Research Article On Intuitionistic Fuzzy Entropy of Order-α

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1 Advaces i Fuzzy Systems, Article ID , 8 pages Research Article O Ituitioistic Fuzzy Etropy of Order-α Rajkumar Verma ad BhuDev Sharma Departmet of Applied Scieces, HMR Istitute of Techology ad Maagemet, Guru Gobid Sigh Idraprastha Uiversity, Hamidpur,Delhi-0036,Idia Departmet of Mathematics, Jaypee Istitute of Ifmatio Techology (Deemed Uiversity), A-0, Sect 6, Noida-0307, Uttar Pradesh, Idia Crespodece should be addressed to Rajkumar Verma; rkver83@gmail.com Received 9 Jauary 04; Revised 5 July 04; Accepted 7 July 04; Published 3 September 04 Academic Edit: Nig Xiog Copyright 04 R. Verma ad B. Sharma. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the igial wk is properly cited. Usig the idea of Rèyi s etropy, ituitioistic fuzzy etropy of der-α is proposed i the settig of ituitioistic fuzzy sets they. This measure is a geeralized versio of fuzzy etropy of der-α proposed by Bhadari ad Pal ad ituitioistic fuzzy etropy defied by Vlachos ad Sergiadis. Our study of the four essetial ad some other properties of the proposed measure clearly establishes the validity of the measure as ituitioistic fuzzy etropy. Fially, a umerical example is give to show that the proposed etropy measure f ituitioistic fuzzy set is reasoable by comparig it with other existig etropies.. Itroductio I 965, Zadeh [] proposed the otio of fuzzy set (FS) to model ostatistical imprecise vague pheomea. Sice the, the they of fuzzy set has become a vigous area of research i differet disciplies such as egieerig, artificial itelligece, medical sciece, sigal processig, ad expert systems. Fuzziess, a feature of ucertaity, results from the lack of sharp distictio of beig ot beig a member of a set; that is, the boudaries of the set uder cosideratio are ot sharply defied. A measure of fuzziess used ad cited i the literature is fuzzy etropy, also first metioed i 968 by Zadeh []. I 97, De Luca ad Termii [3] first provided axiomatic structure f the etropy of fuzzy sets ad defied a etropy measure of a fuzzy set based o Shao s etropy fuctio [4]. Kaufma [5] itroduced a fuzzy etropy measure by a metric distace betwee the fuzzy set ad that of its earest crisp set. I additio, Yager [6] defiedaetropymeasureofafuzzysetitermsof a lack of distictio betwee fuzzy set ad its complemet. I 989, N. R. Pal ad S. K. Pal[7] proposed fuzzy etropy based o expoetial fuctio to measure the fuzziess called expoetial fuzzy etropy. Further, Bhadari ad Pal [8] proposed fuzzy etropy of der-α crespodig to Rèyi etropy [9]. Recetly, Verma ad Sharma [0] have itroduced a parametric geeralized etropy measure f fuzzy sets called expoetial fuzzy etropy of der-α. Ataassov [, ] itroduced the cocept of ituitioistic fuzzy set (IFS), which is a geeralizatio of the otio of fuzzy set. The distiguishig characteristic of ituitioistic fuzzy set is that it assigs to each elemet a membership degree, a omembership degree, ad the hesitatio degree. Firstly, Burillo ad Bustice [3] defied the etropy o ituitioistic fuzzy sets ad o iterval-valued fuzzy sets. Szmidt ad Kacprzyk [4] usedadifferetapproachfrom that of Burillo ad Bustice to itroduce the etropy measure f IFS based o the geometric iterpretatio of IFS ad the distaces betwee them. Zeg ad Li [5] cosidered the axioms of Szmidt ad Kacprzyk usig the otio of IVFSs addiscussedtherelatioshipbetweesimilaritymeasure ad etropy. Hug ad Yag [6] gave their axiomatic defiitio of etropy f IFSs by exploitig the cocept of probability. Vlachos ad Sergiadis [7] proposed a measure of ituitioistic fuzzy etropy ad revealed a ituitive ad mathematical coectio betwee the otios of etropy f fuzzy set ad ituitioistic fuzzy set. Zhag ad Jiag

2 Advaces i Fuzzy Systems [8] proposed ituitioistic (vague) fuzzy etropy by meas of itersectio ad uio of the membership degree ad omembership degree of the ituitioistic (vague) fuzzy set. Havig come to rather rigid measures, it is atural to see other possible measures ad those havig parametric geeralizatio that covers existig measures ad leads to may me. The parameters ivolved give flexibility i applicatios ad their values ca be determied from the data itself experimet. I this paper, a ew etropy measure f IFSs is defied.thisiscalled ituitioistic fuzzy etropy of der-α. The paper is gaized as follows. I Sectio, some basic defiitios related to fuzzy set they ad ituitioistic fuzzy set they are briefly give. I Sectio 3,aewetropy measure called ituitioistic fuzzy etropy of der-α is proposed, ad its axiomatic justificatio is established. Some mathematical properties of the proposed measure are also studiedithissectio.isectio 4, a umerical example is give to demostrate the effectiveess of the proposed measure of ituitioistic fuzzy etropy with existig ituitioistic fuzzy etropies [3 8] by the compariso. Brief coclusios are preseted i Sectio 5.. Prelimiaries I the followig, some eeded basic cocepts ad defiitios related to fuzzy sets ad ituitioistic fuzzy sets are itroduced. Defiitio (fuzzy set []). A fuzzy set A i a fiite uiverse of discourse X{x,x,...,x } is give by A { x, μ A (x) x X}, () where μ A : X [0, ] is the membership fuctio of A. The umber μ A (x) describes the degree of belogigess of x Xi A. De Luca ad Termii [3] defied fuzzy etropy f a fuzzy set A crespodig to Shao s etropy [4]as H( A) [μ A (x i) log (μ A (x i)) +( μ A (x i)) log ( μ A (x i))]. Later, Bhadari ad Pal [8] made a survey of ifmatio measures o fuzzy sets ad proposed some ew measures of fuzzy etropy. Crespodig to Rèyi etropy [9], they defied the followig measure: H α ( A) ( α) log [μ α A (x i)+( μ A (x i)) α ]; α, α>0. Ataassov [, ], as metioed earlier, geeralized Zadeh s idea of fuzzy sets, by what is called ituitioistic fuzzy sets, defied as follows. () (3) Defiitio (ituitioistic fuzzy set []). A ituitioistic fuzzy set A i a fiite uiverse of discourse X{x,x,..., x } is give by where with the coditio A { x, μ A (x), ] A (x) x X}, (4) μ A :X [0, ], ] A :X [0, ] (5) 0 μ A (x) + ] A (x), x X. (6) I this defiitio umbers μ A (x) ad ] A (x), respectively, deote the degree of membership ad degree of omembership of x Xto the set A. F each IFS A i X,ifπ A (x) μ A (x) ] A (x), x X, the π A (x) represets the degree of hesitace of x Xto the set A. π A (x) is also called ituitioistic idex. Obviously, whe π A (x) 0,thatis,] A (x) μ A (x) f every x i X, theifsseta becomes fuzzy set. Thus, fuzzy sets are the special cases of IFSs. Defiitio 3 (set operatios o ituitioistic fuzzy sets []). Let IFS(X) deote the family of all IFSs i the uiverse X,ad let A, B IFS(X) give by A { x, μ A (x), ] A (x) x X}, B { x, μ B (x), ] B (x) x X}. The usual set relatios ad operatios are defied as follows: (i) A Bif ad oly if μ A (x) μ B (x) ad ] A (x) ] B (x) f all x X; (ii) ABif ad oly if A Bad B A; (iii) A C { x, ] A (x), μ A (x) x X}; (iv) A B { μ A (x) μ B (x) ad ] A (x) ] B (x) x X}; (v) A B { μ A (x) μ B (x) ad ] A (x) ] B (x) x X}. Szmidt ad Kacprzyk [4] exteded the axioms of De Luca ad Termii [3] f proposig the etropy measure i the settig of IFSs. Defiitio 4 (see [4]). A etropy o IFS(X) is a real-valued fuctioale : IFS(X) [0, ], satisfyig the followig four axioms. (P) E(A) 0 if ad oly if A is a crisp set; that is, μ A (x i ) 0, ] A (x i ) μ A (x i ),] A (x i )0f all x i X. (P) E(A) if ad oly if μ A (x i )] A (x i ) f all x i X. (P3) E(A) E(B) if ad oly if A B,thatis,ifμ A (x i ) μ B (x i ) ad ] A (x i ) ] B (x i ),fμ B (x i ) ] B (x i ), if μ A (x i ) μ B (x i ) ad ] A (x i ) ] B (x i ),fμ B (x i ) ] B (x i ) f ay x i X. (P4) E(A) E(A C ). (7)

3 Advaces i Fuzzy Systems 3 Crespodig to (),thedelucaadtermii[3] etropy, Vlachos ad Sergiadis[7] itroduced a measure of ituitioistic fuzzy etropy as E VS (A) [μ A (x i ) log μ A (x i )+] A (x i ) log ] A (x i ) ( π A (x i )) log ( π A (x i )) π A (x i )]. Throughout this paper, we will deote the set of all ituitioistic fuzzy sets i X by IFS(X) ad by FS(X) the set of all fuzzy sets defied i X. With these backgroud ideas ad cocepts, we, i the ext sectio, itroduce a ew measure called ituitioistic fuzzy etropy of der-α f ituitioistic fuzzy sets, which has a parameter. 3. Ituitioistic Fuzzy Etropy of Order-α Defiitio 5. The ituitioistic fuzzy etropy of der-α f ituitioistic fuzzy set A is defied as E α (A) ( α) log [(μ α A (x i)+] α A (x i)) (μ A (x i )+] A (x i )) α (8) + α π A (x i )], α > 0, α. (9) (μ α A (x i)+] α A (x i)) (μ A (x i )+] A (x i )) α + α π A (x i ) x i X (μ α A (x i)+] α A (x i)) (μ A (x i )+] A (x i )) α α (μ A (x i )+] A (x i )) α (μ A (x i )+] A (x i )) x i X [ (μα A (x i)+] α A (x i)) (μ A (x i )+] A (x i )) α α ] α, x i X. () Sice α>0, α, therefe ()willholdolyifμ A (x i )0, ] A (x i ),μ A (x i ),] A (x i )0,fallx i X. Hece, E α (A) 0 if ad oly if μ A (x i )0,] A (x i ), μ A (x i ),] A (x i )0,fallx i X.Thisproves(P). (P) First let μ A (x i )] A (x i ) f all x i X.Thefrom (9)wehave E α (A) ( α) ( α) log [(μ α A (x i)+μ α A (x i)) (μ A (x i )+μ A (x i )) α + α ( μ A (x i )) ] log [ α μ A (x i )+ α α μ A (x i )] ( α) ( α). (3) Theem 6. The E α (A) measure i (9) of the ituitioistic fuzzy etropy of der-α is a etropy measure f IFSs; that is, it satisfies all the axioms give i Defiitio 4 above. Proof. (P)LetA be a crisp set with membership values beig either 0 f all x i X.Thefrom(9)weobtaithat Next, if E α (A) 0,thatis, E α (A) 0. (0) Next, let E α (A) ;thatis, log [(μ α A (x i)+] α A (x i)) (μ A (x i )+] A (x i )) α + α π A (x i )]( α) log [(μ α A (x i)+] α A (x i)) (μ A (x i )+] A (x i )) α + α π A (x i )]( α) x i X (4) (5) ( α) log [(μ α A (x i)+] α A (x i)) (μ A (x i )+] A (x i )) α + α π A (x i )]0 () [(μ α A (x i)+] α A (x i)) (μ A (x i )+] A (x i )) α + α ( μ A (x i ) ] A (x i )) ] α x i X (6)

4 4 Advaces i Fuzzy Systems (μ A (x i )+] A (x i )) [ μα A (x i)+] α A (x i) Equatio (7)willholdif μ A (x i )+] A (x i )0 μ A (x i ) [ μα A (x i)+] α A (x i) Now cosider the followig fuctio: ( μ α A (x i )+] A (x i ) ) ]0. ] A (x i )0 x i X x i X. (7) (8) ( μ α A (x i )+] A (x i ) ) ]0. (9) f (z) z α where z [0, ]. (0) Differetiatig (0)withrespecttoz,weget f (z) αz α, f (x) α(α ) z α. () Sice f (z) > 0 whe α>ad f (z) < 0 whe α<, f(z) is covex ( ) cocave ( ) fuctio accdig to α> α<. Therefe, f ay two poits z ad z i [0, ], the followig iequalities hold: f(z )+f(z ) f(z )+f(z ) f( z +z ) 0 f α>, f( z +z ) 0 f α< () with the equality holdig i ()olyfz z. Therefe, from (8), (9), ad (), we coclude that (7) holdsolyif μ A (x i )] A (x i ) f all x i X. (P3) I der to show that (9) satisfies (P3), it suffices to prove that the fuctio g(x,y) ( α) log [(xα +y α )(x+y) α + α ( x y)], (3) where x, y [0, ], is a icreasig fuctio with respect to x ad decreasig with respect to y. Takigthepartial derivatives of g with respect to x ad y, respectively, yields x ( α)(x + y) α (x α +y α )+α(x+y) α x α α ( α)(x + y) α, (x α +y α )+ α ( x y) (4) y ( α)(x + y) α (x α +y α )+α(x+y) α y α α ( α)(x + y) α. (x α +y α )+ α ( x y) (5) F critical poit of g, we set g(x, y)/ x 0 ad g(x, y)/ y 0.Thisgives xy. (6) Also, from (4)ad(6), we have 0 whe x y, 0<α<as also f α>, x 0 whe x y, 0<α<as also f α>, x (7) f ay x, y [0, ].Thus g(x, y) is icreasig with respect to x f x yad decreasig whe x y. Similarly, we see that y y 0, whe x y, 0<α<as also f α>, 0, whe x y, 0<α<as also f α>. (8) Let us ow cosider two sets A, B IFS(X), witha B. Assume that the fiite uiverse of discourse X {x,x,...,x } is partitioed ito two disjoit sets X ad X with X X X. Let us further suppose that all x i X are domiated by the coditio μ A (x i ) μ B (x i ) ] B (x i ) ] A (x i ), (9) while f all x i X satisfy μ A (x i ) μ B (x i ) ] B (x i ) ] A (x i ). (30) The from the mootoicity of the fuctio g ad (9), we obtai that E α (A) E α (B) whe A B. (P4) ItisclearthatA C { x,] A (x), μ A (x) x X} f x i X;thatis μ A C (x i ) ] A (x i ), ] A C (x i ) μ A (x i ). (3)

5 Advaces i Fuzzy Systems 5 The, from (9), we straightfwardly have E α (A) E α (A C ). (3) Hece, E α (A) is a measure of ituitioistic fuzzy etropy. This proves the theem. The proposed ituitioistic fuzzy etropy of der-α satisfies the followig additioal properties. Theem 7. Let A ad B be two ituitioistic fuzzy sets defied i X{x,x,...,x },wherea{ x i,μ A (x i ), ] A (x i ) x i X}, B{ x i,μ B (x i ), ] B (x i ) x i X}, such that they satisfy f ay x i Xeither A B A B; the oe has E α (A B) +E α (A B) E α (A) +E α (B). (33) Proof. Let us separate X ito two parts X ad X,where X {x i X:A B}, X {x i X:A B}. (34) That is, f all x i X, ad, f all x i X, μ A (x i ) μ B (x i ), ] A (x i ) ] B (x i ) (35) μ A (x i ) μ B (x i ), ] A (x i ) ] B (x i ). (36) From (9), we the have E α (A B) ( α) log [(μ α A B (x i)+] α A B (x i)) (μ A B (x i )+] A B (x i )) α + α ( μ A B (x i ) ] A B (x i )) ] ( α) [{ x X log [(μ α B (x i)+] α B (x i)) (μ B (x i )+] B (x i )) α + α ( μ B (x i ) ] B (x i )) ] } +{ x X log [(μ α A (x i)+] α A (x i)) (μ A (x i )+] A (x i )) α + α ( μ A (x i ) ] A (x i )) ] }], E α (A B) ( α) log [(μ α A B (x i)+] α A B (x i)) (μ A B (x i )+] A B (x i )) α + α ( μ A B (x i ) ] A B (x i )) ] ( α) [{ x X log [(μ α A (x i)+] α A (x i)) Now from (37), we get (μ A (x i )+] A (x i )) α + α ( μ A (x i ) ] A (x i )) ] } +{ x X log [(μ α B (x i)+] α B (x i)) (μ B (x i )+] B (x i )) α + α ( μ B (x i ) ] B (x i )) ] }]. (37) E α (A B) +E α (A B) E α (A) +E α (B). (38) This proves the theem. Collary 8. F ay A IFS(X) ad A C the complemet of ituitioistic fuzzy set A, E α (A) E α (A C )E α (A A C )E α (A A C ). (39) Theem 9. E α (A) attais the maximum value whe the set is most ituitioistic fuzzy set ad the miimum value whe the set is crisp set; meover, maximum ad miimum values are idepedet of α. Proof. It has already bee proved that E α (A) is maximum if ad oly if A is most ituitioistic fuzzy set, that is, μ A (x i ) ] A (x i ),fallx i X,admiimumwheA is a crisp set. So, it is eough to prove that the maximum ad miimum values are idepedet of α. Let A be the most ituitioistic fuzzy set; that is, μ A (x i ) ] A (x i ),fallx i X.The E α (A) ( α) log [(] α A (x i)+] α A (x i)) (] A (x i )+] A (x i )) α + α ( ] A (x i ) ] A (x i )) ]

6 6 Advaces i Fuzzy Systems ( α) log [ α ], (40) which is idepedet of α. O the other had, if A is a crisp set, that is, μ A (x i ) ad ] A (x i )0 μ A (x i )0ad ] A (x i ),fallx i X, the E α (A) 0 f ay value of α. This proves the theem. Particular ad Limitig Cases () Whe α,themeasurei(9) reduces to measures i (3). () Itmaybeoticedthatifaituitioisticfuzzysetisa diary fuzzy set, that is, f all x i X, ] A (x i ) μ A (x i ), the the ituitioistic fuzzy etropy of der- α reduces to fuzzy etropy of der-α [3]. I the ext sectio, we cosider a example to demostrate the perfmace of proposed ituitioistic fuzzy etropy of der-α by comparig with other existig measures of ituitioistic fuzzy etropy. 4. Numerical Example Let A {(x i,μ A (x i ), ] A (x i )) x i X} be a IFS i X {x,x,...,x }.Faypositiverealumber,Deetal.[9] defied the IFS A as follows: A {(x i,(μ A (x i )), ( ] A (x i )) ) x i X}. (4) Usig the above operatio, they also defied the cocetratio ad dilatio of A give by Cocetratio: CON (A) A, Dilatio: DIL (A) A /. (4) Like fuzzy sets, CON(A) ad DIL(A) may be treated as very (A) ad meless(a), respectively. Example. Let us cosider a IFS A i X {6,7,8,9,0}, defied by De et al. [9], as A{(6, 0., 0.8), (7, 0.3, 0.5), (8, 0.5, 0.4), (9, 0.9, 0.0), (0,.0, 0.0)}. (43) Usig the operatio defied i (4), we ca geerate the followig IFSs: A /, A, A 3, A 4. (44) By takig ito accout the characterizatio of liguistic variables, we regarded A as LARGE i X. Usig the above operatio, we have A / {(6, 0.36, 0.558), (7, , 0.99), (8, 0.707, 0.54), (9, , 0.0), (0,.0, 0.0)}, A {(6, 0.000, ), (7, , ), (8, 0.500, ), (9, 0.800, 0.000), (0,.0, 0.0)}, A 3 {(6, 0.000, 0.990), (7, 0.070, ), (8, 0.50, ), (9, 0.790, 0.000), (0,.0, 0.0)}, A 4 {(6, 0.000, ), (7, 0.008, ), (8, 0.065, ), (9, 0.659, 0.000), (0,.0, 0.0)}. (45) The levels represeted by the above ituitioistic fuzzy sets are described as follows: A / may be treated as Me less LARGE, A maybetreatedas VeryLARGE, A 3 may be treated as Quite very LARGE, A 4 may be treated as Very very LARGE. Now we use these IFSs to compare the ituitioistic fuzzy etropy of der-α ad other existig measures of ituitioistic fuzzy etropy. F the purpose of compariso, we first metio here some etropy measures f ituitioistic fuzzy sets defied by various researchers as Burillo ad Bustice s etropy [3]: E BB (A) Zeg ad Li s etropy [5]: π A (x i ), (46) E ZL (A) μ A (x i ) ] A (x i ), (47) Szmidt ad Kacprzyk s etropy [4]: E SK (A) Zhag ad Jiag s etropy [8]: E ZJ (A) ( μ A (x i ) ] A (x i )+π A (x i ) ), (48) μ A (x i ) ] A (x i )+π A (x i ) ( μ A (x i ) ] A (x i ) ), (49) μ A (x i ) ] A (x i )

7 Advaces i Fuzzy Systems 7 Table : Values of the differet etropy measures uder A /,A, A,A 3,A 4. A / A A A 3 A 4 E BB E ZL E SK E VS E ZJ E hc E / r E E E E E E E E ad Hug ad Yag etropies [6]: E hc (A) E / r (A) ( (μ A (x i)+] A (x i)+π A (x i))), log (μ / A (x i)+] / A (x i)+π / A (x i)). (50) From the viewpoit of mathematical operatios, the etropy values of the above defied IFSs, A /, A, A, A 3, A 4,havethe followig requiremet: E(A / )>E(A) >E(A )>E(A 3 )>E(A 4 ). (5) The values of differet etropy measures uder A /, A, A, A 3,adA 4 are show i Table. Based o Table, we see that the etropy measures E ZL, E SK, E VS, E 0.3, E 0.8, E, E 5, E 0, E 5,adE 00 satisfy the requiremet give i (5), but E BB, E ZJ, E hc,ade/ r do ot satisfy the requiremet. Therefe, the behavi of ituitioistic fuzzy etropy of der-α, E α (A), isgoodftheviewpoitofstructured liguistic variables. 5. Coclusios This wk itroduces a ew etropy measure called ituitioistic fuzzy etropy of der-α i the settig of ituitioistic fuzzy set they. Some properties of this measure have bee also studied. This measure geeralizes Bhadari ad Pal [8] fuzzy etropy of der-α ad Vlachos ad Sergiadis [7] logarithmic ituitioistic fuzzy etropy. Itroductio of parameter provides ew flexibility ad wider applicatio of ituitioistic fuzzy etropy to differet situatios. Coflict of Iterests The auths declare that there is o coflict of iterests regardig the publicatio of this paper. Refereces [] L. A. Zadeh, Fuzzy sets, Ifmatio ad Computatio, vol. 8, pp , 965. [] L. A. Zadeh, Probability measures of fuzzy evets, Joural of Mathematical Aalysis ad Applicatios,vol.3,o.,pp.4 47, 968. [3] A. De Luca ad S. Termii, A defiitio of a oprobabilistic etropy i the settig of fuzzy sets they, Ifmatio ad Computatio,vol.0,o.4,pp.30 3,97. [4] C. E. Shao, A mathematical they of commuicatio, The Bell System Techical Joural,vol.7,pp ,948. [5] A. Kaufma, Itroductio to the They of Fuzzy Subsets, Academic Press, New Yk, NY, USA, 975. [6] R. R. Yager, O the measure of fuzziess ad egatio part I: membership i the uit iterval, Iteratioal Joural of Geeral Systems,vol.5,o.4,pp. 9,979. [7] N.R.PaladS.K.Pal, Object-backgroudsegmetatiousig ew defiitios of etropy, IEE Proceedigs E: Computers ad Digital Techiques,vol.36,o.4,pp.84 95,989. [8] D. Bhadari ad N. R. Pal, Some ew ifmatio measures f fuzzy sets, Ifmatio Scieces, vol. 67, o. 3, pp. 09 8, 993. [9] A. Réyi, O measures of etropy ad ifmatio, i Proceedigs of the 4th Berkeley Symposium o Mathematical Statistics ad Probability, pp ,UiversityofCalifia Press, Berkeley, Calif, USA, 96. [0] R. Verma ad B. D. Sharma, O geeralized expoetial fuzzy etropy, Egieerig ad Techology, vol. 5, pp , 0. [] K. T. Ataassov, Ituitioistic fuzzy sets, Fuzzy Sets ad Systems,vol.0,o.,pp.87 96,986. [] K. T. Ataassov, Ituitioistic Fuzzy Sets, Physica, Heidelberg, Germay, 999. [3] P. Burillo ad H. Bustice, Etropy o ituitioistic fuzzy sets ad o iterval-valued fuzzy sets, Fuzzy Sets ad Systems,vol. 78,o.3,pp ,996. [4] E. Szmidt ad J. Kacprzyk, Etropy f ituitioistic fuzzy sets, Fuzzy Sets ad Systems,vol.8,o.3,pp ,00. [5] W. Zeg ad H. Li, Relatioship betwee similarity measure ad etropy of iterval valued fuzzy sets, Fuzzy Sets ad Systems,vol.57,o.,pp ,006. [6] W. L. Hug ad M. S. Yag, Fuzzy etropy o ituitioistic fuzzy sets, Iteratioal Joural of Itelliget Systems, vol., o. 4, pp , 006. [7] I. K. Vlachos ad G. D. Sergiadis, Ituitioistic fuzzy ifmatio applicatios to patter recogitio, Patter Recogitio Letters, vol. 8, o., pp , 007.

8 8 Advaces i Fuzzy Systems [8] Q. Zhag ad S. Jiag, A ote o ifmatio etropy measures f vague sets ad its applicatios, Ifmatio Scieces, vol. 78, o., pp , 008. [9]S.K.De,R.Biswas,adA.R.Roy, Someoperatioso ituitioistic fuzzy sets, Fuzzy Sets ad Systems,vol.4,o.3, pp ,000.

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