Skip to main content
SpringerLink
Log in
Book cover

Natural Intelligence for Scheduling, Planning and Packing Problems pp 195–217Cite as

A Coevolutionary Particle Swarm Algorithm for Bi-Level Variational Inequalities: Applications to Competition in Highway Transportation Networks

  • Chapter

Part of the book series: Studies in Computational Intelligence ((SCI,volume 250))

Abstract

A climate of increasing deregulation in traditional highway transportation, where the private sector has an expanded role in the provision of traditional transportation services, provides a background for practical policy issues to be investigated. One of the key issues of interest, and the focus of this chapter, would be the equilibrium decision variables offered by participants in this market. By assuming that the private sector participants play a Nash game, the above problem can be described as a Bi-Level Variational Inequality (BLVI). Our problem differs from the classical Cournot-Nash game because each and every player’s actions are constrained by another variational inequality describing the equilibrium route choice of users on the network. In this chapter, we discuss this BLVI and suggest a heuristic coevolutionary particle swarm algorithm for its resolution. Our proposed algorithm is subsequently tested on example problems drawn from the literature. The numerical experiments suggest that the proposed algorithm is a viable solution method for this problem.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   169.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Abido, M., Abido, M.: Optimal design of power system stabilizers using particle swarm optimization. IEEE Transactions on Energy Conversion 17(4), 406–413 (2002)

    Article  Google Scholar 

  2. Beckmann, M., McGuire, C., Winsten, C.: Studies in the Economics of Transportation. Yale University Press, New Haven (1956)

    Google Scholar 

  3. Bureau of Public Roads, Traffic Assignment Manual. Urban Planning Division, US Dept. of Commerce, Washington DC (1964)

    Google Scholar 

  4. Cardell, J., Hitt, C., Hogan, W.: Market power and strategic interaction in electricity networks. Resource and Energy Economics 19(1-2), 109–137 (1997)

    Article  Google Scholar 

  5. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)

    MATH  Google Scholar 

  6. Clerc, M., Kennedy, J.: The particle swarm - explosion, stability, and convergence in a multidimensional complex space. IEEE Transactions on Evolutionary Computation 6(1), 58–73 (2002)

    Article  Google Scholar 

  7. Cockshott, A., Hartman, B.: Improving the fermentation medium for Echinocandin B production, Part II: Particle swarm optimization. Process Biochemistry 36(7), 661–669 (2001)

    Article  Google Scholar 

  8. Cournot, A.A.: Recherches sur les Principes Mathematiques de la Theorie des Richesses. Hachette, Paris (1838); English trans. in: Researches into the Mathematical Principles of the Theory of Wealth. A M Kelley, New York (1971)

    Google Scholar 

  9. Dafermos, S.C.: Traffic Equilibrium and Variational Inequalities. Transportation Science 14(1), 42–54 (1980)

    Article  MathSciNet  Google Scholar 

  10. Deb, K.: Multi-objective Optimization using Evolutionary Algorithms. John Wiley, Chichester (2001)

    MATH  Google Scholar 

  11. Ehrlich, P.R., Raven, P.H.: Butterflies and plants: a study in coevolution. Evolution-International Journal of Organic Evolution 18(4), 586–608 (1964)

    Google Scholar 

  12. Engel, E., Fischer, R., Galetovic, A.: A New Approach to Private Roads. Regulation 25(3), 18–22 (2002)

    Google Scholar 

  13. Engelbrecht, A.P.: Computational Intelligence: An Introduction. John Wiley, New York (2002)

    Google Scholar 

  14. Ficici, S.G., Pollack, J.B.: Game theory and the Simple Coevolutionary Algorithm. In: Deb, K., Rudolph, G., Lutton, E., Merelo, J.J., Schoenauer, M., Schwefel, H.-P., Yao, X. (eds.) PPSN 2000. LNCS, vol. 1917, pp. 467–476. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  15. Fisher, G., Babbar, S.: Private Financing of Toll Roads. RMC Discussion Paper Series, vol. 117. World Bank, Washington DC (1996)

    Google Scholar 

  16. Fisk, C.: Game theory and transportation systems modeling. Transportation Research Part B 18(4/5), 301–313 (1984)

    Article  MathSciNet  Google Scholar 

  17. Fourie, P., Groenwold, A.: The particle swarm optimization algorithm in size and shape optimization. Structural and Multidisciplinary Optimization 23(4), 259–267 (2002)

    Article  Google Scholar 

  18. Gabay, D., Moulin, H.: On the uniqueness and stability of Nash-equilibria in non cooperative games. In: Bensoussan, A., Dorfer, P.K., Tapiero, C. (eds.) Applied Stochastic Control in Econometrics and Management Science: Contributions to Economic Analysis, vol. 130, pp. 271–293. North-Holland, Amsterdam (1980)

    Google Scholar 

  19. Goldberg, D.E.: Genetic Algorithms in Search, Optimization, and Machine Learning. Addison-Wesley, Boston (1989)

    MATH  Google Scholar 

  20. Henderson, J.M., Quandt, R.E.: Microeconomic Theory: A Mathematical Approach. McGraw Hill, Auckland (1980)

    Google Scholar 

  21. Harker, P.T.: A variational inequality approach for the determination of Oligopolistic Market Equilibrium. Mathematical Programmming 30(1), 105–111 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  22. Hobbs, B.F., Metzler, C.B., Pang, J.S.: Strategic gaming analysis for electric power networks: An MPEC approach. IEEE Transactions on Power Systems 15(2), 638–645 (2000)

    Article  Google Scholar 

  23. Kennedy, J., Eberhart, R.C.: Particle swarm optimization. In: Proceedings of the IEEE International Conference on Neural Networks, pp. 1942–1948 (1995)

    Google Scholar 

  24. Kennedy, J., Eberhart, R.C., Shi, Y.: Swarm Intelligence. Morgan Kaufmann Series in Artificial Intelligence, San Francisco (2001)

    Google Scholar 

  25. Koh, A.: Solving Transportation bi-level programs with Differential Evolution. In: Proceedings of the IEEE Congress on Evolutionary Computation, pp. 2243–2250 (2007)

    Google Scholar 

  26. Koh, A., Shepherd, S.P.: Tolling, Capacity Selection and Equilibrium Problems with Equilibrium Constraints. In: Proceedings 3rd Kuhmo-Nectar Conference, Free University of Amsterdam, July 3-4 (2008)

    Google Scholar 

  27. Koh, A., Shepherd, S.P., Sumalee, A.: Second Best Toll and Capacity Optimisation in Networks: Solution Algorithm and Policy Implications. Transportation 36(2), 147–165 (2009)

    Article  Google Scholar 

  28. Koza, J.R.: Genetic evolution and coevolution of computer programs. In: Langton, C.G., Taylor, C., Farmer, J.D., Rasmussen, S. (eds.) Artificial Life II, pp. 603–629. Addison-Wesley, Redwood City (1990)

    Google Scholar 

  29. Leyffer, S., Munson, T.: Solving Multi-Leader-Follower Games. Preprint Argonne National Laboratory/MCS-P1243-0405 (2005)

    Google Scholar 

  30. Lockwood, S.: Public private partnerships in U.S. highway finance: ISTEA and beyond. Transportation Quarterly 49(l), 5–26 (1995)

    Google Scholar 

  31. Luo, Z.Q., Pang, J.S., Ralph, D.: Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge (1996)

    Google Scholar 

  32. Mordukhovich, B.S.: Equilibrium problems with equilibrium constraints via multiobjective optimization. Optimization Methods and Software 19(5), 479–492 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  33. Mordukhovich, B.S.: Optimization and equilibrium problems with equilibrium constraints. Omega-International Journal of Management Science 33(5), 379–384 (2005)

    Article  Google Scholar 

  34. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation. II: Applications. Grundlehren Series in Fundamental Principles of Mathematical Sciences, vol. 331. Springer, Berlin (2006)

    Google Scholar 

  35. Nagurney, A.: Network Economics: a Variational Inequality Approach. Kluwer, Boston (1999)

    Google Scholar 

  36. Nash, J.: Equilibrium points in N-person games. Proceedings of the National Academy of Science 36(1), 48–49 (1950)

    Article  MATH  MathSciNet  Google Scholar 

  37. Ortuzar, J.D., Willumsen, L.: Modelling Transport. Wiley, Chichester (1998)

    Google Scholar 

  38. Ourique, C., Biscala, E., Pinto, J.: The use of particle swarm optimization for dynamical analysis in chemical processes. Computers and Chemical Engineering 26(12), 1783–1793 (2002)

    Article  Google Scholar 

  39. de Palma, A., Lindsey, R.: Private roads: competition under various ownership regimes. Annals of Regional Science 34(1), 13–35 (2000)

    Article  Google Scholar 

  40. Parsopoulos, K.E., Vrahatis, M.N.: Parameter selection and adaptation in Unified Particle Swarm Optimization. Mathematical and Computer Modelling 46(1-2), 198–213 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  41. Perez, R., Behdinan, K.: Particle swarm approach for structural design optimization. Computers and Structures 85(19-20), 1579–1588 (2007)

    Article  Google Scholar 

  42. Potter, M.A., De Jong, K.: A cooperative coevolutionary approach for function optimization. In: Davidor, Y., Männer, R., Schwefel, H.-P. (eds.) PPSN 1994. LNCS, vol. 866, pp. 249–257. Springer, Heidelberg (1994)

    Google Scholar 

  43. Pyndyck, R., Rubinfeld, S.: Microeconomics. Macmillan, New York (1992)

    Google Scholar 

  44. Ralph, D., Smeers, Y.: EPECs as models for electricity markets. In: Proceedings of IEEE Power Systems Conference and Exposition, pp. 74–80 (2006)

    Google Scholar 

  45. Robinson, S.M.: Strong regularity and the sensitivity analysis of traffic equilibria: A comment. Transportation Science 40(4), 540–542 (2006)

    Article  Google Scholar 

  46. Rockafellar, R.T., Wets, J.B.: Variational Analysis. Grundlehren Series in Fundamental Principles of Mathematical Sciences, vol. 317. Springer, Berlin (1998)

    MATH  Google Scholar 

  47. Rosin, C.D., Belew, R.K.: Methods for Competitive Co-Evolution: Finding Opponents Worth Beating. In: Eshelman, L.J. (ed.) Proceedings of the 6th International Conference on Genetic Algorithms, Pittsburgh, USA, July 15-19, 1995, pp. 373–381 (1995)

    Google Scholar 

  48. Rosin, C.D., Belew, R.K.: New methods for competitive coevolution. Evolutionary Computation 5(1), 1–29 (1997)

    Article  Google Scholar 

  49. Sheffi, Y.: Urban Transportation Networks: Equilibrium Analysis with Mathematical Programming Methods. Prentice Hall, Englewood Cliffs (1985)

    Google Scholar 

  50. Smith, M.J.: The existence, uniqueness and stability of traffic equilibria. Transportation Research Part B 13(4), 295–304 (1979)

    Article  MathSciNet  Google Scholar 

  51. von Stackelberg, H.: The Theory of the Market Economy (1934); English trans. in: Peacock, A. Oxford University Press, New York (1952)

    Google Scholar 

  52. Storn, R., Price, K.: Differential Evolution – A Simple and Efficient Heuristic for Global Optimization over Continuous Spaces. Journal of Global Optimization 11(4), 341–359 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  53. Stough, R., Rietveld, P.: Institutional issues in transport systems. Journal of Transport Geography 5(3), 207–214 (1997)

    Article  Google Scholar 

  54. Su, C.: Equilibrium Problems with Equilibrium Constraints: Stationarities, Algorithms, and Applications. PhD Thesis, Department of Management Science and Engineering, Stanford University (2005)

    Google Scholar 

  55. Thompson, J.N.: The Coevolutionary Process. University of Chicago Press, Chicago (1994)

    Google Scholar 

  56. Viton, P.: Private Roads. Journal of Urban Economics 37(3), 260–289 (1995)

    Article  MATH  Google Scholar 

  57. Wardrop, J.G.: Some theoretical aspects of road traffic research. Proceedings of the Institution of Civil Engineers - Transport 1(2), 325–378 (1952)

    Google Scholar 

  58. Yang, H., Xiao, F., Huang, H.: Private Road Competition and Equilibrium with Traffic Equilibrium Constraints. Journal of Advanced Transportation 43(1), 21–45 (2009)

    Article  Google Scholar 

  59. Ye, J.J., Zhu, Q.J.: Multiobjective optimization problems with variational inequality constraints. Mathematical Programming Series A 96(1), 139–160 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  60. Yin, Y.: Genetic Algorithms-Based Approach for Bilevel Programming Models. ASCE Journal of Transportation Engineering 126(2), 115–120 (2000)

    Article  Google Scholar 

  61. You, S., Baldick, R.: Hybrid coevolutionary programming for Nash equilibrium search in games with local optima. IEEE Transactions on Evolutionary Computation 8(4), 305–315 (2004)

    Article  Google Scholar 

  62. Zubeita, L.: A network equilibrium model for oligopolistic competition in city bus services. Transportation Research Part B 32(6), 413–422 (1998)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute for Transport Studies, University of Leeds, Leeds, LS2 9JT, United Kingdom

    Andrew Koh

Authors
  1. Andrew Koh
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. School of Computing & Design, Swinburne University of Technology (Sarawak Campus), Jalan Simpang Tiga, 93350, Kuching, Sarawak, Malaysia

    Raymond Chiong  & Sandeep Dhakal  & 

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Koh, A. (2009). A Coevolutionary Particle Swarm Algorithm for Bi-Level Variational Inequalities: Applications to Competition in Highway Transportation Networks. In: Chiong, R., Dhakal, S. (eds) Natural Intelligence for Scheduling, Planning and Packing Problems. Studies in Computational Intelligence, vol 250. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04039-9_8

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-04039-9_8

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-04038-2

  • Online ISBN: 978-3-642-04039-9

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation