An Application of Fixed Point Algorithm to Traffic Network Equilibrium Problem

Abstract

Traffic equilibrium problems become very complex when multi-type of users are on networks in which the effects of a certain type of user on other users are heterogeneous. Since transportation cost function on each link in a network should be defined as the vector function, the usual mathematical optimization formulation is no longer useful except for a special case. For such a general class of traffic equilibrium problems, only either the variational inequality approach or the fixed point approach can be applicable. Almost all of the approaches recently developed use the variational inequality approach. Although the variational inequality approach has become the primary mode of analysis in the area of traffic equilibrium and very general as a model, convergence results for algorithms for solving this general model often impose restrictions on the model that go beyond the monotonicity assumptions required in the equivalent convex optimization approach. This paper presents the theory of a new algorithm for the network equilibrium model that works in the space of path flows using a label and pivot technique in a fixed point approach. The idea of a pivot method that is extended to the traffic equilibrium problem in this paper has been motivated by a classical model of equilibrium in an exchange economy. However, the algorithm presented here is different from the economic equilibrium application in the following ways: (1) We work not on a price simplex but in the space of proportionate flows on path joining origin-destination pairs. (2) An appropriate labelling that produces a traffic assignment equilibrium is constructed. The calculation method is constructed on the facts that the labelling method appropriate to the traffic equilibria is to assign the number of paths available with the highest travel cost to each vertex on the simplex generated by changing the proportion of path flows and that if each vertex of a subdivided simplex has a differet number to each other, that simplex includes an equilibrium solution. Furthermore, the method is generalized to produce accurate solution in corresponding with the requirement of accurate prediction of network flows. © 1990, JAPAN SECTION OF THE REGIONAL SCIENCE ASSOCIATION INTERNATIONAL. All rights reserved.