Dynamic user equilibrium with side constraints for a traffic network: Theoretical development and numerical solution algorithm

R. X. Zhong, Agachai Sumalee, T. L. Friesz, Hing Keung William Lam

Research output: Journal article publicationJournal articleAcademic researchpeer-review

40 Citations (Scopus)


This paper investigates a traffic volume control scheme for a dynamic traffic network model which aims to ensure that traffic volumes on specified links do not exceed preferred levels. The problem is formulated as a dynamic user equilibrium problem with side constraints (DUE-SC) in which the side constraints represent the restrictions on the traffic volumes. Travelers choose their departure times and routes to minimize their generalized travel costs, which include early/late arrival penalties. An infinite-dimensional variational inequality (VI) is formulated to model the DUE-SC. Based on this VI formulation, we establish an existence result for the DUE-SC by showing that the VI admits at least one solution. To analyze the necessary condition for the DUE-SC, we restate the VI as an equivalent optimal control problem. The Lagrange multipliers associated with the side constraints as derived from the optimality condition of the DUE-SC provide the traffic volume control scheme. The control scheme can be interpreted as additional travel delays (either tolls or access delays) imposed upon drivers for using the controlled links. This additional delay term derived from the Lagrange multiplier is compared with its counterpart in a static user equilibrium assignment model. If the side constraint is chosen as the storage capacity of a link, the additional delay can be viewed as the effort needed to prevent the link from spillback. Under this circumstance, it is found that the flow is incompressible when the link traffic volume is equal to its storage capacity. An algorithm based on Euler's discretization scheme and nonlinear programming is proposed to solve the DUE-SC. Numerical examples are presented to illustrate the mechanism of the proposed traffic volume control scheme.
Original languageEnglish
Pages (from-to)1035-1061
Number of pages27
JournalTransportation Research Part B: Methodological
Issue number7
Publication statusPublished - 1 Jan 2011


  • Dynamic user equilibrium (DUE)
  • Lagrange multiplier
  • Pontryagin minimum principle
  • Traffic assignment with side constraint
  • Traffic management and control
  • Traffic volume control

ASJC Scopus subject areas

  • Transportation
  • Management Science and Operations Research


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