Bilevel Optimal Tolls Problems with Nonlinear Costs: A Heuristic Solution Method

  • Vyacheslav KalashnikovEmail author
  • José Guadalupe Flores Muñiz
  • Nataliya Kalashnykova
Part of the Studies in Computational Intelligence book series (SCI, volume 835)


We consider a bilevel programming problem modeling the optimal toll assignment as applied to an abstract network of toll and free highways. A public governor or a private lease company run the toll roads and make decisions at the upper level when assigning the tolls with the aim of maximizing their profits. The lower level decision makers (highway users), however, search an equilibrium among them while trying to distribute their transportation flows along the routes that would minimize their total travel costs subject to the satisfied demand for their goods/passengers. Our model extends the previous ones by adding quadratic terms to the lower level costs thus reflecting the mutual traffic congestion on the roads. Moreover, as a new feature, the lower level quadratic costs aren’t separable anymore, i.e., they are functions of the total flow along the arc (highway). In order to solve the bi-level programming problem, a heuristic algorithm making use of the sensitivity analysis techniques for quadratic programs is developed. As a remedy against being stuck at a local maximum of the upper level objective function, we adapt the well-known “filled function” method which brings us to a vicinity of another local maximum point. A series of numerical experiments conducted on test models of small and medium size shows that the new algorithm is competitive enough.



The authors’ research activity was financially supported by the SEP-CONACYT (Mexico) grants CB-2013-01-221676 and FC-2016-01-1938.


  1. 1.
    J.C.G. Boot, On sensitivity analysis in convex quadratic programming problems. Oper. Res. 11, 771–786 (1963)MathSciNetCrossRefGoogle Scholar
  2. 2.
    L. Brotcorne, Operational and strategic approaches to traffic routers’ problems (in French). Ph.D. dissertation, Université Libre de Bruxelles (1998)Google Scholar
  3. 3.
    L. Brotcorne, F. Cirinei, P. Marcotte, G. Savard, An exact algorithm for the network pricing problem. Discret. Optim. 8(2), 246–258 (2011)MathSciNetCrossRefGoogle Scholar
  4. 4.
    L. Brotcorne, F. Cirinei, P. Marcotte, G. Savard, A Tabu search algorithm for the network pricing problem. Comput. Oper. Res. 39(11), 2603–2611 (2012)MathSciNetCrossRefGoogle Scholar
  5. 5.
    S. Dempe, T. Starostina, Optimal toll charges: fuzzy optimization approach, in Methods of Multicriteria Decision - Theory and Applications, ed. by F. Heyde, A. Lóhne, C. Tammer (Shaker Verlag, Aachen, 2009), pp. 29–45Google Scholar
  6. 6.
    S. Dempe, V.V. Kalashnikov, G.A. Pérez, N.I. Kalashnykova, Bilevel Programming Problems: Theory, Algorithms and Applications to Energy Networks (Springer, Berlin-Heidelberg, 2015)CrossRefGoogle Scholar
  7. 7.
    S. Dempe, A.B. Zemkoho, Bilevel road pricing; theoretical analysis and optimality conditions. Ann. Oper. Res. 196(1), 223–240 (2012)MathSciNetCrossRefGoogle Scholar
  8. 8.
    M. Didi-Biha, P. Marcotte, G. Savard, Path-based formulation of a bilevel toll setting problem, in Optimization with Multi-Valued Mappings: Theory, ed. by S. Dempe, V.V. Kalashnikov (Applications and Algorithms, Springer Science, Boston, MA, 2006), pp. 29–50Google Scholar
  9. 9.
    J.G. Flores-Muñiz, V.V. Kalashnikov, V. Kreinovich, N.I. Kalashnykova, Gaussian and Cauchy functions in the filled function method why and what next: on the example of optimizing road tolls. Acta Polytecnica Hung. 14(13), 237–250 (2017)Google Scholar
  10. 10.
    A.G. Hadigheh, O. Romanko, T. Terlaky, Sensitivity analysis in convex quadratic optimization: Simultaneous perturbation of the objective and right-hand-side vectors. Algorithmic Oper. Res. 2, 94–111 (2007)MathSciNetzbMATHGoogle Scholar
  11. 11.
    B. Jansen, Interior Point Techniques in Optimization: Complementarity, Sensitivity and Algorithms, (Dordrecht, The Netherlands: Springer-Science+Business Media, B.V, 1997)Google Scholar
  12. 12.
    V.V. Kalashnikov, F. Camacho, R. Askin, N.I. Kalashnykova, Comparison of algorithms solving a bilevel toll setting problem. Int. J. Innov. Comput. Inf. Control 6(8), 3529–3549 (2010)Google Scholar
  13. 13.
    V.V. Kalashnikov, R.C. Herrera, F. Camacho, N.I. Kalashnykova, A heuristic algorithm solving bilevel toll optimization problems. Int. J. Logist. Manag. 27(1), 31–51 (2016)CrossRefGoogle Scholar
  14. 14.
    V.V. Kalashnikov, N.I. Kalashnykova, R.C. Herrera, Solving bilevel toll optimization problems by a direct algorithm using sensitivity analysis, in Proceedings of the 2011 New Orleans International Academic Conference, (New Orleans, LA, March 21–23, 2011) pp. 1009–1018Google Scholar
  15. 15.
    V.V. Kalashnikov, V. Kreinovich, J.G. Flores-Muñiz, N.I. Kalashnykova, Structure of filled functions: why Gaussian and Cauchy templates are most efficient, to appear in Int. J. Comb. Optim. Probl. Inform. 7 (2017)Google Scholar
  16. 16.
    M. Labbé, P. Marcotte, G. Savard, A bilevel model of taxation and its applications to optimal highway pricing. Manag. Sci. 44(12), 1608–1622 (1998)CrossRefGoogle Scholar
  17. 17.
    M. Labbé, P. Marcotte, G. Savard, On a class of bilevel programs, in Nonlinear Optimization and Related Topics, ed. by G. Di Pillo, F. Giannessi (Kluwer Academic Publishers, Dordrecht, 2000), pp. 183–206Google Scholar
  18. 18.
    S. Lohse, S. Dempe, Best highway toll assigning models and an optimality test (in German), Preprint, TU Bergakademie Freiberg, Nr. 2005-6, Fakultt fr Mathematik und Informatik, Freiberg (2005)Google Scholar
  19. 19.
    T.L. Magnanti, R.T. Wong, Network design and transportation planning: models and algorithms. Transp. Sci. 18(1), 1–55 (1984)CrossRefGoogle Scholar
  20. 20.
    P. Marcotte, Network design problem with congestion effects: a case of bilevel programming. Math. Program. 34(2), 142–162 (1986)MathSciNetCrossRefGoogle Scholar
  21. 21.
    G.E. Renpu, A filled function method for finding a global minimizer of a function of several variables. Math. Program. 46(1), 191–204 (1990)MathSciNetCrossRefGoogle Scholar
  22. 22.
    S. Roch, G. Savard, P. Marcotte, Design and analysis of an algorithm for Stackelberg network pricing. Networks 46(1), 57–67 (2005)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Z. Wan, L. Yuan, J. Chen, A filled function method for nonlinear systems of equalities and inequalities. Comput. Appl. Math. 31(2), 391–405 (2012)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Z.Y. Wu, M. Mammadov, F.S. Bai, Y.J. Yang, A filled function method for nonlinear equations. Appl. Math. Comput. 189(2), 1196–1204 (2007)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Z.Y. Wu, F.S. Bai, Y.J. Yang, M. Mammadov, A new auxiliary function method for general constrained global optimization. Optimization 62(2), 193–210 (2013)MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  • Vyacheslav Kalashnikov
    • 1
    • 2
    • 3
    Email author
  • José Guadalupe Flores Muñiz
    • 4
  • Nataliya Kalashnykova
    • 4
  1. 1.Tecnológico de Monterrey (ITESM)MonterreyMexico
  2. 2.Central Economics and Mathematics Institute (CEMI)MoscowRussia
  3. 3.Sumy State UniversitySumyUkraine
  4. 4.Universidad Autónoma de Nuevo León (UANL)San Nicols de los GarzaMexico

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