The lack of uniqueness constitutes a serious concern for any analysis that relies on class-specific traffic assignment results, such as understanding the impact of a transport policy on the welfare of travelers from different income groups, sometimes known the vertical equity analysis.  Entropy maximization is a standard approach to consistently selecting a unique class-specific solution for multi-class traffic assignment.

Here, we show the conventional maximum entropy formulation fails to strictly observe the multi-class bi-criteria user equilibrium condition, because a class-specific solution matching the total equilibrium link flow may violate the equilibrium condition. We propose to fix the problem by requiring the class-specific solution, in addition to matching the total equilibrium link flow, also match the objective function value at the equilibrium.  This leads to a new formulation that is solved using an exact algorithm based on dualizing the hard, equilibrium-related constraints.

Our numerical experiments highlight the superior stability of the maximum entropy solution, in that it is affected by a perturbation in inputs much less than an untreated benchmark multi-class assignment solution.  In addition to instability, the benchmark solution also exhibits varying degrees of arbitrariness, potentially rendering it unsuitable for assessing distributional effects across different groups, a capability crucial in applications concerning vertical equity and environmental justice. The proposed formulation and algorithm offer a practical remedy for these shortcomings.

This is the third paper completed by the first author, Qianni Wang, who officially joined my group last year.

The paper was currently under review at Transportation Research Part B.  You may download a preprint here, or read the abstract below.

Abstract: Entropy maximization is a standard approach to consistently selecting a unique class-specific solution for multi-class traffic assignment. Here, we show the conventional maximum entropy formulation fails to strictly observe the multi-class bi-criteria user equilibrium condition, because a class-specific solution matching the total equilibrium link flow may violate the equilibrium condition. We propose to fix the problem by requiring the class-specific solution, in addition to matching the total equilibrium link flow, also match the objective function value at the equilibrium. This leads to a new formulation that is solved using an exact algorithm based on dualizing the hard, equilibrium-related constraints. Our numerical experiments highlight the superior stability of the maximum entropy solution, in that it is affected by a perturbation in inputs much less than an untreated benchmark multi-class assignment solution. In addition to instability, the benchmark solution also exhibits varying degrees of arbitrariness, potentially rendering it unsuitable for assessing distributional effects across different groups, a capability crucial in applications concerning vertical equity and environmental justice. The proposed formulation and algorithm offer a practical remedy for these shortcomings.