On location-allocation problems for dimensional facilities

TL;DR

This paper introduces a bilevel model for location-allocation problems involving complex-shaped dimensional facilities, providing theoretical existence results, discretization methods, and heuristics for large instances.

Contribution

It presents a general model for dimensional facilities with arbitrary shapes, proves existence of solutions, and offers practical algorithms including discretization and heuristics.

Findings

Explicit solution structure derived using optimal transport theory

Discretization approach approximates solutions with arbitrary accuracy

GRASP heuristic performs well on large instances

Abstract

This paper deals with a bilevel approach of the location-allocation problem with dimensional facilities. We present a general model that allows us to consider very general shapes of domains for the dimensional facilities and we prove the existence of optimal solutions under mild, natural assumptions. To achieve these results we borrow tools from optimal transport mass theory that allow us to give explicit solution structure of the considered lower level problem. We also provide a discretization approach that can approximate, up to any degree of accuracy, the optimal solution of the original problem. This discrete approximation can be optimally solved via a mixed-integer linear program. To address very large instance sizes we also provide a GRASP heuristic that performs rather well according to our experimental results. The paper also reports some experiments run on test data.