EAI Endorsed Transactions on Industrial Networks and Intelligent Systems Research Article 1 Applying algorithm finding shortest path in the multiple- weighted graphs to find maximal flow in extended linear multicomodity multicost network Chien Tran Quoc 1 , Hung Ho Van 2 1 University of Da Nang, tqchien@dce.udn.vn 2 Quang Nam University, hovanhung@qnamuni.edu.vn Abstract The shortest path finding algorithm is used in many problems on graphs and networks. This article will introduce the algorithm to find the shortest path between two vertices on the extended graph. Next, the algorithm finds the shortest path between the pairs of vertices on the extended graph with multiple weights is developed. Then, the shortest path finding algorithms is used to find the maximum flow on the multicommodity multicost extended network is developed in the article [12]. Key word: Graph; Network; Multicommodity Multicost flow; Optimization; Linear Programming. Received on 12 October 2017, accepted on 7 December 2017, published on 21 December 2017 Copyright © 2017 Chien Tran Quoc and Hung Ho Van et al., licensed to EAI. This is an open access article distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/3.0/), which permits unlimited use, distribution and reproduction in any medium so long as the original work is properly cited. doi: 10.4108/eai.21-12-2017.153499 ________________________ 2 Corresponding author: Hung Ho Van. Email: hovanhung@qnamuni.edu.vn 1. Introduction The shortest path finding algorithm is used in many problems on graphs and networks. This article will introduce the algorithm to find the shortest path between two vertices on the extended graph. Next, the algorithm finds the shortest path between the pairs of vertices on the extended graph with multiple weights is developed. Then, the shortest path finding algorithms is used to find the maximum flow on the multicommodity multicost extended network is developed in the article [12]. 2. The problem of finding the shortest path in extended graph Given extended graph G = (V, E) with a set of vertices V and a set of edges E, where edges can be directed or undirected. Each edge eE is assigned a weight we(e). For each vertex vV, we denote Ev the set of edges incident vertex v. For each vertex vV and each of pair of edges (e,e’)EvEv, ee’ is assigned switch weight wv(v,e,e’). The sets (V, E, we, wv) are called extended graph. Let p be a path from a vertex u to a vertex v through the edges ei, i = 1, …, h+1, and vertices ui, i = 1, …, h, as following: p = [u, e1, u1, e2, u2, …, eh, uh ,eh+1, v] (1) Define the length of the path p, denoted l(p), as following: l(p) =    1 1 ) ( h j j e we +    h j j j j e e u wv 1 1 ) , , ( (2) • The problem of finding the shortest path Given extended graph G = (V, E, we, wv) and vertices s, tV . Find the shortest path from s to t. • Algorithm ◊ Input. The extended graph G = (V, E, we, wv) and vertices s, tV. ◊ Output. l(t) is the length of the shortest path from s to t, and the shortest path (if l(t)<+). EAI Endorsed Transactions on Industrial Networks and Intelligent Systems 02 2017 - 12 2017 | Volume 4 | Issue 11 | e1