Modeling and solving the multimodal car- and ride-sharing problem

We introduce the multimodal car- and ride-sharing problem (MMCRP), in which a pool of cars is used to cover set ride requests while uncovered are assigned other modes transport (MOT). A car’s route consists one or more trips. Each trip must have specific but non-predetermined driver, start depot finish (possibly different) depot. Ride-sharing between users allowed, even when two rides do not same origin and/or destination. user has always option using according an individual list preferences. The can be formulated as vehicle scheduling problem. In order solve problem, auxiliary graph constructed each starting ending depot, covering possible ride-shares, modeled arc time-space graph. propose two-layer decomposition algorithm based on column generation, where master ensures that request only covered at most once, pricing generates new promising routes by solving kind shortest-path network. Computational experiments realistic instances reported. benchmark demographic, spatial, economic data Vienna, Austria. large with generation approach near optimality reasonable time, we further investigate various exact heuristic schemes.