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An augmented lagrangean dual algorithm for link capacity side constrained traffic assignment problems


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  • Larsson, Torbjörn
  • Patriksson, Michael
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    As a means to obtain a more accurate description of traffic flows than that provided by the basic model of traffic assignment, there have been suggestions to impose upper bounds on the link flows. This can be done either by introducing explicit link capacities or by employing travel time functions with asymptotes at the upper bounds. Although the latter alternative has the disadvantage of inherent numerical ill-conditioning, the capacitated assignment model has been studied and applied to a limited extent, the main reason being that the solutions can not be characterized by the classical Wardrop equilibrium conditions; they may, however, be characterized as Wardrop equilibria in terms of a well-defined, natural generalized travel cost. The introduction of link capacity side constraints makes the problem computationally more demanding. The availability of efficient algorithms for the basic model of traffic assignment motivates the use of dualization approaches for handling the capacity constraints. We propose and evaluate an augmented Lagrangean dual method in which the uncapacitated traffic assignment subproblems are solved with the disaggregate simplicial decomposition algorithm. This algorithm fully exploits the subproblem's structure and has very favourable reoptimization capabilities; both these properties are necessary for achieving computational efficiency in iterative dualization schemes. The dual method exhibits a linear rate of convergence under a standard nondegeneracy assumption. The efficiency of the overall algorithm is demonstrated through experiments with capacitated versions of well-known test problems, with the conclusion that the introduction of link capacities increases the computing times with no more than a factor of four. The introduction of capacities and the algorithm suggested can be used to derive tolls for the reduction of flows on overloaded links. The solution strategy can be applied also to other types of traffic assignment models where side constraints have been added in order to refine a descriptive or prescriptive assignment model.

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    Article provided by Elsevier in its journal Transportation Research Part B: Methodological.

    Volume (Year): 29 (1995)
    Issue (Month): 6 (December)
    Pages: 433-455

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    Handle: RePEc:eee:transb:v:29:y:1995:i:6:p:433-455

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    1. Smith, M. J., 1979. "The existence, uniqueness and stability of traffic equilibria," Transportation Research Part B: Methodological, Elsevier, vol. 13(4), pages 295-304, December.
    2. Jeff Kennington & Mohamed Shalaby, 1977. "An Effective Subgradient Procedure for Minimal Cost Multicommodity Flow Problems," Management Science, INFORMS, vol. 23(9), pages 994-1004, May.
    3. Hearn, Donald W. & Ribera, Jaime, 1981. "Convergence of the Frank-Wolfe method for certain bounded variable traffic assignment problems," Transportation Research Part B: Methodological, Elsevier, vol. 15(6), pages 437-442, December.
    4. Yang, Hai & Yagar, Sam, 1994. "Traffic assignment and traffic control in general freeway-arterial corridor systems," Transportation Research Part B: Methodological, Elsevier, vol. 28(6), pages 463-486, December.
    5. Hearn, Donald W. & Lawphongpanich, Siriphong & Nguyen, Sang, 1984. "Convex programming formulations of the asymmetric traffic assignment problem," Transportation Research Part B: Methodological, Elsevier, vol. 18(4-5), pages 357-365.
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    Cited by:
    1. Correa, José R. & Schulz, Andreas S. & Stier-Moses, Nicolás E., 2008. "A geometric approach to the price of anarchy in nonatomic congestion games," Games and Economic Behavior, Elsevier, vol. 64(2), pages 457-469, November.
    2. Correa, Jose R. & Schulz, Andreas S. & Stier Moses, Nicolas E., 2003. "Selfish Routing in Capacitated Networks," Working papers 4319-03, Massachusetts Institute of Technology (MIT), Sloan School of Management.
    3. Huang, Hai-Jun & Xu, Gang, 1998. "Aggregate scheduling and network solving of multi-stage and multi-item manufacturing systems," European Journal of Operational Research, Elsevier, vol. 105(1), pages 52-65, February.
    4. Seungkyu Ryu & Anthony Chen & Xiangdong Xu & Keechoo Choi, 2014. "A Dual Approach for Solving the Combined Distribution and Assignment Problem with Link Capacity Constraints," Networks and Spatial Economics, Springer, vol. 14(2), pages 245-270, June.
    5. Chen, H. K. & Chang, M. S. & Wang, C. Y., 2001. "Dynamic capacitated user-optimal departure time/route choice problem with time-window," European Journal of Operational Research, Elsevier, vol. 132(3), pages 603-618, August.
    6. Li, Zhi-Chun & Lam, William H.K. & Wong, S.C. & Fu, Xiaowen, 2010. "Optimal route allocation in a liberalizing airline market," Transportation Research Part B: Methodological, Elsevier, vol. 44(7), pages 886-902, August.
    7. Zhong, R.X. & Sumalee, A. & Friesz, T.L. & Lam, William H.K., 2011. "Dynamic user equilibrium with side constraints for a traffic network: Theoretical development and numerical solution algorithm," Transportation Research Part B: Methodological, Elsevier, vol. 45(7), pages 1035-1061, August.
    8. Jahn, Olaf & Möhring, Rolf & Schulz, Andreas & Stier Moses, Nicolás, 2004. "System-Optimal Routing of Traffic Flows with User Constraints in Networks with Congestion," Working papers 4394-02, Massachusetts Institute of Technology (MIT), Sloan School of Management.
    9. Ferrari, Paolo, 2002. "Road network toll pricing and social welfare," Transportation Research Part B: Methodological, Elsevier, vol. 36(5), pages 471-483, June.
    10. Alfredo Garcia & Mingyi Hong & Jorge Barrera, 2012. "“Cap and Trade” for Congestion Control," Dynamic Games and Applications, Springer, vol. 2(3), pages 280-293, September.
    11. Larsson, Torbjörn & Patriksson, Michael & Rydergren, Clas, 2004. "A column generation procedure for the side constrained traffic equilibrium problem," Transportation Research Part B: Methodological, Elsevier, vol. 38(1), pages 17-38, January.
    12. Ferrari, Paolo, 1997. "Capacity constraints in urban transport networks," Transportation Research Part B: Methodological, Elsevier, vol. 31(4), pages 291-301, August.
    13. Yildirim, Mehmet Bayram & Hearn, Donald W., 2005. "A first best toll pricing framework for variable demand traffic assignment problems," Transportation Research Part B: Methodological, Elsevier, vol. 39(8), pages 659-678, September.
    14. Hamdouch, Younes & Florian, Michael & Hearn, Donald W. & Lawphongpanich, Siriphong, 2007. "Congestion pricing for multi-modal transportation systems," Transportation Research Part B: Methodological, Elsevier, vol. 41(3), pages 275-291, March.


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