A multiclass, multicriteria traffic network equilibrium model with elastic demand

Abstract

In this paper, we develop a multiclass, multicriteria traffic network equilibrium model in which travelers of a class perceive their travel disutility or generalized cost on a route as a weighting of travel time and travel cost, each of which is flow-dependent. In addition, the weights are not only class-dependent but also link-dependent. The model is an elastic demand model and allows the demand function for each class and origin/destination (O/D) pair to depend, in general, upon the disutilities of all classes at all O/D pairs. The formulation of the governing equilibrium conditions, as well as the qualitative analysis, and the computational procedure, are based on finite-dimensional variational inequality theory. The model provides an alternative to existing multimodal and multicriteria traffic network equilibrium models and has location choice implications as well.

Introduction

Multicriteria traffic network models were introduced by Quandt (1967) and Schneider (1968) and explicitly consider that travelers may be faced with several criteria, notably, travel time and travel cost, in selecting their optimal routes of travel. The ideas were further developed by Dial (1979) who proposed an uncongested model and Dafermos (1981) who introduced congestion effects and derived an infinite-dimensional variational inequality formulation of her multiclass, multicriteria traffic network equilibrium problem, along with some qualitative properties.

Recently, there has been renewed interest in the formulation, analysis, and computation of multicriteria traffic network equilibrium problems. For the convenience of the reader, we have identified in Table 1 some principal contributions, since those of Quandt (1967) and Schneider (1968), to the multicriteria traffic network equilibrium literature which reflect conceptual/modeling/methodological advances. We believe that such a tabularization is useful since some of the contributions appear as unpublished manuscripts or in proceedings volumes and, hence, may not be as accessible to the general audience.

In particular, Table 1 provides a chronology of citations, which highlights: the number of criteria treated, typically, travel time and travel cost; whether or not these functions are allowed to be flow-dependent or not, and the form (separable or general) handled. In addition, Table 1 notes the type of demand considered, that is, fixed or elastic, and whether the demand is class-dependent, and, if elastic, what form the demand functions take. Moreover, Table 1 provides the type of methodology used in the formulation and analysis such as, for example, an optimization approach, a finite-dimensional variational inequality (fin.-dim. VI) approach, or infinite-dimensional (inf.-dim. VI) approach, along with whether the citation contains algorithmic contributions and/or qualitative ones. We note that, in the case of infinite-dimensional variational inequality formulations, the number of classes is, usually, infinite, whereas in the case of finite-dimensional formulations, the number of classes is assumed to be finite.

Additional citations, including literature exploring multicriteria traffic models used in practice, may be found in the book chapter by Leurent (1998).

This paper, in turn, focuses on the development of a new multiclass, multicriteria network equilibrium model with elastic demand. The model has the following novel and what we believe are significant features:

• It includes weights associated with the two criteria of travel time and travel cost which are not only class-dependent but also, explicitly, link-dependent. Such weights may incorporate such subjective factors as the relative safety or risk associated with particular links, the relative comfort, or even the view. None of the citations in Table 1 have this feature.

• It treats demand functions (rather than their inverses) which are very general and not separable functions. Specifically, the demand associated with a class and origin/destination (O/D) pair can depend not only on the travel disutility of different classes traveling between the particular O/D pair but can also be influenced by the disutilities of the classes traveling between other O/D pairs. Hence, the model has implications for locational choice (see, e. g., Beckmann et al., 1956; Boyce, 1980; Boyce et al., 1983). Not one of the citations in Table 1 considers such general demand functions. Moreover, for completeness, we include the case of known O/D pair travel disutility (or inverse demand) functions at a similar level of generality.

This paper not only develops such a model, but provides qualitative properties, as well as a computational procedure, accompanied by convergence results and numerical examples.