Geometrical properties of the Fermat-Weber problem

Durier, Roland; Michelot, Christian

doi:10.1016/0377-2217(85)90006-2

Cited by 133 publications

(71 citation statements)

References 8 publications

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“…Traditionally the Weber problem [10,11] has been addressed in its point version. However, since the early seventies one can find the régional approach in different papers.…”

Section: Introductionmentioning

confidence: 99%

The generalized Weber problem with expected distances

Carrizosa¹,

Conde²,

Munõz-Márquez³

et al. 1995

RAIRO-Oper. Res.

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“…Traditionally the Weber problem [10,11] has been addressed in its point version. However, since the early seventies one can find the régional approach in different papers.…”

Section: Introductionmentioning

confidence: 99%

The generalized Weber problem with expected distances

Carrizosa¹,

Conde²,

Munõz-Márquez³

et al. 1995

RAIRO-Oper. Res.

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“…The classical location theory is dedicated to the problem of locating a new facility such that all relevant distances, e.g. to customers, are minimized [7,8,11,13,19,35,42]. Nevertheless, since some facilities also cause negative effects like noise, stench or even danger, such facilities need to be established as far away as possible from nature reserves and residential areas.…”

Section: Application and Numerical Resultsmentioning

confidence: 99%

Solving DC programs with a polyhedral component utilizing a multiple objective linear programming solver

Löhne¹,

Wagner²

2017

J Glob Optim

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A class of non-convex optimization problems with DC objective function is studied, where DC stands for being representable as the difference f = g − h of two convex functions g and h. In particular, we deal with the special case where one of the two convex functions g or h is polyhedral. In case g is polyhedral, we show that a solution of the DC program can be obtained from a solution of an associated polyhedral projection problem. In case h is polyhedral, we prove that a solution of the DC program can be obtained by solving a polyhedral projection problem and finitely many convex programs. Since polyhedral projection is equivalent to multiple objective linear programming (MOLP), a MOLP solver (in the second case together with a convex programming solver) can be used to solve instances of DC programs with polyhedral component. Numerical examples are provided, among them an application to locational analysis.

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“…In doing that, we recall the concept of an elementary convex set (see Durier and Michelot [4]). Through this section we will assume that F Ã is different from 0 and þ1, otherwise it will be stated explicitly.…”

Section: Geometrical Description Of Optimal Setsmentioning

confidence: 99%

“…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.…”

mentioning

confidence: 99%

Single facility location problems with unbounded unit balls

Hinojosa

Puerto

2003

Mathematical Methods of Operations Research (ZOR)

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Abstract. In this paper we consider a new class of continuous location problems where the ''distances'' are measured by gauges of closed (not necessarily bounded) convex sets. These distance functions do not satisfy the definiteness property and therefore they can be used to model those situations where there exist zero-distance regions. We prove a geometrical characterization of these measures of distance as the length of shortest paths between points using only a subset of directions of their unit balls. We also characterize the complete set of optimal solutions for this class of continuous single facility location problems and we give resolution methods to solve them. Our analysis allows to consider new models of location problems and generalizes previously known results.

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Geometrical properties of the Fermat-Weber problem

Cited by 133 publications

References 8 publications

The generalized Weber problem with expected distances

The generalized Weber problem with expected distances

Solving DC programs with a polyhedral component utilizing a multiple objective linear programming solver

Single facility location problems with unbounded unit balls

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