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Submission to the DTA2012 Special Issue: Approximating Time Delays in Solving Continuous-Time Dynamic User Equilibria

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Abstract

We develop an approximation scheme, called “pseudo-derivative (PD)” to solving the dynamic user equilibrium (DUE) problem. The PD approximation can convert time-varying, state dependent delays usually involved in DUE to a constant time delay. We study the properties of the proposed PD and the resulting approximate DUE (ADUE) problem after applying the approximation. Some issues of the ADUE, such as the possible violation of the flow conservation at network nodes are also discussed and resolved. It turns out that the original DUE problem can be solved iteratively with an ADUE solved in each iteration. Numerical results are shown on a small test network and the Sioux Falls network. The results show that the iterative algorithm can converge to some reasonable solution, although a formal convergence proof result is not established in the paper.

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Notes

  1. We notice here that some theoretical results are provided in Friesz et al. (2001) such as the necessary conditions of the optimal control problem with time delays related to DUE, by assuming the time-delay term is differentiable with respect to the state variable. However we still lack a practical tool that can directly analyze and solve such a problem with time-varying, state dependent delays.

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Acknowledgments

The authors would like to thank the anonymous referees for their helpful suggestions on an earlier version of the paper. The work of Rui Ma and Xuegang (Jeff) Ban is based on research supported by the National Science Foundation under Grant EFRI 1024647. The work of Jong-Shi Pang is based on research supported by the National Science Foundation under Grants EFRI 1024984 and CMMI 0969600. The work of Henry X. Liu is based on research supported by the National Science Foundation under Grant EFRI 1024604. Any opinions, findings, and conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the National Science Foundation.

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Authors and Affiliations

  1. Department of Civil and Environmental Engineering, 1001 Ghausi Hall, University of California, One Shields Avenue, Davis, CA, 15616, USA

    Rui Ma

  2. Department of Civil and Environmental Engineering, Rensselaer Polytechnic Institute (RPI), 110 Eighth Street, Room JEC 4034, Troy, NY, 12180-3590, USA

    Xuegang (Jeff) Ban

  3. Department of Industrial and Systems Engineering, University of Southern California, University Park, Los Angeles, CA, 90089-1450, USA

    Jong-Shi Pang

  4. Department of Civil Engineering, University of Minnesota, Minneapolis, MN, 55455, USA

    Henry X. Liu

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  1. Rui Ma
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  2. Xuegang (Jeff) Ban
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  3. Jong-Shi Pang
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  4. Henry X. Liu
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Corresponding author

Correspondence to Xuegang (Jeff) Ban.

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Submittal to Networks and Spatial Economics (Special Issue for DTA2012 Symposium)

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Ma, R., Ban, X.(., Pang, JS. et al. Submission to the DTA2012 Special Issue: Approximating Time Delays in Solving Continuous-Time Dynamic User Equilibria. Netw Spat Econ 15, 443–463 (2015). https://doi.org/10.1007/s11067-014-9240-z

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  • DOI: https://doi.org/10.1007/s11067-014-9240-z

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