Skip to main content

Column generation for the equilibrium route-flow traffic assignment problem


Today efficient traffic management seems to be impossible without the support of the artificial intelligence systems based on mathematical models of traffic flow assignment since a modern road network is a large-scale system with huge amounts of elements. The present paper is devoted to the route-flow traffic assignment problem, which solution is the most valuable from decision-making perspectives. The paper aims to fill the gap in the relation between the column generation process and the uniqueness of the equilibrium route-flow traffic assignment pattern. The independence of routes is showed to be highly significant when travel time functions are arc-additive. Indeed, on the one hand, the independence of routes is proved to guarantee the uniqueness of the equilibrium route-flow traffic assignment pattern. On the other hand, the independence of routes appears to be crucial for column generation when solving the route-flow traffic assignment problem since the equilibrium travel time is proven to be decreased only by adding independent candidate route. Obtained results contribute to the development of algorithms for route-flow traffic assignment based on travel times equilibration procedure.

This is a preview of subscription content, access via your institution.


  1. Bar-Gera, H., Luzon, A.: Differences among route flow solutions for the user-equilibrium traffic assignment problem. J. Transp. Eng. 133(4), 232–239 (2007)

    Article  Google Scholar 

  2. Bar-Gera, H., Luzon, A.: Non-unique route flow solutions for user-equilibrium assignments. Traffic Eng. Control 48(9), 408–412 (2007)

    Google Scholar 

  3. Beckmann, M., McGuire, C., Winsten, C.: Studies in the Economics of Transportation. Yale University Press, New Haven Conn (1956)

    Google Scholar 

  4. Bertsekas, D.P.: Nonlinear Programming, 2nd edn. Athena Scientific, Belmont (1999)

    Google Scholar 

  5. Chao, S., Chenga, L., Xub, T.: Range of user-equilibrium route flow with applications. Procedia – Soc. Behav. Sci. 138, 86–96 (2014)

    Article  Google Scholar 

  6. Chen, A., Jayakrishnan, R.: A path-based gradient projection algorithm: Effects of equilibration with a restricted path set under two flow update policies. In: Transportation Research Board Annual Meeting, pp. 1–19 (1998)

  7. Dafermos, S., Sparrow, F.: The traffic assignment problem for a general network. J. Res. Natl. Bur. Stand. 73B(2), 91–118 (1969)

    MathSciNet  Article  Google Scholar 

  8. Di Lorenzo, D., Galligari, A., Sciandrone, M.: A convergent and efficient decomposition method for the traffic assignment problem. Comput. Optim. Appl. 60(1), 151–170 (2015)

    MathSciNet  Article  Google Scholar 

  9. Fisk, C.: A nonlinear equation framework for solving network equilibrium problems. Environ. Plan. 16A(1), 67–80 (1984)

    Article  Google Scholar 

  10. Fisk, C., Nguyen, S.: A unified approach for the solution of network equilibrium problems Publication, vol. 169. Centre de sur les transports, Université de Montréal, Montréal (1980)

    Google Scholar 

  11. Florian, M., Constantin, I., Florian, D.: A new look at projected gradient method for equilibrium assignment. Transp. Res. Rec. 2090, 10–16 (2009)

    Article  Google Scholar 

  12. Galligari, A., Sciandrone, M.: A convergent and fast path equilibration algorithm for the traffic assignment problem. Optim. Methods Softw. 33(2), 354–371 (2018)

    MathSciNet  Article  Google Scholar 

  13. Galligari, A., Sciandrone, M.: A computational study of path-based mathods for optimal traffic assignment with both inelastic and elastic demand. Comput. Oper. Res. 103, 158–166 (2019)

    MathSciNet  Article  Google Scholar 

  14. Jayakrishnan, R., Tsai, W., Prashker, J., Rajadhyaksha, S.: A faster path-based algorithm for traffic assignment. Transp. Res. Rec. 1443, 75–83 (1994)

    Google Scholar 

  15. Krylatov, A.: Network flow assignment as a fixed point problem. J. Appl. Ind. Math. 10(2), 243–256 (2016)

    MathSciNet  Article  Google Scholar 

  16. Krylatov, A.: Reduction of a minimization problem for a convex separable function with linear constraints to a fixed point problem. J. Appl. Ind. Math. 12 (1), 98–111 (2018)

    MathSciNet  Article  Google Scholar 

  17. Krylatov, A., Raevskaya, A.: Travel times equilibration procedure for route-flow traffic assignment problem. Lect. Notes Comput. Sci., 12096 (2020)

  18. Krylatov, A., Zakharov, V., Tuovinen, T.: Optimization Models and Methods for Equilibrium Traffic Assignment. Springer International Publishing (2020)

  19. Kumar, A., Peeta, S.: An improved social pressure algorithm for the static deterministic user equilibrium traffic assignment problem. In: Washington DC: Transportation Research Board of the National Academics (2011)

  20. Kumar, A., Peeta, S.: Entropy weighted average method for the determination of a single representative path flow solution for the static user equilibrium traffic assignment problem. Transp. Res. B 71, 213–229 (2015)

    Article  Google Scholar 

  21. Kupsizewska, D., Van Vliet, D.: 101 uses for path-based assignment. In: European Transport Conference (1999)

  22. Larsson, T., Patriksson, M.: Simplicial decomposition with disaggregated representation for the traffic assignment problem. Transp. Sci. 26(1), 4–17 (1992)

    Article  Google Scholar 

  23. LeBlanc, L.: Mathematical programming algorithms for large scale network equilibrium and network design problem. Department of Industrial Engineering and Management Science Northwestern University (1973)

  24. LeBlanc, L., Morlok, E., Pierskalla, W.: An efficient approach to solving the road network equilibrium traffic assignment problem. Transp. Res. 9, 309–318 (1975)

    Article  Google Scholar 

  25. Leventhal, T., Nemhauser, G., Trotter, L.J.: A column generation algorithm for optimal traffic assignment. Transp. Sci. 7(2), 168–176 (1973)

    MathSciNet  Article  Google Scholar 

  26. Nguyen, S.: A mathematical programming approach to equilibrium methods of traffic assignment with fixed demands. Publication 138. Départment d’Informatique et de Recherche Opérationelle, Université de Montréal, Montréal (1973)

    Google Scholar 

  27. Patriksson, M.: The Traffic Assignment Problem: Models and Methods. VSP, Utrecht (1994)

    MATH  Google Scholar 

  28. Perederieieva, O., Ehrgott, M., Raith, A., Wang, J.: A framework for and empirical study of algorithms for traffic assignment. Comput. Oper. Res. 54, 90–107 (2015)

    MathSciNet  Article  Google Scholar 

  29. Sender, J., Netter, M.: Équilibre offre-demande et tarification sur un réseau de transport Rapport de recherche, vol. 3. Département Economie, Institut de Recherche des Transports, Arcueil, France (1970)

    Google Scholar 

  30. Sheffi, Y.: Urban Transportation Networks: Equilibrium Analysis with Mathematical Programming Methods. Prentice-Hall, Inc, Englewood Cliffs (1985)

    Google Scholar 

  31. Sun, C., Jayakrishnan, R., Tsai, W.: Computational study of a path-based algorithm and its variants for static traffic assignment. Transp. Res. Rec. 1537(1), 106–115 (1996)

    Article  Google Scholar 

  32. Wardrop, J.G.: Some theoretical aspects of road traffic research. In: Proc. Institution of Civil Engineers, vol. 2, pp. 325–378 (1952)

Download references


The work was jointly supported by a grant from the Russian Science Foundation (No. 19-71-10012 Multi-agent systems development for automatic remote control of traffic flows in congested urban road networks).

Author information

Authors and Affiliations


Corresponding author

Correspondence to Alexander Krylatov.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Krylatov, A. Column generation for the equilibrium route-flow traffic assignment problem. Ann Math Artif Intell 90, 697–713 (2022).

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • Nonlinear optimization
  • Column generation
  • Traffic assignment problem
  • User-equilibrium of Wardrop

Mathematics Subject Classification (2010)

  • 90C30
  • 90C33
  • 90C35