Complementarity Formulations for User Equilibrium and Ideal System State in Dynamic Transportation Networks

A key fundamental research question in real-time guidance in dynamic transportation networks is: Can this paper characterize and compute system states which are close to system-optimal state but does not make any individual worse off as compared to the equilibrium problem? Within the static network assignment context this question has recently been addressed by Jahn et al. (2005), where they motivate a route-guidance system based on a constrained system optimal formulation that maximizes the total travel time subject to certain user constraints. Unlike static traffic assignment, dynamic traffic assignment does not have a well-accepted analytical formulation with realistic traffic flow models. The present paper provides an analytical formulation of dynamic traffic equilibrium problem with a realistic traffic flow model – the cell transmission model. We develop an analytical formulation using complementarity constraints for the cell transmission model. The dynamic user equilibrium (DUE) state is formulated as a Mixed Linear Complementarity problem. We subsequently define an ideal system state that ensures the best possible system state such that no user is worse off compared to the user equilibrium state. The ideal system state is modeled as a Linear Program with Linear Complementarity Constraints. The paper provides a strong theoretical modeling framework for prescriptive travel guidance systems that seek to improve overall system payoff without negatively impacting any user.