Analyzing the make-to-stock queue in the supply chain and eBusiness settings

Abstract

This thesis presents two applications of the prototypical make-to-stock queue model that are mainly motivated by supply chain management and e-commerce issues. In the first part, we consider the decentralized version of the make-to-stock model. Two different agents that we call the supplier and the retailer control production and finish goods inventory level independently. The retailer carries finished goods inventory to service an exogenous demand and specifies a policy for replenishing his/her inventory from the upstream supplier. The supplier, on the other hand, chooses the capacity of his manufacturing facility. 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 based on backorder, inventory and capacity levels coordinates the system in the absence of participation constraints.

(cont.) We also derive explicit formulas to assess the inefficiency of the Nash equilibrium, compare the agents' decision variables and the customer service level of the centralized versus Nash solutions, and identify conditions under which a coordinating contract is desirable for both agents. In the second part, we return to the centralized version of the make-to-stock model and analyze the situation where the price that the end customers are willing to pay for the good changes dynamically and stochastically over time. We also as- sume that demand is fully backlogged and that holding and backordering costs are linearly incurred by the manufacturer. In this setting, we formulate the stochastic control problem faced by the manager. That is, at each moment of time and based on the current inventory position, the manager decides (i) whether or not to accept an incoming order and (ii) whether or not to idle the machine. We use the expected long-term average criteria to compute profits. Under heavy traffic conditions, we approximate the problem by a dynamic diffusion control problem and derive optimality (Bellman) conditions. Given the mathematical complexity of the Bellman equations, numerical and approximated solutions are presented as well as a set of computational experiments showing the quality of the proposed policies.