Scheduling trains and containers with due dates and dynamic arrivals

We consider the problem of scheduling trains and containers (or trucks and pallets) between a depot and a destination. Goods arrive at the depot dynamically over time and have distinct due dates at the destination. There is a fixed-charge transportation cost for each vehicle, and each vehicle has the same capacity. The cost of holding goods may differ between the depot and the destination. The goal is to minimize the sum of transportation and holding costs. For the case in which all goods have the same holding costs, we consider two variations: one in which the holding cost at the destination is less than that at the origin, and one in which the relationship is reversed. For the first variation, we derive properties of the optimal solution which provide the basis for an O(T²) solution procedure. For the second variation, we introduce a new definition of a regeneration state, derive strong characterizations of the shipment schedule within a regeneration interval, and develop an O(T⁴) procedure. We also analyze two multi-item scenarios. In the first, for each item, the holding cost at the origin is less than that at the destination; in the second, the relationship is reversed for all items. We generalize several of the structural results for the single-item problem to the corresponding multi-item case. We also show that the optimal vehicle schedule can be obtained by solving a related single-item problem in which the item demands are aggregated in a particular way. The optimal assignment of customer orders to vehicles can then be found by solving a linear program.