Advertisement

Mathematical Programming

, Volume 34, Issue 2, pp 142–162 | Cite as

Network design problem with congestion effects: A case of bilevel programming

  • P. Marcotte
Article

Abstract

Recently much attention has been focused on multilevel programming, a branch of mathematical programming that can be viewed either as a generalization of min-max problems or as a particular class of Stackelberg games with continuous variables. The network design problem with continuous decision variables representing link capacities can be cast into such a framework. We first give a formal description of the problem and then develop various suboptimal procedures to solve it. Worst-case behaviour results concerning the heuristics, as well as numerical results on a small network, are presented.

Key words

Network Design Bilevel Programming Variational Inequalities Stackelberg Games 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    H.A. Aashtiani and T.L. Magnanti, “Equilibria on a congested transporation network“,SIAM Journal on Algebraic and Discrete Methods 2 (1981) 213–226.Google Scholar
  2. [2]
    M. Abdulaal and L.J. LeBlanc, “Continuous equilibrium network design models“,Transportation Research B 13B (1979) 19–32.Google Scholar
  3. [3]
    J.F. Bard, “An Algorithm for solving the general bilevel programming problem“,Mathematics of Operations Research 8 (1983) 260–272.Google Scholar
  4. [4]
    J.F. Bard and J.E. Falk, “An explicit solution to the multilevel programming problem“,Computers and Operations Research 9 (1982) 77–100.Google Scholar
  5. [5]
    D. Bertsekas and E.M. Gafni, “Projection methods for variational inequalities with application to the traffic assignment problem“,Mathematical Programming Study 17 (1982) 139–159.Google Scholar
  6. [6]
    W.F. Bialas and M.H. Karwan, “On two-level optimization“,IEEE Transactions on Automatic Control AC-27 (1982) 211–214.Google Scholar
  7. [7]
    J.W. Blankenship and J.E. Falk, “Infinitely constrained optimization problems“,Journal of Optimization Theory and Applications 19 (1976) 261–281.Google Scholar
  8. [8]
    W. Candler and R.J. Townsley, “A linear two-level programming problem“,Computers and Operations Research 9 (1982) 59–76.Google Scholar
  9. [9]
    S.C. Dafermos, “Traffic assignment and resource allocation in transportation networks”, Ph.D. Dissertation, Johns Hopkins University (Baltimore, Maryland, 1968).Google Scholar
  10. [10]
    S.C. Dafermos, “Traffic equilibrium and variational inequalities“,Transportation Science 14 (1980) 42–54.Google Scholar
  11. [11]
    C. Daganzo, “Stochastic network equilibrium with multiple vehicle types and asymmetric, indefinite link cost Jacobians“,Transportation Science 17 (1983) 282–300.Google Scholar
  12. [12]
    G.B. Dantzig et al., “Formulating and solving the network design problem by decomposition“,Transportation Research B 13B (1979) 5–18.Google Scholar
  13. [13]
    R. Dionne and M. Florian, “Exact and approximate algorithms for optimal network design“,Networks 9 (1979) 37–50.Google Scholar
  14. [14]
    M. Florian, “An improved linear approximation algorithm for the network equilibrium (Packet Switching) problem”,Proceedings of the IEEE Conference on Decision and Control (1977), 812–818.Google Scholar
  15. [15]
    A. Haurie and P. Marcotte, “On the relationship between Nash and Wardrop equilibrium“,Networks 15 (1985) 295–308.Google Scholar
  16. [16]
    H.H. Hoang, “A computational approach to the selection of an optimal network“,Management Science 19 (1973) 488–498.Google Scholar
  17. [17]
    L.J. LeBlanc, “Mathematical programming algorithms for large scale network equilibrium and network design problems”, Ph.D. Dissertation, Northwestern University (Evanston, IL, 1973).Google Scholar
  18. [18]
    M. Los, “A discrete-convex programming approach to the simultaneous optimization of land-use and transportation“,Transportation Research B 13B (1979) 33–48.Google Scholar
  19. [19]
    P. Marcotte, “Network optimization with continuous control parameters“,Transportation Science 17 (1983) 181–197.Google Scholar
  20. [20]
    P. Marcotte, “Design optimal d'un réseau de transport en présence d'effets de congestion”, Ph.D. Thesis, Université de Montréal (Montréal, Canada, 1981).Google Scholar
  21. [21]
    S. Nguyen, “An algorithm for the traffic assignment problem“,Transportation Science 8 (1974) 203–216.Google Scholar
  22. [22]
    G. Papavassilopoulos, “Algorithms for leader-follower games”,Proceedings of the 18th Annual Allerton Conference on Communication Control and Computing (1980) 851–859.Google Scholar
  23. [23]
    G. Papavassilopoulos, “Algorithms for static Stackelberg games with linear costs and polyhedral constraints”,Proceedings of the 21st IEEE Conference on Decisions and Control (1982) 647–652.Google Scholar
  24. [24]
    M.J. Smith, “The existence, uniqueness and stability of traffic equilibrium“,Transportation Research B 13B (1979) 295–304.Google Scholar
  25. [25]
    P.A. Steenbrink,Optimization of transportation networks (Wiley, New York, 1974).Google Scholar
  26. [26]
    H.N. Tan, S.B. Gershwin and M. Athans, “Hybrid optimization in urban traffic networks”, MIT Report DOT-TSC-RSPA-79–7 (1979).Google Scholar
  27. [27]
    J.G. Wardrop, “Some theoretical aspects of road traffic research“,Proceedings of the Institute of Civil Engineers, Part II 1 (1952) 325–378.Google Scholar
  28. [28]
    T.L. Magnanti and R.T. Wong, “Network design and transporation planning—Models and algorithms“,Transportation Science 18 (1984) 1–55.Google Scholar

Copyright information

© The Mathematical Programming Society, Inc. 1986

Authors and Affiliations

  • P. Marcotte
    • 1
    • 2
  1. 1.Ecole des Hautes Etudes CommercialesGERADMontréalCanada
  2. 2.CRT, Université de MontréalMontréalCanada

Personalised recommendations