An optimal method for solving the (Generalized) Multi-Weber Problem

The Single Weber Problem and the Multi-Weber Problem have troubled generations of Mathematicians, Economists, and Geographers. By enumerating all feasible elements of a solution (non-overlapping convex hulls) linear programming can be employed to obtain optimal solutions to the Multi-Weber Problem and the Generalized Multi-Weber Problem in Euclidean space. First lists of all feasible convex hulls are generated. These are the universe of potential elements of an optimal solution. Second these convex hulls are used to cover the fixed points of the problem. A linear programme, based on the Set Covering Problem is then used to cover the fixed points minimizing the total (weighted) distance. Computational experience is presented. This method guarantees the optimality of the solution.