Ranking Fuzzy Numbers with a Distance Method using Circumcenter of Centroids and an Index of Modality

Ranking fuzzy numbers are an important aspect of decision making in a fuzzy environment. Since their inception in 1965, many authors have proposed different methods for ranking fuzzy numbers. However, there is no method which gives a satisfactory result to all situations. Most of the methods proposed so far are nondiscriminating and counterintuitive. This paper proposes a new method for ranking fuzzy numbers based on the Circumcenter of Centroids and uses an index of optimism to reflect the decision maker's optimistic attitude and also an index of modality that represents the neutrality of the decision maker. This method ranks various types of fuzzy numbers which include normal, generalized trapezoidal, and triangular fuzzy numbers along with crisp numbers with the particularity that crisp numbers are to be considered particular cases of fuzzy numbers.

[1]  Madan M. Gupta,et al.  Approximate reasoning in decision analysis , 1982 .

[2]  Shyi-Ming Chen,et al.  Fuzzy risk analysis based on ranking generalized fuzzy numbers with different heights and different spreads , 2009, Expert Syst. Appl..

[3]  Emmanuel Valvis,et al.  A new linear ordering of fuzzy numbers on subsets of $${{\mathcal F}({\pmb{\mathbb{R}}}})$$ , 2009, Fuzzy Optim. Decis. Mak..

[4]  Marc Roubens,et al.  Ranking and defuzzification methods based on area compensation , 1996, Fuzzy Sets Syst..

[5]  Ching-Hsue Cheng,et al.  A new approach for ranking fuzzy numbers by distance method , 1998, Fuzzy Sets Syst..

[6]  E. Lee,et al.  Ranking of fuzzy sets based on the concept of existence , 1994 .

[7]  P. Anand Raj,et al.  Ranking alternatives with fuzzy weights using maximizing set and minimizing set , 1999, Fuzzy Sets Syst..

[8]  Hsuan-Shih Lee,et al.  The revised method of ranking fuzzy numbers with an area between the centroid and original points , 2008, Comput. Math. Appl..

[9]  Zhong-xin Wang,et al.  Ranking Fuzzy Numbers Based on Ideal Solution , 2008, ACFIE.

[10]  F. Choobineh,et al.  An index for ordering fuzzy numbers , 1993 .

[11]  J. J. Saade,et al.  Ordering fuzzy sets over the real line: an approach based on decision making under uncertainty , 1992 .

[12]  Huijun Sun,et al.  A new approach for ranking fuzzy numbers based on fuzzy simulation analysis method , 2006, Appl. Math. Comput..

[13]  Lucien Duckstein,et al.  Comparison of fuzzy numbers using a fuzzy distance measure , 2002, Fuzzy Sets Syst..

[14]  Miin-Shen Yang,et al.  Convergence Rate of the Fuzzy Generalized Nearest Neighbor Rule , 1994 .

[15]  Ying Luo,et al.  Area ranking of fuzzy numbers based on positive and negative ideal points , 2009, Comput. Math. Appl..

[16]  Ramesh Jain,et al.  DECISION MAKING IN THE PRESENCE OF FUZZY VARIABLES , 1976 .

[17]  Xinwang Liu,et al.  Ranking fuzzy numbers with preference weighting function expectations , 2005 .

[18]  Etienne E. Kerre,et al.  Reasonable properties for the ordering of fuzzy quantities (II) , 2001, Fuzzy Sets Syst..

[19]  K. Kim,et al.  Ranking fuzzy numbers with index of optimism , 1990 .

[20]  T. Chu,et al.  Ranking fuzzy numbers with an area between the centroid point and original point , 2002 .

[21]  Soheil Sadi-Nezhad,et al.  Ranking fuzzy numbers by preference ratio , 2001, Fuzzy Sets Syst..

[22]  B. Asady,et al.  The revised method of ranking LR fuzzy number based on deviation degree , 2010, Expert Syst. Appl..

[23]  Saeid Abbasbandy,et al.  Ranking of fuzzy numbers by sign distance , 2006, Inf. Sci..

[24]  D. Dubois,et al.  The mean value of a fuzzy number , 1987 .

[25]  Ronald R. Yager,et al.  A procedure for ordering fuzzy subsets of the unit interval , 1981, Inf. Sci..

[26]  Yufei Yuan.  Criteria for evaluating fuzzy ranking methods , 1991 .

[27]  M. Vila,et al.  A procedure for ranking fuzzy numbers using fuzzy relations , 1988 .

[28]  R. Yager.  Ranking fuzzy subsets over the unit interval , 1978, 1978 IEEE Conference on Decision and Control including the 17th Symposium on Adaptive Processes.

[29]  G. Bortolan,et al.  A review of some methods for ranking fuzzy subsets , 1985 .

[30]  Shyi-Ming Chen,et al.  Fuzzy risk analysis based on fuzzy numbers with different shapes and different deviations , 2008, Expert Syst. Appl..

[31]  Etienne E. Kerre,et al.  Reasonable properties for the ordering of fuzzy quantities (II) , 2001, Fuzzy Sets Syst..

[32]  Shan-Huo Chen.  Ranking fuzzy numbers with maximizing set and minimizing set , 1985 .

[33]  Shyi-Ming Chen,et al.  Fuzzy risk analysis based on the ranking of generalized trapezoidal fuzzy numbers , 2007, Applied Intelligence.

[34]  K. Nakamura.  Preference relations on a set of fuzzy utilities as a basis for decision making , 1986 .

[35]  J. Adamo.  Fuzzy decision trees , 1980 .

[36]  Ramesh C. Jain.  A procedure for multiple-aspect decision making using fuzzy sets , 1977 .

[37]  Qi Liu,et al.  Ranking fuzzy numbers with an area method using radius of gyration , 2006, Comput. Math. Appl..

[38]  B. Asady,et al.  RANKING FUZZY NUMBERS BY DISTANCE MINIMIZATION , 2007 .

[39]  E. Lee,et al.  Comparison of fuzzy numbers based on the probability measure of fuzzy events , 1988 .

[40]  Jing-Shing Yao,et al.  Ranking fuzzy numbers based on decomposition principle and signed distance , 2000, Fuzzy Sets Syst..

[41]  Huibert Kwakernaak,et al.  Rating and ranking of multiple-aspect alternatives using fuzzy sets , 1976, Autom..

[42]  J. Baldwin,et al.  Comparison of fuzzy sets on the same decision space , 1979 .

[43]  Didier Dubois,et al.  Ranking fuzzy numbers in the setting of possibility theory , 1983, Inf. Sci..

[44]  María Teresa Lamata,et al.  A Modification of the Index of Liou and Wang for Ranking Fuzzy Number , 2007, Int. J. Uncertain. Fuzziness Knowl. Based Syst..

[45]  Mao-Jiun J. Wang,et al.  Ranking fuzzy numbers with integral value , 1992 .

[46]  Stanisław Heilpern,et al.  The expected value of a fuzzy number , 1992 .

[47]  Saeid Abbasbandy,et al.  A new approach for ranking of trapezoidal fuzzy numbers , 2009, Comput. Math. Appl..

[48]  L. M. D. C. Ibáñez,et al.  A subjective approach for ranking fuzzy numbers , 1989 .

[49]  Liang-Hsuan Chen,et al.  An approximate approach for ranking fuzzy numbers based on left and right dominance , 2001 .

[50]  R. Yager.  ON CHOOSING BETWEEN FUZZY SUBSETS , 1980 .

[51]  Hui-Chin Tang,et al.  Ranking nonnormal p-norm trapezoidal fuzzy numbers with integral value , 2008, Comput. Math. Appl..

[52]  Deng-Feng Li,et al.  A ratio ranking method of triangular intuitionistic fuzzy numbers and its application to MADM problems , 2010, Comput. Math. Appl..

[53]  Ying-Ming Wang,et al.  Centroid defuzzification and the maximizing set and minimizing set ranking based on alpha level sets , 2009, Comput. Ind. Eng..

[54]  H. Maeda,et al.  Fuzzy decision analysis in the development of centralized regional energy control systems , 1983 .

[55]  Bo Feng,et al.  Ranking L-R fuzzy number based on deviation degree , 2009, Inf. Sci..

[56]  Waldemar Koodziejczyk.  Orlovsky's concept of decision-making with fuzzy preference relation—Further results , 1986 .