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Principles of Wardrop for Traffic Assignment in a Road Network

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

Abstract

In this chapter is devoted to user equilibrium and system optimum of Wardrop. Discussion on the mathematical formulation of traffic assignment problems with regard to their meaning is available in the Sect. 2.1. The specification of necessary basic statements completes this discussion further. The dual traffic assignment problem with travel times between all origins and destinations as dual variables is considered in the Sect. 2.2. The practical significance of such dual formulation is shown to become evident due to the wide spread of online traffic services. The route-flow assignment problem and link-flow assignment problem are reduced to fixed-point problems with explicit operators in the Sect. 2.3 and Sect. 2.4 respectively. Proofs of corresponding theorems are fully given.

References

  1. 1.
    Wardrop JG (1952) Some theoretical aspects of road traffic research. Proc Inst Civil Eng 2:325–378Google Scholar
  2. 2.
    Pang J-S, Fukushima M (2005) Quasi-variational inequalities, generalized Nash equilibria, and multileader-follower games. Comput Manag Sci 2(1):21–56MathSciNetCrossRefGoogle Scholar
  3. 3.
    Xie J, Yu N, Yang X (2013) Quadratic approximation and convergence of some bush-based algorithms for the traffic assignment problem. Transp Res Part B 56:15–30CrossRefGoogle Scholar
  4. 4.
    Sheffi Y (1985) Urban transportation networks: equilibrium analysis with mathematical programming methods. Prentice-Hall Inc., Englewood Cliffs, NJGoogle Scholar
  5. 5.
    Beckmann MJ, McGuire CB, Winsten CB (1956) Studies in the economics of transportation. Yale University Press, New Haven, CTGoogle Scholar
  6. 6.
    Dafermos SC, Sparrow FT (1969) The traffic assignment problem for a general network. J Res Nat Bureau Stand 73B:91–118MathSciNetCrossRefGoogle Scholar
  7. 7.
    Dafermos SC (1968) Traffic assignment and resource allocation in transportation networks. PhD thesis. Johns Hopkins University, Baltimore, MDGoogle Scholar
  8. 8.
    Patriksson M (2015) The traffic assignment problem: models and methods. Dover Publications Inc., New YorkGoogle Scholar
  9. 9.
    Shen W, Zhang HM (2009) On the morning commute problem in a corridor network with multiple bottlenecks: its system-optimal traffic flow patterns and the realizing tolling scheme. Transp Res Part B 43:267–284CrossRefGoogle Scholar
  10. 10.
    Ahuja RK, Magnanti TL, Orlin JB (1993) Network flows: theory, algorithms, and applications. Prentice-Hall, Englewood Cliffs, NJzbMATHGoogle Scholar
  11. 11.
    Bertsekas DP (1999) Nonlinear programming, 2nd edn. Athena Scientific, Belmont, MAzbMATHGoogle Scholar
  12. 12.
    Krylatov AY, Shirokolobova AP, Zakharov VV (2016) OD-matrix estimation based on a dual formulation of traffic assignment problem. Informatica (Slovenia) 40(4):393–398MathSciNetGoogle Scholar
  13. 13.
    Fisk C (1984) A nonlinear equation framework for solving network equilibrium problems. Environ Plan 16A:67–80CrossRefGoogle Scholar
  14. 14.
    Fisk C, Nguyen S (1980) A unified approach for the solution of network equilibrium problems. Publication 169. Centre de rechercher sur les transports, Universite de Montreal, Montreal (1980)Google Scholar
  15. 15.
    Chen R-J, Meyer RR (1988) Parallel optimization for traffic assignment. Math Program 42:327–345MathSciNetCrossRefGoogle Scholar
  16. 16.
    Patriksson M (1993) A unified description of iterative algorithms for traffic equilibria. Eur J Oper Res 71:154–176CrossRefGoogle Scholar
  17. 17.
    Krylatov AY (2016) Network flow assignment as a fixed point problem. J Appl Ind Math 10(2):243–256MathSciNetCrossRefGoogle Scholar
  18. 18.
    Swamy MNS, Thulasiraman K (1981) Graphs, networks, and algorithms. Wiley, New YorkGoogle Scholar
  19. 19.
    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
  20. 20.
    Gantmacher F (1959) Theory of matrices. AMS Chelsea Publishing, New YorkzbMATHGoogle 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

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