Antonio M. Rodríguez-Chía

A flexible approach to location problems

Abstract. When dealing with location problems we are usually 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 defined 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 the formulation of completely new location models. Efficient algorithms as well as structural results for this algebraic approach to location problems are presented. A complexity analysis and extensions to the multifacility and restricted case are also considered.

An Approach to Location Models Involving Sets as Existing Facilities

In this paper, we deal with single facility location problems in a general normed space in which the existing facilities are represented by convex sets of points. The criterion to be satisfied by the service facility is the minimization of an increasing, convex function of the distances from the service facility to the closest point of each demand set. We obtain a geometrical characterization of the set of optimal solutions for this problem. Two remarkable cases--the classical Weber problem and the minimax problem with demand sets--are studied as particular instances of our problem. Finally, for the planar polyhedral case, we give an algorithm to find the solution set of the considered problems.

Closest assignment constraints in discrete location problems

The objective of this paper is to identify the most promising sets of closest assignment constraints in the literature of Discrete Location Theory, helping the authors in the field to model their problems when clients must be assigned to the closest plant inside an Integer Programming formulation. In particular, constraints leading to weak Linear Programming relaxations should be avoided if no other good property supports their use. We also propose a new set of constraints with good theoretical properties.

Weber problems with alternative transportation systems

Abstract In this paper we address a planar p -facility location problem where, together with a metric induced by a gauge, there exists a series of rapid transit lines, which can be used as alternative transportation system to reduce the total transportation cost. The location problem is reduced to solving a finite number of (multi)-Weber problems, from which localization results are obtained. In particular, it is shown that, if the gauge in use is polyhedral, then the problem is reduced to finding a p -median.

Conditional location of path and tree shaped facilities on trees

In this paper we deal with the location of extensive facilities on trees, both discrete and continuous, under the condition that existing facilities are already located. We require that the selected new server is a subtree, although we also specialize to the case of paths. We study the problem with the two most widely used criteria in Location Analysis: center and median. Our main results under the center criterion are nestedness properties of the solution and subquadratic algorithms for the location of paths and subtrees. For the case of the median criterion we prove that unlike the case where there is no existing facility, the continuous conditional median subtree problem is NP-hard and we develop a corresponding fully polynomial approximation algorithm. We also present subquadratic algorithms for almost all other models.

Minimax Regret Single-Facility Ordered Median Location Problems on Networks

We consider the single-facility ordered median location problem with uncertainty in the parameters (weights) defining the objective function. We study two cases. In the first case, the uncertain weights belong to a region with a finite number of extreme points, and in the second case, they must satisfy some order constraints and belong to some box (convex case). To deal with the uncertainty, we apply the minimax regret approach, providing strongly polynomial time algorithms to solve these problems. Finally, we also extend the proposed methodology to other problems with order constraints, which are not necessarily convex.

A comparison of formulations and solution methods for the minimum-envy location problem

We consider a discrete facility location problem with a new form of equity criterion. The model discussed in the paper analyzes the case where demand points only have strict preference order on the sites where the plants can be located. The goal is to find the location of the facilities minimizing the total envy felt by the entire set of demand points. We define this new total envy criterion and provide several integer linear programming formulations that reflect and model this approach. The formulations are illustrated by examples. Extensive computational tests are reported, showing the potentials and limits of each formulation on several types of instances. Finally, some improvements for all the formulations previously presented are developed, obtaining in some cases much better resolution times.

MCDM Location Problems

In this chapter, we provide a broad overview of the most representative multicriteria location problems as well as of the most relevant achievements in this field, indicating the relationship between them whenever possible. We consider a large number of references which have been classified in three sections depending on the type of decision space where the analyzed models are stated. Therefore, we distinguish between continuous, network, and discrete multicriteria location problems.

On the structure of the solution set for the single facility location problem with average distances

This paper analyzes continuous single facility location problems where the demand is randomly defined by a given probability distribution. For these types of problems that deal with the minimization of average distances, we obtain geometrical characterizations of the entire set of optimal solutions. For the important case of total polyhedrality on the plane we derive efficient algorithms with polynomially bounded complexity. We also develop a discretization scheme that provides