The dial-a-ride problem with private fleet and common carrier

Dial-a-ride problems aim to design the least-costly door-to-door vehicle routes for transporting individual users, subject to several service constraints like time windows, service and route durations, and ride-time. In some cases, providers cannot meet the demand and may outsource some requests. In this paper, we introduce, model, and solve the dial-a-ride problem with private fleet and common carrier (DARP-PFCC) that makes it possible to transfer the demand unmet by the provider to mobility-on-demand services and taxis. All outsourced vehicles are assumed to be available at any instant of the day and have unlimited capacity, enabling to satisfy all user requests, particularly during peak times. We implement a branch-and-cut (B&C) algorithm based on an exact method from the literature to solve the DARP-PFCC, and we develop a near parameter-free parallel metaheuristic to handle large instances. Our metaheuristic combines the Biased Random-key Genetic Algorithm (BRKGA) and the Q-learning (QL) method into the same framework (BRKGA-QL), in which an agent helps to use feedback information to dynamically choose the parameters of BRKGA during the search to select the most appropriate configuration to solve a specific problem instance. Both algorithms are flexible enough to solve the classical DARP, and extensive computational experiments demonstrate the efficiency of our methods. For the DARP instances, the B&C proved optimality for 41 of the 42 instances tested in a reasonable computational time, and the BRKGA-QL found the best-known solution for these instances within a matter of seconds. These results indicate that our metaheuristic performs equally well than state-of-the-art DARP algorithms. In the DARP-PFCC experiments on a set of 504 small-size instances, B&C proved optimality for 497 instances, while BRKGA-QL found 452 optimal solutions, totaling 90.94% of the instances solved to optimality. Finally, we present the results for a real case study for the DARP-PFCC, where BRKGA-QL solved very large problem instances containing up to 713 transportation requests. We also derive some managerial analyses to assess the effects of vehicle capacity reduction, for example due to the COVID-19 pandemic, on shared transportation. The results point to the benefits of combining the private fleet and common carriers in dial-a-ride problems, both for the provider and for the users.