Analysis of a Decentralized Production-Inventory System

We model an isolated portion of a competitive supply chain as a M/M/1 make-tostock queue. The retailer carries finished goods inventory to service a Poisson demand process, and specifies a policy for replenishing his inventory from an upstream supplier. The supplier chooses the service rate, i.e., capacity, of his manufacturing facility, which behaves as a single-server queue with exponential service times. Demand is backlogged and both agents share the backorder cost. In addition, a linear inventory holding cost is charged to the retailer, and a linear cost for building production capacity is incurred by the supplier. The inventory level, demand rate and cost parameters are common knowledge to both agents. Under the continuous-state approximation that the M/M/1 queue has an exponential rather than geometric steady-state distribution, we characterize the optimal centralized and Nash solutions, and show that a contract with linear transfer payments replicates a cost sharing agreement and coordinates the system. We also compare the total system costs, the agents’ decision variables and the customer service levels of the centralized versus Nash versus Stackelberg solutions.