Document Type : Research Paper


1 Department of Statistics, School of Mathematics, University of Kashan, Kashan,Iran.

2 Department of Mathematical Sciences, Payame Noor University, Tehran, Iran.


In this paper, we propose a novel method for ranking a set of fuzzy numbers. In this method a preference index is proposed based on $\alpha$-optimistic values of a fuzzy number. We propose a new ranking method by adopting a level of credit in the ordering procedure. Then, we investigate some desirable properties of the proposed ranking method.


Main Subjects

[1] S. Abbasbandy, and B. Asady, Ranking of fuzzy numbers by sign distance, Inform. Sciences., 176 (2006), pp. 2405-24016.
[2] M. Adamo, Fuzzy decision trees, Fuzzy. Set. Syst., 4 (1980), pp. 207-219.
[3] B. Asady and A. Zendehnam, Ranking fuzzy numbers by distance minimization, Appl. Math. Model., 31 (2007), pp. 2589-2598.
[4] J.F. Baldwin and N.C.F. Guild, Comparison of fuzzy sets on the same decision space, Fuzzy. Set. Syst., 2 (1979), pp.213-231.
[5] G. Bortolan and R.A. Degani, Review of some methods for ranking fuzzy subsets, Fuzzy. Set. Syst., 15 (1985), pp. 1-19.
[6] P.T. Chang and E.S. Lee, Ranking of fuzzy sets based on the concept of existence, Comput. Math. Appl., 27 (1994), pp. 1-21.
[7] S.H. Chen, Ranking fuzzy numbers with maximizing set and minimizing set, Fuzzy. Set. Syst., 17 (1985), pp. 113-129.
[8] S.J. Chen and S.M. Chen, Fuzzy risk analysis based on the ranking of generalized trapezoidal fuzzy numbers, Appl. Intell., 26 (2007), pp. 1-11.
[9] L.H. Chen and H.W. Lu, An approximate approach for ranking fuzzy numbers based on left and right dominance, Comput. Math. Appl., 41 (2001), pp. 1589-1602.
[10] C.H. Cheng, A new approach for ranking fuzzy numbers by distance method, Fuzzy. Set. Syst., 95 (1998), pp. 307-317.
[11] T.C. Chu and C.T. Tsao, Ranking fuzzy numbers with an area between the centroid point and original point, Comput. Math. Appl., 43 (2002), pp. 111-117.
[12] L. de Campos and G.A Munoz, A subjective approach for ranking fuzzy numbers, Fuzzy. Set. Syst., 29(1989), pp. 145-153.
[13] M. Delgado, J.L. Verdegay, and M.A. Vila, A procedure for ranking fuzzy numbers using fuzzy relations, Fuzzy. Set. Syst., 26 (1988), pp. 49-62.
[14] Y. Deng, Z. Zhenfu, and L. Qi, Ranking fuzzy numbers with an area method using radius of gyration, Comput. Math. Appl., 51 (2006), pp. 11271136.
[15] D. Dubois and H. Prade, Ranking fuzzy numbers in the setting of possibility theory, Inform. Sciences., 30 (1983), pp. 183-224.
[16] R. Ezzatia, T. Allahviranloob, S. Khezerlooa, and M. Khezerloob, An approach for ranking of fuzzy numbers, Expert. Syst. Appl., 39 (1983), pp. 690-695.
[17] P. Fortemps and M. Roubens, Ranking and defuzzification methods based on area compensation, Fuzzy. Set. Syst., 82 (1996), pp. 319-330.
[18] M. Hanns, Applied Fuzzy Arithmetic: An Introduction with Engineering Applications, Springer: New York, 2005.
[19] M. Inuiguchi, J. Ramik, T. Tanino, and M. Vlach, Satisficing solutions and duality in interval and fuzzy linear programming, Fuzzy. Set. Syst., 135 (2003), pp. 151-177.
[20] R. Jain, Decision making in the presence of fuzzy variables, IEEE. T. Syst. Man and Cyb., 6 (1976), pp. 698-703.
[21] K. Kim and K.S. Park, Ranking fuzzy numbers with index of optimism, Fuzzy. Set. Syst., 35 (1990), pp. 143-150.
[22] W. Kolodziejczyk, Orlovsky’s concept of decision-making with fuzzy preference relation-further results, Fuzzy. Set. Syst., 19 (1986), pp. 11-20.
[23] K.H. Lee, First Course on Fuzzy Theory and Applications, Springer-Verlag: Berlin, 2005.
[24] E.S. Lee and R.J. Li  Comparison of fuzzy numbers based on the probability measure of fuzzy events, Comput. Math. Appl., 15 (1988), pp. 887-896.
[25] D.F. Li, A ratio ranking method of triangular intuitionistic fuzzy numbers and its application to MADM problems, Comput. Math. Appl., 60 (2010), pp. 1557-1570.
[26] T.S. Liou and M.J. Wang, Ranking fuzzy numbers with integral value, Fuzzy. Set. Syst., 50 (1992), pp. 247-255.
[27] B. Liu, Uncertainty Theory, Springer-Verlag: Berlin, 2004.
[28] X.W. Liu and S.L. Han, Ranking fuzzy numbers with preference weighting function expectations, Comput. Math. Appl., 49 (2005), pp. 1731-1753.
[29] M. Modarres and S.S. Nezhad, Ranking fuzzy numbers by preference ratio, Fuzzy. Set. Syst., 118 (2001), pp. 429-436.
[30] K. Nakamura, Preference relations on a set of fuzzy utilities as a basis for decision making, Fuzzy. Set. Syst., 20 (1986), pp. 147-162.
[31] J. Peng and, B. Liu, Some properties of optimistic and pessimistic values of fuzzy, IEEE. Int. Conf. Fuzzy., 2 (2004), pp. 745-750.
[32] P.A. Raj and D.N. Kumar, Ranking alternatives with fuzzy weights using maximizing set and minimizing set, Fuzzy. Set. Syst., 105 (1999), pp. 365-375.
[33] J. Saade and H. Schwarzlander, Ordering fuzzy sets over the real line: an approach based on decision making under uncertainty, Fuzzy. Set. Sys., 50 (1992), pp.237-246.
[34] H. Sun and J. Wu, A new approach for ranking fuzzy numbers based on fuzzy simulation analysis method, Appl. Math. Comput., 174 (2006), pp. 755-767.
[35] L. Tran and L. Duckstein, Comparison of fuzzy numbers using a fuzzy distance measure, Fuzzy. Set. Syst., 130 (2002), pp. 331-341.
[36] E. Valvis, A new linear ordering of fuzzy numbers on subsets of F(R), Fuzzy. Optim. Decis. Ma., 8 (2009), pp. 141-163.
[37] Y.M. Wang, Centroid defuzzification and the maximizing set and minimizing set ranking based on alpha level sets, Comput. Ind. Eng., 57 (2009), pp. 228-236.
[38] X. Wang, A class of approaches to ordering alternatives, M.S. thesis, Taiyuan University Technology, 1987.
[39] X. Wang and E.E. Kerre, Reasonable properties for the ordering of fuzzy quantities (I), Fuzzy. Set. Syst., 118 (2001), pp. 375-385.
[40] X. Wang and E.E. Kerre, Reasonable properties for the ordering of fuzzy quantities (II), Fuzzy. Set. Syst., 118 (2001), pp. 387405.
[41] Y.J. Wang and H.S. Lee, The revised method of ranking fuzzy numbers with an area between the centroid and original points, Comput. Math. Appl., 55 (2008), pp. 2033-2042.
[42] Y.M. Wang and Y. Luo, Area ranking of fuzzy numbers based on positive and negative ideal points, Comput. Math. Appl., 58 (2009), pp. 1769-1779.
[43] Z.X. Wang, Y.J. Liu, Z.P. Fan, and B. Feng, Ranking LR-fuzzy number based on deviation degree, Inform. Sciences., 179 (2009), pp. 2070-2077.
[44] Z.X. Wang, and Y.N. Mo, Ranking fuzzy numbers based on ideal solution, Adv. Intel. Soft. Compu., 2 (2010), pp. 27-36.
[45] P. Xu, X. Su, J. Wu, , X. Sun, Y. Zhang, and Y. Deng, A note on ranking generalized fuzzy numbers, Expert. Syst. Appl., 39 (2012), pp. 6454-6457.
[46] R.R. Yager, A procedure for ordering fuzzy subsets of the unit interval, Inform. Sciences., 24 (1981), pp. 143-161.
[47] R.R. Yager, On choosing between fuzzy subsets, Kybernetes, 9 (1980), pp. 151-154.
[48] R.R. Yager, Ranking fuzzy subsets over the unit interval, IEEE. Conf. Decis. Contr. P., pp. 14351437, Albuquerque, NM, USA, 1978.
[49] J.S. Yao and K. Wu, Ranking fuzzy numbers based on decomposition principle and signed distance, Fuzzy. Set. Syst., 116 (2000), pp. 275-288.
[50] Y. Yuan, Criteria for evaluating fuzzy ranking methods, Fuzzy. Set. Syst., 43 (1991), pp. 139-157.