A variational inequality formulation for inferring dynamic origin-destination travel demands

Yu Nie*, H. M. Zhang

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

42 Scopus citations


In this paper, we develop a relaxation strategy to the dynamic O-D estimation problem (DODE) problem. Cast as a variational inequality (VI), the DODE problem endogenizes the determination of the dynamic path-link incidence relationship (i.e., the dynamic assignment matrix) and takes users' response to traffic congestion into account. In our formulation, traffic dynamics on road links can be modeled by the Lighthill, Whitham and Richards theory, a delay-function model, or a point-queue model, coupled with CTM-like flow distribution models at nodes. Which model to use depends, of course, on specific modeling situations. Different from numerous previous studies, our formulation avoids the bi-level structure that poses analytical and numerical difficulties. This is achieved by balancing the path cost and the path deviation (the latter measures the difference between estimated and measured traffic conditions), weighed by a dispersion parameter which determines the extent to which users' behavior is respected. We prove the equivalence between the VI problem and the derived dynamic DODE optimality conditions, and establish the conditions under which a solution to the VI problem exists. A column generation algorithm is proposed to solve the VI problem. Numerical results based on synthetic data are also presented.

Original languageEnglish (US)
Pages (from-to)635-662
Number of pages28
JournalTransportation Research Part B: Methodological
Issue number7-8
StatePublished - Jan 1 2008


  • Bi-level
  • Column generation
  • Dynamic O-D estimation
  • Solution existence
  • Variational inequality

ASJC Scopus subject areas

  • Transportation
  • Management Science and Operations Research

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