An Optimal Toll Design Problem with Improved Behavioural Equilibrium Model: The Case of the Probit Model

Abstract

This paper considers the optimal toll design problem that uses the Probit model to determine travellers’ route-choices. Under probit, the route flow solution to the resulting stochastic user equilibrium (SUE) is unique and can be stated implicitly as a function of tolls. This reduces the toll design problem to an optimization problem with only nonnegativity constraints. Additionally, the gradient of the objective function can be approximated using the chain rule and the first order Taylor approximation of the equilibrium condition. To determine SUE, this paper considers two techniques. One uses Monte-Carlo simulation to estimate route choice probabilities and the method of successive averages with its prescribed step length. The other relies on the Clark approximation and computes an optimal step length. Although both are effective at solving the toll design problem, numerical experiments show that the technique with the Clark approximation is more robust on a small network.