Revisiting a Cornuéjols-Nemhauser-Wolsey formulation for the p-median problem

We revisit a formulation for the simple plant facility location and p-median problems introduced by Cornuéjols, Nemhauser and Wolsey (1980). Despite being the smallest known formulation regarding the number of variables, this formulation is barely used or cited in the literature. Here, we reintroduce the formulation for the p-median problem from a different perspective, resulting from the intersection of a selection problem with an additional family of optimality constraints to define the costs correctly. An alternative proof that the linear relaxation of the formulation is equivalent to the linear relaxation of the well-known classical formulation is provided. By exploring the optimality constraints we discuss approaches to derive bounds for large-size instances. These approaches are based on relaxations obtained by eliminating optimality constraints and can be seen as a simple matheuristic to solve large size instances. In particular, we characterize relaxations which provide the optimal solution, and therefore, can be seen as new formulations for the p-median problem. Computational tests are reported showing that the renewed formulation can be used efficiently to solve p-median instances.