THE P-CENTER PROBLEM ON FUZZY NETWORKS AND REDUCTION OF COST

Abstract

Here we consider the p-center problem on dierent types of fuzzy networks. In particular, we are interested in the networks with interval and triangular fuzzy arc lengths and vertex-weights. A methodology to obtain the best satisfaction level of the decision maker who wishes to reduce the cost within the tolerance limits is proposed. Illustrative examples are provided. The p-center problem is a well-known facility location problem. It arises in the following way. Suppose some demand points are given by the vertices of a network, and the weight of each vertex represents the demand at that point, and there is a path between every pair of vertices in order to transport products from one vertex to another. The decision maker is to locate p locations within the network where the facilities are to be located such that the maximum cost to transport products to each demand point from the nearest facility is minimized. It is assumed that these costs are directly proportional to the distances to be covered and the quantities of product to be transported. If the facilities are considered to be located at the vertices only, the p-center problem can be defined as follows. Let G = (V,E) be a connected undirected network, where V = {v1,v2,...,vn} is the set of vertices of G and E is the set of edges. As G is connected, there exists a path between every pair of vertices. The distance d(vi,vj) between two vertices vi and vj is denoted by dij, the length of the shortest path joining the vertices vi and vj. The weight wi is associated with the vertex vi for all i = 1,2,...,n. Let us consider all the subsets of {1,2,...,n} with p number of elements. Let those subsets be A1,A2,...,Am. So for a given set covering Ai, the distance to be covered in order to serve the vertex vj is given by ij = min