Stochastic analysis of the Webber problem on the sphere

This paper deals with the location of one facility on the surface of the sphere (glove) that minimizes the weighted sum of distances to a given set of demand points on the surface of the sphere. It assumes that demand points are randomly generated on the sphere, and so are the weights. The paper proves that when the number of demand points increases to infinity, then the ratio between the maximum possible value of the objective function and the minimum possible value of the objective function converges to one. It also shows that the expected number of demand points that are a local minimum is approximately one when there are a large number of demand points. Some computational experiments are presented.