Solving a minisum single facility location problem in three regions with different norms

The single facility location problem in multiple regions with different norms (SMDN) generalizes the well-known Weber Problem. The SMDN consists of finding the optimum location of a single facility in the plane which is partitioned into multiple regions where the distance to travel in each region is measured or approximated with different norms. In this study, we specifically focus on the SMDN considering three regions with either rectilinear or Euclidean norms. We first introduce some analytical properties of this problem. Then, we devise a specially tailored branch-and-bound algorithm, i.e. a Big Square Small Square algorithm (BSSS), and two heuristics, named as Discrete Approximation algorithm (DA) and Modified Weiszfeld procedure (MW). The performance of the proposed approaches are tested using both standard test instances from the literature and randomly generated instances. According to our extensive computational experiments, the BSSS stands out to be a suitable exact solution approach in terms of both accuracy and efficiency when commercial mixed integer nonlinear solvers are not applicable. Besides, we also observe that the DA yields quite accurate solutions at the expense of high computation times while the MW arises to be the most efficient method with the least accuracy.