REVISION OF SIGN DISTANCE METHOD FOR RANKING OF FUZZY NUMBERS

Document Type : Research Paper

Authors

1 Department of Mathematics, Science and Research Branch, Is- lamic Azad University, Tehran, Iran

2 Department of Mathematics, South Tehran Branch, Islamic Azad University, Tehran, Iran

3 Department of Mathematics, Aliabad Katoul Branch, Islamic Azad University, Aliabad Katoul, Iran

Abstract

Recently, Abbasbandy and Asady have been proposed a modification
of the distance based approach, namely ``sign distance method''.
However, in this paper, it is shown that this method has some drawbacks, i.e.,
the result is not consistent with human intuition for special
cases and this method cannot always logically infer ranking
order of the images of the fuzzy numbers. In this paper, we
present a revised method which can avoid these problems for
ranking fuzzy numbers. Also, we present several properties
for revised sign distance method while the original method does not have some of
them.

Keywords


\bibitem{ABB1} S. Abbasbandy and B. Asady, {\it Ranking of fuzzy numbers by sign distance},
Information Sciences, \textbf{176} (2006), 2405--2416.

\bibitem{ABB2} S. Abbasbandy and T. Hajjari, {\it A new approach for ranking of trapezoidal fuzzy
numbers}, Computers and Mathematics with  Applications, \textbf{57} (2009), 413--419.

\bibitem{ADA1} M. Adamo, {\it Fuzzy decision trees}, Fuzzy Sets and Systems, \textbf{4} (1980), 207--219.

\bibitem{ASA1} B. Asady, {\it The revised method of ranking LR fuzzy number based on deviation
degree}, Expert Systems with Applications, \textbf{37} (2010), 5056--5060.

\bibitem{ASA3} B. Asady, {\it Revision of distance minimization method for ranking of fuzzy
numbers}, Applied Mathematical Modelling, \textbf{35} (2011), 1306--1313.

\bibitem{ASA2}  B. Asady and  M. Zendehnam, {\it Ranking of fuzzy numbers by
distance minimization}, Applied Mathematical Modelling, \textbf{31} (2007),
2589--2598.

\bibitem{BAS1} S. Bass and H. Kwakernaak, {\it Rating and ranking of multiple-aspect
alternatives using fuzzy sets}, Automatica, \textbf{13} (1977), 47--58.

\bibitem{BOR1} G. Bortolan and R. Degani, {\it A review of some methods for ranking
fuzzy numbers}, Fuzzy Sets and Systems, \textbf{15} (1985), 1--19.

\bibitem{CHE2} S. J. Chen and C. L. Hwang, {\it Fuzzy multiple
attribute decision making}, Springer, New York, 1992.

\bibitem{CHE1} C. H. Cheng, {\it A new approach for ranking fuzzy numbers by distance method}, Fuzzy Sets and
Systems, \textbf{95} (1998), 307--317.

\bibitem{CHO1}F. Choobineh and H. Li, {\it An index for ordering fuzzy numbers}, Fuzzy Sets and Systems, \textbf{54} (1993), 287--294.

\bibitem{CHU1} T. Chu and C. Tsao, {\it Ranking fuzzy numbers with an area
between the centroid point and original point}, Computers and
Mathematics with  Applications, \textbf{43} (2002), 111--117.

\bibitem{DIA1} P. Diamond and P. Kloeden, {\it Metric spaces of fuzzy sets}, Fuzzy Sets and Systems, \textbf{35} (1990), 241--249.

\bibitem{DUB2} D. Dubios and H. Prade, {\it Operations on fuzzy numbers}, Int. J. System Sci., \textbf{9} (1978), 626--631.

\bibitem{DUB1} D. Dubois and H. Prade, {\it Fuzzy sets and systems: theory and application}, Academic Press, New York, 1980.

\bibitem{FOR1}  P. Fortemps and M. Roubens, {\it Ranking and defuzzification methods
based on area compensation}, Fuzzy Sets and Systems, \textbf{82} (1996), 319--330.

\bibitem{GRZ3}  P. Grzegorzewski, {\it Metrics and orders in space of fuzzy numbers},
Fuzzy Sets and Systems, \textbf{97}(1998), 83--94.

\bibitem{GRZ1}  P. Grzegorzewski, {\it Nearst interval approximation of a fuzzy
number}, Fuzzy Sets and Systems, \textbf{130} (2002), 321--330.

\bibitem{JAI1} R. Jain, {\it Decision-making in the presence of fuzzy variable},
IEEE Trans. Syst. Man Cybernet., \textbf{6} (1976), 698--703.

\bibitem{JAI2} R. Jain, {\it A procedure for multi-aspect decision making using fuzzy sets},
Int. J. Syst. Sci., \textbf{8} (1978), 1--7.

\bibitem{KAU1} A. Kauffman and M. M. Gupta, {\it Introduction to fuzzy arithmetic: theory and application}, Van
Nostrand Reinhold, New York, 1991.

\bibitem{KUM1} A. Kumar, A. Gupta and M. K. Sharma, {\it Application of tabu search for solving the bi-objective
warehouse problem in a fuzzy environment}, Iranian Journal of Fuzzy Systems,  \textbf{9(1)} (2012), 1--19.

\bibitem{LEE2} K. M. Lee, C. H. Cho and H. Lee-Kwang, {\it Ranking fuzzy values with
satisfaction function}, Fuzzy Sets and Systems, \textbf{64} (1994), 295--311.

\bibitem{LEE1} E. Lee and R. J. Li, {\it Comparison of fuzzy numbers based
on the probability measure of fuzzy events}, Computers and
Mathematics with Applications,  \textbf{15(10}) (1988), 887--896.


\bibitem{LI1} J. Li, W. Li and X. Kong, {\it A New Algorithm Model for Solving Fuzzy Linear
Systems}, Southeast Asian Bulletin of Mathematics, \textbf{34} (2010), 121--132.

\bibitem{MA1} M. Ma, M. Friedman and A. Kandel, {\it A new fuzzy arithmetic}, Fuzzy Sets and Systems, \textbf{108} (1999), 83--90.

\bibitem{MAH1} I. Mahdavi, N. Mahdavi-Amiri and S. Nejati, {\it Algorithms for biobjective shortest path problems in fuzzy networks},
Iranian Journal of Fuzzy Systems , \textbf{8(4)} (2011), 9--37.

\bibitem{MOH1} A. Mohamadi Nejad and  M. Mashinchi, {\it Ranking fuzzy
numbers based on the areas on the left and right sides of fuzzy
number}, Computers  and Mathematics with Applications, \textbf{61(2)} (2011), 731--442.

\bibitem{SEV1} P. Sevastianov, {\it Numerical methods for interval and fuzzy number comparison based
on the probabilistic approach and Dempster-Shafer theory}, Information Sciences, \textbf{177} (2007), 4645--4661.

\bibitem{WAN1} X. Wang and E. E. Kerre, {\it Reasonable properties for the
ranking of fuzzy quantities I}, Fuzzy Sets and Systems, \textbf{118} (2001), 375--385.

\bibitem{WAN2} X. Wang and E. E. Kerre, {\it Reasonable properties for the
ranking of fuzzy quantities II}, Fuzzy Sets and Systems, \textbf{118} (2001), 387--405.

\bibitem{WAN5} Y. J. Wang and S. H. Lee, {\it The revised method of ranking fuzzy numbers
with an area between the centroid and original points}, Computers
and Mathematics with Applications, \textbf{55} (2008), 2033--2042.

\bibitem{ZXW1} Z. X. Wang, Y. J. Liu, Z. P. Fan and B. Feng, {\it Ranking L-R fuzzy number based on deviation
degree}, Information Sciences, \textbf{179} (2009), 2070--2077.

\bibitem{WAN3} Y. M. Wang and Y. Luo, {\it Area ranking of fuzzy numbers based on positive and negative ideal
points}, Computers  and Mathematics with Applications, \textbf{58} (2009), 1769--1779.

\bibitem{WAN4} Y. M. Wang, J. B. Yang, D. L. Xu and  K. S. Chin, {\it On the centroids of
fuzzy numbers}, Fuzzy Sets and Systems, \textbf{157} (2006), 919--926.

\bibitem{YAG2} R. R. Yager, {\it On choosing between fuzzy subsets}, Kybernetes, \textbf{9} (1980), 151--154.

\bibitem{YAG1}  R. R. Yager, {\it A procedure for ordering fuzzy subsets of the unit interval},
Information Sciences, \textbf{24} (1981), 143--161.

\bibitem{ZAD1} L. A. Zadeh, {\it Fuzzy sets}, Information and  Control, \textbf{8} (1965), 338--353.