Solving the Asymmetric Traac Assignment Problem with a Non-monotonic Cost Operator

In this paper, we demonstrate when and why existing algorithms fail to give coherent results to the asymmetric traac assignment problem when the model does not admit a monotonic cost operator. Through an analysis of why these problems occur, a new algorithm is developed that is eeective precisely when these restrictions are not satissed. Numerical tests are then performed to illustrate the functioning of the new algorithm, both on a simple network as well as on a realistic, medium-sized network. One of the rst to discuss the inability of multi-class traac assignment models to treat realistic transport scenarios was Netter (1972). Netter illustrated, on a simple two-arc network, that the presence of two classes, each with its own cost function, led to an indeterminate situation. Deening linear cost functions for each class, he showed the presence of three distinct equilibria, indicating, furthermore, that one of the three equilibria may be unstable. This result follows directly from the non-monotonicity of the travel cost operator. The implication of this is quite important to the future of multi-class, or asymmetric, traac assignment modeling. While often cited in more recent work of other researchers, the crux of this issue has never been addressed in the literature. Instead, work on the asymmetric traac assignment problem has, since that time, followed two more-or-less parallel paths. On one path, further research on the algorithmic side has concentrated on developing methods that can be shown to converge under monotonicity restrictions on the cost functions. On an almost parallel route were attempts to develop realistic models which incorporate multiple ow dependencies into the link cost functions. The extended SATURN model is of this type; it explicitly treats the ow interactions at traac junctions and multiple user classes (DRIVE 1990). The SATURN model was developed heuristically; although theoretical work already existed which permitted proofs of existence and uniqueness of the model solutions and of the algorithm convergence, the constraints imposed to ensure these properties were too stringent to allow adequate modeling of the traac phenomena. Indeed, the calibrated cost operators of the SATURN model are non-monotonic. When the two paths did meet, the results have been generally unsatisfying{both from a theoretical as well as from a practical point of view. Fisk and Nguyen (1982) tested a number of convergent algorithms to treat the asymmetric traac assignment problem, which they applied successfully to a simple contrived network with aane cost functions. …