A lagrangean relaxation approach for a two-stage capacitated facility location problem with choice of depot size

In this paper, we study a two-stage capacitated facility location problem with choice of depot size (TSCFLP-CD). Given a set of potential sites for plants in the first echelon, a set of potential sites for capacitated depots associated each with several possible sizes in the second echelon and a set of customers with given demands, the TSCFLP-CD aims to determine the locations of plants and depots, the size of depots, the product flows from plants to depots, and then to the customers under the single-sourcing constraints, so that all customers' demands are satisfied with the minimum sum of the fixed opening costs of facilities, the product handling costs and the logistics costs. A Lagrangean relaxation approach is proposed to achieve a lower bound and an upper bound of the TSCFLP-CD. Numerical experiments on randomly generated instances demonstrated the effectiveness and efficiency of the proposed Lagrangean relaxation approach with the average gap of upper bound over lower bound around 1% in a reasonable time.