The Piecewise Constant/Linear Solution for Dynamic User Equilibrium

František Kolovský1, Ivana Kolingerová2
1Department of Geomatics, University of West Bohemia, Pilsen, Czech Republic
2Department of Computer Science and Engineering, University of West Bohemia, Pilsen, Czech Republic

Tóm tắt

The aim of this work is to increase the precision of the solution of dynamic user equilibrium assuming that the computation takes a reasonable time so that the solution is useful in practice. The proposed method replaces the classical grid-based solution by a near continuous-time solution based on piecewise linear/constant functions that removes a lot of disadvantage of discretization. The testing shows that the precision of the solution can be easily driven by a few approximation parameters that can be changed during computation. Using the proposed method, the near continuous time solution of the dynamic user equilibrium for a real size network can be computed in reasonable computation time.

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Tài liệu tham khảo

Bliemer MCJ, Raadsen MPH (2019) Continuous-time general link transmission model with simplified fanning, Part I: Theory and link model formulation. Trans Res Part B-Meth 126:442–470. https://doi.org/10.1016/j.trb.2018.01.001, http://www.sciencedirect.com/science/article/pii/S0191261517301479 Corthout R, Himpe W, Viti F, Frederix R, Tampère CM (2014) Improving the efficiency of repeated dynamic network loading through marginal simulation. Transp Res Part C Emerg Technol 41:90–109. https://doi.org/10.1016/j.trc.2013.12.009, https://linkinghub.elsevier.com/retrieve/pii/S0968090X1300274X Daganzo CF (1994) The cell transmission model: A dynamic representation of highway traffic consistent with the hydrodynamic theory. Transp Res Part B-Meth 28(4):269–287. https://doi.org/10.1016/0191-2615(94)90002-7, http://linkinghub.elsevier.com/retrieve/pii/0191261594900027 Daganzo CF (1995) The cell transmission model, part II: Network traffic. Transp Res Part B-Meth 29(2):79–93. https://doi.org/10.1016/0191-2615(94)00022-R, http://linkinghub.elsevier.com/retrieve/pii/019126159400022R Dean BC (2004) Shortest paths in FIFO time-dependent networks: Theory and algorithms. Rapport technique, Massachusetts Institute of Technology Dehne F, Omran MT, Sack JR (2012) Shortest paths in time-dependent FIFO networks. Algorithmica 62(1-2):416–435. https://doi.org/10.1007/s00453-010-9461-6, http://link.springer.com/10.1007/s00453-010-9461-6 Ding B, Yu JX, Qin L (2008) Finding time-dependent shortest paths over large graphs. In: Proceedings of the 11th international conference on Extending database technology: Advances in database technology, ACM Press, p 205, https://doi.org/10.1145/1353343.1353371, http://portal.acm.org/citation.cfm?doid=1353343.1353371 Facchinei F, Pang JS (eds) (2004) Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer Series in Operations Research and Financial Engineering, Springer New York, New York, NY. https://doi.org/10.1007/b97543, http://link.springer.com/10.1007/b97543 Foschini L, Hershberger J, Suri S (2014) On the Complexity of Time-Dependent Shortest Paths. Algorithmica 68(4):1075–1097. https://doi.org/10.1007/s00453-012-9714-7, https://link.springer.com/article/10.1007/s00453-012-9714-7 Friesz TL, Han K (2019) The mathematical foundations of dynamic user equilibrium. Transp Res Part B-Meth 126:309–328. https://doi.org/10.1016/j.trb.2018.08.015, https://www.sciencedirect.com/science/article/pii/S0191261517301960 Friesz TL, Bernstein D, Smith TE, Tobin RL, Wie BW (1993) A variational inequality formulation of the dynamic network user equilibrium problem. Oper Res 41(1):179–191 Friesz TL, Kim T, Kwon C, Rigdon MA (2011) Approximate network loading and dual-time-scale dynamic user equilibrium. Trans Res Part B-Meth 45(1):176–207 Geisberger R (2010) Engineering Time-dependent One-To-All Computation. http://arxiv.org/abs/1010.0809 Geisberger R, Sanders P (2010) Engineering time-dependent many-to-many shortest paths computation. In: OASIcs-OpenAccess Series in Informatics, Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, vol 14 Gentile G (2015) Using the general link transmission model in a dynamic traffic assignment to simulate congestion on urban networks. Trans Res Procedia 5:66–81. https://doi.org/10.1016/j.trpro.2015.01.011, http://linkinghub.elsevier.com/retrieve/pii/S2352146515000125 Gentile G (2016) Solving a dynamic user equilibrium model based on splitting rates with gradient projection algorithms. Transp Res Part B-Meth 92:120–147. https://doi.org/10.1016/j.trb.2016.02.005, http://linkinghub.elsevier.com/retrieve/pii/S0191261516000370 Han D, Lo HK (2004) Solving non-additive traffic assignment problems: A descent method for co-coercive variational inequalities. Eur J Oper Res 159(3):529–544. https://doi.org/10.1016/S0377-2217(03)00423-5, https://linkinghub.elsevier.com/retrieve/pii/S0377221703004235 Han K, Friesz TL, Szeto WY, Liu H (2015a) Elastic demand dynamic network user equilibrium: Formulation, existence and computation. Transp Res Part B-Meth 81:183–209 Han K, Szeto W, Friesz TL (2015b) Formulation, existence, and computation of boundedly rational dynamic user equilibrium with fixed or endogenous user tolerance. Transp Res Part B-Meth 79:16–49 Han K, Piccoli B, Friesz TL (2016) Continuity of the path delay operator for dynamic network loading with spillback. Transp Res Part B-Meth 92:211–233. https://doi.org/10.1016/j.trb.2015.09.009, http://linkinghub.elsevier.com/retrieve/pii/S0191261515002039 Han K, Eve G, Friesz TL (2019) Computing dynamic user equilibria on large-scale networks with software implementation. Netw Spatial Econ 19(3):869–902 Himpe W (2016) Integrated algorithms for repeated dynamic traffic assignments. PhD thesis, KU Leuven, Leuven. https://www.mech.kuleuven.be/en/cib/traffic/downloads/PhDtextWilemHimpe Himpe W, Corthout R, Tampére MC (2016) An efficient iterative link transmission model. Transp Res Part B-Meth 92:170–190. https://doi.org/10.1016/j.trb.2015.12.013, https://linkinghub.elsevier.com/retrieve/pii/S0191261515002805 Hoogendoorn SP, Bovy PHL (2001) State-of-the-art of vehicular traffic flow modelling 215:21 Huang HJ, Lam WH (2002) Modeling and solving the dynamic user equilibrium route and departure time choice problem in network with queues. Transp Res Part B-Meth 36(3), 253–273 Jang W, Ran B, Choi K (2005) A discrete time dynamic flow model and a formulation and solution method for dynamic route choice. Transp Res Part B-Meth 39(7):593–620. https://doi.org/10.1016/j.trb.2004.07.005, http://linkinghub.elsevier.com/retrieve/pii/S0191261504001043 Kolovský F, Ježek J, Kolingerová I (2019a) The e-approximation of the time-dependent shortest path problem solution for all departure times. ISPRS Int J Geo-Inform 8(12):538. https://doi.org/10.3390/ijgi8120538, https://www.mdpi.com/2220-9964/8/12/538, number: 12 Publisher: Multidisciplinary Digital Publishing Institute Kolovský F, Kolingerová I, Ježek J (2019b) Algorithms in transportation: The State of the Art and Concept of PhD Thesis. https://kolovsky.cz/static/teze_final_web.pdf Konno H, Kuno T (1988) Best piecewise constant approximation of a function of single variable. Oper Res Lett 7(4):205–210. https://doi.org/10.1016/0167-6377(88)90030-2 http://www.sciencedirect.com/science/article/pii/0167637788900302 Lighthill MJ, Whitham GB (1955) On kinematic waves II. A theory of traffic flow on long crowded roads. Proc R Soc A: Math Phys Eng Sci 229(1178):317–345 Lo HK, Szeto WY (2002) A cell-based variational inequality formulation of the dynamic user optimal assignment problem. Transp Res Part B:Meth 36(5):421–443 Long J, Huang HJ, Gao Z, Szeto WY (2013) An Intersection-Movement-Based Dynamic User Optimal Route Choice Problem. Oper Res 61(5):1134–1147. https://doi.org/10.1287/opre.2013.1202, http://pubsonline.informs.org/doi/abs/10.1287/opre.2013.1202 Nezamuddin N, Boyles SD (2015) A continuous DUE algorithm using the link transmission model. Netw Spatial Econ15(3):465–483. https://doi.org/10.1007/s11067-014-9234-x, https://doi.org/10.1007/s11067-014-9234-x Omran M, Sack JR (2014) Improved approximation for time-dependent shortest paths. In: International Computing and Combinatorics Conference, Springer, pp 453–464 Orda A, Rom R (1990) Shortest-path and minimum-delay algorithms in networks with time-dependent edge-length. Journal of the ACM (JACM) 37(3):607–625 Raadsen MPH, Bliemer MCJ (2019) Continuous-time general link transmission model with simplified fanning, Part II: Event-based algorithm for networks. Transp Res Part B-Meth 126:471–501. https://doi.org/10.1016/j.trb.2018.01.003, http://www.sciencedirect.com/science/article/pii/S0191261517301492 Raadsen MPH, Bliemer MCJ, Bell MGH (2016) An efficient and exact event-based algorithm for solving simplified first order dynamic network loading problems in continuous time. Transp Res Part B-Meth 92:191–210. https://doi.org/10.1016/j.trb.2015.08.004, http://www.sciencedirect.com/science/article/pii/S0191261515001794 Ran B, Hall RW, Boyce DE (1996) A link-based variational inequality model for dynamic departure time/route choice. Transp Res Part B-Meth 30(1):31–463. https://doi.org/10.1016/0191-2615(95)00010-0, http://linkinghub.elsevier.com/retrieve/pii/0191261595000100 Richards PI (1956) Shock waves on the highway. Oper Res 4(1):42–51 Song W, Han K, Wang Y, Friesz T, del Castillo E (2017) Statistical metamodeling of dynamic network loading. Transportation Research Procedia 23:263–282. https://doi.org/10.1016/j.trpro.2017.05.016, https://linkinghub.elsevier.com/retrieve/pii/S2352146517302934 Szeto W, Lo HK (2004) A cell-based simultaneous route and departure time choice model with elastic demand. Transp Res Part B-Meth 38(7):593–612. https://doi.org/10.1016/j.trb.2003.05.001, http://linkinghub.elsevier.com/retrieve/pii/S0191261503000924 Szeto WY, Lo HK (2006) Dynamic traffic assignment: properties and extensions. Transportmetrica 2(1):31–52. https://doi.org/10.1080/18128600608685654, http://www.tandfonline.com/doi/abs/10.1080/18128600608685654 Tian LJ, Huang HJ, Gao ZY (2012) A cumulative perceived value-based dynamic user equilibrium model considering the travelers’ risk evaluation on arrival time. Netw Spatial Econ 12(4):589–608. https://doi.org/10.1007/s11067-011-9168-5 , http://link.springer.com/10.1007/s11067-011-9168-5 Transportation Networks for Research Core Team (2021) Transportation networks for research. https://github.com/bstabler/TransportationNetworks, https://github.com/bstabler/TransportationNetworks Ukkusuri SV, Han L, Doan K (2012) Dynamic user equilibrium with a path based cell transmission model for general traffic networks. Transp Res Part B-Meth 46(10):1657–1684. https://doi.org/10.1016/j.trb.2012.07.010, http://linkinghub.elsevier.com/retrieve/pii/S0191261512001038 van der Gun JPT, Pel AJ, van Arem B (2017) Extending the Link Transmission Model with non-triangular fundamental diagrams and capacity drops. Transp Res Part B-Meth 98:154–178. https://doi.org/10.1016/j.trb.2016.12.011, http://www.sciencedirect.com/science/article/pii/S0191261516309572 Wardrop JG (1952) Road paper. some theoretical aspects of road traffic research. Proceed Ins Civil Eng 1(3):325–362 Wu X, Liu HX (2011) A shockwave profile model for traffic flow on congested urban arterials. Transp Res Part B-Meth 45(10):1768–1786. https://doi.org/10.1016/j.trb.2011.07.013, http://www.sciencedirect.com/science/article/pii/S0191261511001135 Yperman I (2007) The Link Transmission Model for Dynamic Network Loading. Katholieke Universiteit Leuven, Belgium