Finding second-best toll locations and levels by relaxing the set of first-best feasible toll vectors

This paper provides a framework for optimizing toll locations and levels in congestion pricing schemes for large urban road networks, with the objective to maximize the social surplus. This optimization problem is referred to as the toll location and level setting problem (TLLP) and is both non-convex, non-smooth and involves binary decision variables, and is therefore considered as a hard problem to solve. In this paper a solution approach is provided which instead of directly solving the TLLP, makes use of the first-best toll level solution, in which no restrictions are imposed on toll locations or levels. A first-best pricing scheme can be obtained by solving a convex program, and it has previously been shown that for the used routes in the network, the first-best toll levels on a route level are unique. By formulating an optimization problem, which instead of maximizing the social surplus, tries to find the link toll levels which minimize the deviation from first-best route tolls, a mixed integer linear program is obtained, and if the toll locations are predetermined the resulting optimization problem is a linear program. The approach of minimizing the deviation from first--best route tolls is applied for two different network models, and results are provided to show the applicability of the approach, as well as to compare with other approaches. Also, it is shown that for the Stockholm network, virtually the first-best level of social surplus can be obtained with a significantly reduced number of located tolls