doi: 10.3934/jimo.2022104
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Algorithms for the Pareto solution of the multicriteria traffic equilibrium problem with capacity constraints of arcs

College of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing 400074, China

*Corresponding author: Zhi Lin

Received  October 2021 Revised  March 2022 Early access June 2022

Fund Project: The first author is supported by Joint Training Base Construction Project for Graduate Students in Chongqing (JDLHPYJD2021016) and Group Building Scientific Innovation Project for universities in Chongqing (CXQT21021)

We focus on the multicriteria traffic equilibrium problem with capacity constraints of arcs. First, we generalize Beckmann's formula to deal with multicriteria traffic equilibrium problems with capacity constraints of arcs and prove that the solution of the mathematical programming problem is a Pareto traffic equilibrium flow with capacity constraints of arcs. Furthermore, we present a restricted algorithm for computing the Pareto traffic equilibrium flow with capacity constraints of arcs. Using the restricted algorithm, one does not need to know the set of available paths joining origin-destination pairs. This proves very helpful for complex traffic networks. Finally, for the algorithms of the Pareto traffic equilibrium flow, we give two examples to exemplify calculation processes.

Citation: Zhi Lin, Zaiyun Peng. Algorithms for the Pareto solution of the multicriteria traffic equilibrium problem with capacity constraints of arcs. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022104
References:
[1]

L. Q. Anh and N. V. Hung, Levitin-Polyak well-posedness for strong bilevel vector equilibrium problems and applications to traffic network problems with equilibrium constraints, Positivity, 22 (2018), 1223-1239.  doi: 10.1007/s11117-018-0569-2.

[2]

M. J. Beckmann, C. B. McGuire and C. B. Winsten, Studies in the Economics of Transportation, New Haven: Yale University Press, 1956.

[3]

J. D. CaoR. X. LiW. HuangJ. H. Guo and Y. Wei, Traffic network equilibrium problems with demands uncertainty and capacity constraints of arcs by scalarization approaches, Sci. China Tech. Sci., 61 (2018), 1642-1653. 

[4]

E. W. Dijkstra, A note on two problems in connexion with graphs, Numer. Math., 1 (1959), 269-271.  doi: 10.1007/BF01386390.

[5]

N. V. Hung, On the stability of the solution mapping for parametric traffic network problems, Indagationes Math., 29 (2018), 885-894.  doi: 10.1016/j.indag.2018.01.007.

[6]

N. V. Hung and A. A. Keller, Painleve-Kuratowski convergence of the solution sets for controlled systems of fuzzy vector quasi-optimization problems with application to controlling traffic networks under uncertainty, Comput. Appl. Math., 40 (2021), 1-21.  doi: 10.1007/s40314-021-01415-8.

[7]

N. V. HungV. Novo and V. M. Tam, Error bound analysis for vector equilibrium problems with partial order provided by a polyhedral cone, J. Glob. Optim., 82 (2022), 139-159.  doi: 10.1007/s10898-021-01056-5.

[8]

N. V. HungV. M. TamE. Koebis and J. C. Yao, Existence of solutions and algorithm for generalized vector quasi-complementarity problems with application to traffic network problems, J. Nonlinear Convex Anal., 20 (2019), 1751-1775. 

[9]

N.V. HungV.V. Tri and D. O'Regan, Existence conditions for solutions of bilevel vector equilibrium problems with application to traffic network problems with equilibrium constraints, Positivity, 25 (2021), 213-228.  doi: 10.1007/s11117-020-00759-5.

[10]

N. V. Hung and V. V. Tri, Stability analysis for parametric symmetric vector quasi-equilibrium problems with application to traffic network problems, J. Nonlinear Convex Anal., 21 (2020), 2207-2223. 

[11]

Z. Lin, On existence of vector equilibrium flows with capacity constraints of arcs, Nonlinear Anal., 72 (2010), 2076-2079.  doi: 10.1016/j.na.2009.10.007.

[12]

Z. Lin, The study of traffic equilibrium problems with capacity constraints of arcs, Nonlinear Anal. RWA, 11 (2010), 2280-2284.  doi: 10.1016/j.nonrwa.2009.07.002.

[13]

Z. Lin, An algorithm for traffic equilibrium flow with capacity constraints of arcs, J. Transp. Tech., 5 (2015), 240-246. 

[14]

T. T. T. Phuong and D. T. Luc, Equilibrium in multi-criteria supply and demand networks with capacity constraints, Math. Methods Oper. Res., 81 (2015), 83-107.  doi: 10.1007/s00186-014-0487-4.

[15]

J. Wardrop, Some theoretical aspects of road traffic research, in Proceedings of the Institute of Civil Engineers, Part II, 1, 1952,325–376. doi: 10.1680/ipeds.1952.11362.

[16]

H. WeiC. Chen and B. Wu, Vector network equilibrium problems with uncertain demands and capacity constraints of arcs, Optim. Lett., 15 (2021), 1113-1131.  doi: 10.1007/s11590-020-01610-2.

[17]

Y. D. Xu and S. J. Li, Vector network equilibrium problems with capacity constraints of arcs and nonlinear scalarization methods, Appl. Anal., 93 (2014), 2199-2210.  doi: 10.1080/00036811.2013.875160.

show all references

References:
[1]

L. Q. Anh and N. V. Hung, Levitin-Polyak well-posedness for strong bilevel vector equilibrium problems and applications to traffic network problems with equilibrium constraints, Positivity, 22 (2018), 1223-1239.  doi: 10.1007/s11117-018-0569-2.

[2]

M. J. Beckmann, C. B. McGuire and C. B. Winsten, Studies in the Economics of Transportation, New Haven: Yale University Press, 1956.

[3]

J. D. CaoR. X. LiW. HuangJ. H. Guo and Y. Wei, Traffic network equilibrium problems with demands uncertainty and capacity constraints of arcs by scalarization approaches, Sci. China Tech. Sci., 61 (2018), 1642-1653. 

[4]

E. W. Dijkstra, A note on two problems in connexion with graphs, Numer. Math., 1 (1959), 269-271.  doi: 10.1007/BF01386390.

[5]

N. V. Hung, On the stability of the solution mapping for parametric traffic network problems, Indagationes Math., 29 (2018), 885-894.  doi: 10.1016/j.indag.2018.01.007.

[6]

N. V. Hung and A. A. Keller, Painleve-Kuratowski convergence of the solution sets for controlled systems of fuzzy vector quasi-optimization problems with application to controlling traffic networks under uncertainty, Comput. Appl. Math., 40 (2021), 1-21.  doi: 10.1007/s40314-021-01415-8.

[7]

N. V. HungV. Novo and V. M. Tam, Error bound analysis for vector equilibrium problems with partial order provided by a polyhedral cone, J. Glob. Optim., 82 (2022), 139-159.  doi: 10.1007/s10898-021-01056-5.

[8]

N. V. HungV. M. TamE. Koebis and J. C. Yao, Existence of solutions and algorithm for generalized vector quasi-complementarity problems with application to traffic network problems, J. Nonlinear Convex Anal., 20 (2019), 1751-1775. 

[9]

N.V. HungV.V. Tri and D. O'Regan, Existence conditions for solutions of bilevel vector equilibrium problems with application to traffic network problems with equilibrium constraints, Positivity, 25 (2021), 213-228.  doi: 10.1007/s11117-020-00759-5.

[10]

N. V. Hung and V. V. Tri, Stability analysis for parametric symmetric vector quasi-equilibrium problems with application to traffic network problems, J. Nonlinear Convex Anal., 21 (2020), 2207-2223. 

[11]

Z. Lin, On existence of vector equilibrium flows with capacity constraints of arcs, Nonlinear Anal., 72 (2010), 2076-2079.  doi: 10.1016/j.na.2009.10.007.

[12]

Z. Lin, The study of traffic equilibrium problems with capacity constraints of arcs, Nonlinear Anal. RWA, 11 (2010), 2280-2284.  doi: 10.1016/j.nonrwa.2009.07.002.

[13]

Z. Lin, An algorithm for traffic equilibrium flow with capacity constraints of arcs, J. Transp. Tech., 5 (2015), 240-246. 

[14]

T. T. T. Phuong and D. T. Luc, Equilibrium in multi-criteria supply and demand networks with capacity constraints, Math. Methods Oper. Res., 81 (2015), 83-107.  doi: 10.1007/s00186-014-0487-4.

[15]

J. Wardrop, Some theoretical aspects of road traffic research, in Proceedings of the Institute of Civil Engineers, Part II, 1, 1952,325–376. doi: 10.1680/ipeds.1952.11362.

[16]

H. WeiC. Chen and B. Wu, Vector network equilibrium problems with uncertain demands and capacity constraints of arcs, Optim. Lett., 15 (2021), 1113-1131.  doi: 10.1007/s11590-020-01610-2.

[17]

Y. D. Xu and S. J. Li, Vector network equilibrium problems with capacity constraints of arcs and nonlinear scalarization methods, Appl. Anal., 93 (2014), 2199-2210.  doi: 10.1080/00036811.2013.875160.

Figure 1.  The traffic network $ \aleph $
Figure 2.  The weighted network $ \hat{\aleph}^{1} $
Figure 3.  The weighted network $ \hat{\aleph}^{2} $
Figure 4.  The weighted network $ \hat{\aleph}^{3} $
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