Lagrangean-based solution approaches for the generalized problem of locating capacitated warehouses

The traditional capacitated warehouse location problem consists of determining the number and the location of capacitated warehouses on a predefined set of potential sites such that the demands of a set of customers are met. A very common assumption made in modeling this problem in almost all of the existing research is that the total capacity of all potential warehouses is sufficient to meet the total demand. Whereas this assumption facilitates to define a well-structured problem from the mathematical modeling perspective, it is in fact restrictive, not realistic, and hence rarely held in practice. The modeling approach presented in this paper breaks away from the existing research in relaxing this very restrictive assumption. This paper therefore investigates the generalized problem of locating warehouses in a supply chain setting with multiple commodities with no restriction on the total capacity and the demand. A new integer programming formulation for this problem is presented, and an algorithm based on Lagrangean relaxation and decomposition is described for its solution. Three Lagrangean heuristics are proposed. Computational results indicate that reasonably good solutions can be obtained with the proposed algorithms, without having to use a general purpose optimizer.