The dial-a-ride problem with split requests and profits

In this paper we introduce the dial-a-ride problem with split requests and profits (DARPSRP). Users place transportation requests, specifying a pickup location, a delivery location, and a time window for either of the two. Based on maximum user ride time considerations, the second time window is generated. A given fleet of vehicles, each with a certain capacity, is available to serve these requests, and maximum route duration constraints have to be respected. Each request is associated with a revenue and the objective is to maximize the total profit, that is, the total revenue minus the total costs. Transportation requests involving several persons may be split if it is beneficial to do so. We formulate the DARPSRP as a mixed-integer program using position variables and in terms of a path-based formulation. For the solution of the latter, we design a branch-and-price algorithm. The largest instance solved to optimality, when applied to available instances from the literature, has 40 requests; when applied to newly generated instances, the largest instance solved to optimality consists of 24 requests. To solve larger instances a variable neighborhood search algorithm is developed. We investigate the impact of request splitting under different geographical settings, assuming favorable settings for request splitting in terms of the number of people per request. The largest benefits from request splitting are obtained for problem settings exhibiting clustered customer locations.