Graphs are excellent mathematical tools applied in many fields such as transportation, communication, informatics, economy,…. A network and a flow network is a useful device to solve many problems in many fields in reality. However, most of the network applications in traditional graphs have only considered the weights of edges and vertexes independently, in which the length of a path is the sum of weights of the edges and the vertexes on the path. However, in many practical problems, weights at a vertex are not the same for all paths passing the vertex, but depend on the edges coming to and leaving the vertex. For example, the transit time on the transport network depends on the direction of transportation: turn right, turn left or go straight, even some directions are forbidden. Furthermore, on a network, there are many types of commodities, each of which are at different costs. Types of commodities share the capacity of edges and vertexes. Therefore, it is necessary to study a network with multiple commodities at multiple costs. The article builds a model of extended multi-commodity multi-cost network in order to modelise practical problems more exactly and effectively. The maximal concurent multi-commodity multi-cost flow limited cost problems, that are modelized by implicit linear programming problems. On the basis of duality theory in linear programming, an effective polynomial approximation algorithm is developed.
| Published in | American Journal of Applied Mathematics (Volume 8, Issue 3) |
| DOI | 10.11648/j.ajam.20200803.11 |
| Page(s) | 74-82 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2024. Published by Science Publishing Group |
Network, Graph, Multi-cost Multi-commodity Flow, Linear Optimization, Approximation
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APA Style
Ho Van Hung, Tran Quoc Chien. (2020). Maximal Concurent Limited Cost Flow Problems on Extended Multi-commodity Multi-cost Network. American Journal of Applied Mathematics, 8(3), 74-82. https://doi.org/10.11648/j.ajam.20200803.11
ACS Style
Ho Van Hung; Tran Quoc Chien. Maximal Concurent Limited Cost Flow Problems on Extended Multi-commodity Multi-cost Network. Am. J. Appl. Math. 2020, 8(3), 74-82. doi: 10.11648/j.ajam.20200803.11
AMA Style
Ho Van Hung, Tran Quoc Chien. Maximal Concurent Limited Cost Flow Problems on Extended Multi-commodity Multi-cost Network. Am J Appl Math. 2020;8(3):74-82. doi: 10.11648/j.ajam.20200803.11
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Download@article{10.11648/j.ajam.20200803.11,
author = {Ho Van Hung and Tran Quoc Chien},
title = {Maximal Concurent Limited Cost Flow Problems on Extended Multi-commodity Multi-cost Network},
journal = {American Journal of Applied Mathematics},
volume = {8},
number = {3},
pages = {74-82},
doi = {10.11648/j.ajam.20200803.11},
url = {https://doi.org/10.11648/j.ajam.20200803.11},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20200803.11},
abstract = {Graphs are excellent mathematical tools applied in many fields such as transportation, communication, informatics, economy,…. A network and a flow network is a useful device to solve many problems in many fields in reality. However, most of the network applications in traditional graphs have only considered the weights of edges and vertexes independently, in which the length of a path is the sum of weights of the edges and the vertexes on the path. However, in many practical problems, weights at a vertex are not the same for all paths passing the vertex, but depend on the edges coming to and leaving the vertex. For example, the transit time on the transport network depends on the direction of transportation: turn right, turn left or go straight, even some directions are forbidden. Furthermore, on a network, there are many types of commodities, each of which are at different costs. Types of commodities share the capacity of edges and vertexes. Therefore, it is necessary to study a network with multiple commodities at multiple costs. The article builds a model of extended multi-commodity multi-cost network in order to modelise practical problems more exactly and effectively. The maximal concurent multi-commodity multi-cost flow limited cost problems, that are modelized by implicit linear programming problems. On the basis of duality theory in linear programming, an effective polynomial approximation algorithm is developed.},
year = {2020}
}
TY - JOUR T1 - Maximal Concurent Limited Cost Flow Problems on Extended Multi-commodity Multi-cost Network AU - Ho Van Hung AU - Tran Quoc Chien Y1 - 2020/04/29 PY - 2020 N1 - https://doi.org/10.11648/j.ajam.20200803.11 DO - 10.11648/j.ajam.20200803.11 T2 - American Journal of Applied Mathematics JF - American Journal of Applied Mathematics JO - American Journal of Applied Mathematics SP - 74 EP - 82 PB - Science Publishing Group SN - 2330-006X UR - https://doi.org/10.11648/j.ajam.20200803.11 AB - Graphs are excellent mathematical tools applied in many fields such as transportation, communication, informatics, economy,…. A network and a flow network is a useful device to solve many problems in many fields in reality. However, most of the network applications in traditional graphs have only considered the weights of edges and vertexes independently, in which the length of a path is the sum of weights of the edges and the vertexes on the path. However, in many practical problems, weights at a vertex are not the same for all paths passing the vertex, but depend on the edges coming to and leaving the vertex. For example, the transit time on the transport network depends on the direction of transportation: turn right, turn left or go straight, even some directions are forbidden. Furthermore, on a network, there are many types of commodities, each of which are at different costs. Types of commodities share the capacity of edges and vertexes. Therefore, it is necessary to study a network with multiple commodities at multiple costs. The article builds a model of extended multi-commodity multi-cost network in order to modelise practical problems more exactly and effectively. The maximal concurent multi-commodity multi-cost flow limited cost problems, that are modelized by implicit linear programming problems. On the basis of duality theory in linear programming, an effective polynomial approximation algorithm is developed. VL - 8 IS - 3 ER -
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