Continuous location under the effect of ‘refraction’

Abstract

In this paper we address the problem of locating a new facility on a d-dimensional space when the distance measure ($\ell _p$- or polyhedral-norms) is different at each one of the sides of a given hyperplane $\mathcal {H}$. We relate this problem with the physical phenomenon of refraction, and extend it to any finite dimensional space and different distances at each one of the sides of any hyperplane. An application to this problem is the location of a facility within or outside an urban area where different distance measures must be used. We provide a new second order cone programming formulation, based on the $\ell _p$-norm representation given in Blanco et al. (Comput Optim Appl 58(3):563–595, 2014) that allows to solve the problem in any finite dimensional space with second order cone or semidefinite programming tools. We also extend the problem to the case where the hyperplane is considered as a rapid transit media (a different third norm is also considered over $\mathcal {H}$) that allows the demand to travel, whenever it is convenient, through $\mathcal {H}$ to reach the new facility. Extensive computational experiments run in Gurobi are reported in order to show the effectiveness of the approach. Some extensions of these models are also presented.