Abstract
The competition among a finite number of firms who must transport the fixed volume of traffic over a prescribed planning horizon is considered on a congested transportation network with one origin-destination pair connected by parallel routes. It is assumed that each firm attempts to minimize individual transportation cost by making a sequence of simultaneous decisions of departure time, route, and departure flow rate based on the trade-off between arc traversal time and schedule delay penalty. The model is formulated as anN-person nonzero-sum discrete-time dynamic game. A Cournot-Nash network equilibrium is defined under the open-loop information structure. Optimality conditions are derived using the Kuhn-Tucker theorem and given economic interpretation as a dynamic game theoretic generalization of Wardrop’s second principle which requires equilibration of certain marginal costs. A computational algorithm based on the augmented Lagrangian method and the gradient method is proposed and a numerical example is provided. Future extensions of the model and the algorithm are also discussed.
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References
Basar, T. and G.J. Olsder (1982).Dynamic Noncooperative Game Theory. Academic Press, New York.
Bazararaa, M.S. and C.M. Shetty (1979).Nonlinear Programming. John Wiley & Sons, New York.
Bertsekas, D.P. (1982).Constrained Optimization and Lagrange Multiplier Methods, Academic Press, New York.
Bryson, A.E. and Y-C Ho (1975).Applied Optimal Control, revised version, John Wiley & Sons, New York.
de Palma, A., M. Ben-Akiva, C. Lefevre, and N. Litina (1983). Stochastic Equilibrium Model of Peak Period Traffic Congrestion,Transportation Science, 17, 430–453.
Friesz T.L., F.J. Luque, R.L. Tobin, and B.W. Wie (1989). Dynamic Network Traffic Assignment Considered as a Continuous Time Optimal Control Problem.Operations Research, 37, 893–901.
Glad S.T. (1979). A Combination of Penalty Function and Multiplier Methods for Solving Optimal Control Problems,Journal of Optimization Theory and Applications, 28, 303–329.
Haurie, A. and P. Marcotte (1985). On the Relationship Between Nash-Cournot and Wardrop Equilibria.Networks, 15, 295–308.
Hestenes, M.R. (1969). Multiplier and Gradient Methods,Journal of Optimization Theory and Applications, 4, 303–320.
Merchant, D.K. and G.L. Nemhauser (1978a). A Model and An Algorithm for the Dynamic Traffic Assignment Problems,Transportation Science, 12, 62–77.
Merchant, D.K. and G.L. Nemhauser (1978b). Optimality Conditions for a Dynamic Traffic Assignment Model.Transportation Science, 12, 200–207.
Powell, M.J.D. (1969). A Method for Nonlinear Constraints in Minimization Problems, InOptimization (R. Fletcher, ed.), pp. 283–298, Academic Press, New York.
Rupp, R.D. (1972). A Method for Solving a Quadratic Optimal Control Problem,Journal of Optimization Theory and Applications, 9, 238–250.
Wardrop, J.G. (1952). Some Theoretical Aspects of Road Traffic Research.Proceedings, Institution of Civil Engineers, II, 325–378.
Wie, B.W. (1988).Dynamic Models of Network Traffic Assignment: A Control Theoretic Approach. Ph.D. Dissertation, Department of City and Regional Planning, University of Pennsylvania.
Wie, B.W. (1993). A Differential Game Model of Nash Equilibrium on a Congested Traffic Network,Networks, 23, 557–565.
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The manuscript for this paper was submitted for review on May 29, 2000.
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Wie, BW., Choi, K. The computation of dynamic cournot-nash traffic network equilibria in discrete time. KSCE J Civ Eng 4, 239–248 (2000). https://doi.org/10.1007/BF02823972
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DOI: https://doi.org/10.1007/BF02823972