Skip to main content
SpringerLink
Log in

Equilibrium in Multi-criteria Transportation Networks

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

We develop a new method to generate the set of equilibrium flows of a multi-criteria transportation network. To this end, we introduce two optimization problems by using a vector version of the Heaviside step function and the distance function to Pareto minimal elements and show that the optimal solutions of these problems are exactly the equilibria of the network. We study the objective functions by establishing their generic differentiability and local calmness at equilibrium solutions. Then we present an algorithm to generate a discrete representation of equilibrium solutions by using a modified Frank–Wolfe reduced gradient method and prove its convergence. We give some numerical examples to illustrate our algorithm and show its advantage over a popular method by using linear scalarization.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Wardrop, J.: Some theoretical aspects of road traffic research. In: Proceedings of the Institute of Civil Engineers, vol. 1, pp. 325–378 (1952)

  2. Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows: Theory, Algorithms, and Applications. Prentice-Hall, Englewood Cliffs (1993)

    MATH  Google Scholar 

  3. Beckmann, M., Mcguire, C.B., Wisten, C.B.: Studies in the Economics of Transportation. Yale University Press, New Haven (1956)

    Google Scholar 

  4. Gendreau, M., Marcotte, P. (eds.): Transportation and Network Analysis: Current Trends. Kluwer Academic, Dordrecht (2002)

    MATH  Google Scholar 

  5. Nagurney, A.: Network Economics: A Variational Inequality Approach. Kluwer Academic, Dordrecht (1999)

    Book  MATH  Google Scholar 

  6. Sheffi, Y.: Urban Transportation Networks. Prentice-Hall, Englewood (1985)

    Google Scholar 

  7. Quandt, R. E.: A probabilistic abstract mode model. In: Studies in Travel Demand VIII, pp. 127–149. Mathematica Inc., Princeton (1967)

  8. Schneider, M.: Access and land development, urban development models. Highw. Res. Board Spec. Rep. 97, 164–177 (1968)

    Google Scholar 

  9. Dafermos, S.: A Multicrieria Route-Mode Choice Traffic Equilibrium Model. Lefschetz Center for Dynamical Systems. Brown University, Providence (1981)

    Google Scholar 

  10. Dial, R.B.: A model and algorithms for multi-criteria route-mode choice. Transp. Res. 13B, 311–316 (1979)

    Article  Google Scholar 

  11. Chen, G.Y., Goh, C.J., Yang, X.Q.: Vector network equilibrium problems and nonlinear scalarization methods. Math. Methods Oper. Res. 49, 239–253 (1999)

    MathSciNet  MATH  Google Scholar 

  12. Chen, G. Y., Yen, N. D.: On the Variational Inequality Model for Network Equilibrium. Internal Report 3.196 (724). Department of Mathematics, University of Pisa, Pisa (1993)

  13. Goh, C.J., Yang, X.Q.: Vector equilibrium problem and vector optimization. Eur. J. Oper. Res. 116, 615–628 (1999)

    Article  MATH  Google Scholar 

  14. Li, S., Chen, G.: Multiclass, two-criteria traffic network equilibrium models and vector variational inequalities. Syst. Eng. Theory Pract. 28, 141–145 (2008)

    Article  Google Scholar 

  15. Lin, Z.: On existence of vector equilibrium flows with capacity constraints of arcs. Nonlinear Anal. Theory Methods Appl. 72, 2076–2079 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Luc, D.T.: Multi-product supply demand networks with elementary flows. Acta Math. Vietnam. 36, 299–317 (2011)

    MathSciNet  MATH  Google Scholar 

  17. Luc, D.T., Rocca, M., Papalia, M.: Equilibrium in a vector supply-demand network with capacity constraints. Appl. Anal. 90, 1029–1045 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Nagurney, A.: A multiclass, multi-criteria traffic network equilibrium model. Math. Comput. Model. 32, 393–411 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  19. Nagurney, A., Dong, J.: A multiclass, multi-criteria traffic network equilibrium model with elastic demand. Transp. Res. B 36, 445–469 (2002)

    Article  Google Scholar 

  20. Tan, Z., Yang, H., Guo, R.: Pareto efficiency of reliability-based traffic equilibria and risk-taking behavior of travelers. Transp. Res. B Methodol. 66, 16–31 (2014)

    Article  Google Scholar 

  21. Tian, X.Q., Xu, Y.D.: Traffic network equilibrium problems with capacity constraints of arcs and linear scalarization methods. J. Appl. Math. (2012). doi:10.1155/2012/612142

    MathSciNet  MATH  Google Scholar 

  22. Wang, J.Y.T., Ehrgott, M.: Modeling route choice behavior in a tolled road network with a time surplus maximisation bi-objective user equilibrium model. Procedia Soc. Behav. Sci. 80, 266–288 (2013)

    Article  Google Scholar 

  23. Cheng, T.C.E., Wu, Y.N.: A multiclass, multi-criteria supply-demand network equilibrium model. Oper. Res. 53, 544–554 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Daniele, P., Maugeri, A., Oettli, W.: Variational inequalities and time-dependent traffic equilibria. C. R. Acad. Sci. Paris Ser. I Math. 326, 1059–1062 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  25. Konnov, I.V.: Vector network equilibrium problems with elastic demands. J. Glob. Optim. 57, 521–531 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  26. Li, S., Chen, G.: On relations between multiclass, multi-criteria traffic network equilibrium models and vector variational inequalities. J. Syst. Sci. Eng. 15, 284–297 (2006)

    Article  Google Scholar 

  27. Li, S.J., Teo, K.L., Yang, X.Q.: Vector equilibrium problems with elastic demands and capacity constraints. J. Glob. Optim. 37, 647–660 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  28. Li, S.J., Teo, K.L., Yang, X.Q.: A remark on a standard and linear vector network equilibrium problems with capacity constraints. Eur. J. Oper. Res. 184, 13–23 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  29. Li, S.J., Yang, X.Q., Chen, G.Y.: A note on vector network equilibrium principles. Math. Methods Oper. Res. 64, 327–334 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  30. Oettli, W.: Necessary and sufficient conditions of Wardrop type for vectorial traffic equilibria. In: Giannessi, F., Maugeri, A., Pardalos, P. (eds.) Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Models, Nonconvex Optimization and Applications, vol. 58, pp. 223–228. Springer, Berlin (2001)

    Google Scholar 

  31. Phuong, T.T.T., Luc, D.T.: Equilibrium in multi-criteria supply and demand networks with capacity constraints. Math. Methods Oper. Res. 81, 83–107 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  32. Raciti, F.: Equilibrium conditions and vector variational inequalities: a complex relation. J. Glob. Optim. 40, 353–360 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  33. Raith, A., Ehrgott, M.: On vector equilibria, vector optimization and vector variational inequalities. J. Multi-Criteria Decis. Anal. 18, 39–54 (2011)

    Article  Google Scholar 

  34. Wu, Y., Cheng, T.C.E.: Benson efficiency of a multi-criterion network equilibrium model. Pac. J. Optim. 5, 443–458 (2009)

    MathSciNet  MATH  Google Scholar 

  35. Giannessi, F.: Vector Variational Inequalities and Vector Equilibria. Kluwer Academic, Dordrecht (2000)

    Book  MATH  Google Scholar 

  36. Luc, D.T.: Theory of Vector Optimization. Lecture Notes Economics and Maths Systems, vol. 319. Springer, Berlin (1989)

    Google Scholar 

  37. Wierzbicki, A.P.: A mathematical basis for satisficing decision making. Math. Model. 3, 391–405 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  38. Aubin, J.-P., Frankowska, H.: Set-Valued Analysis. Birkhauser, Boston (1990)

    MATH  Google Scholar 

  39. Dontchev, A., Rockafellar, R.T.: Implicit Functions and Solution Mappings: A View From Variational Analysis. Springer, New York (2009)

    Book  MATH  Google Scholar 

  40. Bigi, G., Panicucci, B.: A successive linear programming algorithm for nonsmooth monotone variational inequalities. Optim. Methods Softw. 25, 29–35 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  41. Han, D., Sun, W.: A new modified Goldstein-Levitin-Polyak projection method for variational inequality problems. Comput. Math. Appl. 47, 1817–1825 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  42. Mastroeni, G.: Gap functions and descent methods for Minty variational inequality. Optim. Control Appl. 96, 529–547 (2005)

    MathSciNet  MATH  Google Scholar 

  43. Solodov, M.V., Svaiter, B.F.: A new projection method for variational inequality problems. SIAM J. Control Optim. 37, 765–776 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  44. Wang, M., Bertsekas D.P.: Incremental Constraint Projection Methods for Variational Inequalities. Massachusetts Institute of Technology, Cambridge, MA, Laboratory for Information and Decision Systems, Report LIDS-P-2898 (2012)

Download references

Acknowledgments

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant Number 101.01-2013.10.

Author information

Authors and Affiliations

  1. LMA, Avignon University, Avignon, France

    Dinh The Luc

  2. Center for Information and Computing, VAST, Hanoi, Vietnam

    Dinh The Luc

  3. Quy Nhon University, Quy Nhon, Vietnam

    Truong Thi Thanh Phuong

Authors
  1. Dinh The Luc
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Truong Thi Thanh Phuong
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dinh The Luc.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Luc, D.T., Phuong, T.T.T. Equilibrium in Multi-criteria Transportation Networks. J Optim Theory Appl 169, 116–147 (2016). https://doi.org/10.1007/s10957-016-0876-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-016-0876-3

Keywords

Mathematics Subject Classification

Search

Navigation