VOL. 10, NO. 11, JUNE 2015 ISSN 1819-6608 ARPN Journal of Engineering and Applied Sciences © 2006-2015 Asian Research Publishing Network (ARPN). All rights reserved. www.arpnjournals.com 5034 NETWORK FLOW WITH FUZZY ARC LENGTHS USING HAAR RANKING S. Dhanasekar 1 , S. Hariharan 2 , P. Sekar 2 and Kalyani Desikan 3 1 Vellore Institute of Technology, Chennai Campus, Chennai, India 2 CKN College for Men, Chennai, India 3 Institute of Technology, Chennai Campus, Chennai, India ABSTRACT Shortest path problem is a classical and the most widely studied phenomenon in combinatorial optimization. In a classical shortest path problem, the distance of the arcs between different nodes of a network are assumed to be certain. In some uncertain situations, the distance will be calculated as a fuzzy number depending on the number of parameters considered. This article proposes a new approach based on Haar ranking of fuzzy numbers to find the shortest path between nodes of a given network. The combination of Haar ranking and the well-known Dijkstra’s algorithm for finding the shortest path have been used to identify the shortest path between given nodes of a network. The numerical examples ensure the feasibility and validity of the proposed method. Keywords: Haar ranking, fuzzy numbers, ranking techniques, Dijkstra’s algorithm, fuzzy shortest path problem. 1. INTRODUCTION Solving mathematical models by using network fascinates many researchers and problems like assignment, transportation and shortest path based on networks have certainly garnered much attention. This article primarily focuses on the fuzzy based shortest path problem which is essential in many applications like communication, transportation, scheduling and routing. The objective of the shortest path problem is to find the path of minimum length between any pair of vertices. The arc length of the network may represent the real life quantities such as time, cost etc.. Algorithms for finding the shortest path in a network have been studied for a long time. The algorithms are still being improved [15 - 17]. Most of the time the parameters used in the algorithms are assumed to be deterministic. This may not be the case in the real time models. Fuzzy concepts [12] help us to overcome this difficulty and Dubois [1] was the first one to study the fuzzy shortest path problem. Klen [2] and Madhavi et al. [7] proposed a dynamic programming technique to solve the fuzzy shortest path problem. Chern [3] proposed an algorithm which concentrates on finding the important arc in the network. Li et al. [4] and Gen et al. [14] proposed algorithms involving neural networks and genetic algorithm respectively. Liu & Kao [5] investigated the network flows using fuzzy parameters. Hernandes et al. [6] used generic ranking to order the fuzzy numbers in the fuzzy shortest path problem. Ali Tajdin et al. [8] proposed an algorithm for the fuzzy shortest path problem in a mixed network. The choice of ranking technique plays a vital role in decision making problems. The shortest path may vary depending on the ranking technique used. There is no unique ranking methodology of fuzzy numbers to decide on the shortest path of a given network. Furthermore, some of them give different ranking orders using different -cuts. Yuan [9] formulated four criteria such as distinguish ability, rationality, fuzzy or linguistic presentation and robustness to evaluate the fuzzy ranking methods. Wang and Kerre [10] added three more properties for evaluating fuzzy ranking methods. Chien and et al. [11] extended the necessary characteristics of fuzzy ranking methods to eight. Decision makers must consider the various characteristics of ranking methods in determining whether the chosen fuzzy ranking method can support the features of the decision making problems. While studying fuzzy shortest path problem, the main difficulty is to find the suitable ordering of fuzzy numbers. We can avoid this by choosing Haar ranking [13] because it not only gives the Haar tuple to make use of the crisp value to decide the shortest path but also gives back the original fuzzy number using defuzzification to calculate the fuzzy distance of the shortest path. In this paper we utilized the Haar ranking which converts a given fuzzy number into a Haar tuple and applied the Dijkstra’s algorithm in order to identify the shortest path of a given network. The fuzzzification from the defuzzied value is very easy using Haar ranking method and is unique. This paper is organised as follows: section 2 deals with the preliminaries that are required for formulating fuzzy shortest path problem, section 3 discusses the proposed algorithm and section 4 deals with numerical examples of the fuzzy shortest path problems. Finally, the conclusion part is given in section 5. 2. PREMILINARIES Fuzzy set: The Fuzzy set can be mathematically constructed by assigning to each possible individual in the universe of discourse a value representing its grade of membership in the fuzzy set [12]. Fuzzy number: The Fuzzy number is a fuzzy set whose membership function satisfies the following condition [12] a) is a piecewise continuous function. b) is a convex function. c) is normal.( Triangular fuzzy number: A Fuzzy number with membership function of the form