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Properties of the Equilibrium State in Transportation Networks

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  • Michael A. Hall

    (Bell Laboratories, Piscataway, New Jersey)

Abstract

The Wardrop equilibrium problem for urban road networks is considered. It is shown that the (unique) equilibrium total flow vector is a continuous function of the input traffic flows. Under fairly weak conditions it is proven that the total origin to destination travel costs are also continuous functions of the input traffic flows. It is then shown that each origin to destination cost is a monotonically nondecreasing function of its own input flow when other inputs are held fixed. Finally, it is demonstrated by means of simple examples that the equilibrium total flow vector and origin to destination travel cost functions are not differentiable at certain (possibly difficult to predict) points in the set of feasible input flow vectors and that the cost functions do not in general possess such other potentially useful properties as convexity or concavity. These results are important to an understanding of the sensitivity of the equilibrium state to variations in input data.

Suggested Citation

  • Michael A. Hall, 1978. "Properties of the Equilibrium State in Transportation Networks," Transportation Science, INFORMS, vol. 12(3), pages 208-216, August.
    • Handle: RePEc:inm:ortrsc:v:12:y:1978:i:3:p:208-216
    DOI: 10.1287/trsc.12.3.208
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    Citations

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    Cited by:

    1. C Fisk, 1984. "A Nonlinear Equation Framework for Solving Network Equilibrium Problems," Environment and Planning A, , vol. 16(1), pages 67-80, January.
    2. Codina, Esteve & Barcelo, Jaume, 2004. "Adjustment of O-D trip matrices from observed volumes: An algorithmic approach based on conjugate directions," European Journal of Operational Research, Elsevier, vol. 155(3), pages 535-557, June.
    3. 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.
    4. Patrick Maillé & Nicolás E. Stier-Moses, 2009. "Eliciting Coordination with Rebates," Transportation Science, INFORMS, vol. 43(4), pages 473-492, November.
    5. Wang, Shuaian & Meng, Qiang & Liu, Zhiyuan, 2013. "Fundamental properties of volume–capacity ratio of a private toll road in general networks," Transportation Research Part B: Methodological, Elsevier, vol. 47(C), pages 77-86.
    6. Watling, David, 1998. "Perturbation stability of the asymmetric stochastic equilibrium assignment model," Transportation Research Part B: Methodological, Elsevier, vol. 32(3), pages 155-171, April.
    7. 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.
    8. Yang, Hai, 1998. "Multiple equilibrium behaviors and advanced traveler information systems with endogenous market penetration," Transportation Research Part B: Methodological, Elsevier, vol. 32(3), pages 205-218, April.
    9. Michael Patriksson, 2004. "Sensitivity Analysis of Traffic Equilibria," Transportation Science, INFORMS, vol. 38(3), pages 258-281, August.
    10. 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.

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