Due to the increasing pressure for remaining competitive in the global market place, optimizing inventories across the supply chain has become an emerging challenge for the process industries to reduce costs and improve the customer service.[1,2] This challenge requires integrating inventory management with supply chain network design, so that decisions on the locations to stock the inventory and the associated amount of inventory in each stocking location can be determined simultaneously for lower costs and higher customer service level. However, the integration is usually nontrivial for multi-echelon supply chains and their associated inventory systems in the presence of uncertain customer demands.[3]

In this work we develop a joint multi-echelon supply chain design and inventory model for simultaneously optimizing the transportation, inventory and network structure in a supply chain under demand uncertainty. By assuming that each node in the supply chain network operates with a base-stock policy for a bounded normally distributed demand, [4] we capture the stochastic nature of the problem and develop an equivalent deterministic optimization model. The model determines the supply chain design decisions such as the locations of distribution centers (DCs), assignments of customers to DCs, assignments of DCs to plants, shipment levels from plants to the DCs and from DCs to customers, and inventory decisions such as the working inventory, pipeline inventory and safety stock in each node of the supply chain network. The model also captures risk-pooling effects [5] by consolidating the safety stock inventory of downstream nodes to the upstream nodes in the multi-echelon supply chain.

We formulate this problem as a mixed-integer nonlinear program (MINLP) with a nonconvex objective function including bilinear, trilinear and square root terms. By exploiting the properties of the basic model, we reformulate the problem as a separable concave minimization program. A tailored hierarchical decomposition algorithm based on Lagrangean relaxation is developed to obtain near global optimal solutions with reasonable computational expense. Three heuristics based on the model properties are also proposed to speed up the computation and are incorporated into the Lagrangean relaxation and decomposition algorithm. Computational examples for chemical and industrial gases supply chains with up to 20 plants, 200 potential distribution centers and 250 customers are presented to illustrate the application of the model and the performance of the algorithm.

[1] Grossmann, I. E., Enterprise-wide Optimization: A New Frontier in Process Systems Engineering. AIChE Journal, 2005, 51, 1846.

[2] Chopra, S.; Meindl, P., Supply Chain Management: Strategy, Planning and Operation. Prentice Hall: Saddle River, NJ, 2003.

[3] Zipkin, P. H., Foundations of Inventory Management. McGraw-Hill: Boston, MA, 2000.

[4] Graves, S. C.; Willem, S. P., Optimizing Strategic Safety Stock Placement in Supply Chains. Manufacturing & Service Operations Management, 2000, 2, 68.

[5] Eppen, G., Effects of centralization on expected costs in a multi-echelon newsboy problem. Management Science, 1979, 25, (5), 498.