A Multi-Class Multi-Mode Variable Demand Network Equilibrium Model with Hierarchical Logit Structures

  • Michael Florian
  • Jia Hao Wu
  • Shuguang He
Part of the Applied Optimization book series (APOP, volume 63)


We consider a multi-class multi-mode variable demand network equilibrium model where the mode choice model is given by aggregate hierarchical logit structures and the destination choice is specified as a multi-proportional entropy type trip distribution model. The travel time of transit vehicles depends on the travel time of other vehicles using the road network. A variational inequality formulation captures all the model components in an integrated form. A solution algorithm, based on a Block Gauss-Seidel decomposition approach coupled with the method of successive averages results in an efficient algorithm which successively solves network equilibrium models with fixed demands and multi-dimensional trip distribution models. Numerical results obtained with an implementation of the model with the EMME/2 software package are presented based on data originating from the city of Santiago, Chile.


Variational Inequality Mode Choice Travel Mode Network Equilibrium Traffic Equilibrium 
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  1. Abrahamsson, T. and Lundqvist, L. (1999). “Formulation and Estimation of Combined Network Equilibrium Models with Applications to Stockholm,” Transportation Science, 33, 1, 80–100.CrossRefGoogle Scholar
  2. Beckmann, M., McGuire, C.B. and Winsten, C.B. (1956). Studies in the Economics of Transportation. New Haven, CT: Yale University Press.Google Scholar
  3. Boyce, D. (1998). “Long-Term Advances in the State of the Art of Travel Forecasting Methods”, Equilibrium and Advanced Transportation Modeling, Edited by Marcotte, P. and Nguyen, S., Kluwer Acacemic Publishers.Google Scholar
  4. Dafermos, S. (1980). “Traffic Equilibria and Variational Inequalities,” Transportation Science, 14, 42–54.CrossRefGoogle Scholar
  5. Dafermos, S. (1982). “Relaxation Algorithms for the General Asymmetric Traffic Equilibrium Problem,” Transportation Science, 16 (2), 231–240.CrossRefGoogle Scholar
  6. ESTRAUS, (1998). Internal report.Google Scholar
  7. Evans, S.P. (1976). “Derivation and Analysis of Some Models for Combining Trip Distribution and Assignment,” Transportation Research, 10, 37–57.CrossRefGoogle Scholar
  8. Fernandez, E., De Cea, J., Florian, M. and Cabrera, E. (1994). “Network Equilibrium Models with Combined Modes,” Transportation Science, 3, 182–192.CrossRefGoogle Scholar
  9. Fisk, C.S. and Nguyen, S. (1982). “Solution Algorithms for Network Equilibrium Models with Asymmetric User Costs,” Transportation Science, 3, 361–381.CrossRefGoogle Scholar
  10. Fisk, C.S. and Boyce, D.E. (1983). “Alternative Variational Inequality Formulations of the Network Equilibrium-Travel Choice Problem,” Transportation Science, 4, 454–463.CrossRefGoogle Scholar
  11. Florian, M. (1977). “A Traffic Equilibrium Model of Travel by Car and Public Transit Modes,” Transportation Science, 2, 166–179.CrossRefGoogle Scholar
  12. Florian, M., Ferland, J. and Nguyen, S. (1975). “On the Combined Distributed-assignment of Traffic,” Transportation Science, 9, 45–53.Google Scholar
  13. Florian, M. and Nguyen, S. (1978). “A Combined Trip Distribution Modal Split and Trip Assignment Model,” Transportation Research, 12, 241–246.CrossRefGoogle Scholar
  14. Florian, M. and Spiess, H. (1982). “The Convergence of Diagonalization Algorithms for Asymmetric Network Equilibrium Problems,” Transportation Research, 16B, 447–483.CrossRefGoogle Scholar
  15. Frank, M. and Wolfe, P. (1956). “An Algorithm for Quadratic Programming,” Naval Research Logistics Quarterly, 3, 95–110.CrossRefGoogle Scholar
  16. Friesz, T.L. (1981). “An Equivalent Optimization Problem for Combined Multi-class Distribution, Assignment and Modal Split which Obviates Symmetry Restrictions.” Transportation Research, 15B, 5, 361–369.CrossRefGoogle Scholar
  17. Garrett, M. and Wachs, M. (1996). Transportation Planning on Trail: The Clean Air Act and Travel Forecasting. Thousand Oaks, CA: Sage Publications.Google Scholar
  18. Lam, W.H.K. and Hai-Jun, H. (1992). “A Combined Trip Distribution and Assignment Model for Multiple User Classes,” Transportation Research, 26B, 4, 275–287.CrossRefGoogle Scholar
  19. Oppenheim, N. (1995). Urban Travel Demand Modeling. A Wiley-Interscience Publication, John Wiley & Sons, Inc.Google Scholar
  20. Ortuzar, J. de D. and Willumsen, L.G. (1996). Modeling Transport. John Wiley & Sons, Second Edition.Google Scholar
  21. Pang, J.S. and Chan, D. (1982). “Iterative Methods for Variational and Complementarity Problems,” Mathematical Programming, 24, 284–313.CrossRefGoogle Scholar
  22. Safwat, K. and Magnanti, T. (1988). “A Combined Trip Generation, Trip Distribution, Modal Split, and Trip Assignment Model,” Transportation Science, 1, 14–30.CrossRefGoogle Scholar
  23. Safwat, K., Nabil, A. and Walton, C.M. (1988). “Computational Experience with an Application of a Simultaneous Transportation Equilibrium Model to Urban Travel in Austin, Texas,” Transportation Research, 22B, 6, 457–467.CrossRefGoogle Scholar
  24. Smith, M.J. (1979). “The Existence, Uniqueness and Stability of Traffic Equilibria,” Transportation Research, 13B, 289–294.Google Scholar

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© Springer Science+Business Media Dordrecht 2002

Authors and Affiliations

  • Michael Florian
  • Jia Hao Wu
  • Shuguang He

There are no affiliations available

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