A flexible model and efficient solution strategies for discrete location problems

Abstract

Flexible discrete location problems are a generalization of most classical discrete locations problems like p-median or p-center problems. They can be modeled by using so-called ordered median functions. These functions multiply a weight to the cost of fulfilling the demand of a customer, which depends on the position of that cost relative to the costs of fulfilling the demand of other customers. In this paper a covering type of model for the discrete ordered median problem is presented. For the solution of this model two sets of valid inequalities, which reduces the number of binary variables tremendously, and several variable fixing strategies are identified. Based on these concepts a specialized branch & cut procedure is proposed and extensive computational results are reported

A comparative study of formulations and solution methods for the discrete ordered p-median problem

“…Further, if clients and facility locations coincide and if the allocation cost of a client to itself is equal to zero (the so-called free self service), then instances with up to 100 clients could be solved by Marín et al [10,11]. We observe that in all previously considered formulations the gaps with respect to the linear programming relaxations of those models are rather large, as mentioned in all those papers.…”

“…Given a vector of weights, this function consists in an ordered weighted average of the allocation costs, namely it sorts these costs in non-decreasing sequence and then it performs the scalar product of this so obtained sorted cost vector and the given vector of weights. This objective function has been widely applied in the field of location analysis and distribution models (Marín et al [10], Kalcsics et al [9] and Puerto et al [17,18]). In addition, it has the potential to yield new models for order statistics embedded within mathematical programming formulations; thus enlarging the applications of optimization tools to the resolution of statistical problems in data analysis.…”

When centers can fail: A close second opportunity

“…[24,25,29]. In particular, a formulation based on these variables was used in [14] for the p-center problem, providing very good lower bounds.…”

“…Additionally, even if constraints (24) are removed from the formulation, z-variables will take binary values in any optimal solution.…”

The ordered capacitated facility location problem

“…Structural results as well as a specially tailored branch and bound procedure can be found in Boland et al (2006). By using a different approach, following the principal idea of Elloumi et al (2004), an improved branch and cut algorithm is developed in Marín et al (2006Marín et al ( , 2009. This way, it is now possible to optimally solve DOMPs with more than 100 clients in reasonable time.…”