Abstract
Traffic network models tend to become very large even for medium-size static assignment problems. Adding a time dimension, together with time-varying flows and travel times within links and queues, greatly increases the scale and complexity of the problem. In view of this, to retain tractability in dynamic traffic assignment (DTA) formulations, especially in mathematical programming formulations, additional assumptions are normally introduced. In particular, the time varying flows and travel times within links are formulated as so-called whole-link models. We consider the most commonly used of these whole-link models and some of their limitations.
In current whole-link travel-time models, a vehicle's travel time on a link is treated as a function only of the number of vehicles on the link at the moment the vehicle enters. We first relax this by letting a vehicle's travel time depend on the inflow rate when it enters and the outflow rate when it exits. We further relax the dynamic assignment formulation by stating it as a bi-level program, consisting of a network model and a set of link travel time sub-models, one for each link. The former (the network model) takes the link travel times as bounded and assigns flows to links and routes. The latter (the set of link models) does the reverse, that is, takes link inflows as given and finds bounds on link travel times. We solve this combined model by iterating between the network model and link sub-models until a consistent solution is found. This decomposition allows a much wider range of link flow or travel time models to be used. In particular, the link travel time models need not be whole-link models and can be detailed models of flow, speed and density varying along the link. In our numerical examples, algorithms designed to solve this bi-level program converged quickly, but much remains to be done in exploring this approach further. The algorithms for solving the bi-level formulation may be interpreted as traveller learning behaviour, hence as a day-to-day traffic dynamics. Thus, even though in our experiments the algorithms always converged, their behaviour is still of interest even if they cycled rather than converged. Directions for further research are noted. The bi-level model can be extended to handle issues and features similar to those addressed by other DTA models.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Adamo, V., Astarita, V., Florian, M., Mahut, M., and Jia Hao Wu (1999). “Modelling the Spill-Back of Congestion in Link Based Dynamic Network Loading Models: A Simulation Model with Application.” 14th International Symposium on Theory of Traffic Flow, Jerusalem July 1999. Elsevier, pp. 555–573.
Addison, J.D. and B.G. Heydecker. (1995). “Traffic Models for Dynamic Traffic Assignment.” In N.H. Gartner and G. Improta (eds.), Urban Traffic Networks: Dynamic Flow Modelling and Control. London: Springer-Verlag, pp. 213–231.
Ali, A.I. and J.L. Kennington. (1989). MODFLO User's Guide. Technical Report 89-OR-03. Department of Operations Research and Engineering, Southern Methodist University, Dallas, TX.
Astarita, V. (1995). “Flow Propagation Description in Dynamic Network Loading Models.” Proceedings of IV International Conference on Application of Advanced Technologies in Transportation Engineering (AATT), 27–30 June 1995. Capri. Published by American Society of Civil Engineers, pp. 599–603.
Astarita, V. (1996). “A Continuous Time Link Model for Dynamic Network Loading Based on Travel Time Function.” 13th International Symposium on Theory of Traffic Flow, Lyon July 1996. Elsevier, pp. 79–102.
Boyce, D.E., B. Ran, and L.J. LeBlanc. (1995). “Solving an Instantaneous Dynamic User Optimal Route Choice Model.” Transportation Science 29(2), 128–142.
Carey, M. (1984). “Exploring Time Varying Flows on Congested Networks.” Carnegie Mellon University, Pittsburgh, PA. Presented at the TIMS/ORSA Conference, Session TA18, San Francisco, CA, 14–16 May.
Carey, M. (1986). “A Constraint Qualification for a Dynamic Traffic Assignment Model.” Transportation Science 20(1).
Carey, M. (1987). “Optimal Time-Varying Flows on Congested Networks.” Operations Research 35(1), 56–69.
Carey, M. (1990). “Extending and Solving a Multi-Perod Congested Network Flow Model.” Computers and Operations Research 17(5), 495–507.
Carey, M. (1992). “Nonconvexity of the Dynamic Traffic Assignment Problem.” Transportation Research 26B(2), 127–133.
Carey, M. (1997a). “Time-Varying Inflows, Outflows and Trip Times on Single Links.” Research Report, Faculty of Business and Management, University of Ulster, Northern Ireland.
Carey, M. (1997b). “Dynamic Modelling for Congestion Pricing and Related Strategies.” Final Report to the UK Engineering and Science Research Council (EPSRC) on EPSRC Grant no. GR/J53171. October. Faculty of Business and Management, University of Ulster.
Carey, M. (1998). “A Framework for User Equilibrium Dynamic Traffic Assignment.” Research Report, Faculty of Business and Management, University of Ulster. Under revision for Transportation Research.
Carey, M. (1999). “A Framework for System Optimal Dynamic Traffic Assignment.” Research Report, Faculty of Business and Management, University of Ulster, Northem Ireland.
Carey, M. and M. McCartney. (1999a). “Behaviour of a Whole-Link Travel Time Model used in Dynamic Traffic Assignment.” Forthcoming in Transportation Research.
Carey, M. and M. McCartney. (2000). “A New Whole-Link Travel-Time Model with Desirable Properties.” Research Report, Faculty of Business and Management, University of Ulster, Northern Ireland. Under revision for publication.
Carey, M. and A. Srinivasan. (1988). “Congested Traffic Networks: Time-Varying Flows and Start-Time Policies.” European Journal of Operational Research 36, 227–240.
Carey, M. and A. Srinivasan. (1993). “Externalities, Average and Marginal Costs, and Tolls on Congested Networks with Time-Varying Flows.” Operations Research 41(1), 217–231.
Carey, M. and E. Subrahmanian. (2000). “An Approach to Modeling Time-Varying Flows on Congested Networks.” Trunsportation Research B 34(3), 157–183.
Daganzo, C.F. (1995). “Properties of Link Travel Times under Dynamic Load.” Transportation Research 29B(2), 95–98.
Daganzo, C.F. (1997). Fundamentals of Transportation and Traffic Operations. Elsevier Science.
Fox, K. (ed.) (1997). Review of Micro-Simulation Models, http://www.its.leeds.ac.uk/smartest/. Deliverable No. 3 of the SMARTEST (Simulation Modelling Applied to Road Transport European Scheme Tests) Project. Institute of Transport Studies, Leeds University, Leeds, England.
Friesz, T.L., Luque, J., Tobin, R.L., and Wie, B.-Y. (1989). “Dynamic Network Traffic Assignment Considered as a Continuous Time Optimal Control Problem.” Operations Research 37(6), 893–901.
Friesz, T.L., Bernstein, D., Smith, T.E., Tobin, R.L. and Wie, B.W. (1993). “A Variational Inequality Formulation of the Dynamic Network User Equilibrium Problem.” Operations Research 41, 179–191.
Gartner, N., C.J. Messer, and A.K. Rathi. (eds.) (1997). Traffic Flow Theory: A State of the Art Report. http://stargate.ornl.gov/trb/tft.html. An update and expansion of Traffic Flow Theory, Special Report 165. Transportation Research Board, USA.
Heydecker, B.G. and J.D. Addison. (1998). “Analysis of Traffic Models for Dynamic Equilibrium Traffic Assignment.” In M.G.H. Bell (ed), Transportation Networks: Recent Methological Advances. Oxford: Pergamon, pp. 35–49.
Hurdle, V.F. (1986). “Technical Note on a Paper by Andre de Palma, Moshe Ben-Akiva, Claude Lefevre, and Nicholas Litinas entitled 'stochastic equilibrium model of peak period traffic congestion'.” Transportation Science 20(4), 287–289.
Jayakrishnan, R., H.S. Mahmassani, and T.Y. Hu. (1994). “An Evaluation Tool for Advanced Traffic Information and Management Systems in Urban Networks.” Transportation Research 3C, 129–147.
Kennington, J. (1995). Modflo2: A Programme for Solving Network Flow Problems. Department of Computer Science & Eng., Southern Methodist University, Dallas, Texas.
Leonard, D.R., J.B. Tough, and P.C. Baguely. (1978). CONTRAM: A Traffic Assignment Model for Predicting Flows and Queues during Peak Periods. TRL Report RR 841. Transport Research Laboratory, Crowthorne, UK.
Leonard, D.R., P. Gower, and N.B. Taylor. (1989). CONTRAM: Structure of the Model. TRL Report RR 178. Transport Research Laboratory, Crowthorne, UK.
Lighthill, M.J. and G.B. Whitham. (1955). “On Kinematic Waves. I: Flow Movement in Long Rivers II: A Theory of Traffic Flow on Long Crowded Roads.” Proceedings of the Royal Society A 229, 281–345.
Mahmassani, H.S. and S. Peeta. (1995). “System Optimal Dynamic Assignment for Electronic Route Guidance in a Congested Traffic Network.” In N.H. Gartner and G. Improta (eds.), Urban Traffic Networks: Dynamic Flow Modelling and Control. London: Springer-Verlag.
Mahmassani, H.S., Peta, S., Chang, G. and Junchaya, T. (1992). “A Review of Dynamic Assignment and Traffic Simulation Models for ADIS/ATMS Applications.” Technical Report DTFH61–90-R-00074-1. Center for Transportation Research, University of Texas at Austin.
May, A.D. (1990). Traffic Flow Fundamentals. Englewood Cliffs, NJ: Prentice Hall.
Merchant, D.K. (1974). “A Study of Dynamic Traffic Assignment and Control.” Ph.D. Thesis, Cornell University.
Merchant, D.K. and G.L. Nemhauser. (1978a). “A Model and an Algorithm for the Dynamic Traffic Assignment Problem.” Transportation Science 12(3), 183–199.
Merchant, D.K. and G.L. Nemhauser. (1978b). “Optimality Conditions for a Dynamic Traffic Assignment Model.” Transportation Science 12(3), 200–207.
Ran, B. and D. Boyce. (1994). Dynamic Urban Transportation Network Models—Theory and Implications for Intelligent Vehicle-Highway Systems. Lecture notes in Economics and Mathematical Systems 417. Heidelber: Springer-Verlag.
Ran, B. and D. Boyce. (1996). Modelling Dynamic Transportation Networks. Heidelber: Springer-Verlag.
Ran, B., D.E. Boyce, and L.J. LeBlanc. (1993). “A New Class of Instantaneous Dynamic User-Optimal Traffic Assignment Models.” Operations Research 41, 192–202.
Ran, B., Rouphail, N., Tarko, A. and Boyce, D.E. (1997). “Towards a Class of Link Travel Time Functions for Dynamic Assignment Models of Signalized Networks.” Transportation Research 31B(4), 277–290.
Richards, P.I. (1956). “Shock Waves on the Highway.” Operations Research 4, 42–51.
Taylor, N.B. (1990). CONTRAM5: “An Enhanced Traffic Assignment Model.” TRL Research Report 249. Transport Research Laboratory, Crowthorne, Essex, England.
TRL (2000). http://www.contram.com. Web site for CONTRAM (CONtinuous TRaffic Assignment Model) Version 7, released January 2000, by the TRL (Transport Research Laboratory), Crowthorne, Essex, England, and Mott MacDonald transport consultants.
Wie, B.W., T.L. Friesz, and R.L. Tobin. (1990). “Dynamic User Optimal Traffic Assignment on Congested Multidestination Networks.” Transportation Research 24B, 431–442.
Wie, B.W., Tobin, R.L., Bernstein, D. and Friesz, T.L. (1995). “A Comparison of System Optimum and User Equilibrium Traffic Assignments with Schedule Delay.” Transportation Research 3C, 389–411.
Wu, J.H., Y. Chen, and M. Florian. (1995). “The Continuous Dynamic Network Loading Problem: A Mathematical Formulation and Solution Method.” Presented at the 3rd EURO WORKING GROUP Meeting on Urban Traffic and Transportation Barcelona 27–29 September.
Wu, J.H., Y. Chen, and M. Florian. (1998). “The Continuous Dynamic Network Loading Problem: A Mathematical Formulation and Solution Method.” Transportation Research 32B, 173–187.
Xu, Y.W., Wu, J.H., Florian, M., Marcotte, P. and Zhu, D.L. (1999). “Advances in the Continuous Dynamic Network Loading Problem.” Transportation Science 33(4), 341–353.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Carey, M. Dynamic Traffic Assignment with More Flexible Modelling within Links. Networks and Spatial Economics 1, 349–375 (2001). https://doi.org/10.1023/A:1012848329399
Issue Date:
DOI: https://doi.org/10.1023/A:1012848329399