Further Analysis of the Weber Problem

Abstract

The most basic location problem is the Weber problem, that is a basis to many advanced location models. It is finding the location of a facility which minimizes the sum of weighted distances to a set of demand points. Solution approaches have convergence issues when the optimal solution is at a demand point because the derivatives of the objective function do not exist on a demand point and are discontinuous near it. In this paper we investigate the probability that the optimal location is on a demand point, create example problems that may take millions of iterations to converge to the optimal location, and suggest a simple improvement to the Weiszfeld solution algorithm. One would expect that if the number of demand points increases to infinity, the probability that the optimal location is on a demand point converges to 1 because there is no “space" left to locate the facility not on a demand point. Consequently, we may experience convergence issues for relatively large problems. However, it was shown that for randomly generated points in a circle the probability converges to zero, which is counter intuitive. In this paper we further investigate this probability. Another interesting result of our experiments is that FORTRAN is much faster than Python for such simulations. Researchers are advised to apply old fashioned programming languages rather than newer software for simulations of this type.