Optimal ECR Policy in a Single-Depot System

This chapter considers the ECR problem in a single depot such as an inland terminal or a seaport facing random demand and supply. We seek the optimal dynamic ECR policy over a planning horizon to minimize the expected cost consisting of container repositioning costs, inventory holding costs, and container leasing costs. Two situations are investigated. First, we consider the discrete-time sequential ECR decision-making situation. We formulate the problem as a discrete-time stochastic dynamic programming model. We prove that the optimal ECR policy can be characterized by two threshold parameters at each period in the form of (s, S) inventory control. Second, we consider the continuous-time continuous-state sequential ECR decision-making situation. Under the assumption of a two-state Markov demand process, we derive the closed-form solution to the optimal ECR problem. We use a fluid-flow model to characterize the underlying dynamics and stochasticity of the system. The qualitative structural properties of the value functions and the optimal control policies are established. The Hamilton–Jacobi-Bellman (HJB) equations are solved analytically, which leads to the closed-forms of the value functions. The fluid-flow model is then extended to the situations with multiple-state Markov demand process. Numerical examples are given to illustrate the results.