ALGORITHMS FOR SOLVING THE PROBIT PATH-BASED STOCHASTIC USER EQUILIBRIUM TRAFFIC ASSIGNMENT PROBLEM WITH ONE OR MORE USER CLASSES

摘要

The paper has considered different analytical approximation methods for the solution of the probit choice function and addressed their application for the calculation of the path-based Stochastic User Equilibrium traffic pattern with one or more user classes. The approximations of Clark (1961), Mendell-Elston (1974), Langdon (1984a, b), Tang and Melchers (1987) have been reviewed and their accuracy and computational cost assessed in two series of numerical experiments in which the choice probabilities obtained with each method were compared with those calculated with the numerical integration method of Genz (1992). The results of the experiments show the tendency to produce large percentage errors for low choice probabilities of the two possible implementations of the method of Clark (although at a smaller scale for the improved Clark method), the good accuracy of the Langdon method and the very good accuracy of the methods of Tang and Melchers and of Mendell and Elston. The Mendell and Elston approximation offers the best trade-off between accuracy and calculation time for different numbers of options. Therefore it seems particularly suitable for applications where the choice calculations need be performed a large number of times as in traffic assignment. In general the experiments proved that the analytical approach to solving the probit choice function is feasible and accurate. The numerical integration method, here used only to provide an accurate comparator, constitutes also a precise, although more time consuming, alternative to the use of approximations. The application of the analytical approximation methods to solve the probit path-based SUE traffic assignment problem using the objective function of Sheffi and Powell (1982) has been investigated exploring the performance of different solution algorithms in cases with paths enumerated before the assignment. Path-based implementations of the optimisation algorithms proposed by Maher and Hughes (1997) have been found to be much more efficient then the traditional MSA, consistent with the result of Maher and Hughes. A heuristic to obtain the value of the satisfaction for use in the optimisation algorithms for the approximations that do not calculate it directly has been proposed. False position and rational function interpolation methods to estimate the optimal step length have been proposed and found to be more efficient than the MSA and typically as efficient as the methods proposed by Maher and Hughes (1997).