Solving the p-median problem on regular and lattice networks

Abstract

The p-median problem is one of the true classic problems of location science and has been applied in many ways. It involves the location of p-facilities on a network where the objective is to minimize the weighted distance of serving all demand. This problem was originally proposed by Hakimi (1964, 1965) where the facilities were telephone switching centers and the connections represent wire stretched between each customer and their closest facility. It has since been viewed as the quintessential public facility location problem as it involves placing facilities as close to as possible on the average to each demand. This problem was originally formulated as an integer-programming problem by ReVelle and Swain (1970). Their formulation has withstood the test of time as most of the approaches to optimally solve the p-median problem involve a form of this model. There are several notable exceptions to the use of the classical formulation which take advantage of the underlying distance matrix defined by regular networks (Elloumi 2010; García et al., 2011), such as grid-defined networks. We demonstrate that inherent properties of the distance matrix defined for regular networks can be taken into account, resulting in a reduced, frugal form of the classic p-median model of ReVelle and Swain (1970). This new model called CARS is tested and compared to a form of the original model, recent computational experience presented by Daskin and Maass (2015) and to a form of the model used by García et al. (2011). This test demonstrates that this new, simple model is very competitive to other approaches in solving sizable p-median problems using off-the-shelf commercial software.