A Global Optimization Method For Weber’s Problem With Attraction And Repulsion

Weber’s problem involves the optimum location of a single facility on a plane in such a way that the weighted sum of the Euclidean distances of the facility from n given points be at the global minimum. Each point can either have an attractive or repulsive effect on the location of the facility depending on whether the corresponding weight is positive or negative respectively. Because attractive contributions correspond to convex functions and repulsive contributions to concave ones, the total expression for the weighted sum of Euclidean distances is nonconvex. In this paper, two global optimization algorithms are proposed, one based on a concave and one on a concave + convex lower bounding operation. Both of these algorithms utilize efficient rectangular subdivision processes. The proposed approaches attain ∈-convergence to the global minimum in a finite number of iterations. Computational results are presented for problems involving as many as n = 10,000 points.