On the Existence of System-Optimum–Inducing Tolling Functions for Transportation Networks

A category of congestion pricing that charges travelers based on their trips’ some particular measurable attribute has been proposed in the literature to avoid the difficulty of pricing individual links on dense urban networks. To dig into such pricing’s theoretical performance, given any readily measurable attribute, this paper aims to propose a general mathematical framework for determining the existence of the tolling functions that can induce the system optimum of a transportation network. Specifically, we first formulate tolled network equilibrium as variational inequality. Then, in the elastic demand case, we prove that determining the existence of the system-optimum–inducing tolling functions is equivalent to solving a linear programming problem. In the fixed demand case, we provide a mixed integer linear programming formulation and show that the existence is confirmed if and only if its optimal objective function value equals zero. To enhance computational efficiency, we also derive valid inequalities for the formulation. Furthermore, we identify a special case in which determining the existence in the fixed demand case can be reduced to solving a linear programming problem similarly to the one in the elastic demand case. Finally, numerical examples, using distance- and emission-based tolls, are presented to derive important insights and demonstrate the proposed approaches’ potential of solving large-scale problems.