The One-Dimensional Dynamic Dispatch Waves Problem

We study same-day delivery (SDD) systems by formulating the Dynamic Dispatch Waves Problem(DDWP), which models a depot where delivery requests arrive dynamically throughout a service day.At any dispatch epoch (wave), the information available to the decision maker is (1) a set of known,open requests which remain unfulfilled, and (2) a set of potential requests that may arrive later in theservice day. At each wave, the decision maker decides whether or not to dispatch a vehicle, and if so,which subset of open requests to serve, with the objective of minimizing expected vehicle operatingcosts and penalties for unserved requests. We consider the DDWP with a single delivery vehicle andrequest destinations on a line, where vehicle operating times and costs depend only on the distancebetween points. We propose an efficient dynamic programming approach for the deterministic variant,and leverage it to design an optimal a priori policy with predetermined routes for the stochastic case.We then show that fully dynamic policies may perform arbitrarily better than a priori ones, and proposeheuristics and dual bounds for this case.