Optimal ECR Policy in Two-Depot Shuttle Systems: Continuous Review

This chapter addresses the ECR problem in a two-depot shuttle service system over an infinite time horizon with the focus of deriving optimal stationary ECR policies. Empty containers are required at each depot to meet random customer demands that derive laden container movements between two depots. Customer demands must be satisfied by either owned empty containers or leasing from lessors. The system is based on continuous review and discrete state, where the system state represents the inventory level of containers at two depots. An event-driven model is formulated, where the ECR decisions are made at each event when the system state changes. Under the assumption of Poisson arrival process of laden containers and an exponential distribution of empty container transfer times, we convert the continuous-time Markov decision process into an equivalent discrete-time Markov decision process by using the uniformization technique. It is shown that the optimal ECR policy is a threshold policy, characterized by two control parameters, in both the discounted cost and the long-run average cost cases. The closed-form of the optimal discounted cost function is derived by using the characteristic equation method. The closed-form of the optimal long-run average cost function is derived by calculating the stationary distribution under the threshold control policy. Numerical examples are provided to demonstrate the analytical results and test the sensitivity of the model to the distribution assumption. Finally, the models are extended to the cases with random supply and demand of empty containers at both depots, where empty containers may exit and enter the two-depot shuttle system randomly.