new facilities. In the classical approaches weights are assigned to existing facilities expressing the importance of the new facilities for the existing ones. In this paper, we consider a pointwise de ned objective function where the weights are assigned to the existing facilities depending on the location of the new facility. This approach is shown to be a generalization of the median, center and centdian objective functions. In addition, this approach allows to formulate completely new location models. E cient algorithms as well as structural results for this algebraic approach for location problems are presented. A complexity analysis and extensions to the multifacility and restricted case are also considered.

Single facility location problems with unbounded unit balls

“…Notice that similar LP-formulations have been obtained for other location problems with polyhedral gauges of compact convex sets. The interested reader is referred to Rodriguez-Chia et al [18] and the references therein.…”

“…In the last years, there are also a number of papers that consider the use of gauges defined by the Minkowski functional of a compact convex set (not necessarily symmetric) containing the origin in its interior (see e.g. Durier and Michelot [4]; Fliege [7]; or RodriguezChia et al [18]). These functions have been used in Location Theory to model situations where the symmetry property of a norm does not make sense.…”

Ordered median problem with demand distribution weights

“…It originates from early models [19,29] of compensatory extensions of the lexicographic center approach, thus representing weighted sum of the ordered outcomes (distances). The ordered median location problems (OMP) were formulated for locations on networks [17], on the plane [30] and for general discrete location problems [16]. Some special classes of the ordered solution concepts such as k-centrum and conditional median were independently developed for location problems [27,31].…”

Multifacility ordered median problems on networks: A further analysis

“…The starting point of this development may be considered the node-dominance result of Hakimi [7] and the extensions by Hooker et al [8], which we will show to be essential. In a series of previous papers, a new type of objective function in location theory was introduced and analyzed (see Nickel and Puerto [15], Puerto and Fernández [18], Rodríguez-Chía et al [20], Francis et al [6], Nickel [14], and Kalcsics et al [10]). In this paper, we will analyze the p-facility version of this new type of objective function called ordered median function which is a generalization of the most popular objective functions: median, center, centdian, k-centrum, among many others.…”

“…Note that this function is defined pointwise. In the context of continuous location theory, a similar objective function was introduced in Puerto and Fernández [18] and later studied in Francis et al [6], Rodríguez-Chía et al [20], and Puerto and Fernández [19].…”