Equilibrium route flow assignment in linear network as a system

A. Yu. Krylatov ; A. P. Shirokolobova

Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, Tome 14 (2018) no. 2, pp. 103-115

Résumé

Decision making requires possibilities to influence the object. In urban traffic area it is crucial to influence traffic flows. However, first of all, decision maker needs comprehensive information about traffic flows. From a practical perspective, the most valuable is information about route flows, unlike information about link flows. In this paper, a route flow traffic assignment model in a linear network is studied. Linear road network (linear link performance function) gives a chance to reduce traffic assignment problem to a system of linear equations and conditions in the form of linear inequalities. The directed graph represents road network. Route flow traffic assignment problem is presented as a nonlinear constrained problem. The theorem about the reduction of a route flow traffic assignment problem in linear road network to the system of linear equations is proved. Implementation of developed approach to an example of the linear road network is disassembled in details.

Détail

Keywords: constrained nonlinear optimization, user equilibrium of Wardrop, route flow traffic assignment.

Bibliographie

[1] Wardrop J. G., “Some theoretical aspects of road traffic research”, Proc. Institution of Civil Engineers, 2 (1952), 325–378 | DOI

[2] Beckmann M. J., McGuire C. B., Winsten C. B., Studies in the economics of transportation, Yale University Press, New Haven, CT, 1956, 359 pp.

[3] Sheffi Y., Urban transportation networks: equilibrium analysis with mathematical programming methods, Prentice-Hall Inc., Englewood Cliffs, New Jersey, 1985, 416 pp.

[4] Yang H., Huang H.-J., “The multi-class, multi-criteria traffic network equilibrium and systems optimum problem”, Transportation Research. Pt B: Methodological, 38 (2004), 1–15 | DOI

[5] V. A. Druzhinina, I. A. Volkova (eds.), Introduction to mathematical modelling of traffic flows, Moscow phys. and techn. University Publ., M., 2010, 360 pp. (In Russian)

[6] Di X., Liu H. X., “Boundedly rational route choice behavior: A review of models and methodologies”, Transportation Research. Pt B: Methodological, 85 (2016), 142–179 | DOI

[7] Sheleykhovskiy G. V., Transportation basis for urban planning, Giprogor Publ., L., 1936, 150 pp. (In Russian)

[8] Bregman L. M., “The proof of convergence of the Sheleykhovskiy's method for the problem with transportation constraints”, Computational Mathematics and Mathematical Physics, 7:1 (1967), 147–156 (In Russian)

[9] Wilson A. G., Entropy in urban and regional modelling, Pion, London, 1970, 166 pp.

[10] Fedorov V. P., Losin L. A., “Mathematical modelling methods in designing urban transport system on pre-network level”, Transport of Russian Federation, 2012, no. 2(39), 30–33 (In Russian)

[11] Fedorov V. P., “Working out the method for forming variations for development of urban transport networks”, Transport of Russian Federation, 2012, no. 3-4(40-41), 17–21 (In Russian)

[12] Patriksson M., “A mixed user-equilibrium and system-optimal traffic flow for connected vehicles stated as a complementarity problem”, Computer-Aided Civil and Infrastructure Engineering, 32:7 (2017), 562–580 | DOI

[13] Konnov I. V., “On auction equilibrium models with network applications”, Netnomics, 16 (2015), 107–125 | DOI

[14] Konnov I. V., “Vector network equilibrium problems with elastic demands”, Journal of Global Optimization, 57 (2013), 521–531 | DOI | MR | Zbl

[15] Mazalov V. V., Melnik A. V., “Equilibrium prices and flows in the passenger traffic problem”, International Game Theory Review, 18:1 (2016) (accessed: 04.04.2018 g.) https://www.worldscientific.com/doi/abs/10.1142/S0219198916500018 | DOI | MR

[16] Lien J. W., Mazalov V. V., Melnik A. V., Zheng J., “Wardrop equilibrium for networks with the BPR latency function”, Lecture Notes in Computer Science, 9869, 2016, 37–49 | DOI | MR | Zbl

[17] Krylatov A. Yu., “Network flow assignment as a fixed point problem”, Journal of Applied and Industrial Mathematics, 10:2 (2016), 243–256 | DOI | MR | Zbl

[18] Krylatov A. Yu., Shirokolobova A. P., “Projection approach versus gradient descent for network's flows assignment problem”, Lecture Notes in Computer Science, 10556, 2017, 345–350 | DOI | MR

[19] Zakharov V. V., Krylatov A. Yu., “Competitive routing of traffic flows by navigation providers”, Automation and Remote Control, 77:1 (2016), 179–189 | DOI | MR | Zbl

[20] Krylatov A. Y., Zakharov V. V., Malygin I. G., “Competitive traffic assignment in Road Networks”, Transport and Telecommunication, 17:3 (2016), 212–221 | DOI | MR

[21] Patriksson M., The traffic assignment problem: models and methods, VSP, Utrecht, The Netherlands, 1994, 223 pp.

[22] Krylatov A. Yu., “Optimal strategies for traffic flow management on the transportation network of parallel links”, Vestnik of Saint Petersburg University. Series 10. Applied Mathematics. Computer Science. Control Processes, 2014, no. 2, 121–130 (In Russian)