Trial‐and‐error method for congestion pricing scheme under side‐constrained probit‐based stochastic user equilibrium conditions

A toll pattern that can restrict link flows on the tolled links to some predetermined thresholds is named as effective toll solution, which can be theoretically obtained by solving a side‐constraint traffic assignment problem. Considering the practical implementation, this paper investigates availability of an engineering‐oriented trial‐and‐error method for the effective toll pattern of cordon‐based congestion pricing scheme, under side‐constrained probit‐based stochastic user equilibrium (SUE) conditions. The trial‐and‐error method merely requires the observed traffic counts on each entry of the cordon. A minimization model for the side‐constrained probit‐based SUE problem with elastic demand is first proposed and it is shown that the effective toll solution equals to the product of value of time and optimal Lagrangian multipliers with respect to the side constraints. Then, employing the Lagrangian dual formulation of the minimization method, this paper has built a convergent trial‐and‐error method. The trial‐and‐error method is finally tested by a numerical example developed from the cordon‐based congestion pricing scheme in Singapore.