A comparative study of different formulations for the capacitated discrete ordered median problem

Highlights

• Different formulations for the capacitated discrete ordered median problem.

• Development of preprocessing phases for fixing variables.

• Valid inequalities have been proposed for strengthening the formulations.

• The so called block formulations result very competitive.

Abstract

This paper deals with the capacitated version of discrete ordered median problems. We present different formulations considering three-index variables or covering variables to address the order requirements in this problem. Some preprocessing phases for fixing variables and some valid inequalities are developed to enhance the initial formulations. Finally, an extensive computational analysis is addressed with data taken from the OR-library and AP-library showing the efficiency of the formulations and the improvements presented in the paper.

Introduction

A very important aspect of most optimization problems is the correct choice of their objective functions. If the considered problem models an actual application, the objective function plays a crucial role because it has to describe the utility (disutility) to be optimized in order to achieve a reasonable outcome once a solution is found. Therefore, modelling flexible families of objective functions that are applicable in contemporary Operations Research, and more specifically in Location Analysis, is a fruitful area of research. Apart from the choice of the optimization criterion, another crucial aspect in the literature on facility location, is the assumption of capacity constraints. This assumption implies more realistic models at the price of increasing its complexity with respect to their uncapacitated counterparts. In many cases new formulations are needed and a more specialized analysis is often required to solve even smaller sizes than those previously addressed for the uncapacitated versions of the same problems. This paper tries to contribute to this area of research providing new insights, models and algorithms to solve the so called capacitated discrete ordered median problem (CDOMP), see Nickel and Puerto (2005).

The Discrete Ordered Median Problem (DOMP) is a flexible model that provides a common framework to cast most popular problems in discrete location analysis. These models share in common a similar multiparametric objective function called ordered median function that was introduced for the first time in Nickel and Puerto, 1999, Puerto and Fernández, 2000. An ordered median function is an ordered weighted average of the arguments of that function. Therefore, when it is applied to a location problem it results in the ordered weighted average of the distances or allocation costs from clients to service facilities. Given a set of clients, a set of candidate locations and assuming that the allocation costs of clients to facilities are known, DOMP consists in choosing p facility locations and assigning each client to a chosen facility with the smallest allocation cost in order to minimize the ordered weighted average of these costs. The ordered weighted average sorts the allocation costs in a non-decreasing sequence and then it performs the scalar product of this so-obtained sorted cost vector with a given vector of weights, $\lambda$-vector.

This idea of sorting elements and exploiting radii is in the basis of current state of the art formulation in location theory. Specifically, state of the art models for the p-median are those by Elloumi (2010) and García et al. (2011) that solve the largest instances in the literature. These models are specifications of the general idea of radius formulations that is the basis of ordered median formulations. Actually, these formulations can be seen as particular instances of the latter whenever all lambdas are equal to one. The same happens for the p-center problem where the best formulations, namely those by Elloumi et al. (2004) and Calik and Tansel (2013) are also based on radius formulations that appear from ordered median ones with particular choices of lambdas. Therefore, the use of the rationale behind ordered median models has given rise to improve the efficiency of model solving for the p-median and p-center models and beyond, it has allowed to consider more sophisticated objective functions not considered before. More examples of similar behavior on alternative models can be found, for instance, in Albareda-Sambola et al., 2019, Benati and García, 2014, Benati et al., 2018, Ponce et al., 2018, among others.

Ordered median location problems were first introduced in the 90’s to be applied to problems in networks (Nickel and Puerto, 1999) and continuous spaces (Puerto and Fernández, 2000), and later, they were extended to the discrete setting by Nickel, 2001, Boland et al., 2006. Since those early days, these problems have attracted a number of researchers and one can find many papers devoted to the study of different aspects of DOMP. This interest has been not only theoretical but also by its applications. In this regard, DOMP has been applied to discrete facility location in Boland et al., 2006, Marín et al., 2009, Marín et al., 2010, Nickel, 2001, Puerto, 2008, Puerto et al., 2009; to location on networks in Nickel and Puerto, 1999, Puerto and Rodríguez-Chía, 2005; to network design problems in Puerto et al., 2011, Puerto et al., 2013, Puerto et al., 2016; to combinatorial optimization problems with ordering in Fernández et al., 2013, Fernández et al., 2014, Fernández et al., 2017, to determine values in cooperative game theory in Perea and Puerto (2013); and to voting problems in Ponce et al. (2018), to mention a few. The reader is referred to the recent monography by Puerto and Rodríguez-Chía (2015) for some other applications.

From a more theoretical point of view, there are several valid formulations for DOMP that exploit specific features of the problem, as for instance free self-service, ties in the matrix of costs or null elements in the vector of weights (see e.g. Boland et al., 2006, Deleplanque et al., 2020, Marín et al., 2009, Marín et al., 2010, Puerto, 2020, Puerto et al., 2013, Labbé et al., 2017 and the references therein). For general cost coefficients (with no ties), all these formulations have a very large (cubic) number of binary variables and therefore, instances with more than 100 clients are very hard to solve.

Turning to the capacitated versions of these problems one can find remarkably less number of references Kalcsics et al., 2010a, Kalcsics et al., 2010b, Puerto, 2008, Puerto et al., 2016. To the best of our knowledge, the first paper dealing with the capacitated discrete ordered median problem is Puerto (2008). It considered a formulation based on a coverage approach and compared its performance with respect to previously known formulations for the uncapacitated problem. However, the methods in that paper were able to solve only small instances. The paper (Kalcsics et al., 2010b) describes three different points of view of a location problem in a logistics system. These models are extensions of the basic DOMP but it is assumed that demands are divisible and thus, they can be split. Kalcsics et al. (2010a) can be considered as the first attempt to deal with a capacitated strategic location problems with order requirements where the number of facilities to be located is not given in advance and the demands are also divisible. This is an important difference with respect to the CDOMP, where the number of new facilities is fixed a priori and the demand cannot be split. Finally, Puerto et al. (2016) deals with a hub network design problems including capacities.

From the above literature review, we realize that there is an important gap to be filled concerning the analysis of the standard DOMP including capacities. This is exactly the goal of this paper: to analyze the CDOMP. In our approach, we try to derive structural properties and formulations for solving the CDOMP. Some of these formulations will be adapted from previously known ones for the uncapacitated versions of the problem. Nevertheless, some others are new and have been obtained exploiting the intrinsic characteristics of the considered model.

The rest of the paper is organized as follows. In Section 2 different formulations for the CDOMP are presented. Section 3 is devoted to test the previous formulations on data taken from the benchmark instances of different libraries. We try to improve the most promising formulations using different strategies which are computationally compared in Section 4. We close the paper with some conclusions.