A semi-analytical approach for solving the bottleneck model with general user heterogeneity

Yang Liu, Yu Marco Nie*, Jonathan Hall

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

47 Scopus citations


This paper proposes a novel semi-analytical approach for solving the dynamic user equilibrium (DUE) of a bottleneck model with general heterogeneous users. The proposed approach makes use of the analytical solutions from the bottleneck analysis to create an equivalent assignment problem that admits closed-form commute cost functions. The equivalent problem is a static and asymmetric traffic assignment problem, which can be formulated as a variational inequality problem (VIP). This approach provides a new tool to analyze the properties of the bottleneck model with general heterogeneity, and to design efficient solution methods. In particular, the existence and uniqueness of the DUE solution can be established using the P-property of the Jacobian matrix. Our numerical experiments show that a simple decomposition algorithm is able to quickly solve the equivalent VIP to high precision. The proposed VIP formation is also extended to address simultaneous departure time and route choice in a single O-D origin-destination network with multiple parallel routes.

Original languageEnglish (US)
Pages (from-to)56-70
Number of pages15
JournalTransportation Research Part B: Methodological
StatePublished - Jan 1 2015


  • Bottleneck model
  • Dynamic user equilibrium
  • General heterogeneity
  • P-property
  • Variational inequality problem

ASJC Scopus subject areas

  • Civil and Structural Engineering
  • Transportation


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