Weber problem

Abstract: Given n demand points in the plane, the p-center problem is to find p supply points (anywhere in the plane) so as to minimize the maximum distance from a demand point to its respective nearest supply point. The p-median problem is to minimize the sum of distances from demand points to their respective nearest supply points. We prove that the p-center and the p-median problems relative to both the Euclidean and the rectilinear metrics are NP-hard. In fact, we prove that it is NP-hard even to approximate the p-center problems sufficiently closely. The reductions are from 3-satisfiability.

Abstract: Spatial economics in the past has been hampered in its progress by certain peculiarities which are inherent in the functional relationships among its variables. The frequency of corner solutions in its extremum problems, a common tendency for the constraints upon these problems to be stated as inequalities, and the importance of discontinuous functions in the field, are examples of these peculiarities. It is not fortuitous that the origins and development of programming techniques are so closely tied to the problems of spatial economics, and these techniques have resulted in an advance of breakthrough proportions in obtaining solutions to formerly insoluble spatial problems.

Abstract: The problem considered is the choice of locations form sources so as to minimize the sum of weighted distances betweenn fixed sinks and the source closest to each sink. The weights represent the amounts to be shipped between the sinks and their respective sources; the allowable source locations are free of restriction. An algorithm for the approximate solution of the problem, and computational experience with it, are discussed first. A branch-and-bound algorithm for exact solution of the problem is then developed, and computational experience with it is described.