Advertisement

Journal of Global Optimization

, Volume 67, Issue 1–2, pp 325–342 | Cite as

A modified active set algorithm for transportation discrete network design bi-level problem

  • Ximing WangEmail author
  • Panos M. Pardalos
Article

Abstract

Transportation discrete network design problem (DNDP) is about how to modify an existing network of roads and highways in order to improve its total system travel time, and the candidate road building or expansion plan can only be added as a whole. DNDP can be formulated into a bi-level problem with binary variables. An active set algorithm has been proposed to solve the bi-level discrete network design problem, while it made an assumption that the capacity increase and construction cost of each road are based on the number of lanes. This paper considers a more general case when the capacity increase and construction cost are specified for each candidate plan. This paper also uses numerical methods instead of solvers to solve each step, so it provides a more direct understanding and control of the algorithm and running procedure. By analyzing the differences and getting corresponding solving methods, a modified active set algorithm is proposed in the paper. In the implementation of the algorithm and the validation, we use binary numeral system and ternary numeral system to avoid too many layers of loop and save storage space. Numerical experiments show the correctness and efficiency of the proposed modified active set algorithm.

Keywords

Discrete network design Bi-level problem Binary variable Modified active set algorithm Binary and ternary numeral system 

Notes

Acknowledgments

We are grateful to Yafeng Yin for providing the problem and the original active set algorithm, and also the valuable discussions with him. P. M. Pardalos was partially supported by LATNA Laboratory, NRU HSE, RF government grant, ag. 11.G34.31.0057, National Research University Higher School of Economics, Nizhny Novgorod, Russia.

References

  1. 1.
    Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network flows: theory, algorithms, and applications. Tech. rep., DTIC Document (1988)Google Scholar
  2. 2.
    Beckmann, M., McGuire, C., Winsten, C.B.: Studies in the economics of transportation. Tech. rep. (1956)Google Scholar
  3. 3.
    Bertsekas, D.P.: Nonlinear programming, 2nd edn. Athena Scientific, Belmont (1999)Google Scholar
  4. 4.
    CPLEX, I.I.: V12. 1: Users manual for CPLEX. Int. Bus. Mach. Corp. 46(53), 157 (2009)Google Scholar
  5. 5.
    Dijkstra, E.W.: A note on two problems in connexion with graphs. Numer. Math. 1(1), 269–271 (1959)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Drud, A.S.: Conopt a large-scale GRG code. ORSA J. Comput. 6(2), 207–216 (1994)CrossRefzbMATHGoogle Scholar
  7. 7.
    Frank, M., Wolfe, P.: An algorithm for quadratic programming. Nav. Res. Logist. Q. 3(1–2), 95–110 (1956)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Friesz, T.L., Bernstein, D., Smith, T.E., Tobin, R.L., Wie, B.: A variational inequality formulation of the dynamic network user equilibrium problem. Oper. Res. 41(1), 179–191 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Fukushima, M.: A modified frank-wolfe algorithm for solving the traffic assignment problem. Transp. Res. Part B Methodol. 18(2), 169–177 (1984)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Larsson, T., Patriksson, M.: Side constrained traffic equilibrium models analysis, computation and applications. Transp. Res. Part B Methodol. 33(4), 233–264 (1999)CrossRefGoogle Scholar
  11. 11.
    Rosenthal, R.E.: Gams—a user’s guide. GAMS Development Corporation, Washington, DC (2004)Google Scholar
  12. 12.
    Seref, O., Ahuja, R.K., Orlin, J.B.: Incremental network optimization: theory and algorithms. Oper. Res. 57(3), 586–594 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Sheffi, Y.: Urban transportation networks: Equilibrium Analysis with Mathematical Programming Methods. Prentice-Hall, Inc., Englewood Cliffs (1984)Google Scholar
  14. 14.
    Skiena, S.: Dijkstra’s Algorithm. Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Addison-Wesley, Reading (1990)zbMATHGoogle Scholar
  15. 15.
    Yin, Y., Lawphongpanich, S.: A robust approach to continuous network designs with demand uncertainty. In: Transportation and Traffic Theory 2007. Papers Selected for Presentation at ISTTT17 (2007)Google Scholar
  16. 16.
    Zhang, L., Lawphongpanich, S., Yin, Y.: An active-set algorithm for discrete network design problems. In: Lam, W.H.K., Wong, H., Lo, H.K. (eds.) Transportation and Traffic Theory 2009: Golden Jubilee, pp. 283–300. Springer, Berlin (2009)Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of Industrial and Systems EngineeringUniversity of FloridaGainesvilleUSA

Personalised recommendations