1209 - Proceedings of the Eastern Asia Society for...

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AN ALGORITHM FOR LOGIT NETWORK LOADING PROBLEM BY TOPOLOGICAL SORTING Jun LI Dr. Eng. School of Engineering Sun Yat-sen University 135 Xingangxi Lu, Guangzhou, Guangdong 510275, China Fax: +86-20-84113689 E-mail: stslijun@zsu.edu.cn Songxin XIN School of Engineering Sun Yat-sen University 135 Xingangxi Lu, Guangzhou, Guangdong 510275, China Fax: +86-20-84113689 E-mail: 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 E-mail: 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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(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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1209 - Proceedings of the Eastern Asia Society for...

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