Abstract
A two direction method of solving variable demand equilibrium models is considered. Throughout an equilibration using this method, if the first direction fails to reduce disequilibrium, use the second. A proof of convergence under fairly weak conditions is provided. The paper shows how the method works on models without and with responsive signal controls. In the former case the method may replace the iteration of an assignment and a demand model. In the latter case the method may replace the iteration of an assignment and a control model; and may then be utilized to design new fixed time signal timings suitable for a variety of situations. In both cases there will be a reasonable convergence guarantee if the two-direction method is utilized.
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References
Aashtiani, H. and Magnanti, T. (1983). Equilibrium on a congested transport network. SIAM Journal of Algebraic and Discrete Methods, 2, 213-216.
Allsop, R.E. (1974). Some possibilities for using traffic control to influence trip distribution and route choice. Proceedings of the Sixth International Symposium on Transportation and Traffic Theory, 345-374.
Armijo, L. (1966). Minimization of functions having Lipschitz-continuous first partial derivatives. Pacific Journal of Mathematics, 16, 1-3.
Bar-Gera, H., and Boyce, D. (2003). Origin-based algorithms for combined travel forecasting models. Transportation Research Part B, 37, 405-422.
Bar-Gera, H. and Boyce, D. (2006). Solving a non-convex combined travel forecasting model by the method of successive averages with constant step sizes.Transportation Research Part B, 40, 351-367.
Beckmann, M., McGuire, C.B. and Winsten, C.B. (1956). Studies in the Economics of Transportation. Yale University Press, New Haven, CT.
Bertsekas, D.P. (1982). Constrained Optimisation and Lagrange Multiplier Methods.Academic Press, New York.
Cantarella, G.E. (1997). A general fixed-point approach to multimodel multi-user equilibrium assignment with Elastic Demand. Transportation Science,31 (2), 107-128.
Charnes, A. and Cooper, W.W. (1958). Multicopy traffic network models. Proceedings of the Symposium on the Theory of Traffic Flow, 1958.
Commission for Integrated Transport. (2004). A Rreview of Transport Appraisal: Advice from the Commision for Integrated Transport.
COMSIS. (1996). Incorporating Feedback in Travel Forecasting Methods, Pitfalls and Common Concerns, Travel Model Improvement Program. Report for the US Department of Transportation.
DfT. (2005). Variable Demand Modelling - Scope of the Model;TAG Unit 3.10.2: www.webtag.org.uk/webdocuments/3Expert/10VariableDemandModelling/3.10.2.htm
Eckstein, J. and Bertsekas, D.P. (1992). On the Douglas-Rachford splitting method and the proximal point method for monotone operators. Mathematical Programming Series A, 55, 293-318.
Evans, S.P. (1976). Derivation and analysis of some models for combining trip distribution and assignment. Transportation Research, 10, 37-57.
Han, D.R. and Lo, H.K. (2002). New alternating direction method for a class of nonlinear variational inequality problems. Journal of Optimization Theory and its Applications, 112(3), 549-560.
Lv, Q., Salis, A., Skrobanski, G., Smith, M.J., Springham, J., Woods, A. and Gordon, A. (2007). Two-direction methods for variable demand traffic assignment. Mathematics in Transport,71-96.
Lyapunov, A.M. (1907). Probleme general de la stabilite de movement. Annals of Mathematic Studies,17, 1949.
Minty (1962), Monotone (nonlinear) operators in Hilbert space. Duke Mathematical Journal,29, 341-346.
Payne, H.J. and Thompson, W.A. (1975). Traffic assignment on transportation networks with capacity constraints and queueing. Presented at the 47 th National ORSA/TIMS North American Meeting.
Robertson, D.I. (1969). TRANSYT: A Traffic Network Study Tool. RRL Lab. report LR253, Road Research Laboratory, Crowthorne, UK.
Rockafellar, R.T. (1976). Monotone operators and the proximal point algorithm. SIAM J. Control Optimization,14, 877-898.
Sheffi, Y. (1985). Urban Transportation Networks: Equilibrium Analysis with Mathematical Programming Methods. Prentice Hall (New Jersey).
Smith, M.J. (1980). A local traffic control policy which automatically maximises the overall travel capacity of an urban road network. Traffic Engineering and Control, 21, 298-302.
Smith, M.J. (1984a). A descent algorithm for solving a variety of monotone equilibrium problems. Proceedings of the Ninth International Symposium on Transportation and Traffic Theory, 273-297.
Smith, M.J. (1984b). A descent method for solving monotone variational inequalities and monotone complementarity problems. Journal of Optimization Theory and Applications, 44, 485-496.
Smith, M.J. (1987). Traffic control and traffic assignment in a signal-controlled network with queueing. Proceedings of the Tenth International Symposium on Transportation and Traffic Theory, 61-68.
Smith, M.J. and van Vuren, T. (1993). Traffic equilibrium with responsive traffic control. Transportation Science, 27, 118-132.
Smith, M.J., Clegg, J. and Yarrow, R. (2001). Modelling traffic signal control. Handbook of Transport Systems and Traffic Control,503-525.
Van Vliet, D. (2003). Combined variable demand and traffic assignment: a hyper network formulation. Preprint.
Wardrop, J.G. (1952). Some theoretical aspects of road traffic research, Proceedings of the Institute of Civil Engineers Part II, 1, 325-378.
Webster, F.V. (1958). Traffic Signal Settings, Department of Transport, Road Research Technical Paper No. 39, HMSO, London.
Yang, H. and Yagar, S. (1995). Traffic assignment and signal control in saturated road networks. Transportation Research Part A,29, 125-139.
Acknowledgments
I am grateful to all partners in the FREEFLOW project; and especially to Jim Austin and John Polak. The drawing shown in Fig. 1 above originated with Jim and naturally fits the thinking of John when he set up and gained support for the FREEFLOW project. Partners in the FREEFLOW project are: Transport for London, City of York Council, Kent County Council, QinetiQ, Mindsheet, ACIS, Trakm8, Kizoom, Imperial College London, Loughborough University and the University of York. The Ian Routledge Consultancy is a sub-contractor to the University of York.
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Smith, M. (2009). A Two-direction Method of Solving Variable Demand Equilibrium Models with and without Signal Control. In: Lam, W., Wong, S., Lo, H. (eds) Transportation and Traffic Theory 2009: Golden Jubilee. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-0820-9_18
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