Modeling private highways in networks with entry-exit based toll charges

Previous studies on private highways generally involve network equilibrium models with link-specific and hence link-additive toll charges. In reality, toll charges for private highways depend on the entry and exit points, which are not always link-additive. This study formulates and solves the optimal toll design problem of private highways with entry-exit based toll charges using a bi-level programming approach. The lower-level traffic equilibrium problem with entry-exit based toll charges is still formulated as an optimization problem and the Frank-Wolfe algorithm is adapted for finding its solution, where the descent direction-finding sub-problem (all-or-nothing traffic assignment) is solved via a simple network transformation. The proposed method circumvents the difficulty of path enumeration or generation frequently involved in general non-additive traffic assignment problems and, hence, has the potentials to efficiently solve large network problems. Following an exploration of the properties of the lower-level traffic equilibrium sub-problem, the bi-level optimal toll design problem is solved by a recently developed efficient marginal function approach.