Incorporating Inventory and Routing Costs in Strategic Location Models

We consider a stochastic supply chain design problem where the decision maker needs to decide the number and locations of the distribution centers (DCs). The supply chain network is characterized by the explicit inclusion of probabilistic elements. The locations and actual demands of the potential customers are not known, instead, the only information available is the related probability distributions. Each DC maintains a certain amount of safety stock in order to achieve a certain service level for the customers it serves. The objective is to minimize the total cost that includes location costs and inventory costs at the DCs, and distribution costs in the supply chain. We show that this problem can be formulated as a nonlinear integer programming model, for which we propose a Lagrangian relaxation based solution algorithm. By exploring the structure of the problem, we find a low-order polynomial algorithm for the nonlinear integer programming problem that must be solved in solving the Lagrangian relaxation subproblems. We present computational results for several instances of the problem with sizes ranging from 40 to 320 customers. Our solution technique can also be applied to a wide range of other concave cost minimization problems.