In this paper, we analyze how drones can be combined with regular delivery vehicles to improve same‐day delivery performance. To this end, we present a dynamic vehicle routing problem with heterogeneous fleets. Customers order goods over the course of the day. These goods are delivered either by a drone or by a regular transportation vehicle within a delivery deadline. Drones are faster, but have a limited capacity as well as require charging after use. In the same‐day context, vehicle capacity is not a constraint, but vehicles are slow due to urban traffic. To decide whether an order is delivered by a drone or by a vehicle, we present a policy function approximation based on geographical districting. Our computational study reveals two major implications. First, geographical districting is highly effective increasing the expected number of same‐day deliveries. Second, a combination of drone and vehicle fleets may significantly reduce the required delivery resources.

Parcel services route vehicles to pick up parcels in the service area. Pickup requests occur dynamically during the day and are unknown before their actual request. Due to working hour restrictions, service vehicles only have limited time to serve dynamic requests. As a result, not all requests can be confirmed. To achieve an overall high number of confirmed requests, dispatchers have to budget their time effectively by anticipating future requests. To determine the value of a decision, i.e., the expected number of future confirmations given a point of time and remaining free time budget, we present an anticipatory time budgeting heuristic (ATB) drawing on methods of approximate dynamic programming. ATB frequently simulates problem's realization to subsequently approximate the values for every vector of point of time and free time budget to achieve an approximation of an optimal decision policy. Since the number of vectors is vast, we introduce the dynamic lookup table (DLT), a general approach adaptively partitioning the vector space to the approximation process. Compared with state-of-the-art benchmark heuristics, ATB allows an effective use of the time budget resulting in anticipatory decision making and high solution quality. Additionally, the DLT significantly strengthens and accelerates the approximation process.

We consider a stochastic dynamic pickup and delivery problem in which a fleet of drivers delivers food from a set of restaurants to ordering customers. The objective is to dynamically control a fleet of drivers in a way that avoids delays with respect to customers’ deadlines. There are two sources of uncertainty in the problem. First, the customers are unknown until they place an order. Second, the time at which the food is ready at the restaurant is unknown. To address these challenges, we present an anticipatory customer assignment (ACA) policy. To account for the stochasticity in the problem, ACA postpones the assignment decisions for selected customers, allowing more flexibility in assignments. In addition, ACA introduces a time buffer to reduce making decisions that are likely to result in delays. We also consider bundling, which is the practice of assigning multiple orders at a time to a driver. Based on real-world data, we show how ACA is able to improve service significantly for all stakeholders compared with current practice.

Although increasing amounts of transaction data make it possible to characterize uncertainties surrounding customer service requests, few methods integrate predictive tools with prescriptive optimization procedures to meet growing demand for small-volume urban transport services. We incorporate temporal and spatial anticipation of service requests into approximate dynamic programming (ADP) procedures to yield dynamic routing policies for the single-vehicle routing problem with stochastic service requests, an important problem in city-based logistics. We contribute to the routing literature as well as to the field of ADP. We combine offline value function approximation (VFA) with online rollout algorithms resulting in a high-quality, computationally tractable policy. Our offline-online policy enhances the anticipation of the VFA policy, yielding spatial and temporal anticipation of requests and routing developments. Our combination of VFA with rollout algorithms demonstrates the potential benefit of using offline and online methods in tandem as a hybrid ADP procedure, making possible higher-quality policies with reduced computational requirements for real-time decision-making. Finally, we identify a policy improvement guarantee applicable to VFA-based rollout algorithms, showing that base policies composed of deterministic decision rules lead to rollout policies with performance at least as strong as that of their base policy.

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