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Maximal Concurent Limited Cost Flow Problems on Extended Multi-commodity Multi-cost Network
American Journal of Applied Mathematics
Volume 8, Issue 3, June 2020, Pages: 74-82
Received: Apr. 1, 2020; Accepted: Apr. 20, 2020; Published: Apr. 29, 2020
Authors
Ho Van Hung, Faculty of Information Technology, Quangnam University, Tamky, Vietnam
Tran Quoc Chien, The University of Education, University of Danang, Danang, Vietnam
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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.
Keywords
Network, Graph, Multi-cost Multi-commodity Flow, Linear Optimization, Approximation
Ho Van Hung, Tran Quoc Chien, Maximal Concurent Limited Cost Flow Problems on Extended Multi-commodity Multi-cost Network, American Journal of Applied Mathematics. Vol. 8, No. 3, 2020, pp. 74-82. doi: 10.11648/j.ajam.20200803.11
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