Solving the step-tolled bottleneck model with general user heterogeneity

Hongyu Chen, Yang Liu, Yu Marco Nie*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

20 Scopus citations


Two new numerical methods, a semi-analytical method and an exact method, are proposed in this paper for solving the step-tolled user equilibrium problem for the bottleneck model with general user heterogeneity. The semi-analytical method transforms the step-tolled user equilibrium problem into a static traffic assignment problem with side constraints, which is formulated and solved as a variational inequality problem (VIP). Existence and uniqueness results are obtained by examining the properties of the VIP. The exact method, on the other hand, locates the equilibrium solution by enumerating all possible combinations of user departure orders. To improve the computational efficiency of the exact method, a simple heuristic is developed to reduce the number of solutions to be enumerated. Examples of up to 1000 classes are tested. The results confirm that ignoring user heterogeneity may lead to sub-optimal design of step tolls. In all experiments, the semi-analytical method is found to be more efficient than the heuristic version of the exact method by an order of magnitude, albeit they consistently produce nearly identical equilibrium solutions.

Original languageEnglish (US)
Pages (from-to)210-229
Number of pages20
JournalTransportation Research Part B: Methodological
StatePublished - Nov 1 2015


  • Bottleneck model
  • General user heterogeneity
  • Semi-analytical method
  • Step toll
  • Variational inequality problem

ASJC Scopus subject areas

  • Civil and Structural Engineering
  • Transportation


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