Warehouse–retailer network design problem

In this paper, we study the distribution network design problem integrating transportation and infinite horizon multiechelon inventory cost function. We consider the trade-off between inventory cost, direct shipment cost, and facility location cost in such a system. The problem is to determine how many warehouses to set up, where to locate them, how to serve the retailers using these warehouses, and to determine the optimal inventory policies for the warehouses and retailers. The objective is to minimize the total multiechelon inventory, transportation, and facility location costs. To the best of our knowledge, none of the papers in the area of distribution network design has explicitly addressed the issues of the 2-echelon inventory cost function arising from coordination of replenishment activities between the warehouses and the retailers. We structure this problem as a set-partitioning integer-programming model and solve it using column generation. The pricing subproblem that arises from the column generation algorithm gives rise to a new class of the submodular function minimization problem. We show that this pricing subproblem can be solved in O(n log n) time, where n is the number of retailers. Computational results show that the moderate size distribution network design problem can be solved efficiently via this approach.