The Facility Location Model with Risk Pooling (LMRP) extends the uncapacitated fixed charge model to incorporate inventory decisions at the distribution centers (DCs). In this paper, we introduce a capacitated version of the LMRP that handles inventory management at the DCs such that the capacity limitations at the DCs are not exceeded. We consider a logistics system in which a single plant ships one type of product to a set of retailers, each with uncertain demand. Each DC serves as the direct intermediary between the plant and the retailers for the shipment of the product. Safety stock is retained at the DCs to provide appropriate service levels. The Capacitated Facility Location Model with Risk Pooling (CLMRP) simultaneously determines DC locations, shipment sizes and frequencies from the plant to the DCs, the working inventory and safety stock levels at the DCs and the assignment of retailers to the DCs, to minimize the fixed facility location costs, transportation costs and inventory carrying costs. The capacity constraints are defined based on how the inventory is managed. Thus, the relationship between the capacity of a DC and the inventory levels are embedded in the model. CLMRP is capable of evaluating the tradeoff between having more DCs to have sufficient system capacity versus ordering more frequently through the definition of capacity. The model is formulated as a non-linear integer-programming problem. A Lagrangian relaxation solution algorithm is proposed. The Lagrangian subproblem is also a non-linear integer program. An efficient algorithm is obtained for the linear relaxation of this subproblem.

## Citation

Northwestern University, August 2003