A generalized Weiszfeld method for the multi-facility location problem

The Fermat–Weber location problem (also single facility location problem) is to locate a facility that will serve optimally a set of customers, given by their locations and weights, in the sense of minimizing the weighted sum of distances traveled by the customers. A well known method for solving the problem is the Weiszfeld method [25], a gradient method that expresses and updates the sought center as a convex combination of the data points. The multi–facility location problem (MFLP) is to locate a (given) number of facilities to serve the customers as above. Each customer is assigned to a single facility, and the problem (also called the location–allocation problem) is to determine the optimal locations of the facilities, as well as the optimal assignments of customers (assignment is absent in the single facility case.) MFLP is NP hard, [21]. We approximate it by replacing rigid assignment with probabilistic membership, as in [4], [3] and [8], and propose an iterative method for its solution, a natural generalization of the Weiszfeld method to several facilities.