# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2022104
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

## 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. Cao, R. X. Li, W. Huang, J. 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. Hung, V. 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. Hung, V. M. Tam, E. 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. Hung, V.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. Wei, C. 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. Cao, R. X. Li, W. Huang, J. 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. Hung, V. 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. Hung, V. M. Tam, E. 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. Hung, V.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. Wei, C. 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.
The traffic network $\aleph$
The weighted network $\hat{\aleph}^{1}$
The weighted network $\hat{\aleph}^{2}$
The weighted network $\hat{\aleph}^{3}$
 [1] Annamaria Barbagallo, Rosalba Di Vincenzo, Stéphane Pia. On strong Lagrange duality for weighted traffic equilibrium problem. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1097-1113. doi: 10.3934/dcds.2011.31.1097 [2] Peiyu Li. Solving normalized stationary points of a class of equilibrium problem with equilibrium constraints. Journal of Industrial and Management Optimization, 2018, 14 (2) : 637-646. doi: 10.3934/jimo.2017065 [3] Mohamed Benyahia, Massimiliano D. Rosini. A macroscopic traffic model with phase transitions and local point constraints on the flow. Networks and Heterogeneous Media, 2017, 12 (2) : 297-317. doi: 10.3934/nhm.2017013 [4] Haodong Chen, Hongchun Sun, Yiju Wang. A complementarity model and algorithm for direct multi-commodity flow supply chain network equilibrium problem. Journal of Industrial and Management Optimization, 2021, 17 (4) : 2217-2242. doi: 10.3934/jimo.2020066 [5] Gabriella Bretti, Maya Briani, Emiliano Cristiani. An easy-to-use algorithm for simulating traffic flow on networks: Numerical experiments. Discrete and Continuous Dynamical Systems - S, 2014, 7 (3) : 379-394. doi: 10.3934/dcdss.2014.7.379 [6] Maya Briani, Emiliano Cristiani. An easy-to-use algorithm for simulating traffic flow on networks: Theoretical study. Networks and Heterogeneous Media, 2014, 9 (3) : 519-552. doi: 10.3934/nhm.2014.9.519 [7] Yacine Chitour, Benedetto Piccoli. Traffic circles and timing of traffic lights for cars flow. Discrete and Continuous Dynamical Systems - B, 2005, 5 (3) : 599-630. doi: 10.3934/dcdsb.2005.5.599 [8] Oliver Kolb, Simone Göttlich, Paola Goatin. Capacity drop and traffic control for a second order traffic model. Networks and Heterogeneous Media, 2017, 12 (4) : 663-681. doi: 10.3934/nhm.2017027 [9] Xiaona Fan, Li Jiang, Mengsi Li. Homotopy method for solving generalized Nash equilibrium problem with equality and inequality constraints. Journal of Industrial and Management Optimization, 2019, 15 (4) : 1795-1807. doi: 10.3934/jimo.2018123 [10] Enkhbat Rentsen, Battur Gompil. Generalized Nash equilibrium problem based on malfatti's problem. Numerical Algebra, Control and Optimization, 2021, 11 (2) : 209-220. doi: 10.3934/naco.2020022 [11] Maria Laura Delle Monache, Paola Goatin. A front tracking method for a strongly coupled PDE-ODE system with moving density constraints in traffic flow. Discrete and Continuous Dynamical Systems - S, 2014, 7 (3) : 435-447. doi: 10.3934/dcdss.2014.7.435 [12] Bong Joo Kim, Gang Uk Hwang, Yeon Hwa Chung. Traffic modelling and bandwidth allocation algorithm for video telephony service traffic. Journal of Industrial and Management Optimization, 2009, 5 (3) : 541-552. doi: 10.3934/jimo.2009.5.541 [13] Mary Luz Mouronte, Rosa María Benito. Structural analysis and traffic flow in the transport networks of Madrid. Networks and Heterogeneous Media, 2015, 10 (1) : 127-148. doi: 10.3934/nhm.2015.10.127 [14] Gabriella Bretti, Roberto Natalini, Benedetto Piccoli. Numerical approximations of a traffic flow model on networks. Networks and Heterogeneous Media, 2006, 1 (1) : 57-84. doi: 10.3934/nhm.2006.1.57 [15] Gabriella Bretti, Roberto Natalini, Benedetto Piccoli. Fast algorithms for the approximation of a traffic flow model on networks. Discrete and Continuous Dynamical Systems - B, 2006, 6 (3) : 427-448. doi: 10.3934/dcdsb.2006.6.427 [16] Alberto Bressan, Khai T. Nguyen. Conservation law models for traffic flow on a network of roads. Networks and Heterogeneous Media, 2015, 10 (2) : 255-293. doi: 10.3934/nhm.2015.10.255 [17] Johanna Ridder, Wen Shen. Traveling waves for nonlocal models of traffic flow. Discrete and Continuous Dynamical Systems, 2019, 39 (7) : 4001-4040. doi: 10.3934/dcds.2019161 [18] Tong Li. Qualitative analysis of some PDE models of traffic flow. Networks and Heterogeneous Media, 2013, 8 (3) : 773-781. doi: 10.3934/nhm.2013.8.773 [19] Paola Goatin. Traffic flow models with phase transitions on road networks. Networks and Heterogeneous Media, 2009, 4 (2) : 287-301. doi: 10.3934/nhm.2009.4.287 [20] Rinaldo M. Colombo, Andrea Corli. Dynamic parameters identification in traffic flow modeling. Conference Publications, 2005, 2005 (Special) : 190-199. doi: 10.3934/proc.2005.2005.190

2021 Impact Factor: 1.411