The multi-facility location–allocation problem with polyhedral barriers

In this paper we consider the problem of locating N new facilities with respect to M existing facilities in the plane and in the presence of polyhedral barriers. We assume that a barrier is a region where neither facility location nor traveling is permitted. For the resulting multi-dimensional mixed-integer optimization problem two different alternate location and allocation procedures are developed. Numerical examples show the superiority of a joint treatment of all assignment variables, including those specifying the routes taken around the barrier polyhedra, over a separate iterative solution of the assignment problem and the single-facility location problems in the presence of barriers.