A bilevel approach for compensation and routing decisions in last-mile delivery

In last-mile delivery logistics, peer-to-peer logistic platforms play an important role in connecting senders, customers, and independent carriers to fulfill delivery requests. Since the carriers are not under the platform's control, the platform has to anticipate their reactions, while deciding how to allocate the delivery operations. Indeed, carriers' decisions largely affect the platform's revenue. In this paper, we model this problem using bilevel programming. At the upper level, the platform decides how to assign the orders to the carriers; at the lower level, each carrier solves a profitable tour problem to determine which offered requests to accept, based on her own profit maximization. Possibly, the platform can influence carriers' decisions by determining also the compensation paid for each accepted request. The two considered settings result in two different formulations: the bilevel profitable tour problem with fixed compensation margins and with margin decisions, respectively. For each of them, we propose single-level reformulations and alternative formulations where the lower-level routing variables are projected out. A branch-and-cut algorithm is proposed to solve the bilevel models, with a tailored warm-start heuristic used to speed up the solution process. Extensive computational tests are performed to compare the proposed formulations and analyze solution characteristics.