A Lagrangean relax-and-cut approach for the two-stage capacitated facility location problem

The two-stage capacitated facility location problem is to find the optimal locations of depots to serve customers with a given demand, the optimal assignment of customers to depots and the optimal product flow from plants to depots. In order to solve this problem, a Lagrangean heuristic is proposed which is based on the relaxation of the capacity constraints. The resulting Lagrangean subproblem is an uncapacitated facility location problem with an aggregate capacity constraint and can be solved efficiently by branch-and-bound methods. The best bound available from this relaxation is computed by means of a ‘weighted’ Dantzig–Wolfe decomposition approach. Feasible solutions are constructed from those of the Lagrangean subproblems by applying simple reassignment procedures. Furthermore, the relaxation is strengthened by adding and dualizing valid inequalities which are violated by the fractional primal solution of the dual master program. Computational results and comparisons with some other bounds and heuristics for this problem are presented.