An exact analysis of a joint production-inventory problem in two-echelon inventory systems

We consider a two-echelon inventory system with a manufacturer operating from a warehouse supplying multiple distribution centers (DCs) that satisfy the demand originating from multiple sources. The manufacturer has a finite production capacity and production times are stochastic. Demand from each source follows an independent Poisson process. We assume that the transportation times between the warehouse and DCs may be positive which may require keeping inventory at both the warehouse and DCs. Inventory in both echelons is managed using the base-stock policy. Each demand source can procure the product from one or more DCs, each incurring a different fulfilment cost. The objective is to determine the optimal base-stock levels at the warehouse and DCs as well as the assignment of the demand sources to the DCs so that the sum of inventory holding, backlog, and transportation costs is minimized. We obtain a simple equation for finding the optimal base-stock level at each DC and an upper bound for the optimal base-stock level at the warehouse. We demonstrate several managerial insights including that the demand from each source is optimally fulfilled entirely from a single distribution center, and as the system's utilization approaches 1, the optimal base-stock level increases in the transportation time at a rate equal to the demand rate arriving at the DC.