A planar single facility location and border crossing problem

In this study, we tackle the problem of locating a facility in a region where a fixed line barrier divides the region into two. The resulting subregions communicate with each other through a number of passage points located on the line barrier. Our contribution is threefold. First, we formulate the problem as a Mixed Integer Nonlinear Programming (MINLP) model and provide an optimal solution methodology based on an Outer Approximation (OA) algorithm. Second, we discuss the minimax version of this problem for locating an emergency facility and use the OA algorithm to solve the problem. We then provide simple example problems and extensive computational results for both problems. Finally, we propose a one-infinity approximation approach for the latter problem which yields a linear model. Practical uses of the models have been discussed in the border crossing context.