A New Method for Ordering Fuzzy Number

Volume 4, Issue 3, pp 283--294
• 2923 Views

Authors

S. H. Nasseri - Department of Mathematics, University of Mazandaran, Babolsar, IRAN F. Taleshian - Department of Mathematics, University of Mazandaran, Babolsar, IRAN Z. Alizadeh - Department of Mathematics, University of Mazandaran, Babolsar, IRAN J. Vahidi - Department of Mathematics, Science and Technology of Behshahr, Behshahr, IRAN

Abstract

Ranking fuzzy numbers is an important aspect of decision making in a fuzzy environment. In fuzzy decision making problems, fuzzy numbers must be ranked before an action is taken by a decision maker. This article is about ranking Fuzzy numbers and describes a ranking method for ordering LR fuzzy numbers based on the area of fuzzy numbers. This method is simple in evaluation and can rank various types of LR fuzzy numbers and also crisp numbers which are considered to be a special class of fuzzy numbers.

Keywords

• fuzzy number
• ranking function.
• ranking method

•  03E72
•  93C42

References

• [1] S. Abbasbandy, B. Asady, Ranking of fuzzy numbers by sign distance, Inform. Sci., 176 (2006), 2405--2416

• [2] S. Abbasbandy, T. Hajjari, A new approach for ranking of trapezoidal fuzzy numbers, Computers and Mathematics with Applications, 57 (2009), 413--419

• [3] J. M. Adamo, Fuzzy decision trees, Fuzzy Sets and Systems, 4 (1980), 207--219

• [4] P. Anand Raj, D. Nagesh Kumar, Ranking alternatives with fuzzy weights using maximizing set and minimizing set, Fuzzy Sets and Systems, 105 (1999), 365--375

• [5] B. Asady, The revised method of ranking LR fuzzy number based on deviation degree, Expert Systems with Applications, 37 (2010), 5056--5060

• [6] B. Asady, A. Zendehnam, Ranking fuzzy numbers by distance minimization, Applied Mathematical Modelling, 31 (2007), 2589--2598

• [7] S. M. Baas, H. Kwakernaak, Rating and ranking of multiple-aspect alternatives using fuzzy sets, Automatica, 13 (1977), 47--58

• [8] J. F. Baldwin, N. C. F. Guild, Comparison of fuzzy sets on the same decision space, Fuzzy Sets and Systems, 2 (1979), 213--231

• [9] G. Bortolan, R. Degani, A review of some methods for ranking fuzzy subsets, Fuzzy Sets and Systems, 15 (1985), 1--19

• [10] W. Chang, Ranking of fuzzy utilities with triangular membership functions, Proceedings of International Conference on Policy Analysis and Systems, 1981 (1981), 263--272

• [11] P. T. Chang, E. S. Lee, Ranking of fuzzy sets based on the concept of existence, Computers and Mathematics with Applications, 27 (1994), 1--21

• [12] S. H. Chen, Ranking fuzzy numbers with maximizing set and minimizing set, Fuzzy Sets and Systems, 17 (1985), 113--129

• [13] S. J. Chen, S. M. Chen, Fuzzy risk analysis based on the ranking of generalized trapezoidal fuzzy numbers, Applied Intelligence, 26 (2007), 1--11

• [14] S. M. Chen, J. H. Chen, Fuzzy risk analysis based on ranking generalized fuzzy numbers with different heights and different spreads, Expert Systems with Applications, 36 (2009), 6833--6842

• [15] L. H. Chen, H. W. Lu, An approximate approach for ranking fuzzy numbers based on left and right dominance, Computers and Mathematics with Applications, 41 (2001), 1589--1602

• [16] C. C. Chen, H. C. Tang, Ranking nonnormal p-norm trapezoidal fuzzy numbers with integral value, Computers and Mathematics with Applications, 56 (2008), 2340--2346

• [17] C. H. Cheng, A new approach for ranking fuzzy numbers by distance method, Fuzzy Sets and Systems, 95 (1998), 307--317

• [18] F. Choobineh, H. Li, An index for ordering fuzzy numbers, Fuzzy Sets and Systems, 54 (1993), 287--294

• [19] T. C. Chu, C. T. Tsao, Ranking fuzzy numbers with an area between the centroid point and original point, Computers and Mathematics with Applications, 43 (2002), 111--117

• [20] L. De Campos, G. A. Muñoz, A subjective approach for ranking fuzzy numbers, Fuzzy Sets and Systems, 29 (1989), 145--153

• [21] M. Delgado, J. L. Verdegay, M. A. Vila, A procedure for ranking fuzzy numbers using fuzzy relations, Fuzzy Sets and Systems, 26 (1988), 49--62

• [22] Y. Deng, Z. Zhenfu, L. Qi, Ranking fuzzy numbers with an area method using radius of gyration, Computers and Mathematics with Applications, 51 (2006), 1127--1136

• [23] D. Dubois, H. Prade, Ranking fuzzy numbers in the setting of possibility theory, Inform. Sci., 30 (1983), 183--224

• [24] D. Dubois, H. Prade, The mean value of a fuzzy number, Fuzzy Sets and Systems, 24 (1987), 279--300

• [25] P. Fortemps, M. Roubens, Ranking and defuzzification methods based on area compensation , Fuzzy Sets and Systems, 82 (1996), 319--330

• [26] M. S. Garcia, M. T. Lamata, A modification of the index of liou and wang for ranking fuzzy number, International Journal of Uncertainty, Fuzziness and Knowlege-Based Systems, 15 (2007), 411--424

• [27] S. Heilpern, The expected value of a fuzzy number, Fuzzy Sets and Systems, 47 (1992), 81--86

• [28] R. Jain, A procedure for multi-aspect decision making using fuzzy sets, International Journal of Systems Science, 8 (1978), 1--7

• [29] R. Jain, Decision making in the presence of fuzzy variables, IEEE Transactions on Systems, Man and Cybernetics, 6 (1976), 698--703

• [30] E. Kerre, The use of fuzzy set theory in electrocardiological diagnostics, Approximate Reasoning in Decision-Analysis, 20 (1982), 277--282

• [31] W. Kołodziejczyk, S. Orlovsky, Orlovsky's concept of decision-making with fuzzy preference relation-further results, Fuzzy Sets and Systems, 19 (1986), 11--20

• [32] K. Kim, K. S. Park, Ranking fuzzy numbers with index of optimism, Fuzzy Sets and Systems, 35 (1990), 143--150

• [33] L. W. Lee, S. M. Chen, Fuzzy risk analysis based on fuzzy numbers with different shapes and different deviations , Expert Systems with Applications, 34 (2008), 2763--2771

• [34] E. S. Lee, R. J. Li, Comparison of fuzzy numbers based on the probability measure of fuzzy events, Computers and Mathematics with Applications, 15 (1988), 887--896

• [35] D. F. Li, A ratio ranking method of triangular intuitionistic fuzzy numbers and its application to MADM problems, Computers and Mathematics with Applications, 60 (2010), 1557--1570

• [36] T. S. Liou, M. J. Wang, Ranking fuzzy numbers with integral value, Fuzzy Sets and Systems, 50 (1992), 247--255

• [37] X. W. Liu, S. L. Han, Ranking fuzzy numbers with preference weighting function expectations, Computers and Mathematics with Applications, 49 (2005), 1731--1753

• [38] N. Mahdavi-Amiri, S. H. Nasseri, A. Yazdani Cherati, Fuzzy primal simplex algorithm for solving fuzzy linear programming problems, Iranian Journal of Operational Research, 2 (2009), 68--84

• [39] M. Modarres, S. S. Nezhad, Ranking fuzzy numbers by preference ratio, Fuzzy Sets and Systems, 118 (2001), 429--436

• [40] S. Murakami, H. Maeda, S. Imamura, Fuzzy decision analysis on the development of centralized regional energy control system , Proceedings of the IFAC on Fuzzy Information, Knowledge Representation and Decision Analysis, 16 (1983), 353--358

• [41] K. Nakamura, Preference relations on a set of fuzzy utilities as a basis for decision making, Fuzzy Sets and Systems, 20 (1986), 147--162

• [42] S. H. Nasseri, A. Ebrahimnejad, A fuzzy primal simplex algorithm and its application for solving flexible linear programming problems, European Journal of Industrial Engineering, 4 (2010), 327--389

• [43] S. H. Nasseri, H. Attari, A. Ebrahimnejad, Revised simplex method and its application for solving fuzzy linear programming problems, European Journal of industrial Engineering, 6 (2012), 259--280

• [44] S. H. Nasseri, N. Mahdavi-Amiri, Some duality results on linear programming problems with symmetric fuzzy numbers, Fuzzy Information and Engineering, 1 (2009), 59--66

• [45] J. J. Saade, H. Schwarzlander, Ordering fuzzy sets over the real line: an approach based on decision making under uncertainty, Fuzzy Sets and Systems, 50 (1992), 237--246

• [46] H. Sun, J. Wu, A new approach for ranking fuzzy numbers based on fuzzy simulation analysis method, Applied Mathematics and Computation, 174 (2006), 755--767

• [47] L. Tran, L. Duckstein, Comparison of fuzzy numbers using a fuzzy distance measure, Fuzzy Sets and Systems, 130 (2002), 331--341

• [48] E. Valvis, A new linear ordering of fuzzy numbers on subsets of F (R), Fuzzy Optimization and Decision Making, 8 (2009), 141--163

• [49] X. Wang, A class of approaches to ordering alternatives, M.S. thesis (Taiyuan University Technology), China (1987)

• [50] Y. M. Wang, Centroid defuzzification and the maximizing set and minimizing set ranking based on alpha level sets, Computers and Industrial Engineering, 57 (2009), 228--236

• [51] X. Wang, E. E. Kerre, Reasonable properties for the ordering of fuzzy quantities (I), Fuzzy Sets and Systems, 118 (2001), 375--385

• [52] X. Wang, E. E. Kerre, Reasonable properties for the ordering of fuzzy quantities (II), Fuzzy Sets and Systems, 118 (2001), 387--405

• [53] Y. J. Wang, H. S. Lee, The revised method of ranking fuzzy numbers with an area between the centroid and original points, Computers and Mathematics with Applications, 55 (2008), 2033--2042

• [54] Z. X. Wang, Y. J. Liu, Z. P. Fan, B. Feng, Ranking LR fuzzy number based on deviation degree, Inform. Sci., 179 (2009), 2070--2077

• [55] Y. M. Wang, Y. Luo, Area ranking of fuzzy numbers based on positive and negative ideal points, Computers and Mathematics with Applications, 58 (2009), 1769--1779

• [56] Z. X. Wang, Y. N. Mo, Ranking fuzzy numbers based on ideal solution, Fuzzy Information and Engineering, 2 (2010), 27--36

• [57] R. R. Yager, A procedure for ordering fuzzy subsets of the unit interval, Inform. Sci., 24 (1981), 143--161

• [58] R. R. Yager, On choosing between fuzzy subsets, Kybernetes, 9 (1980), 151--154

• [59] R. R. Yager, Ranking fuzzy subsets over the unit interval, Proceedings of the IEEE Conference on Decision and Control , 1979 (1979), 1435--1437

• [60] J. S. Yao, K. Wu, Ranking fuzzy numbers based on decomposition principle and signed distance, Fuzzy Sets and Systems, 116 (2000), 275--288

• [61] Y. Yuan, Criteria for evaluating fuzzy ranking methods, Fuzzy Sets and Systems, 43 (1991), 139--157

• [62] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338--353