Multicommodity flows and Benders decomposition for restricted continuous location problems

The restricted continuous facility location problem arises when there is a need to locate a number of facilities to serve a discrete set of demand points, and where the location of a facility can be anywhere on the plane except for in restricted regions. The problem finds applications in urban planning, disaster management, and healthcare logistics. The restricted regions can occur randomly or are known in advance. The paper describes a new model for the problem that is based on multicommodity flows with unknown destinations and defined on a discretization of the plane. The model and discretization are applied to both the deterministic and the stochastic continuous restricted location problem, where the latter is converted into a deterministic equivalent problem by minimizing the expected value of the objective function weighted by the probabilities of scenarios. The paper also describes a Benders decomposition algorithm to optimally solve the model. Extensive computational results are presented on both benchmark instances from the literature and new instances, on both the deterministic and stochastic variant of the problem. The results indicate that the proposed algorithm is superior to an off-the-shelf solver in terms of computational time. To the best of the authors' knowledge, the exact algorithm described here is the first to address both the deterministic and the stochastic variants of continuous restricted location problems with any number of facilities.