A p-median problem with distance selection

This paper introduces an extension of the p-median problem and its application to clustering, in which the distance/dissimilarity function between units is calculated as the distance sum on the q most important variables. These variables are to be chosen from a set of m elements, so a new combinatorial feature has been added to the problem, that we call the p-median model with distance selection. This problem has its origin in cluster analysis, often applied to sociological surveys, where it is common practice for a researcher to select the q statistical variables they predict will be the most important in discriminating the statistical units before applying the clustering algorithm. Here we show how this selection can be formulated as a non-linear mixed integer optimization mode and we show how this model can be linearized in several different ways. These linearizations are compared in a computational study and the results outline that the radius formulation of the p-median is the most efficient model for solving this problem.