A Decomposition Algorithm for a Discrete Location-Allocation Problem

This paper considers a discrete location-allocation problem for simultaneously determining the location of a given number of capacitated facilities on certain predesignated sites, and the allocation of their products among a fixed number of customers. The objective is to determine the location and subsequent allocation that minimizes the total cost of construction or location, production, and transportation. The formulation of this model is a specially structured, generally nonconvex problem. However, we specify certain sufficient conditions for the solution to two network-structured linear programs to provide an optimal solution to this problem. For other situations, we present an implicit enumeration algorithm within the framework of Benders' decomposition method, which fully exploits the structure of the problem. This method is an improvement over traditional Lagrangian approaches. Computational experience is provided to demonstrate the practicality of this algorithm for reasonable size problems, as well as its superiority over alternative implicit enumeration schemes. This paper also sheds light on the general implementation of Benders' decomposition technique.