Single facility location problem with region-dependent distance metrics

A single facility location problem where the distance is measured differently in different regions on the plane is considered. For example, if some demand points are in a city with streets located as horizontal or vertical lines on the map and if other demand points are outside the city where travel in a straight line is possible using, e.g. helicopters, we obtain a mixed distance problem and the current model becomes applicable. We first formulate the problem as a mixed integer non-linear programming problem. Next, we prove the non-convexity of the cost function by showing that it is discontinuous along the line that divides the two regions. Bounds on the value of the cost function are provided. We propose a heuristic, as a modified version of the Weiszfeld algorithm, to solve the problem and compare its performance with a global optimization method. A numerical example and sensitivity analyses are discussed comparing the efficiency of the modified algorithm with the results of the global optimization method