On solving a large-scale problem on facility location and customer assignment with interaction costs along a time horizon

Abstract

A 0-1 quadratic programming model is presented for solving the strategic problem of timing the location of facilities and the assignment of customers to facilities in a multi-period setting. It is assumed that all parameters are known and, on the other hand, the quadratic character of the objective function is due to considering the interaction cost incurred by the joint assignment of customers belonging to different categories to a facility at a period. The plain use of a state-of-the-art MILP engine with capabilities for dealing with quadratic terms does not give any advantage over the matheuristic algorithm proposed in this work. In fact, the MILP engine was frequently running out of memory before reaching optimality for the equivalent mixed 0-1 linear formulation, being its best lower bound at that time instant too far from the incumbent solution for the large-sized instances which we have worked with. As an alternative, a fix-and-relax algorithm is presented. A deep computational comparison between MILP alternatives is performed, such that fix-and-relax provides a solution value very close to (and, frequently, a better than) the one provided by the MILP engine. The time required by fix-and-relax is very affordable, being frequently two times smaller than the time required by the MILP engine.

Appendix

The appendix presents in Tables 5 and 6 the sequence of tighter lower bounds $\underline{Z}_{x\mathrm{FR}}^\ell$ and $\underline{Z}_{y\mathrm{FR}}^\ell$ on the optimal solution of the original MILP model (31) and the related elapsed times obtained by the models $\underline{\mathrm{MILP}}_{x\mathrm{FR}}^\ell$ (35) and $\underline{\mathrm{MILP}}_{y\mathrm{FR}}^\ell$ (36), respectively, for the FR levels, $\ell =1,\ldots ,5$. It can be observed that the yFR bound for $\ell =5$ is still computationally affordable while comparing it with the time required by the plain use of CPLEX, at least.