Probit based methods in traffic assignment and discrete choice modelling

Many phenomena studied in transportation are the result of choices made by individuals, such as the choice of route through a road network or the choice of mode to travel with, and are therefore modelled using choice models. This thesis addresses some aspects of the application of the multinomial probit (MNP) choice model, a well known model that, although theoretically sound, is not widely used as it is difficult to calculate since its choice function cannot be written in closed form. In particular, this thesis investigates the solution of the multinomial probit choice function by means of analytical approximations (that allow us to overcome the computational difficulties of the model) and addresses the application of MNP analytical approximations to the solution of the traffic assignment problem and some of its extensions. The use of MNP approximations to solve the model calibration problem is also addressed. The feasibility of solving the multinomial probit choice function by analytical approximation, rather than by simulation as typically done in applications, is explored by reviewing a number of approximation methods, some already used in transportation and some borrowed from other disciplines. Their precision is investigated by comparing their results and those obtained by numerical integration on a large number of test cases designed to replicate choice situations encountered in traffic assignment. Their computational cost is assessed by comparing the calculation times needed to solve the same set of test cases. The application of analytical approximations to solve the Stochastic User Equilibrium (SUE) traffic assignment problem is studied by describing and testing solution algorithms based on solving analytically the choice model and on the link-based formulation of the problem due to Sheffi and Powell (1982) but keeping track of the solution both in terms of path and of link flows. A first series of algorithms considered use the search direction of the method of successive averages (MSA) and are based on those developed by Maher and Hughes (1997a). A second series of algorithms employ alternative search vectors devised by considering the MSA search direction as a preconditioned steepest descent direction. Numerical tests, carried out on fixed sets of paths using two of the MNP approximations investigated, show that some of the algorithms of the second series outperform consistently the previous ones and those in the literature. The inclusion of multiple classes of users (MUC) with different perceptions of the network costs is treated using the formulation put forward by Daganzo (1982) and extending the algorithms investigated for the SUE case. The cases of SUE with elastic demand and with both multiple user classes and elastic demand are studied proposing formulations of the problems that are extensions of those for SUE and MUC SUE and allow us to develop efficient algorithms that are extensions of those proposed for SUE. Finally, the solution of the multinomial probit calibration problem is investigated by describing and applying a method to obtain analytical derivatives of the log-likelihood function by exploiting the structure of the approximation employed. The work reported focuses on the approximation of Mendell and Elston but the calculation structure suggested can be used also with other approximations. The efficiency of the approach proposed is tested by comparing the computational effort it requires against that required using numerical derivatives.