Generalization of Beckmann’s Transformation for Traffic Assignment Models with Asymmetric Cost Functions

Abstract

An optimization model is developed to solve the deterministic traffic assignment problem under congested transport networks with cost functions that have an asymmetric Jacobian. The proposed formulation is a generalization of Beckmann’s transformation that can incorporate network links with multivariate vector cost functions to capture the asymmetric interactions between the flows and costs of the different links. The objective function is built around a line integral that generalizes the simple definite integral in Beckmann’s transformation and is parameterised to ensure the solution of the new problem satisfies Wardrop’s first principle of network equilibrium. It is shown that this method is equivalent to the variational inequality approach. Our new approach could be extended to supply-demand equilibria models in other markets than transportation, with complementary or substitute goods/services in which there are asymmetric interactions between prices.