## Manifestations of SUE - a Comparison and Evaluation of Different Methods of Stochastic User Equilibrium

### Authors

MAYER M and HUGHES P, Napier University, UK

### Description

It is commonly observed in empirical studies of driver route choice that a variety of routes are chosen between any origin - destination (O-D) pair. Traditionally, traffic assignment models fall into two classes, depending on the form of mechanism used to

### Abstract

It is commonly observed in empirical studies of driver route choice that a variety of routes are chosen between any origin - destination (O-D) pair. Traditionally, traffic assignment models fall into two classes, depending on the form of mechanism used to produce this "multi-routeing" or "route spreading" effect.

The first class consists of models which incorporate the effect of congestion through the use of capacity restraint and which aim to find the User Equilibrium (LIE) solution, in accordance with Wardrop's First Principle (Wardrop (1952)). They are deterministic in nature, assuming that drivers are perfectly rational and identical, and have complete and perfect knowledge of network conditions. Most implementations ofUE minimise the Beckmann (Beckmann et al (1956)) objective function, using the method due to Frank and Wolfe (1956).

The second class consists of probabitistic or stochastic methods, which aim to model the variations in driver perceptions or preferences, and reflect the imperfect knowledge drivers have of conditions throughout the network; drivers choose different routes through the network based on these different perceptions. Pure stochastic methods do not include any capacity restraint. There have been two main approaches to stochastic assignment - logit-based models, due originally to Dial (1971), and probit-based models, which postulate Normal link cost distributions.

UE models are widely used in application to congested urban networks, but perform less well in less congested, inter-urban networks. Stochastic methods are more suitable for lightly congested networks, typically inter-urban networks. However, it would be far better to have one method which incorporated both congestion and stochastic effects, so that this method could be applied equally well to all types of network. Such a model would then unify the field of static assignment modelling. As a consequence, several authors have, in recent years, developed techniques for the solution of the stochastic user equilibrium (SUE) problem, and Van Vuren (1994) in particular argued for its importance in practical terms.

Daganzo and Sheffi (1977) first formulated SUE, showing it as a generalisation of user equilibrium, and Sheffi and Powell (1982) showed that the SUE problem could be posed as a mathematical programming problem. Fisk (1980) proposed an alternative formulation for the logit-based problem. Most implementations of SUE have used the Method of Successive Averages (MSA), though Chen and Alfa (1991), Bell (1994), Leurent (1994) and Akamatsu (1995) have each made significant progress towards the development of algorithms which involve the evaluation and minimisation of the Fisk function.

Most authors agree that probit-based models are, ideally, preferable to logit-based ones.

The logit model has the advantage of computational simplicity, but suffers particularly from two disadvantages. Overlapping routes are not modelled, so that the classic bypass/town network produces an unrealistically low assignment on the bypass. Also, the split between any two routes depends on the absolute difference between their costs, rather than the relative difference. The probit model avoids both of these disadvantages, and so ideally should be preferred, but Monte Carlo methods suffer from non-repeatability, and other implementations have in the past been restricted by the need for complete path enumeration.

The SAM model has been described in Maher & Hughes (1995a and 1995b); it is a probit SUE model, which does not rely on path enumeration, but scans through a network; it shares this characteristic, and hence the efficiency, with logit models. However, the calculations involved in estimating splits and correlations according to the Normal distribution are more complex than are needed for the logit model; hence a SAM assignment may still take more computation time than a logit assignment. The purpose of this paper is to undertake some comparisons oflogit and probit assignment, both in detail on a 2-1ink network, and also for a realistic network, to see if the more sophisticated assumptions of the probit model produce significant differences in assignments, and whether any increased computational costs can be seen to be worthwhile.

The next section will use a 2-1ink network to compare the variability parameters for the logit and probit models. As will be explained, for a particular value of the logit parameter, it is not straightforward to find the corresponding value of the probit parameter. Section 3 will examine the other main difference between the models, the non-treatment of correlating routes in the logit model Sections 4 and 5 will describe how the assignment and SUE algorithms have been harmonised, so that only the underlying routeing principle is different. Section 6 will show whether these differences extend to significant differences in assignments in a realistically sized network.

#### Publisher

Association for European Transport