Skip to main content
SpringerLink
Log in

Bilevel Optimisation of Prices and Signals in Transportation Models

  • Chapter
Mathematical and Computational Models for Congestion Charging

Part of the book series: Applied Optimization ((APOP,volume 101))

Summary

We suppose given a variable demand model with some control parameters to represent prices, a smooth function V which measures departure from equilibrium and a smooth function Z which measures overall disbenefit. We suppose that we wish to minimise Z subject to the constraint that the disequilibrium function V is no more than ε, where we think of ε as a small positive number. The paper suggests a simultaneous descent direction to solve this bilevel optimisation problem; such a direction reduces Z and V simultaneously and may often be computed by simply bisecting the angle between −∇Z and −∇V. The paper shows that following a direction Δ which employs the simultaneous descent direction as its central element leads, under natural conditions which preclude edge effects (where a flow may be zero or a price may be maximum), to the set of those approximate equilibria (where V ≤ ε) at which Z is stationary.

Then the method is extended on the one hand to deal with edge effects (allowing a route flow to be zero or a price to be the maximum permitted), by ensuring that the direction Δ followed anticipates nearby edges of the feasible region, using reduced gradients instead of gradients, and on the other hand to deal with signal controls.

Within the optimisation procedure proposed here, optimisation and equilibration move in parallel and the need to compute a sequence of approximate equilibria is avoided.

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

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
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

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Aashtiani, H., Magnanti, T.: Equilibrium on a congested transport network. SIAM Journal of Algebraic and Discrete Methods, 2, 213–216 (1983)

    MathSciNet  Google Scholar 

  2. Abdulaal, M., Leblanc, L.: Continuous network design problems. Transportation Research, 13B, 19–32 (1979)

    Google Scholar 

  3. Allsop, R.E.: Some possibilities for using traffic control to influence trip distribution and route choice. Proceedings of the 7th International Symposium on Transportation and Traffic Theory, 345–374 (1974)

    Google Scholar 

  4. Bar-Gera, H. Origin-based algorithms for the traffic assignment problem. Transportation Science, 36(4), 398–417 (2002)

    Article  MATH  Google Scholar 

  5. Bar-Gera, H., Boyce, D.: Origin-based algorithms for combined travel forecasting models. Transportation Science, 37B(5), 405–422 (2003)

    Google Scholar 

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

    Google Scholar 

  7. Charnes, A., Cooper, W.W.: Multicopy traffic network models. Proceedings of the Symposium on the Theory of Traffic Flow, held at the General Motors Research Laboratories, 1958, Elsevier, Amsterdam (1961)

    Google Scholar 

  8. Chiou, S-W.: Optimisation of area traffic control subject to user equilibrium traffic assignment. Proceedings of the 25th European Transport Forum, Seminar F, Volume II, 53–64 (1997)

    Google Scholar 

  9. Clark, S.D., Watling, D.P.: Sensitivity analysis of the probit-based stochastic user equilibrium assignment model. Transportation Research, 36B, 617–635 (2002)

    Google Scholar 

  10. Clegg, J., Smith, M.J.: Bilevel optimisation of transportation networks. In: Mathematics in Transport Planning and Control, the Proceedings of the Third International IMA Conference on Mathematics in Transport Planning and Control, Pergamon, 29–36 (1998)

    Google Scholar 

  11. Clegg, J., Smith, M.J., Xiang, Y., Yarrow, R.: Bilevel programming applied to optimising urban transportation. Transportation Research, 35B, 41–70 (2001)

    Google Scholar 

  12. Clegg, J., Smith, M.J.: Cone projection versus half-space projection for the bilevel optimisation of transportation networks. Transportation Research, 35B, 71–82 (2001)

    Google Scholar 

  13. Clune, A., Smith, M., Xiang, Y.: A Theoretical Basis for Implementation of a Quantitative Decision Support System Using Bilevel Optimisation. In: Ceder, A. (ed.) Proceedings of the Fourteenth International Symposium on Transportation and Traffic Theory, Jerusalem, Pergamon, 489–514 (1999)

    Google Scholar 

  14. Cohen, G., Quadrat, J-P., Wynter, L.: On the convergence of the algorithm for bilevel programming problems by Clegg and Smith. Transportation Research, 36B, 939–944 (2002)

    Google Scholar 

  15. COMSIS: Incorporating feedback in Travel Forecasting Methods. Pitfalls and Common Concerns, Travel Model Improvement Program, Report for the US Department of Transportation (1996)

    Google Scholar 

  16. Davis, G.A.: Exact Local Solution of the Continuous Network Design Problem via Stochastic User Equilibrium Assignment. Transportation Research, 28B, 61–75 (1994)

    Google Scholar 

  17. Department of the Environment, Transport and the Regions: A New Deal for Transport: Better for Everyone. The Stationery Office (1998)

    Google Scholar 

  18. Evans, S.P.: Derivation and Analysis of some Models for Combining Trip Distribution and Assignment. Transportation Research, 10(1), 37–57 (1976)

    Article  Google Scholar 

  19. Fisk, C.S.: Optimal signal controls on congested networks. In: Volmuller, J., Hammerslag, R. (eds.) Proceedings of the Ninth International Symposium on Transportation and Traffic Theory, Delft, VNU Science Press, Utrecht, 197–216 (1984)

    Google Scholar 

  20. Fletcher, R., Leyffer, S: Nonlinear programming without a penalty function. University of Dundee Numerical Analysis report NA 171 (2000)

    Google Scholar 

  21. Gartner, N.H.: Optimal traffic assignment with elastic demands: A review. Part II: Algorithmic approaches. Transportation Science, 14, 192–208 (1980)

    MathSciNet  Google Scholar 

  22. Gauvin, J., Savard, G.: The steepest descent direction for the nonlinear bilevel programming problem. Operations Research Letters, 15, 265–272 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  23. Luo, Z.Q., Pang, J.S., Ralph, D.: Mathematical programs with equilibrium constraints. Cambridge University Press (1996)

    Google Scholar 

  24. Marcotte, P.: Network Optimisation with Continuous Control Parameters, Transportation Science, 17, 181–197 (1983)

    Article  Google Scholar 

  25. Marcotte, P.: Network Design Problem with Congestion Effects: A Case of Bilevel Programming. Mathematical Programming, 34, 142–162 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  26. Migdalas, A.: Bilevel Programming in Traffic Planning: Models, Methods and Challenge. Journal of Global Optimization, 7, 381–405 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  27. Outrata, J., Zowe, J.: A numerical approach to optimization problems with variational inequality constraints. Mathematical Programming, 68, 105–130 (1995)

    MathSciNet  Google Scholar 

  28. Patriksson, M., Rockafellar, R.T.: A Mathematical model and Descent Algorithm for Bilevel Traffic Management. Transportation Science, 36, 271–291 (2002)

    Article  MATH  Google Scholar 

  29. Payne, H.J., Thompson, W.A.: Traffic assignment on transportation networks with capacity constraints and queueing. Paper presented at the 47th National ORSA/TIMS North American Meeting (1975)

    Google Scholar 

  30. Rodrigues, H.S., Monteiro, M.T.: Solving mathematical programs with complementarity constraints (MPCC) with Nonlinear Solvers. Poster at the 12th French-German-Spanish Conference on Optimization, Avignon (2004)

    Google Scholar 

  31. SACTRA: Transport and the Economy, The Stationery Office (1999)

    Google Scholar 

  32. Smith, M. J.: A descent algorithm for solving a variety of monotone equilibrium problems. Proceedings of the Ninth International Symposium on Transportation and Traffic Theory, The Netherlands, VNU Science Press, Utrecht, 273–297 (1984a)

    Google Scholar 

  33. Smith, M.J.: A Descent Method for Solving Monotone Variational Inequalities and Monotone Complementarity Problems. Journal of Optimization Theory and Applications, 44, 485–496 (1984b)

    Article  MATH  MathSciNet  Google Scholar 

  34. Smith, M.J.: Traffic control and traffic assignment in a signal-controlled network with queueing. In: Gartner, N., Wilson, N.H.M. (eds.) Proceedings of the Tenth International Symposium on Transportation and Traffic Theory, MIT, 319–338 (1987)

    Google Scholar 

  35. Smith, M.J.: Bilevel optimisation of prices in a variey of transportation models. In: Mahmassani, H.S. (ed.) Proceedings of the Sixteenth International Symposium on Transportation and Traffic Theory, University of Maryland, 1–21 (2005a)

    Google Scholar 

  36. Smith, M.J.: Simultaneous descent: some details. Working paper available from the University of York (2005b)

    Google Scholar 

  37. Smith, M.J., Xiang, Y., Yarrow, R.: Bilevel optimisation of signal timings and road prices on urban road networks. Preprints of the IFAC/IFIP/IFORS Symposium, Crete, 628–633 (1997) (available from the University of York)

    Google Scholar 

  38. Smith, M.J., Xiang, Y., Yarrow, R.: Descent Methods of Calculating Locally Optimal Signal Controls and Prices in Multi-Modal and Dynamic Transportation Networks. In: Bell, M.G.H. (ed.) Selected Proceedings of the 4th EURO Transportation Meeting, University of Newcastle, 9–34 (1998)

    Google Scholar 

  39. Smith, M.J., Xiang, Y., Yarrow, R., Ghali, M.O.: Bilevel and Other modelling Approaches to Urban Traffic Management and Control. Paper presented at the 25th Birthday of the Centre de Reserche sur les Transports, University of Montreal (1996), In: Marcotte. P., Nguyen, S. (eds.) Equilibrium and Advanced Transportation Modelling. Kluwer Academic Publishers, Massachusetts, 283–325 (1998)

    Google Scholar 

  40. Tan, H.N., Gershwin, S.B., Athans, M.: Hybrid optimization in urban transport networks. Laboratory for information and Decision Systems, Technical Report DOT-TSC-RSPA-79-7; published by Massachusetts Institute of Technology, Cambridge Massachusetts, USA (1979)

    Google Scholar 

  41. Tobin, R.L., Friesz, T.L.: Sensitivity analysis for equilibrium network flow. Transportation Science, 22, 242–250 (1988)

    MathSciNet  MATH  Google Scholar 

  42. Wardrop, J.G.: Some Theoretical Aspects of Road Traffic Research. Proceedings, Institution of Civil Engineers II, 1, 235–278 (1952)

    Google Scholar 

  43. Yang, H., Yagar, S.: Traffic assignment and traffic control in general freeway-arterial corridor systems. Transportation Research, 28B, 463–486 (1994)

    Google Scholar 

  44. Yang, H. Sensitivity analysis for queueing equilibrium network flow and application to traffic control. Mathematical and Computer Modelling, 22, 247–258 (1996a)

    Google Scholar 

  45. Yang, H.: Sensitivity analysis for the elastic-demand network equilibrium problem with applications, Transportation Research, 31B, 55–70 (1996b)

    Google Scholar 

  46. Yang, H.: Equilibrium network traffic signal setting under conditions of queueing and congestion. In: Stephanedes, Y.J., Filippi, F. (eds.) Applications of Advanced Technologies in transportation Engineering. Proceedings of the 4th International Conference, American Society of Civil Engineers, 578–582 (1996c)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of York, Heslington, York, YO10 5DD, UK

    Michael J Smith

Authors
  1. Michael J Smith
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. University of Florida, Gainesville, Florida, USA

    Siriphong Lawphongpanich  & Donald W. Hearn  & 

  2. The University of York, UK

    Michael J. Smith

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Springer Science and Business Media, Inc.

About this chapter

Cite this chapter

Smith, M.J. (2006). Bilevel Optimisation of Prices and Signals in Transportation Models. In: Lawphongpanich, S., Hearn, D.W., Smith, M.J. (eds) Mathematical and Computational Models for Congestion Charging. Applied Optimization, vol 101. Springer, Boston, MA. https://doi.org/10.1007/0-387-29645-X_8

Download citation

Publish with us

Policies and ethics

Search

Navigation