Methods for Traffic Flow Assignment in Road Networks

  • Alexander KrylatovEmail author
  • Victor Zakharov
  • Tero Tuovinen
Part of the Springer Tracts on Transportation and Traffic book series (STTT, volume 15)


In this chapter is devoted to approaches for solving traffic flow assignment problems. The most popular gradient descent method for solving traffic assignment problems is discussed in the first section. New projection algorithms based on the obtained, explicitly fixed-point operators for the route-flow assignment problem and link-route assignment problem are presented in the third and fourth sections respectively. Obtained operators is proved to be contractive that leads to the linear convergence of provided algorithms. Moreover, under some fairly natural conditions the algorithms converge quadratically. The technique for representing a linear route-flow assignment problem in the form of a system of linear equations is presented in the fourth section. A simple example demonstrates the evident usability of the developed technique for its implementation and further extensions.


  1. 1.
    Frank M, Wolfe P (1956) An algorithm for quadratic programming. Naval Res Logist Quart 3:95–110MathSciNetCrossRefGoogle Scholar
  2. 2.
    Gartner NH (1980) Optimal traffic assignment with elastic demands: a review Part II: algorithmic approaches. Transp Sci 14(2):192–208CrossRefGoogle Scholar
  3. 3.
    Van Vliet D, Dow PDC (1979) Capacity-restrained road assignment. Traffic Eng Control 20:296–305Google Scholar
  4. 4.
    Van Vliet D (1976) Road assignment – I: principles and parameters of model formulation. Transp Res 10:137–143CrossRefGoogle Scholar
  5. 5.
    Larsson T, Patriksson M (1992) Simplicial decomposition with disaggregated representation for the traffic assignment problem. Transp Sci 26:4–17CrossRefGoogle Scholar
  6. 6.
    Rose G, Daskin MS, Koppelman FS (1988) An examination of convergence error in equilibrium traffic assignment models. Transp Res 22B:261–274CrossRefGoogle Scholar
  7. 7.
    Golden BL (1975) A minimum-cost multicommodity network flow problem concerning imports and exports. Networks 5:331–356MathSciNetCrossRefGoogle Scholar
  8. 8.
    LeBlanc LJ, Morlok EK, Pierskalla WP (1974) An accurate and efficient approach to equilibrium traffic assignment on congested networks. Transp Res Record 491:12–23Google Scholar
  9. 9.
    LeBlanc LJ, Morlok EK, Pierskalla WP (1975) An efficient approach to solving the road network equilibrium traffic assignment problem. Transp Res 9:309–318CrossRefGoogle Scholar
  10. 10.
    Nguyen S (1973) A mathematical programming approach to equilibrium methods of traffic assignment with fixed demands. Publication 138. Départment d’Informatique et de Reserche Opérationelle. Université de Montréal, MontréalGoogle Scholar
  11. 11.
    Steenbrink PA (1974) Optimization of transport networks. Wiley, LondonGoogle Scholar
  12. 12.
    Fratta L, Gerla M, Kleinrock L (1973) The flow-deviation method: an approach to store-and-forward computer communication network design. Networks 3:97–133MathSciNetCrossRefGoogle Scholar
  13. 13.
    Klessig RW (1974) An algorithm for nonlinear multicommodity flow problems. Networks 4:343–355MathSciNetCrossRefGoogle Scholar
  14. 14.
    Yaged B Jr (1971) Minimum cost routing for static network models. Networks 1:139–172MathSciNetCrossRefGoogle Scholar
  15. 15.
    Patriksson M (2015) The traffic assignment problem: models and methods. Dover Publications Inc., Mineola, N.Y.Google Scholar
  16. 16.
    Florian M, Nguyen S (1976) An application and validation of equilibrium trip assignment methods. Transp Sci 10:374–390CrossRefGoogle Scholar
  17. 17.
    Dijkstra EW (1959) A note on two problems in connexion with graphs. Numerische Mathematik 1:269–271MathSciNetCrossRefGoogle Scholar
  18. 18.
    Gallo G, Pallottino S (1982) A new algorithm to find the shortest paths between all pairs of nodes. Discrete Appl Math 4:23–35MathSciNetCrossRefGoogle Scholar
  19. 19.
    Gallo G, Pallottino S (1986) Shortest path methods: a unifying approach. Math Progr Study 26:38–64MathSciNetCrossRefGoogle Scholar
  20. 20.
    Halder AK (1970) The method of competing links. Transp Sci 4:36–51CrossRefGoogle Scholar
  21. 21.
    Loubal PS (1967) A network evaluation procedure. Highway Res Record 205:96–109Google Scholar
  22. 22.
    Rodionov VV (1968) The parametric problem of shortest distances. U.S.S.R. Comput Math Math Phys 8:336–343CrossRefGoogle Scholar
  23. 23.
    Wollmer R (1964) Removing arcs from a network. Oper Res 12:934–940MathSciNetCrossRefGoogle Scholar
  24. 24.
    Canon MD, Cullum CD (1968) A tight upper bound on the rate of convergence of the Frank-Wolfe algorithm. SIAM J Control 6:509–516MathSciNetCrossRefGoogle Scholar
  25. 25.
    Dem’janov VF, Rubinov AM (1965) On the problem of minimization of a smooth functional with convex constraints. Soviet Math Doklady 6:9–11Google Scholar
  26. 26.
    Dunn JC (1979) Rates of convergence for conditional gradient algorithms near singular and nonsingular extremals. SIAM J Control Optim 17:187–211MathSciNetCrossRefGoogle Scholar
  27. 27.
    Dunn JC (1980) Convergence rates for conditional gradient sequences generated by implicit step length rules. SIAM J Control Optim 18:473–487MathSciNetCrossRefGoogle Scholar
  28. 28.
    Levitin ES, Polyak BT (1966) Constrained minimization methods. USSR Compu Math Math Phys 6:1–50CrossRefGoogle Scholar
  29. 29.
    Zangwill WI (1969) Nonlinear programming: a unified approach. Prentice-Hall Inc, Englewood Clifs, N.JzbMATHGoogle Scholar
  30. 30.
    Wolfe P (1970) Convergence theory in nonlinear programming. In: Abadie J (ed) Integer and Nonlinear Programming. North-Holland, Amsterdam (1970)Google Scholar
  31. 31.
    Zangwill WI (1969) Convergence conditions for nonlinear programming algorithms. Manag Sci 16:1–13MathSciNetCrossRefGoogle Scholar
  32. 32.
    Lupi M (1986) Convergence of the Frank-Wolfe algorithm in transportation networks. Civil Eng Syst 3:7–15CrossRefGoogle Scholar
  33. 33.
    Janson BN, Zozaya-Gorostiza C (1987) The problem of cyclic flows in traffic assignment. Transp Res 21B:299–310CrossRefGoogle Scholar
  34. 34.
    Konnov IV (2017) An adaptive partial linearization method for optimization problems on product sets. J Optim Theory Appl 175(2):478–501MathSciNetCrossRefGoogle Scholar
  35. 35.
    Krylatov AY (2016) Network flow assignment as a fixed point problem. J Appl Ind Math 10(2):243–256MathSciNetCrossRefGoogle Scholar
  36. 36.
    Marcotte P (1986) Network design problem with congestion effects: a case of bilevel programming. Math Progr 34(2):142–162MathSciNetCrossRefGoogle Scholar
  37. 37.
    Greenshields BD (1934) A study of traffic capacity. Proc (US) Highway Res Board 14:448–494Google Scholar
  38. 38.
    Long J, Gao Z, Zhang H, Szeto WY (2010) A turning restriction design problem in urban road networks. Eur J Oper Res 206(3):569–578MathSciNetCrossRefGoogle Scholar
  39. 39.
    Yang H, Yagar S (1995) Traffic assignment and signal control in saturated road networks. Transp Res Part A 29(2):125–139CrossRefGoogle Scholar
  40. 40.
    Krylatov AY, Shirokolobova AP (2017) Projection approach versus gradient descent for networks flows assignment problem. Lect Notes Comput Sci 10556:345–350CrossRefGoogle Scholar
  41. 41.
    Krylatov AY (2018) Reduction of a minimization problem for a convex separable function with linear constraints to a fixed point problem. J Appl Ind Math 12(1):98–111MathSciNetCrossRefGoogle Scholar
  42. 42.
    Sheffi Y (1985) Urban transportation networks: equilibrium analysis with mathematical programming methods. Prentice-Hall Inc, Englewood Cliffs, N.JGoogle Scholar

Copyright information

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Institute of Transport ProblemsRussian Academy of SciencesSaint PetersburgRussia
  2. 2.Faculty of Applied Mathematics and Control ProcessesSaint Petersburg State UniversitySaint PetersburgRussia
  3. 3.Faculty of Information TechnologyUniversity of JyväskyläJyväskyläFinland

Personalised recommendations