Ridesharing user equilibrium model and its equivalent variational inequality formulation without monotonicity: A decomposition-based approximation approach

Ridesharing (RS), with one driver serving a rider with similar route and schedule, has long been regarded as a charming paradigm for eco-friendly mobility. However, despite the rapid growth of shared mobility platforms, ridesharing demand remains relatively low. To design appropriate network-based policies to stimulate ridesharing, solving the traffic assignment problem in the presence of ridesharing services serves as the first and fundamental step. In a general network with both car owners and non-car owners, this paper models travelers’ mode, ridesharing partner and route choices at ridesharing user equilibrium. The model allows each RS driver to serve riders between any of her/his reachable origin–destination (OD) pairs, ensures each RS rider is served door to door by one RS driver, and endogenously determines the RS driver/rider demand and ridesharing price between each OD pair. The model is formulated into a non-monotone variational inequality (VI) problem. To solve it efficiently on large-scale networks, we propose a tailored Benders decomposition-based approximation approach, which decomposes the VI problem into a subproblem and a non-monotone master problem. The subproblem is transformed into a convex programming problem, which is further equivalent to a Beckmann formulation with side constraints. And for the non-monotone master problem, we apply the constant approximation scheme to approximate it by solving a convex quadratic programming problem with off-the-shelf solvers. The convergence of the proposed algorithm is theoretically established, and the correctness of the model and the efficiency of the algorithm are demonstrated through numerical experiments based on small to large scale networks.