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Title: Alternative method of counting the number of efficient paths in a transportation network
Authors: Zhao, R
Xu, X
Chen, A 
Issue Date: 2021
Source: Transportmetrica. A, Transport science, 2021, Ahead-of-print,
Abstract: The number of efficient paths between an origin-destination (O-D) pair provides the route diversity degree of possibly used paths in a transportation network and has many important applications. The existing counting method was based on the Bell loading method for Logit model to determine the number of efficient paths between any two nodes without path enumeration. However, this method has a high time-complexity and requires many unnecessary computations, which significantly hinder its use in large-scale networks. Inspired by the Dial loading method for Logit model, this paper develops a more computationally attractive method to count not only the number of efficient paths, but also the number of efficient paths using each link/node, and the total/average cost of these paths between each O-D pair. Besides circumventing path enumeration, the proposed method has a much lower time-complexity. Numerical examples are then provided to demonstrate the validity and efficiency of the proposed method.
Keywords: Bell loading
Dial loading
Efficient path
Publisher: Taylor & Francis
Journal: Transportmetrica. A, Transport science 
ISSN: 2324-9935
EISSN: 2324-9943
DOI: 10.1080/23249935.2021.1933255
Rights: © 2021 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group. This is an Open Access article distributed under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives License (, which permits non-commercial re-use, distribution, and reproduction in any medium, provided the original work is properly cited, and is not altered, transformed, or built upon in any way.
The following publication Zhao, R., Xu, X., & Chen, A. (2021). Alternative Method of Counting the Number of Efficient Paths in a Transportation Network. Transportmetrica A: Transport Science, (just-accepted), 1-38 is available at
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