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Mathematical and Computational Models for Congestion Charging pp 219–240Cite as

An Optimal Toll Design Problem with Improved Behavioural Equilibrium Model: The Case of the Probit Model

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

Summary

This paper considers the optimal toll design problem that uses the Probit model to determine travellers’ route-choices. Under probit, the route flow solution to the resulting stochastic user equilibrium (SUE) is unique and can be stated implicitly as a function of tolls. This reduces the toll design problem to an optimization problem with only nonnegativity constraints. Additionally, the gradient of the objective function can be approximated using the chain rule and the first order Taylor approximation of the equilibrium condition. To determine SUE, this paper considers two techniques. One uses Monte-Carlo simulation to estimate route choice probabilities and the method of successive averages with its prescribed step length. The other relies on the Clark approximation and computes an optimal step length. Although both are effective at solving the toll design problem, numerical experiments show that the technique with the Clark approximation is more robust on a small network.

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Author information

Authors and Affiliations

  1. Institute for Transport Studies, University of Leeds, 38 Woodhouse Lane, Leeds, LS2 9JT, UK

    Agachai Sumalee, Richard Connors & David Watling

Authors
  1. Agachai Sumalee
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  2. Richard Connors
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  3. David Watling
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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

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© 2006 Springer Science and Business Media, Inc.

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Sumalee, A., Connors, R., Watling, D. (2006). An Optimal Toll Design Problem with Improved Behavioural Equilibrium Model: The Case of the Probit Model. 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_10

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