This article is cited in 19 papers

Network flow assignment as a fixed point problem

A. Yu. Krylatov<sup>ab

a</sup> Saint Petersburg State University, 7/9 Universitetskaya Nab., 199034 St. Petersburg, Russia
^b Solomenko Institute of Transport Problems of the RAS, 13 12th Line VO, 199178 St. Petersburg, Russia

Abstract: This paper deals with the user equilibrium problem (flow assignment with equal journey time by alternative routes) and system optimum (flow assignment with minimal average journey time) in a network consisting of parallel routes with a single origin-destination pair. The travel time is simulated by arbitrary smooth nondecreasing functions. We prove that the equilibrium and optimal assignment problems for such a network can be reduced to the fixed point problem expressed explicitly. A simple iterative method of finding equilibrium and optimal flow assignment is developed. The method is proved to converge geometrically; under some fairly natural conditions the method is proved to converge quadratically. Bibliogr. 30.

Keywords: user-equilibrium, system optimum, fixed point, network routes.

UDC: 519.8

Received: 31.07.2015
Revised: 24.11.2015

DOI: 10.17377/daio.2016.23.503

 English version: Journal of Applied and Industrial Mathematics, 2016, 10:2, 243–256

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