Ranking Fuzzy Numbers by Using Radius of Gyration

Transcription

1 ustalian Jounal of Basic and pplied Sciences, (): 68-66, 00 ISSN anking Fuzz Numbes b Using adius of Gation. S.H. Nassei, M. Sohabi Depatment of Mathematical Sciences, Mazandaan Univesit, P.O.Bo 7-68, Babolsa, Ian. National Elite Foundation, Tehan, Ian. Febua 7, 009 bstact: anking fuzz numbes plas a ve impotant ole in linguistic decision making and some othe fuzz application sstems. Man methods have been poposed to deal with anking fuzz numbes. ecentl, Deng and his colleagues, pesented a method to ank fuzz numbes. The emploed adius of gation ank fuzz numbes; howeve thee wee some poblems with the anking method. In this pape, we fist indicate the poblems of OG (adius of Gation) method and then popose a evised method which can avoid these poblems fo anking fuzz numbes. Since the evised method is based on the OG method, it is eas to ank fuzz numbes in a wa simila to the oiginal method. Ke wods: fuzz numbe, anking, adius of gation. INTODUCTION anking fuzz numbes is impotant in decision-making, data analsis, atificial intel- ligence, Economic sstems and opeations eseach. Jain (Jain, 976; 978), Dubois and Pade (978) intoduced the elevant concepts of fuzz numbes. Botolan and Degani (98) eviewed some methods to ank fuzz numbes, Chen and Hwang (99) poposed fuzz multiple attibute decision making, Choobineh and Li (99) poposed an inde fo odeing fuzz numbes, Dias (99) anked altenatives b odeing fuzz numbes, Lee et al. (99) anked fuzz numbes with a satisfaction function, equena et al. (99) utilized atificial neual netwoks fo the automatic anking of fuzz numbes, Fotemps and oubens (996) pesented anking and defuzzication methods based on aea compensation, and aj et al. (999) investigated maimizing and minimizing sets to ank fuzz altenatives with fuzz weights. Howeve, Chu and Tsao (00) poposed a method of anking fuzz numbes with an aea between the centoid and oiginal points. Chu and Tsao s method oiginated fom the concepts of Lee, Li (988) and Cheng (998). Lee and Li poposed the compaison of fuzz numbes, fo which the consideed mean and standad deviation values fo fuzz numbes based on the unifom and popotional pobabilit distibutions. Wang and Lee (008) pesented a new method of anking fuzz numbes with using adius of gation. In this pape, we shall popose a new method fo anking fuzz numbes to ovecome the shotcomings of some pevious methods. We pepae ou discussion in sections. In Section, we give some definitions and peliminaies. In Section, we descibe adius of gation method. In Section, we fist give some eamples to state the shotcomings of the pevious methods and then pesent a new method to ovecome these poblems. Moe-ove, we shall compae ou method with some anking methods of fuzz numbes. Finall, we conclude in Section.. Peliminaies: Hee, we eview some basic notations of fuzz sets (taken fom (Wang and Lee, 008)). Definition.. Let U be a univese set. fuzz set of U is defined b a membeship function μ [0, ], whee μ (), U, indicates the degee of in. Definition.. fuzz subset of univese set U is nomal if and onl if sup set. U μ () =, whee U is the univese Coesponding utho: S.H. Nassei, Depatment of Mathematical Sciences, Mazandaan Univesit, P.O.Bo 7-68, Babolsa, Ian. nassei@umz.ac.i (Hadi Nassei) 68

2 ust. J. Basic & ppl. Sci., (): 68-66, 00 Definition.. fuzz subset of univese set U is conve if and onl if μ (ë + ( - ë)) min{μ (), μ ()},, U, ë [0,]. Definition.. fuzz set is a fuzz numbe if and onl if is nomal and conve on U. Definition.. tiangula fuzz numbe is a fuzz numbe with a piecewise linea membeship function μ defined b: which can be denoted as a tiplet (a, a, a ). Definition.6. tapezoidal fuzz numbe is a fuzz numbe with a membeship func- tion μ defined b: which can be denoted as a quatet (a, a, a, a ). The adius of Gation of Fuzz Numbes: adius of gation is a concept in mechanics. Deng and his colleagues gave a new aea method to ank fuzz numbes with the adius of gation (OG) points (see in (Deng et al., 006)). Hee, we keep all discussions of them. The moment of inetia of the aea with espect to the ais, and the moment of inetia of the aea with espect to the ais ae defined, espectivel, as The adius of gation of an aea with espect to the ais is defined as the quantit, that satisfies the elation, () () whee I is the moment of inetia of with espect to the ais. Solving equation fo, concludes that () In a simila wa, the define the adius of gation of an aea with espect to the ais is () () 69

3 ust. J. Basic & ppl. Sci., (): 68-66, 00 Fig. : The mentioned when a genealized fuzz numbe is given, the adius of gation (OG) points of the genealized fuzz numbe is denoted as ( (), ()) whose value can be obtained b equations () and (). Fo an aea made up of a numbe of simple shapes, the moment of inetia of the entie aea is the sum of the moments of inetia of each of the individual aea about the ais desied. Fo eample, the moment of inetia of the genealized tapezoidal fuzz numbe in Figue can be obtained as follows: I = I, I = I. (6) Eample : Detemine the moment of inetia and the adius of gation of the genealized tapezoidal fuzz numbe. Fist, the tapezoid (a, e, f, d) can be divided into thee pats, (a, e, b), (b, e, f, c), and (c, f, d). The moment of inetia of the aea (aeb) with espect to ais, and the moment of inetia of the aea (a, e, b) with espect to ais can be calculated (7) The moment of inetia of aea befc and cfd, with espect to ais and ais, can be obtained, espectivel, as follows: (8) So, the (OG) point of genealized tapezoidal fuzz numbe can be calculated as q 660

4 ust. J. Basic & ppl. Sci., (): 68-66, 00 whee the (I ), (I ), (I ), (I ), (I ), (I ) can be obtained fom equations (7) and (8). The Poblem of OG Method: In this section, we fist give a numeical eample to pesent a poblem of adius of gation method (taken fom (Wang and Lee, 008). We assume that thee ae two tiangula fuzz numbes,, whee = (,,;) and = (9, 0, ;0.). Obviousl, is smalle than. So, b adius of gation method, we can calculate, ( ) = 0.08, ( ) =.0. Theefoe, ( ) = 0.8. lso we obtain ( ) = 0.008, ( ) = Theefoe, ( ) = ccoding to the OG method, we know that is bigge than as >. Howeve, should be smalle than intuitivel. Now, we do this method fo five fuzz numbes which is taken fom (Deng et al., 006)(See also in (Wang and Lee, 008): = (,,7;), = (,,7;0.8), = (,7,9,0;), = (6,7,9,0;0.6), = (7,8,9,0;0.). B adius of gation method, = = =.0687, = = =.66, = = =.988, = = =.6, = = = Theefoe, we have < < < <. ccoding to the aea between the adius and oiginal point, this method is wong. Intuitivel, = (7,8,9,0;0.) should be bigge than = (,,7;0.8), o = (6,7,9,0;0.6) mabebiggethan = (,,7; ). Now, we pesent a new method fo anking fuzz numbes as follows: () () > (B) >B. () () < (B) <B. 66

5 ust. J. Basic & ppl. Sci., (): 68-66, 00 So, b using ou method we can solve fist eample as following: = (,,;), = (9, 0, ; 0.) ( ) =.0, ( ) = 0.008, so, we have. Now we ae going to ode the fuzz numbes as given in the second eample b ou method as follows: = (,,7;), = (,,7;0.8), = (,7,9,0;), = (6,7,9,0;0.6), = (7,8, 9, 0; 0.). Ou method concludes that: ( ) =.06680, ( ) = , ( ) = , ( ) = , ( ) = Theefoe, < < < <. In anothe eample, we want to ode the following fuzz numbes (taken fom (Deng et al., 006)): = (0.,0.,;), = (0.,0.7,;), = (0.,0.9,;). In the following tableau, we give the esult of compaison of ou method with thee convenient methods fo odeing fuzz numbes. method Cheng Chu-Tsao OG ou method eample < < < < < < < < We saw ou method odeed the fuzz numbes, and as well as OG and Chu-Tsao methods. lso we saw the esult of compaison of the mentioned fuzz numbes b Cheng method is wong (see in Fig ). Fig. Now conside the following fuzz numbes: = (,,,6;0.6), = (,,,6;0.9), = (,9,,;0.), = (,9,,;0.7), = (6,8,9,;). The following tableau shows the esult of ou method and some convenient methods fo odeing the above fuzz numbes. 66

6 ust. J. Basic & ppl. Sci., (): 68-66, 00 method Cheng Chu-Tsao OG ou method eample < < < < < < < < < Fig. s we saw ou method obtained the coect esult as well as Cheng method and also we found some shotcomings in OG and Chu-Tsao methods. To sum up, ou method is easonable and effective fo anking fuzz numbes accoding to the above eamples. Conclusion: In this pape, we impoved OG method and pesented a new method fo odeing fuzz numbes. B using this method, we solved some shotcoming of OG method. Fo veifing efficienc of ou method, we gave also used some compaative eamples to illustate the advantage of ou method. CKNOWLEDGMENT The fist autho thanks to the eseach Cente of lgebaic Hpestuctues and Fuzz Mathematics fo its patl suppots. EFEENCES Botolan, G.,. Degani, 98. eview of some methods fo anking fuzz numbes, Fuzz Sets and Sstems, : -9. Chen, S.J., C.L. Hwang, 99. Fuzz Multiple ttibute Decision Making, Spinge, New Yok. Cheng, C.H.,998. new appoach fo anking fuzz numbes b distance method, Fuzz Sets and Sstems, 9: Choobineh, F., H. Li, 99. n inde fo odeing fuzz numbes, Fuzz Sets and Sstems, : Chu, T.C., C.T. Tsao, 00. anking fuzz numbes with an aea between the centoid point and the oiginal point, Computes and Mathematics with pplications, : -7. Deng, Y., Z. Zhenfu, L. Qi, 006. anking fuzz numbes with an aea method using adius of gation, Computes and Mathematics with pplications, : 7-6. Dias, G., 99. anking altenatives using fuzz numbes: computational appoach, Fuzz Sets and Sstems, 6: 7-. Dubois, D., H. Pade, 978. Opeations on fuzz numbes, Intenational Jounal of Sstems Sciences, 9: Fotemps, P., M. oubens, 996. anking and defuzzification methods based on aea compensation, Fuzz Sets and Sstems, 8(): 9-0. Lee, K.M., C.H. Cho, H. Lee-Kwang, 99. anking fuzz values with satisfaction function, Fuzz Sets and Sstems, 6: 9-. Lee, E.S.,.J. Li, 988. Compaison of fuzz numbes based on the pobabilit measue of fuzz events, Computes and Mathematics with pplications, :

7 ust. J. Basic & ppl. Sci., (): 68-66, 00 Jain,., 976. Decision-making in the pesence of fuzz vaiables, IEEE Tansactions on Sstems, Man and Cbenetics, 6: Jain,., 978. pocedue fo multi-aspect decision making using fuzz sets, Intenational Jounal of Sstems Sciences, 8: -7. aj,., D.N. Kuma, 999. anking altenatives with fuzz weights using maimizing set and minimizing set, Fuzz Sets and Sstems, 0: 6-7. equena, I., M. Delgado, J.I. Vedaga, 99. utomatic anking of fuzz numbes with the citeion of decision-make leantb an atificial neual netwok, Fuzz Sets and Sstems, 6: -9. Wang, Y.J., H.S. Lee, 008. The evised method of anking fuzz numbes with an aea between the centoid and oiginal points, Computes and Mathematics with pplications, :