Decomposition of Path Choice Entropy in General Transport Networks

This paper shows that the LOGIT type stochastic assignment / stochastic user equilibrium assignment can be represented as an optimization problem with only link variables. The conventional entropy function defined by path flows in the objective can be decomposed into a function consisting only of link flows. The idea of the decomposed formulation is derived from a consideration ofthe most likely link flow patterns over a network. Then the equivalence ofthe decomposed formulation to LOGIT assignment is proved by using the Markov properties that underlie Dial's algorithm. Through the analyses] some useful properties ofthe entropy function and its conjugate dual function (expected minimum cost function) have been derived. Finally] it is discussed that the derived results have a potential impact on the development of efficient algorithms for the stochastic user equilibrium assignment. requires the prohibitively time-consuming task ofpath enumeration over a general network. The purpose of this paper is to show that the LOGIT-based stochastic assignmentlstochastic user equilibrium assignment can be formulated as the link-based equivalent program, Le., the entropy function in the objective can be decomposed into link variables and does not require path enumeration. The derived program takes the form of the standard link-node formulation, which is convenient for applying the state-of-the-art algorithms for a conventional network programming problem such as New-ton projection method or its variants. Therefore, it seems reasonable to suppose that the derived result in this paper has a potential impact on the development of efficient algorithms to solve the stochastic user equilibrium assignment. The analyses in this paper also disclose that the decomposed formulation of the LOGIT assignment is closely related to the "most likely flow pattern" approach that was originally developed by WILSON (1969) and SASAKI (1969) in the context of trip distribution models. It is also shown that the decomposition of the entropy function stems from the "Markov properties" of the LOGIT-based assignment model that underlie Dial's algorithm (The reader who is not famili~r with Dial's algorithm is advised to read Appendix 2 or the paper by Dial