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AN ALGORITHM FOR LOGIT NETWORK LOADING PROBLEM
BY TOPOLOGICAL SORTING
Jun LI
Dr. Eng.
School of Engineering
Sun Yatsen University
135 Xingangxi Lu, Guangzhou,
Guangdong 510275, China
Fax: +862084113689
Email: stslijun@zsu.edu.cn
Songxin XIN
School of Engineering
Sun Yatsen University
135 Xingangxi Lu, Guangzhou,
Guangdong 510275, China
Fax: +862084113689
Email: stits@zsu.edu.cn
Chunlu LIU
Dr. Eng.
School of Architecture and Building
Geelong Waterfront Campus
Deakin University
1 Gheringhap Street, Geelong
Victoria 3217, Australia
Email: chunlu@deakin.edu.au
Abstract:
This paper presents a topological sorting based algorithm for logit network loading
problem to exclude all cycles by removing certain links from loops. The new algorithm
calculates the link weights and flows according to topological order. It produces the
theoretical results for networks without loops. Numerical examples show that the new
algorithm can reduce errors introduced by the strict definition of “reasonable route” in Dial’s
algorithm.
Key Words:
Logit network loading, Dial’s algorithm, Topological sort
1. INTRODUCTION
For networks with fixed costs, logit and probit stochastic user equilibrium models are two of
the most important models.
The models can be solved through the method of successive
averages proposed by Powell and Sheffi (1982), both of them require a process of network
loading.
Up to now, simulation method remains the most feasible method to load network of
probit model (Sheffi 1985).
Dial (1971) proposed an algorithm for the logit network loading
problem. The algorithm is very efficient and does not require path enumeration.
However, it
is found that the definition of “reasonable” paths in Dial’s algorithm is so strict that it may
lead to some problems.
For example, some paths with smaller cost are excluded, while the
larger ones are included in the reasonable paths (Li et al 2004).
Akamatsu (1996, 1997) also
shows tht Dial’s algorithm fails in the real world application.
Bell (1995) proposed two algorithms for logit network loading problems.
The first method
considers any partial set of all possible paths, including all paths without loops and some of
paths with loops, however the proof for equivalence between the model and logit model is not
given.
The second method, later proved by Akamatsu (1996), considers all possible paths
which those paths with finite and infinite loops.
So the number of paths is infinite.
Li et al
Proceedings of the Eastern Asia Society for Transportation Studies, Vol. 5, pp. 1209  1217, 2005
1209
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View Full Document(2004) proposed an iteration algorithm for this problem.
In addition, Huang and Bell (1998)
proposed an assignment excludes all cyclic flows, but it difficult to be applied to the large
scale networks because the algorithm requires path enumeration.
The remained part of this paper is organized as follows.
Dial’s algorithm is reviewed and the
related issues are discussed in details.
The a new algorithm based on topological sorting is
presented, which excludes all cycles by removing certain links from loops only when it is
necessary.
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 Spring '10
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