Home OALib Journal OALib PrePrints Submit Ranking News My Lib FAQ About Us Follow Us+
 All Title Author Keywords Abstract
 Publish in OALib Journal ISSN: 2333-9721 APC: Only \$99

 Relative Articles More...

# Using Trapezoidal Intuitionistic Fuzzy Number to Find Optimized Path in a Network

 Full-Text   Cite this paper

Abstract:

In real life, information available on situations/issues/problems is vague, inexact, or insufficient and so the parameters involved therein are grasped in an uncertain way by the decision maker. But in real life such uncertainty is unavoidable. One possible way out is to consider the knowledge of experts about the parameters involved as fuzzy data. In a network, the arc length may represent time or cost. In Relevant literature reports there are several methods to solve such problems in network-flow. This paper proposes an optimized path for use in networks, using trapezoidal intuitionistic fuzzy numbers, assigned to each arc length in a fuzzy environment. It proposes a new algorithm to find the optimized path and implied distance from source node to destination node. 1. Introduction Fuzzy network problems have appeared in literature for quite some time, of which the simplest and most often researched mode is the fuzzy shortest path problem. The main objective of the optimized path problem is to find a path with minimum distance. The classical fuzzy shortest path problem seems to have been first introduced by Dubois and Prade [1]. They employed the fuzzy minimum operator to find the shortest path length, but they did not develop any method to decide the shortest path. They have used a fuzzy number instead of a real number assigned to each of the edges. Okada and Soper [2] concentrated on an optimized path problem and introduced the concept of degree of possibility in which an arc is on the optimized path. Takahashi and Yamakami [3] discussed the optimized path from a specified node to every other node on a network. Another algorithm for this problem was presented by Okada and Gen [4, 5], where there is a generalization of Dijkstra’s algorithm. In this algorithm the weight of the arcs is considered to be interval numbers and is defined from a partial order between interval numbers. Klein [6] proposed an improved algorithm that can get the shortest path length as well as the shortest path. However, the assumption made in the algorithm, that each arc with the length between “1” and a fixed integer “M” was not reasonable and practical. Szmidt and Kacprzyk [7] found distance between intuitionistic fuzzy sets using Hamming distance, the normalized Hamming distance, the Euclidean distance, and the normalized Euclidean distance. Kung and Chuang [8] pointed out that there are several methods to solve this kind of problem in the available literature. Przemys？aw Grzegorzewski [9] discussed two families of metrics in space of intuitionistic fuzzy numbers. Nagoor

References

 [1] D. Dubois and H. Prade, Fuzzy Sets and Systems, Academic Press, New York, NY, USA, 1980. [2] S. Okada and T. Soper, “A shortest path problem on a network with fuzzy arc lengths,” Fuzzy Sets and Systems, vol. 109, no. 1, pp. 129–140, 2000. [3] M. T. Takahashi and A. Yamakami, “On fuzzy shortest path problems with fuzzy parameters: an algorithmic approach,” in Proceedings of the Annual Meeting of the North American Fuzzy Information Processing Society (NAFIPS '05), pp. 654–657, June 2005. [4] S. Okada and M. Gen, “Order relation between intervals and its application to shortest path problem,” Computers and Industrial Engineering, vol. 25, no. 1–4, pp. 147–150, 1993. [5] S. Okada and M. Gen, “Fuzzy shortest path problem,” Computers and Industrial Engineering, vol. 27, no. 1–4, pp. 465–468, 1994. [6] C. M. Klein, “Fuzzy shortest paths,” Fuzzy Sets and Systems, vol. 39, no. 1, pp. 27–41, 1991. [7] E. Szmidt and J. Kacprzyk, “Distances between intuitionistic fuzzy sets,” Fuzzy Sets and Systems, vol. 114, no. 3, pp. 505–518, 2000. [8] J.-Y. Kung and T.-N. Chuang, “The shortest path problem with discrete fuzzy arc lengths,” Computers and Mathematics with Applications, vol. 49, no. 2-3, pp. 263–270, 2005. [9] Przemys？aw Grzegorzewski Systems Research Institute and Polish Academy of Sciences, “Distances and orderings in a family of intuitionistic fuzzy numbers,” Newelska 6, 01-447 Warsaw, Poland. [10] A. Nagoor Gani and M. Mohammed Jabarulla, “On searching intuitionistic fuzzy shortest path in a network,” Applied Mathematical Sciences, vol. 4, no. 69-72, pp. 3447–3454, 2010. [11] K.-C. Lin and M.-S. Chern, “The fuzzy shortest path problem and its most vital arcs,” Fuzzy Sets and Systems, vol. 58, no. 3, pp. 343–353, 1993. [12] F. Hernandes, M. T. Lamata, J. L. Verdegay, and A. Yamakami, “The shortest path problem on networks with fuzzy parameters,” Fuzzy Sets and Systems, vol. 158, no. 14, pp. 1561–1570, 2007. [13] A. kumar and M. kaur, “A new algorithm for solving network flow problems with fuzzy arc lengths,” Turkish Journal of Fuzzy Systems, vol. 2, no. 1, 2011. [14] P. Jayagowri and G. Dr. Geetharamani, “On solving network problems using new algorithm with Intuitionistic fuzzy arc length,” in Proceedings of the International Conference on Mathematics in Engineering & Business Management, 2012. [15] D. Yu, “Intuitionistic trapezoidal fuzzy information aggregation methods and their applications to teaching quality evaluation,” Journal of Information and Computational Science, vol. 10, no. 6, pp. 861–869, 2013. [16] L. A. Zadeh, “Fuzzy sets,” Information and Control, vol. 8, no. 3, pp. 338–353, 1965. [17] K. Atanassov, “Intuitionistic fuzzy sets and systems,” Fuzzy sets and Systems, vol. 20, no. 1, pp. 87–96, 1986. [18] W. Jianqiang and Z. Zhong, “Aggregation operators on intuitionistic trapezoidal fuzzy number and its application to multi-criteria decision making problems,” Journal of Systems Engineering and Electronics, vol. 20, no. 2, pp. 321–326, 2009.

Full-Text