A Stochastic Version of the Strategy-Based Congested Transit Assignment Model and a Technique by Smoothing Approximations

Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 572)


This paper develops a stochastic version for the strategy-based congested transit assignment problem stated by Cominetti and Correa (Trans. Sci. 35(3):250–267, 2001). As a distinctive approach, this stochastic version takes into account stochastic mean waiting times of passengers at stops and in-vehicle travel times. The model is formulated as a stochastic variational inequality derived from the formulation of the deterministic version of the problem, also stated as a variational inequality problem, for which only a single solution method is known uptodate. Closely related with the stochastic model, and as a special case of it, a consistent smoothing approximation to the deterministic model is developed and it is shown that this approximation provides an alternative way of solving the deterministic model. It is also shown that both, the stochastic model and the smoothed approximation, can be solved by means of an adaptation of a path based method for the asymmetric traffic assignment problem. Computational tests have been carried out on several medium-large scale networks showing the viability of the method and its applicability to large scale transit models.


Congested transit assignment Stochastic variational inequalities Strategy-based transit equilibrium Smoothing approximation 



Research supported under Spanish Research Projects TRA2008-06782-C02-02, TRA2014-52530-C3-3-P.


  1. 1.
    Auslender, A.: Optimisation: Méthodes Numériques. Masson, Paris (1976)zbMATHGoogle Scholar
  2. 2.
    Birge, J.R., Louveaux, F.: Introduction to Stochastic Programming. Springer Series in Operations Research. Springer, New York (1997)zbMATHGoogle Scholar
  3. 3.
    Cascetta, E.: Transportation Systems Analysis. Models and Applications. Springer Optimization and Its Applications Series, 2nd edn. Springer, New York (2009)CrossRefzbMATHGoogle Scholar
  4. 4.
    Chen, X., Wets, R., Zhang, Y.: Stochastic variational inequalities: residual minimization smoothing sample average approximations. SIAM J. Optim. 22(2), 649–673 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cepeda, M., Cominetti, R., Florian, M.: A frequency-based assignment model for congested transit networks with strict capacity constraints: characterization and computation of equilibria. Transp. Res. B 40, 437–459 (2006)CrossRefGoogle Scholar
  6. 6.
    Chriqui, C., Robillard, P.: Common bus lines. Trans. Sci. 9, 115–121 (1975)CrossRefGoogle Scholar
  7. 7.
    Codina, E.: A variational inequality reformulation of a congested transit assignment model by Cominetti, Correa. Cepeda and Florian. Trans. Sci. 47(2), 231–246 (2013)CrossRefGoogle Scholar
  8. 8.
    Codina, E., Ibáñez, G., Barceló, J.: Applying projection-based methods to the asymmetric traffic assignment problem. Comput. Aided Civ. Infrastruct. Eng. 30(2), 103–119 (2015)CrossRefGoogle Scholar
  9. 9.
    Cominetti, R., Correa, J.: Common-lines and passenger assignment in congested transit networks. Trans. Sci. 35(3), 250–267 (2001)CrossRefzbMATHGoogle Scholar
  10. 10.
    Cortés, C.E., Jara-Moroni, P., Moreno, E., Pineda, C.: Stochastic transit equilibrium. Transp. Res. Part B 51, 29–44 (2013)CrossRefGoogle Scholar
  11. 11.
    De Cea, J., Fernández, E.: Transit assignment for congested public transport systems: an equilibrium model. Trans. Sci. 27(2), 133–147 (1993)CrossRefzbMATHGoogle Scholar
  12. 12.
    Dontchev, A.L., Rockafellar, R.T.: Ample parametrization of variational inclusions. SIAM J. Optim. 12(1), 170–187 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Fu, Q., Liu, R., Hess, S.: A review on transit assignment modelling approaches to congested networks: a new perspective. Procedia Soc. Behav. Sci. 54, 1145–1155 (2012)CrossRefGoogle Scholar
  14. 14.
    Gentile, G., Florian, M., Hamdouch, Y., Cats, O., Nuzzolo, A.: The theory of transit assignment: basic modelling frameworks. In: Gentile, G., Noekel, K. (eds.) Modelling Public Transport Passenger Flows in the Era of Intelligent Transport Systems. Springer Tracts on Transportation and Traffic, vol. 10, pp. 287–386. Springer, Cham (2016). doi: 10.1007/978-3-319-25082-3_6 CrossRefGoogle Scholar
  15. 15.
    Gentile, G., Noekel, K. (eds.): Modelling Public Transport Passenger Flows in the Era of Intelligent Transportation Systems. Springer Tracts on Transportation and Traffic. Springer, Cham (2016)zbMATHGoogle Scholar
  16. 16.
    Jiang, Y., Szeto, W.Y.: Reliability-based stochastic transit assignment: formulations and capacity paradox. Transp. Res. Part B 93, 181–206 (2016)CrossRefGoogle Scholar
  17. 17.
    Jiang, Y., Szeto, W.Y., Ng, T.M., Ho, S.C.: The reliability-based stochastic transit assignment problem with elastic demand. J. Eastern Asia Soc. Transp. Stud. 10, 831–850 (2013)Google Scholar
  18. 18.
    Khobotov, E.N.: Modification of the extragradient method for solving variational inequalities and certain optimization problems. USSR Comput. Math. Math. Phys. 27, 120–127 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Lam, W.H.K., Gao, Z.Y., Chan, K.S., Yang, N.: A stochastic user equilibrium assignment model for congested transit networks. Transp. Res. 33B(5), 351–368 (1999)CrossRefGoogle Scholar
  20. 20.
    Liu, Y., Bunker, J., Ferreira, L.: Transit user’s route-choice modelling in transit assignment: a review. Transp. Rev. 30(6), 753–769 (2010)CrossRefGoogle Scholar
  21. 21.
    Nielsen, O.A.: A stochastic transit assignment model considering differences in passenger utility functions. Transp. Res. Part B 34, 377–402 (2000)CrossRefGoogle Scholar
  22. 22.
    Nguyen, S., Pallotino, S.: Equilibrium traffic assignment in large scale transit networks. Eur. J. Oper. Res. 37(2), 176–186 (1988)CrossRefGoogle Scholar
  23. 23.
    Peng, J.-M.: A smoothing function and its applications. In: Fukushima, M., Qi, L. (eds.) Reformulation: Nonsmooth, Piecewise Smooth and Smoothing Methods. Applied Optimization Series, pp. 293–316. Kluwer Academic Publishers, Dordrecht (1999)Google Scholar
  24. 24.
    Rockafellar, T., Wets, R.J.-B.: Variational Analysis. Grundlehren der mathematischen Wissenschaften, vol. 317. Springer, Heidelberg (1998). doi: 10.1007/978-3-642-02431-3 zbMATHGoogle Scholar
  25. 25.
    Spiess, H.: Contribution à la théorie et aux outils de planification des réseaux de transport urbains. Ph D thesis, Département d’Informatique et Récherche Opérationnelle, Publication 382, CRT, U. de Montréal (1984)Google Scholar

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© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Statistics and Operations Research DepartmentUniversitat Politècnica de CatalunyaBarcelonaSpain

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