Bi-Level Variational Inequalities Model and Solution Algorithm for Link-based Road Pricing

In the recent years road pricing has been recognized as an effective means of managing traffic demand. Under the conventional economic principle, the first-best or second-best pricing scheme seeks to minimize the total system cost by imposing road charges to each network link or a subset of that. Both schemes may not be practically appealing due to the fact that the first priority of traffic management authority is to maintain the level of services for the paid users without specific objectives in terms of reducing total system cost. To maintain the level of service for the paid drivers, traffic management authority often wants to control the volume/capacity ratio of toll links with minimal adjustment of the toll prices. This paper will formulate such problems as bi-level variational inequalities, with the upper level an inverse variational inequality (IVI) with the box-constraints for the lower and upper limits of v/c ratios of toll links, and the lower level a classical variational inequality (VI) for the deterministic user equilibrium. And we further combine the IVI with VI as a single level general variational inequality (GVI). A projection and contraction method based on GVI is developed for solving the bi-level variational inequalities. The proposed formulation and solution algorithm are illustrated with a simple network and the computational results are reported.