Multi-class user equilibrium and system optimum with non-linear value of time

For traffic networks with fixed demand and heterogeneous users differentiated by value of time, when there exists nonlinear relationship between travel cost and travel time, the time-based and cost-based system optimization models and the user equilibrium models are formulated, respectively. If the system performance and user equilibrium conditions are both measured by cost, it is shown that system optimum can be decentralized into multi-class user equilibrium by anonymously charging users on all links. It is further proved that such tolling scheme doesn't exist if the system performance and the users' decision criteria are based on time unit.