The discrete p-Maxian location problem

Given are a set of fixed points and another set of candidate points in some space, and a distance metric. We study the problem of selecting a subset of the candidate points such that the sum of the distances between pairs of fixed and selected points plus the sum of the distances between pairs of selected points is maximized. This model can be used to locate obnoxious new facilities, in the presence of existing facilities. We give a mathematical programming model for this problem and devise a basic implicit enumeration scheme to find an optimal solution. We can solve problems with up to 50 candidate points and 10 selected points on a microcomputer using this procedure. We also describe a heuristic, that has found an optimal solution for 196 of our 200 random test problems.