Containerized liner trades have been growing steadily since the globalization of world economies intensified in the early 1990s. However, these trades are typically imbalanced in terms of the numbers of inbound and outbound containers. As a result, the relocation of empty containers has become one of the major problems faced by liner operators. In this paper, we consider the dynamic empty container allocation problem where we need to reposition empty containers and to determine the number of leased containers needed to meet customers' demand over time. We formulate this problem as a two-stage stochastic network: in stage one, the parameters such as supplies, demands, and ship capacities for empty containers are deterministic; whereas in stage two, these parameters are random variables. We need to make decisions in stage one such that the total of the stage one cost and the expected stage two cost is minimized. By taking advantage of the network structure, we show how a stochastic quasi-gradient method and a stochastic hybrid approximation procedure can be applied to solve the problem. In addition, we propose some new variations of these methods that seem to work faster in practice. We conduct numerical tests to evaluate the value of the two-stage stochastic model over a rolling horizon environment and to investigate the behavior of the solution methods with different implementations.

We consider the class of multistage dynamic networks with random arc capacities, a framework that is well suited to model dynamic eet management problems. We propose a successive convex approximation approach that produces an approximation to the expected recourse function which captures the future e ects of current decisions under uncertainty. This method decomposes the network in each stage into tree subproblems, whose expected recourse functions are easy to obtain. We also compare this method with two alternative methods on a set of dynamic eet management problems. The numerical results show that this method is superior than the two alternative methods.

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