Computers & Operations Research 36 (2009) 2250--2262 Contents lists available at ScienceDirect Computers & Operations Research journal homepage: www.elsevier.com/locate/cor Convex ordered median problem with + p -norms I. Espejo, A.M. Rodríguez-Chía ∗ , C. Valero Department of Statistics and Operations Research, University of Cádiz, Spain ARTICLE INFO ABSTRACT Available online 7 September 2008 Keywords: Continuous location problem Ordered median problem Weizfeld algorithm +p-norms This paper presents a procedure to solve the convex ordered median problem where the distances are measured with + p -norms. In order to do that, we consider an approximated problem and develop an algorithm based on a gradient descent method that generates a sequence with decreasing objective value. We prove its convergence to the optimal solution of the approximated problem. The paper ends with some computational results of the proposed methodology. © 2008 Elsevier Ltd. All rights reserved. 1. Introduction Location analysis is one of the most active fields in Operations Research; in fact, many different models have been developed in the last decades to deal with different real world situations. No- tice that a very important aspect of a location model is the correct choice of the objective function and in most classical location mod- els the objective function is the main differentiator. The median ob- jective is to minimize the sum of the weighted distances from the clients to the server. The center objective is to minimize the max- imum weighted distance from a client to the server. The cent-dian objective is a convex combination of the median and center objec- tives; it aims to keep both the average cost behavior as well as the highest cost in balance. Despite the fact that all three objectives (and some more) are frequently encountered in the literature (see for ex- ample [1]), not much has been done in the direction of a unified framework for handling all of these objectives. The increasing need for location models to better fit different real situations, has made it necessary to develop new and flexible lo- cation models. To that end, [2] introduced a new type of objective function that generalizes the most popular objective functions men- tioned above. This objective function, called ordered median func- tion, applies a penalty to the weighted distance from a client to the server, which is dependent on the position of that weighted distance in the vector of all weighted distances from the clients to the server. For example, a different penalty might be applied to a client if the weighted distance to the server is in the 5th-position rather than in the 2nd-position. It is even possible to neglect some customers by assigning a zero penalty. This adds a “sorting”-problem to the ∗ Corresponding author. Tel.: +34 9560 16087; fax: +34 9560 16050. E-mail address: antonio.rodriguezchia@uca.es (A.M. Rodríguez-Chía). 0305-0548/$ - see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.cor.2008.08.019 underlying facility location problem, making formulation and solu- tion much more challenging. In the last years, these flexible objective functions have attracted the attention of researchers. Puerto and Fernández [2,3] studied char- acterizations of the solution set for the general formulations. For the planar case with polyhedral gauges, [4] develops a polynomial time algorithm and [5] applies these models to semiobnoxious location problems. In network location problems, efforts have been devoted to obtaining finite dominating sets and efficient algorithms to solve this kind of problems [6–10]. Recently, the discrete versions of these models have also been studied in [11–14]. Research in this area also led to a recent monograph; see [15]. However, in continuous location theory, these models have only dealt with smooth norms in [16] for the Euclidean case. In this pa- per, we will consider these formulations when the distances are measured with + p -norms. Notice that the measurement of the dis- tances with + p -norms better fits to some real world situations (see [17,18]). In particular, we will restrict ourselves to convex ordered median problems and p ∈ [1, 2], similarly to other published studies for the median problem, e.g. [19–24]. (Although we can find some papers in the literature dealing with the median problem for p> 2, the solution procedure developed in those papers does not guaran- tee a global convergence to an optimal solution; see [25,26].) For this type of problems, we will develop an iterative procedure based on a modified gradient descent method. Observe that this methodology is complex because the objective function does not have a common expression as sum of the weighted distances from the clients to the server; in fact, it is pointwise defined. On the other hand, this pro- cedure is very robust because we provide a common tool to solve different classical models, for instance, median problems, center problems, cent-dian problems, among others and new ones that can be modelled under this formulation. Moreover, we are also providing a method to solve well-known models for which currently no reso- lution method has been published, such as the k-centrum problem.