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
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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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