About this Journal Submit a Manuscript Table of Contents
Advances in Operations Research
Volume 2011 (2011), Article ID 379505, 10 pages
http://dx.doi.org/10.1155/2011/379505
Research Article

Noniterative Solution of Some Fermat-Weber Location Problems

The Raymond & Beverly Sackler School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel

Received 5 April 2011; Accepted 17 June 2011

Academic Editor: I. L. Averbakh

Copyright © 2011 Reuven Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The Fermat-Weber problem of optimally locating a service facility in the Euclidean continuous two-dimensional space is usually solved by the iterative process first suggested by Weiszfeld or by later versions thereof. The methods are usually rather efficient, but exceptional problems are described in the literature in which the iterative solution is exceedingly long. These problems are such that the solution either coincides with one of the demand points or nearly coincides with it. We describe a noniterative direct alternative, based on the insight that the gradient components of the individual demand points can be considered as pooling forces with respect to the solution point. It is demonstrated that symmetrical problems can thus be optimally solved with no iterations, in analogy to finding the equilibrium point in statics. These include a well-known ill-conditioned problem and its variants, which can now be easily solved to optimality using geometrical considerations.