IDEAS home Printed from https://ideas.repec.org/a/inm/ortrsc/v38y2004i3p258-281.html
   My bibliography  Save this article

Sensitivity Analysis of Traffic Equilibria

Author

Listed:
  • Michael Patriksson

    (Department of Mathematics, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden)

Abstract

The contribution of the paper is a complete analysis of the sensitivity of elastic demand traffic (Wardrop) equilibria. The existence of a directional derivative of the equilibrium solution (link flow, least travel cost, demand) in any direction is given a characterization, and the same is done for its gradient. The gradient, if it exists, is further interpreted as a limiting case of the gradient of the logit-based SUE solution, as the dispersion parameter tends to infinity. In the absence of the gradient, we show how to compute a subgradient. All these computations (directional derivative, (sub)gradient) are performed by solving similar traffic equilibrium problems with affine link cost and demand functions, and they can be performed by the same tool as (or one similar to) the one used for the original traffic equilibrium model; this fact is of clear advantage when applying sensitivity analysis within a bilevel (or mathematical program with equilibrium constraints, MPEC) application, such as for congestion pricing, OD estimation, or network design. A small example illustrates the possible nonexistence of a gradient and the computation of a subgradient.

Suggested Citation

  • Michael Patriksson, 2004. "Sensitivity Analysis of Traffic Equilibria," Transportation Science, INFORMS, vol. 38(3), pages 258-281, August.
    • Handle: RePEc:inm:ortrsc:v:38:y:2004:i:3:p:258-281
    DOI: 10.1287/trsc.1030.0043
    as

    Download full text from publisher

    File URL: http://dx.doi.org/10.1287/trsc.1030.0043
    Download Restriction: no
    File URL: https://libkey.io/10.1287/trsc.1030.0043?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    References listed on IDEAS

    as
    1. Fisk, Caroline, 1980. "Some developments in equilibrium traffic assignment," Transportation Research Part B: Methodological, Elsevier, vol. 14(3), pages 243-255, September.
    2. Yang, Hai & Yagar, Sam, 1995. "Traffic assignment and signal control in saturated road networks," Transportation Research Part A: Policy and Practice, Elsevier, vol. 29(2), pages 125-139, March.
    3. Jiang Qian Ying & Toshihiko Miyagi, 2001. "Sensitivity Analysis for Stochastic User Equilibrium Network Flows—A Dual Approach," Transportation Science, INFORMS, vol. 35(2), pages 124-133, May.
    4. Yan, Hai & Lam, William H. K., 1996. "Optimal road tolls under conditions of queueing and congestion," Transportation Research Part A: Policy and Practice, Elsevier, vol. 30(5), pages 319-332, September.
    5. Dafermos, Stella & Nagurney, Anna, 1984. "On some traffic equilibrium theory paradoxes," Transportation Research Part B: Methodological, Elsevier, vol. 18(2), pages 101-110, April.
    6. Fisk, Caroline, 1979. "More paradoxes in the equilibrium assignment problem," Transportation Research Part B: Methodological, Elsevier, vol. 13(4), pages 305-309, December.
    7. Yuping Qiu & Thomas L. Magnanti, 1989. "Sensitivity Analysis for Variational Inequalities Defined on Polyhedral Sets," Mathematics of Operations Research, INFORMS, vol. 14(3), pages 410-432, August.
    8. Michael Patriksson & R. Tyrrell Rockafellar, 2003. "Sensitivity Analysis of Aggregated Variational Inequality Problems, with Application to Traffic Equilibria," Transportation Science, INFORMS, vol. 37(1), pages 56-68, February.
    9. Larsson, Torbjörn & Patriksson, Michael, 1995. "An augmented lagrangean dual algorithm for link capacity side constrained traffic assignment problems," Transportation Research Part B: Methodological, Elsevier, vol. 29(6), pages 433-455, December.
    10. Michael A. Hall, 1978. "Properties of the Equilibrium State in Transportation Networks," Transportation Science, INFORMS, vol. 12(3), pages 208-216, August.
    11. Janson, Bruce N., 1993. "Most likely origin-destination link uses from equilibrium assignment," Transportation Research Part B: Methodological, Elsevier, vol. 27(5), pages 333-350, October.
    12. Stephen M. Robinson, 1980. "Strongly Regular Generalized Equations," Mathematics of Operations Research, INFORMS, vol. 5(1), pages 43-62, February.
    13. Larsson, Torbjörn & Patriksson, Michael, 1999. "Side constrained traffic equilibrium models-- analysis, computation and applications," Transportation Research Part B: Methodological, Elsevier, vol. 33(4), pages 233-264, May.
    14. Yang, Hai & Bell, Michael G. H., 1997. "Traffic restraint, road pricing and network equilibrium," Transportation Research Part B: Methodological, Elsevier, vol. 31(4), pages 303-314, August.
    15. Davis, Gary A., 1994. "Exact local solution of the continuous network design problem via stochastic user equilibrium assignment," Transportation Research Part B: Methodological, Elsevier, vol. 28(1), pages 61-75, February.
    16. Tam, M. L. & Lam, William H. K., 2000. "Maximum car ownership under constraints of road capacity and parking space," Transportation Research Part A: Policy and Practice, Elsevier, vol. 34(3), pages 145-170, April.
    17. Suh-Wen Chiou, 1999. "Optimization of Area Traffic Control for Equilibrium Network Flows," Transportation Science, INFORMS, vol. 33(3), pages 279-289, August.
    18. Yang, Hai, 1995. "Heuristic algorithms for the bilevel origin-destination matrix estimation problem," Transportation Research Part B: Methodological, Elsevier, vol. 29(4), pages 231-242, August.
    19. Michael Patriksson & R. Tyrrell Rockafellar, 2002. "A Mathematical Model and Descent Algorithm for Bilevel Traffic Management," Transportation Science, INFORMS, vol. 36(3), pages 271-291, August.
    20. Roger L. Tobin & Terry L. Friesz, 1988. "Sensitivity Analysis for Equilibrium Network Flow," Transportation Science, INFORMS, vol. 22(4), pages 242-250, November.
    21. Torbjörn Larsson & Michael Patriksson, 1992. "Simplicial Decomposition with Disaggregated Representation for the Traffic Assignment Problem," Transportation Science, INFORMS, vol. 26(1), pages 4-17, February.
    22. Richard Steinberg & Willard I. Zangwill, 1983. "The Prevalence of Braess' Paradox," Transportation Science, INFORMS, vol. 17(3), pages 301-318, August.
    23. N. D. Yen, 1995. "Lipschitz Continuity of Solutions of Variational Inequalities with a Parametric Polyhedral Constraint," Mathematics of Operations Research, INFORMS, vol. 20(3), pages 695-708, August.
    24. Stephen M. Robinson, 1991. "An Implicit-Function Theorem for a Class of Nonsmooth Functions," Mathematics of Operations Research, INFORMS, vol. 16(2), pages 292-309, May.
    25. Jong-Shi Pang, 1990. "Newton's Method for B-Differentiable Equations," Mathematics of Operations Research, INFORMS, vol. 15(2), pages 311-341, May.
    26. Cho, Hsun-Jung & Smith, Tony E. & Friesz, Terry L., 2000. "A reduction method for local sensitivity analyses of network equilibrium arc flows," Transportation Research Part B: Methodological, Elsevier, vol. 34(1), pages 31-51, January.
    27. Leurent, F., 1998. "Sensitivity and error analysis of the dual criteria traffic assignment model," Transportation Research Part B: Methodological, Elsevier, vol. 32(3), pages 189-204, April.
    28. J. Frédéric Bonnans & Roberto Cominetti & Alexander Shapiro, 1998. "Sensitivity Analysis of Optimization Problems Under Second Order Regular Constraints," Mathematics of Operations Research, INFORMS, vol. 23(4), pages 806-831, November.
    29. Wong, S. C. & Yang, Chao & Lo, Hong K., 2001. "A path-based traffic assignment algorithm based on the TRANSYT traffic model," Transportation Research Part B: Methodological, Elsevier, vol. 35(2), pages 163-181, February.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Josefsson, Magnus & Patriksson, Michael, 2007. "Sensitivity analysis of separable traffic equilibrium equilibria with application to bilevel optimization in network design," Transportation Research Part B: Methodological, Elsevier, vol. 41(1), pages 4-31, January.
    2. Michael Patriksson & R. Tyrrell Rockafellar, 2002. "A Mathematical Model and Descent Algorithm for Bilevel Traffic Management," Transportation Science, INFORMS, vol. 36(3), pages 271-291, August.
    3. Patriksson, Michael, 2008. "On the applicability and solution of bilevel optimization models in transportation science: A study on the existence, stability and computation of optimal solutions to stochastic mathematical programs," Transportation Research Part B: Methodological, Elsevier, vol. 42(10), pages 843-860, December.
    4. Michael Patriksson & R. Tyrrell Rockafellar, 2003. "Sensitivity Analysis of Aggregated Variational Inequality Problems, with Application to Traffic Equilibria," Transportation Science, INFORMS, vol. 37(1), pages 56-68, February.
    5. Shu Lu, 2008. "Sensitivity of Static Traffic User Equilibria with Perturbations in Arc Cost Function and Travel Demand," Transportation Science, INFORMS, vol. 42(1), pages 105-123, February.
    6. Du, Muqing & Chen, Anthony, 2022. "Sensitivity analysis for transit equilibrium assignment and applications to uncertainty analysis," Transportation Research Part B: Methodological, Elsevier, vol. 157(C), pages 175-202.
    7. Connors, Richard D. & Sumalee, Agachai & Watling, David P., 2007. "Sensitivity analysis of the variable demand probit stochastic user equilibrium with multiple user-classes," Transportation Research Part B: Methodological, Elsevier, vol. 41(6), pages 593-615, July.
    8. Seungkyu Ryu & Anthony Chen & Xiangdong Xu & Keechoo Choi, 2014. "A Dual Approach for Solving the Combined Distribution and Assignment Problem with Link Capacity Constraints," Networks and Spatial Economics, Springer, vol. 14(2), pages 245-270, June.
    9. Yang, Hai, 1997. "Sensitivity analysis for the elastic-demand network equilibrium problem with applications," Transportation Research Part B: Methodological, Elsevier, vol. 31(1), pages 55-70, February.
    10. Seungkyu Ryu, 2021. "Mode Choice Change under Environmental Constraints in the Combined Modal Split and Traffic Assignment Model," Sustainability, MDPI, vol. 13(7), pages 1-16, March.
    11. Lu, Shu & (Marco) Nie, Yu, 2010. "Stability of user-equilibrium route flow solutions for the traffic assignment problem," Transportation Research Part B: Methodological, Elsevier, vol. 44(4), pages 609-617, May.
    12. Rinaldi, Marco & Tampère, Chris M.J. & Viti, Francesco, 2018. "On characterizing the relationship between route choice behaviour and optimal traffic control solution space," Transportation Research Part B: Methodological, Elsevier, vol. 117(PB), pages 892-906.
    13. Byung Chung & Hsun-Jung Cho & Terry Friesz & Henh Huang & Tao Yao, 2014. "Sensitivity Analysis of User Equilibrium Flows Revisited," Networks and Spatial Economics, Springer, vol. 14(2), pages 183-207, June.
    14. Clark, Stephen D. & Watling, David P., 2006. "Applications of sensitivity analysis for probit stochastic network equilibrium," European Journal of Operational Research, Elsevier, vol. 175(2), pages 894-911, December.
    15. Clark, Stephen D. & Watling, David P., 2002. "Sensitivity analysis of the probit-based stochastic user equilibrium assignment model," Transportation Research Part B: Methodological, Elsevier, vol. 36(7), pages 617-635, August.
    16. Lundgren, Jan T. & Peterson, Anders, 2008. "A heuristic for the bilevel origin-destination-matrix estimation problem," Transportation Research Part B: Methodological, Elsevier, vol. 42(4), pages 339-354, May.
    17. Takebayashi, Mikio & Kanafani, Adib, 2005. "Network Competition in Air Transportation Markets: Bi-Level Approach," Research in Transportation Economics, Elsevier, vol. 13(1), pages 101-119, January.
    18. S. Dempe & A. Zemkoho, 2012. "Bilevel road pricing: theoretical analysis and optimality conditions," Annals of Operations Research, Springer, vol. 196(1), pages 223-240, July.
    19. Jafari, Ehsan & Boyles, Stephen D., 2016. "Improved bush-based methods for network contraction," Transportation Research Part B: Methodological, Elsevier, vol. 83(C), pages 298-313.
    20. Jong-Shi Pang & Defeng Sun & Jie Sun, 2003. "Semismooth Homeomorphisms and Strong Stability of Semidefinite and Lorentz Complementarity Problems," Mathematics of Operations Research, INFORMS, vol. 28(1), pages 39-63, February.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:inm:ortrsc:v:38:y:2004:i:3:p:258-281. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Chris Asher (email available below). General contact details of provider: https://edirc.repec.org/data/inforea.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.