The p-median problem in fuzzy environment: proving fuzzy vertex optimality theorem and its application

Abstract

The median problem on a graph is an important field of facility location problem in which the aim is to find the best place among vertices and interior points of edges such that the summation of weighted distances are minimized. This point is called absolute median point. Beforehand, it was shown that the absolute median of a graph is always at a vertex of it. This theorem is called vertex optimality theorem. In this paper, a graph with fuzzy weights is considered. Then, the fuzzy vertex optimality theorem is proved for this problem. That is, it is proved that an absolute median of this fuzzy graph is always at a vertex of it. Thereafter, this theorem is proved for the general form problem where the weights of vertices and the lengths of the arcs are all fuzzy numbers. It is worthwhile to note that this is the first time that this theorem is proved for fuzzy location problems. Also, by proving this fuzzy theorem and applying it on existing algorithms for obtaining median point, the computational complexity is reduced. Finally, by solving some numerical examples, the concept of these proved theorems and their proofs are illustrated.