An order aggregation and scheduling problem for meal delivery

We address a single-machine scheduling problem motivated by a last-mile-delivery setting for a food company. Customers place orders, each characterized by a delivery point (customer location) and an ideal delivery time. An order is considered on time if it is delivered to the customer within a time window given by the ideal delivery time ±δ, where δ is the same for all orders. A single courier (machine) is in charge of delivery to all customers. Orders are either delivered individually, or two orders can be aggregated in a single courier trip. All trips start and end at the restaurant, so no routing decisions are needed. The problem is to schedule courier trips so that the number of late orders is minimum. We show that the problem with order aggregation is NP-hard and propose a combinatorial branch and bound algorithm for its solution. The algorithm performance is assessed through a computational study on instances derived by a real-life application and on randomly generated instances. The behavior of the combinatorial algorithm is compared with that of the best ILP formulation known for the problem. Through another set of computational experiments, we also show that an appropriate choice of design parameters allows applying the algorithm to a dynamic context, with orders arriving over time.