Loading and scheduling outbound trucks at a dispatch warehouse

We address the operational planning problem of loading and scheduling outbound trucks at a dispatch warehouse shipping goods to several customers. This entails, first, assigning shipments to out-bound trucks given the trailers' capacities and, second, scheduling the trucks' processing at the dock doors such that the amount of required resources at the terminal (e.g., dock doors and logistics workers) does not exceed the available levels. The trucks should be scheduled as late as possible within their time windows, but no later than the deadlines of the loaded shipments. Such planning problems arise, e.g., at dispatch warehouses of automotive parts manufacturers supplying parts to original equipment manufacturers in a just-in-time or even just-in-sequence manner. We formalize this operational problem and provide a time-indexed mixed-integer linear programming model. Moreover, we develop an exact branch-and-price algorithm, which is shown to perform very well, solving most realistically sized problem instances to optimality within a few minutes. In a numerical study, we also look into the interplay between the time window policy for trucks and just-in-time deliveries. Finally, we find evidence that too small a workforce or too few outbound dock doors in the dispatch warehouse can substantially compromise the punctuality of the deliveries.