An implicit programming approach for the road pricing problem with nonadditive route costs

Road pricing is considered one of the effective means to reduce traffic congestion and environmental damage, and it has been introduced in major highways of most countries. The road pricing problem can be formulated as a mathematical program with equilibrium constraints (MPEC) and the resulting MPEC can be solved efficiently by the implicit programming approach if the user's route costs are additive. However, route costs are generally nonadditive in the real world. In this paper we consider road pricing on the traffic equilibrium problem with nonadditive route costs based on users' disutility functions. We then show that this formulation can be reformulated as a mathematical program with strictly monotone mixed complementarity problem (MCP). Since a strictly monotone MCP has a unique solution for each upper level variable, we can apply the implicit programming approach to solve the resulting reformulation. We establish the differentiability of the resulting implicit function. Numerical experiments using various disutility functions and sample networks are done, and the results show that the implicit programming approach is robust to find a solution of the road pricing problem.