Competitive learning with additional information for solving multi-facility weber problem

Abstract

It is possible to remark that the mathematical formulation of clustering analysis problem is similar to a certain class of location problems. Known as the multi{facility Weber problem, it is a non{convex continuous optimization problem where a prede- termined number of facilities are located on the plane such that demand weighted summation of distances between each customer and its closest facility is minimized. With further investigation, we pointed out that the clustering analysis formula- tion is a special case of this facility location problem mentioned above. Motivated by this relation, we revised two algorithms of competitive learning, namely vector quanti- zation and Kohonen type networks, to use additional information and applied them to solve the Weber problem. The results obtained with the new techniques are superior than those reported previously in the literature. In the second part of the study, we investigated the probabilistic version of the Weber problem. In this version, customer locations are not ¯xed and they are assumed to be distributed randomly. Methods proposed previously are revised to handle probabilistic case as well. To the best of our knowledge, ¯rst experimental results are reported for this type of the problem and traditional techniques are compared with newly proposed methods. Furthermore, we proposed an approximation scheme for dealing with more general probability distributions.