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Modeling and solving continuous-time instantaneous dynamic user equilibria: A differential complementarity systems approach

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  • Ban, Xuegang (Jeff)
  • Pang, Jong-Shi
  • Liu, Henry X.
  • Ma, Rui

Abstract

This paper is the second of a two-part research wherein we undertake a mathematically rigorous investigation of the continuous-time dynamic user equilibrium (DUE) problem using the recently introduced mathematical paradigm of differential complementarity systems (DCSs). Based on the thorough study of continuous-time single-destination point-queue models in the previous part, we first extend this special case to multiple destinations respecting the First-In–First-Out property of travel flows. A DCS with constant time delay is then introduced to formulate the continuous-time model of instantaneous dynamic traffic equilibria (IDUE) with a fixed demand profile. We develop a time decomposition scheme based on link free flow travel times to convert the delay DCS to a series of DCSs without time delays that are solved by a numerical time-stepping method. We provide rigorous numerical treatment of the time-decomposed IDUE model, including solvability of the discrete-time complementarity problems and convergence of the numerical trajectories to a continuous-time solution. We present numerical results to validate the IDUE on a small network and also on the Sioux Falls network.

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  • Ban, Xuegang (Jeff) & Pang, Jong-Shi & Liu, Henry X. & Ma, Rui, 2012. "Modeling and solving continuous-time instantaneous dynamic user equilibria: A differential complementarity systems approach," Transportation Research Part B: Methodological, Elsevier, vol. 46(3), pages 389-408.
    • Handle: RePEc:eee:transb:v:46:y:2012:i:3:p:389-408
    DOI: 10.1016/j.trb.2011.11.002
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    1. Friesz, Terry L. & Mookherjee, Reetabrata, 2006. "Solving the dynamic network user equilibrium problem with state-dependent time shifts," Transportation Research Part B: Methodological, Elsevier, vol. 40(3), pages 207-229, March.
    2. Kuwahara, Masao & Akamatsu, Takashi, 2001. "Dynamic user optimal assignment with physical queues for a many-to-many OD pattern," Transportation Research Part B: Methodological, Elsevier, vol. 35(5), pages 461-479, June.
    3. MERCHANT, Deepak K. & NEMHAUSER, George L., 1978. "Optimality conditions for a dynamic traffic assignment model," LIDAM Reprints CORE 345, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    4. Camlibel, M.K., 2001. "Complementarity methods in the analysis of piecewise linear dynamical systems," Other publications TiSEM c3e08484-56d2-4f1a-9db0-8, Tilburg University, School of Economics and Management.
    5. Friesz, Terry L. & Kim, Taeil & Kwon, Changhyun & Rigdon, Matthew A., 2011. "Approximate network loading and dual-time-scale dynamic user equilibrium," Transportation Research Part B: Methodological, Elsevier, vol. 45(1), pages 176-207, January.
    6. Vickrey, William S, 1969. "Congestion Theory and Transport Investment," American Economic Review, American Economic Association, vol. 59(2), pages 251-260, May.
    7. Daoli Zhu & Patrice Marcotte, 2000. "On the Existence of Solutions to the Dynamic User Equilibrium Problem," Transportation Science, INFORMS, vol. 34(4), pages 402-414, November.
    8. Daganzo, Carlos F., 1995. "Properties of link travel time functions under dynamic loads," Transportation Research Part B: Methodological, Elsevier, vol. 29(2), pages 95-98, April.
    9. Y. W. Xu & J. H. Wu & M. Florian & P. Marcotte & D. L. Zhu, 1999. "Advances in the Continuous Dynamic Network Loading Problem," Transportation Science, INFORMS, vol. 33(4), pages 341-353, November.
    10. Bin Ran & David E. Boyce & Larry J. LeBlanc, 1993. "A New Class of Instantaneous Dynamic User-Optimal Traffic Assignment Models," Operations Research, INFORMS, vol. 41(1), pages 192-202, February.
    11. Carey, Malachy & McCartney, Mark, 2002. "Behaviour of a whole-link travel time model used in dynamic traffic assignment," Transportation Research Part B: Methodological, Elsevier, vol. 36(1), pages 83-95, January.
    12. Terry L. Friesz & David Bernstein & Tony E. Smith & Roger L. Tobin & B. W. Wie, 1993. "A Variational Inequality Formulation of the Dynamic Network User Equilibrium Problem," Operations Research, INFORMS, vol. 41(1), pages 179-191, February.
    13. MERCHANT, Deepak K. & NEMHAUSER, George L., 1978. "A model and an algorithm for the dynamic traffic assignment problems," LIDAM Reprints CORE 346, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    14. Deepak K. Merchant & George L. Nemhauser, 1978. "Optimality Conditions for a Dynamic Traffic Assignment Model," Transportation Science, INFORMS, vol. 12(3), pages 200-207, August.
    15. Heemels, W.P.M.H. & Schumacher, J.M. & Weiland, S., 2000. "Linear complimentarity systems," Other publications TiSEM 6cdf0170-6ea9-4fdc-8cfa-6, Tilburg University, School of Economics and Management.
    16. Lam, William H. K. & Huang, Hai-Jun, 1995. "Dynamic user optimal traffic assignment model for many to one travel demand," Transportation Research Part B: Methodological, Elsevier, vol. 29(4), pages 243-259, August.
    17. Terry L. Friesz, 2010. "Dynamic User Equilibrium," International Series in Operations Research & Management Science, in: Dynamic Optimization and Differential Games, chapter 0, pages 411-456, Springer.
    18. Kuwahara, Masao & Akamatsu, Takashi, 1997. "Decomposition of the reactive dynamic assignments with queues for a many-to-many origin-destination pattern," Transportation Research Part B: Methodological, Elsevier, vol. 31(1), pages 1-10, February.
    19. Deepak K. Merchant & George L. Nemhauser, 1978. "A Model and an Algorithm for the Dynamic Traffic Assignment Problems," Transportation Science, INFORMS, vol. 12(3), pages 183-199, August.
    20. Ban, Xuegang (Jeff) & Liu, Henry X. & Ferris, Michael C. & Ran, Bin, 2008. "A link-node complementarity model and solution algorithm for dynamic user equilibria with exact flow propagations," Transportation Research Part B: Methodological, Elsevier, vol. 42(9), pages 823-842, November.
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    7. Ke Han & Gabriel Eve & Terry L. Friesz, 2019. "Computing Dynamic User Equilibria on Large-Scale Networks with Software Implementation," Networks and Spatial Economics, Springer, vol. 19(3), pages 869-902, September.
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    11. Zhi-Yang Lin & S. C. Wong & Peng Zhang & Keechoo Choi, 2018. "A Predictive Continuum Dynamic User-Optimal Model for the Simultaneous Departure Time and Route Choice Problem in a Polycentric City," Service Science, INFORMS, vol. 52(6), pages 1496-1508, December.
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