Solution of Backup Multifacility Location Problem by Considering the Ideal Radius for each Customer

In this paper we introduce a new facility location model, called backup multifacility location problem by considering the ideal radius for each customer. In this problem the location of clients are given in the plane. A radius is assigned to each client. We should find the location of new facilities, which some of them may fail with a given probability, such that the sum of weighted distances from new facilities to the radius distance of clients and sum of weighted distances between new facilities is minimized. Since in the most instance there dose not exist the location of a new facility such that its distance to each Customers be exactly equal to given radiuses, so we try to minimize the sum of the weighted square errors. We model the problem and propose an iterative method (weiszfeld like algorithm) for solving the presented problem. Then a discussion about convergence of presented method and some numerical examples are given. We show that the optimal solution lies in an extended rectangular hull of the existing points.