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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 the service 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 operating costs and penalties for unserved requests. We consider the DDWP with a single delivery vehicle and request destinations on a line, where vehicle operating times and costs depend only on the distance between 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 propose heuristics and dual bounds for this case.One enhancement to customer service in this sector is same-day delivery (SDD). Several e-retailers and logistics service providers have introduced programs in major US cities; see Table 1. We define SDD as a distribution service where consumers place orders on the same day that they should be delivered. For companies that implement SDD, it is imperative to both offer high levels of customer service and keep logistics costs as low as possible.Providing SDD services requires two core logistics processes: (1) order management at the stocking location, including receiving, picking, and packing orders; and (2) order distribution from the stocking location to delivery locations. To date, two classes of service providers have deployed SDD services: retailers offering items primarily from owned stocks (in distribution centers or retail stores) such as Amazon and Walmart, and logistics providers serving as intermediaries that pick up packages from stocking locations and deliver them to customers, such as Google and Instacart as well as USPS, FedEx and UPS. Retailers must manage both core logistics processes, while logistics providers are typically concerned with the second one. This research effort is a study of primary tradeoffs in SDD distribution. We formulate the Dynamic Dispatch Waves Problem (DDWP) as a Markov Decision Process (MDP). The DDWP models the dynamics of a single dispatch facility (depot) where customer order requests arrive dynamically throughout an operating day. At any decision epoch, which we call a wave, the logistics operator maintains a set of known open requests with known delivery destinations and a set of potential requests that may arrive before the end of the day. At each wave, the operator decides whether or not to dispatch vehicles loaded with known orders, and the vehicle routes for dispatched orders. The objective is to minimize expected operational costs and expect...

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