On the Set of Optimal Points to the Weber Problem: Further Results

Abstract

In a recent paper, Z. Drezner and A. J. Goldman address the problem of determining the smallest set among those containing at least one optimal solution to every Weber problem based on a set of demand points in the plane. In the case of arbitrary mixed gauges (i.e., possibly nonsymmetric norms), the authors have shown that the set of strictly efficient points which are also intersection points always meets the set of Weber solutions. As shown by Drezner and Goldman, this set is optimal with the ℓ1 or ℓx distance but is not optimal with the ℓp distance, 1 < p < ∞. In this latter case they identify the smallest one. This paper generalizes these results for any strictly convex norm or any polyhedral norm. An example shows that in the case of an arbitrary norm there may exist an infinity of smallest sets and some of them contain points which are neither strictly efficient nor intersection points. In the ℓp distance case, we disprove a conjecture of Drezner and Goldman about the possibility of extending their result to more than two dimensions. The paper contains a different view of the problem: whereas Drezner and Goldman use algebraic-analytical approach, the authors use a geometrical approach which permits us to obtain more general results and also clarifies the geometric nature of the problem.