An extended covering model for flexible discrete and equity location problems

Abstract

To model flexible objectives for discrete location problems, ordered median functions can be applied. 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 the other customers. In this paper a reformulated and more compact version of a covering model for the discrete ordered median problem (DOMP) is considered. It is shown that by using this reformulation better solution times can be obtained. This is especially true for some objectives that are often employed in location theory. In addition, the covering model is extended so that ordered median functions with negative weights are feasible as well. This type of modeling weights has not been treated in the literature on the DOMP before. We show that several discrete location problems with equity objectives are particular cases of this model. As a result, a mixed-integer linear model for this type of problems is obtained for the first time

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.…”

“…The equality constraint (11) ensures that there are exactly p facilities to be located. Constraints (12) and (13) state that each client is served by one open facility.…”

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.…”

“…In the case of formulation (F2), we have compared the plain formulation with the formulation reinforced with valid inequalities (16) and (17) (from now on, (F2) 0 ). Regarding the formulations presented in Section 5 (the formulation using covering variables) we have always used in our computational experiments its improved form (F4b) and considered two settings: the plain formulation (F4b) with upper bound, and the reinforcement of (F4b) which incorporates constraints (25), that we will denote by (F4b) 0 . Previous tests discouraged us from studying other alternatives.…”

Inequity averse optimization in operational research

“…Ball et al (2009) investigate a class of models for assigning flights to slots in ground delay problems and discuss the use of Schur-convex aggregation functions as a way of obtaining equitable solutions within this setting. Marín et al (2010) use "ordered median functions" as objective functions of discrete location problems. Ordered median functions are weighted total cost functions, in which the weights are rankdependent.…”