Entropy and Similarity of Intuitionistic Fuzzy Sets

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1 Entropy and Similarity of Intuitionistic Fuzzy Sets Eulalia Szmidt and Janusz Kacprzyk Systems Research Institute Polish Academy of Sciences ul. Newelska 6, Warsaw, Poland Abstract This paper is in a sense a continuation of our previous papers on entropy and similarity measures of the intuitionistic fuzzy sets. We show a sort of connections between the measures on the one hand, and point out that in general there is not a simple way of drawing any conclusions about one measure knowing theother. Wealsorevisesomeconditions for the entropy measures of intuitionistic fuzzy sets. Keywords: Intuitionistic fuzzy sets, entropy, similarity 1 Introduction Fuzziness, a feature of imperfect information, resultsfromthelackofacrispdistinctionbetween the elements belonging and not belongingtoaset (i.e. the boundaries of aset under consideration are not sharply defined). A measure of fuzziness often used and cited in the literature is called an entropy(first mentioned by Zadeh[24]). De Luca and Termini[5] introduced some requirements which capture our intuitive comprehension of a degree of fuzziness. Kaufmann(1975)(cf. [12]) proposed to measure a degreeoffuzzinessofafuzzysetabyametric distance between its membership function and the membership function(charecteristic function) of its nearest crisp set. Another way given by Yager [23] was to view a degree of fuzzinessintermsofalackofdistinctionbetween the fuzzy set and its complement. Higashi and Klir [4] extended Yager s concept to a very general class of fuzzy complements. Yager s approach was also further developed byhuandyu[9]. Indeed,itisthelackofdistinction between sets and their complements that distinguishes fuzzy sets from crisp sets. Thelessthefuzzysetdiffersfromitscomplement, the fuzzier it is. Kosko[11] investigated thefuzzyentropyinrelationtoameasureof subsethood. Fan at al.[6],[7],[8] generalized Kosko s approach. Here we discuss measures of fuzziness for intuitionistic fuzzy sets viewed as a generalization of fuzzy sets. Starting from a geometrical interpretation of intuitionistic fuzzy sets, the meaning of entropy, and the distances for intuitionistic fuzzy sets, we revised the axioms for the entropy and proposed two different formulas for its calculations. Both formulas compare the distances to the nearer and to the farer crisp elements. The first formula (19) compares the distances by subtracting them, the second formula (28) compares the distances by their ratio. We discuss details of the formulas using two types of distances the normalized Hamming distance, and the normalized Euclidean distance. We also present the counterpart similarity measures (25) and (29) for both measures of entropy. The measures considered answer the question if the compared elements/objects are more similar or more dissimilar from each other. The proposed measures take into accounttwokindsofdistances onetoanob-

2 ject to be compared, and one to its complement. We infer about the similarity in two (independent) ways on the basis of a ratio of the two types of distances (the first proposedmeasure)oronthebasisofadifference between the two types of distances (the second proposed measure). One can easily notice the sort of connection between the considered measures of similarity and entropy which does not mean that, in general, knowing one measure one can conclude about the other. For different approaches we refer the reader to Burillo and Bustince[3]. 2 A brief introduction to intuitionistic fuzzy sets AsopposedtoafuzzysetinX (Zadeh[24]), given by A ={<x,µ A (x)> x X} (1) whereµ A (x) [0,1]isthemembershipfunctionofthefuzzysetA,aso-calledintuitionistic fuzzy set (Atanassov [1], [2]) A is given by A={<x,µ A (x),ν A (x)> x X} (2) where: µ A : X [0,1] and ν A : X [0,1] such that 0<µ A (x)+ν A (x)<1 (3) and µ A (x), ν A (x) [0,1] denote a degree of membership and a degree of non-membership ofx A,respectively. Obviously, each fuzzy set may be represented by the following intuitionistic fuzzy set A={<x,µ A (x),1 µ A (x)> x X} (4) ForeachintuitionisticfuzzysetinX,wewill call π A (x)=1 µ A (x) ν A (x) (5) an intuitionistic fuzzy index (or a hesitation margin) of x A, and it expresses a lack of knowledge of whether x belongs to A or not (cf. Atanassov [2]). It is obvious that 0<π A (x)<1,foreachx X. In our further considerations we will use the complementseta C [2] A C ={<x,ν A (x),µ A (x)> x X} (6) The application of intuitionistic fuzzy sets instead of fuzzy sets means the introduction of another degree of freedom into a set description. Such a generalization of fuzzy sets gives us an additional possibility to represent imperfect knowledge what leads to describing manyrealproblemsinamoreadequateway. We refer the interested reader to Szmidt[14], Szmidt and Kacprzyk[21] where the applications of intiutionistic fuzzy sets to group decision making, negotiations and other situations are presented. 3 Geometrical interpretation Having in mind that for each element x belonging to an intuitionistic fuzzy set A, the values of membership, non-membership and the intuitionistic fuzzy index add up to one, i.e. µ A (x)+ν A (x)+π A (x)=1 and that each: membership, nonmembership, and the intuitionistic fuzzy index are from the interval [0,1], we can imagine a unit cube (Figure 1) inside which there is an MNH triangle where the above equations are fulfilled. In other words, the MNH triangle represents a surface where coordinates of any element belonging to an intuitionistic fuzzy set can be represented. Each point belonging to the MNH triangle is described via three coordinates: (µ, ν, π). Points M and N represent crisp elements. Point M(1,0,0) represents elements fully belonging to an intuitionistic fuzzy set as µ = 1. Point N(0,1,0) represents elements fully not belonging to an intuitionistic fuzzy set as ν = 1. Point H(0,0,1) represents elements about which we are not able to say if they belong or not belong to an intuitionistic fuzzy set (intuitionistic fuzzy index π = 1). Such an interpretation is intuitively appealing and provides means for the representation of many aspects of imperfect information. Segment M N (where

3 π H( 0,0,1) x X Baldwin [15, 16], Szmidt and Kacprzyk [17], [22]: the normalized Hamming distance N( 0,1,0) ν x ' O ( 0,0,0) M( 1,0,0) µ l IFS (A,B)= = 1 ( µ A (x i ) µ B (x i ) + ν A (x i ) 2n + ν B (x i ) + π A (x i ) π B (x i ) ) (7) and the normalized Euclidean distance: Figure 1: Geometrical representation π = 0) represents elements belonging to classical fuzzy sets (µ+ν = 1). Any other combination of the values characterizing an intuitionistic fuzzy set can be represented inside the triangle MNH. In other words, each element belonging to an intuitionistic fuzzy set can be represented as a point (µ,ν,π) belonging to the triangle MNH (Figure2). It is worth mentioning that the geometrical interpretation is directly related to the definition of an intuitionistic fuzzy set introduced byatanassov[1],[2],anditdoesnotneedany additional assumptions. By employing the above geometrical representation, we can calculate distances between any two intuitionisticfuzzysetsaandbcontainingnelements. 3.1 Distances between intuitionistic fuzzy sets In Szmidt[14], Szmidt and Baldwin[15, 16], Szmidt and Kacprzyk [17], and especially in Szmidt and Kacprzyk [22] it is shown why when calculating distances between intuitionisticfuzzysetsweshouldtakeintoaccountall three functions describing intuitionistic fuzzy sets. In [22] not only the reasons why we should take into account all three functions are given but also some possible serious problems that can occur while taking into account two functions only. In our further considerations we will use the following distances between fuzzy sets A, B inx={x 1,,...,x n }Szmidt[14],Szmidtand q IFS (A,B)= = ( 1 (µ A (x i ) µ B (x i )) 2 + 2n + (ν A (x i ) ν B (x i )) (π A (x i ) π B (x i )) 2 ) 1 2 (8) For (7), (8) we have: 0<l IFS (A,B)<1, and 0<q IFS (A,B)<1. Clearlythesedistancessatisfy the conditions of the metric. 4 Entropy De Luca and Termini [5] first axiomatized a non-probabilistic entropy. The De Luca and Termini axioms formulated for fuzzy sets are intuitive and have been widely employed in the fuzzy literature. They were formulated in the following way. Let E be a set-to-point mapping E : F(2 x ) [0,1]. Hence E is a fuzzy set defined on fuzzy sets. E is an entropymeasureifitsatisfiesthefourdeluca and Termini axioms: E(A)=0 iff A 2 x (Anon-fuzzy) (9) E(A)=1 iff µ A (x i )=0.5 forall i (10) E(A)<E(B) ifa islessfuzzythanb (11) i.e., µ A (x)<µ B (x) when µ B (x)<0.5 µ A (x) µ B (x) when µ B (x) 0.5 E(A)=E(A c ) (12) Since the De Luca and Termini axioms (9) (12) were formulated for fuzzy sets(given by their membership functions, and depicted by the segment MN in Figures 1 and 2, they

4 D ( 0, 0, 1 ) H(0, 0,1) F M (1, 0,0) G N(0, 1,0) Figure2: ThetriangleMNH (cf. Fig. 1)explaining a distance-based measure of entropy may be expressed for the intuitionistic fuzzy sets as follows: E(A)=0 iff A 2 x (Anon-fuzzy) (13) E(A)=1iffµ A (x i )=ν A (x i )foralli (14) E(A)<E(B) ifa islessfuzzythanb (15) i.e., for each i-th element x i,. from A and B, for which µ A (x i ) ν A (x i ) and µ B (x i ) ν B (x i )thereholds: min[d(m,x i,a ),d(n,x i,a )]< <min[d(m,x i,b ),d(n,x i,b )] where d(.,.) means a distance between points M orn representingcrispelements,andthe elements from intuitionistic fuzzy sets A or B. Finally, the counterpart of (12) remains the same E(A)=E(A c ) (16) In our previous work[18], condition(15) was expressed as µ A (x)<µ B (x) and ν A (x) ν B (x) for µ B (x)<ν B (x) (17) µ A (x) µ B (x) and ν A (x)<ν B (x) for µ B (x) ν B (x) (18) Butitcanbenoticedthat(17)and(18) are not sufficient conditions to assess the entropy for any possible intuitionistic fuzzy elements x i with parameters (µ i,ν i,π i ). For example for two elements x 1 : (0.12,0.4,0.48) and x 2 : (0.1,0.3,0.6)conditions(17)and(18)donot make it possible to conclude about entropies of both elements. Condition(15) does not suffer from such a drawback, and is consistent with the geometrical representation of intuitionistic fuzzy sets, and the very essence of entropy. On the other hand, (15) is weaker than(17) and(18). The fuzziness of a fuzzy set, or its entropy, answers the question: how fuzzy is a fuzzy set? The same question may clearly be posed inthecaseofanintuitionisticfuzzyset. Having in mind our geometric interpretation of intuitionistic fuzzy sets(figure 1), let us concentrateonthetrianglemnh cf. Figure2. A non-fuzzy set (a crisp set) corresponds to point M (representing elements from an intuitionistic fuzzy set fully belonging to it as (µ M,ν M,π M )=(1,0,0))andpointN (representing the elements which fully do not belong to the set as (µ N,ν N,π N )=(0,1,0). Points M and N representing a crisp set have the degree of fuzziness equal 0. AfuzzysetcorrespondstothesegmentMN. WhenwemovefrompointMtowardspointN alongthesegmentmn,wegothroughpoints for which the membership function decreases (from 1 at point M to 0 at point N), the non-membership function increases(from 0 at pointmto1atpointg). ForthemidpointG (Figure 2) the values of both the membership and non-membership functions are equal 0.5. So, the midpoint G represents the elements withthedegreeoffuzzinessequal100%(wedo not know if elements represented by point G belongoriftheydonotbelongtoourset). On thesegmentmgthedegreeoffuzzinessgrows (from0%atmto100%atg). ThesamesituationoccursonthesegmentNG. Thedegree offuzzinessisequal0%atn,growstowards G(hereisequal100%). An intuitionistic fuzzy set is represented by thetrianglemnhanditsinterior. Allpoints which are above the segment MN represent the elements with an intuitionistic fuzzy index greater than 0. The most undefinded is point H. As the intuitionistic fuzzy index for Hisequal1,wecannottellifelementsrepresentedbythispointbelongordonotbelong totheset. ThedistancefromHtoM(fullbelonging) is equal to the distance to N (fully non-belonging). So, the degree of fuzziness

5 forh isequal100%. Butthesamesituation occurs for all elements represented by points x i on the segment HG. For HG we have µ HG (x i )=ν HG (x i ),π HG (x i ) 0(theequalityonlyforpointG),andcertainlyµ HG (x i )+ ν HG (x i )+π HG (x i )=1. For everyx i HG wehave: distance(m,x i )=distance(n,x i ). This geometric representation of an intuitionistic fuzzy set motivates a distance-based measure of fuzziness, i.e. an entropy of fuzzy sets. FirstwewilldefineentropyE(F)fora separate element(represented by point F). Definition 1 E(F)=1 [l IFS (F,F far ) l IFS (F,F near )] (19) where l IFS (F,F near ) is a distance from F to the nearer point F near among (crisp) M and N,l IFS (F,F far )isthedistancefromf tothe farerpointf far amongm andn. From (7), considering separately the cases when µ F < ν F, µ F > ν F, µ F = ν F, we can express(19) in the following way E(F)=1 µ F ν F (20) It can be proved that that (19) satisfies axioms (13) (16). Certainly, (20) is valid for µ F ν F only. Whenµ F =ν F,E(F)=1due to(14). An interpretation of entropy (19) can be as follow. This entropy measures the whole missing information which may be necessary to have no doubts when classifying(an element representedby)thepointftotheareaofconsideration, i.e. to say that(an element representedby)f fullybelongs(pointm)orfully doesnotbelong(pointn)toourset. Formula(19) describes the entropy for a single element belonging to an intuitionistic fuzzy set. For n elements belonging to an intuitionisticfuzzysetwehave E=1 1 n [l IFS (F i,f far ) + l IFS (F i,f near )]= = 1 1 µ Fi ν Fi (21) 2n wheref far andf near asin(19). The following facts are worth mentioning as far as the proposed measure of entropy (21) is concerned: 1. Although while calculating distances for intuitionistic fuzzy sets we should take into account all the three functions(memberships, non-memberships, and the intuitionistic fuzzy indices) seesection3.1,intheentropymeasure(21) the third components(the intuitionistic fuzzy indices) disappear during subtraction. It makes the measure proposed(21) easier to calculate than the measure previously proposed(cf. Szmidt and Kacprzyk[18]). 2. Theproposedmeasureofentropy(21)isa special case of the measure of similarity suggested in(szmidt and Kacprzyk[21]). Having in mind interrelations between an element from an intuitionistic fuzzy set represented by point F, and its complement representedbypointf C,i.e.,thefactthatF = (µ F,ν F,π F ),F C =(ν F,µ F,π F ),from(7),we have l IFS (F i,f C )= = 1 2 ( µ F ν F + ν F µ F + + π F π F )= µ F ν F (22) whichmeansthat(19)canbeexpressedas: E(F)=1 l IFS (F,F C ) (23) So, finally, the measure proposed (21) can be expressed (while using the Hamming distance) as E = 1 1 n [l IFS (F i,f far ) + l IFS (F i,f near )]=1 1 n = 1 1 n µ Fi ν Fi l IFS (F i,f C i ) (24) The measure of entropy proposed reaches by definition its maximum for elements for which the membership function is equal to the non-membership function(see the axiom(14)

6 D ( 0, 0, 1 ) E H (0, 0,1) F c E M (1, 0,0) G N(0, 1,0) Figure 3: The triangle MNH explaining a ratio-based measure of similarity whichisacounterpartofthedelucaandtermini axiom(10)). It is a simple consequence of the definition (the sense) of entropy we are immediately able to indicate the missing amountofknowledgetosayifanelementbelongs or does not belong to an intuitionistic fuzzy set not paying any attention to reasonswhywecannotclassifyanelement(belongingornot). Thefactthatwearenotable tosayifelementsrepresentedbygbelongor not to a set considered is just the same as for point H although G represents elements weknowallabout(µ=ν =0.5)whereasH represents any elements(we know absolutely nothing about them). The entropy introduced(24) is a special case of the similarity measure (see Figure 3) proposed earlier (Szmidt and Kacprzyk [21]). Here we present the similarity measure Sim(F, E) between two elements(represented bypoints)f ande. Definition 2 [21] Sim(F,E)=l IFS (F,E C ) l IFS (F,E)= = ( µ(f) µ(e C ) + ν(f) ν(e C ) + + π(f) π(e C ) ) ( µ(f) µ(e) + + ν(f) ν(e) + π(f) π(e) )= = µ(f) µ(e C ) + ν(f) ν(e C + µ(f) µ(e) ν(f) ν(e) ) (25) where: l IFS (F,E) is a distance from F(µ F,ν F,π F )toe(µ E,ν E,π E ), l IFS (F,E C ) is the distance from F(µ F,ν F,π F )toe C (ν E,µ E,π E ), E C isacomplementofe, thedistancesl IFS (F,E)andl IFS (F,E C )are calculated from(7). In a special case when E = M and E C = N (i.e. when the situation in Figure 3 becomes the situation in Figure 2), the similarity measure (25) becomes closely related to the entropy measure proposed here, namely: E(F) = 1 Sim(F,M) or E(F) = 1 Sim(F,N) that depends on which one from M andn arenearertof. Although Definition 1(of entropy) is general in the sense that any distance could be applied, the detailed formulas were presented for the normalized Hamming distance. Similar formulas can be presented for different distances. For example, for the normalized Euclidean distance(for membership function of the elements considered less than nonmembership functions) we have: E = 1 1 n [q IFS (F i,f far ) 2n + q IFS (F i,f near )]=1 1 2n { (µ Fi 1) 2 +(ν Fi ) 2 +(π Fi ) 2 + (µ Fi ) 2 +(ν Fi 1) 2 +(π Fi ) 2 }= = 1 1 n { (ν Fi ) 2 +π Fi ν Fi +(π Fi ) 2 n (µ Fi ) 2 +π Fi µ Fi +(π Fi ) 2 } (26) Unfortunately for the Euclidean distance we cannotshowthattheentropycanbeequivalently expressed in the terms of the distances between elements and their complements(as for the Hamming distance-(24)). We could introduce the counterpart of (24) butthenewmeasureisnotadirectresultof the distances we consider in Definition 1. E = 1 1 n q IFS (F i,fi C )=1 1 2n 2n (µ Fi ν Fi ) 2 +(ν Fi µ Fi ) 2 = = 1 1 n ν Fi µ Fi (27) n

7 The same conclusion can be drawn for another measure of entropy (28) and its counterpart similarity(29). A ratio-based measure of fuzziness, i.e. when the entropy of an intuitionistic fuzzy element canbegivenas: Definition 3 E(F)= d(f,f near) d(f,f far ) (28) where d(f,f near ) is a distance from F to thenearerpointf near amongm andn,and d(f,f far )isthedistancefromf tothefarer pointf far amongm andn. A ratio-based measure of entropy(28) satisfies axioms (13) (16) and is consistent with the geometrical interpretation of entropy. And evenmore,(28)isaspecialcaseofthefollowing similarity measure (29) (cf. Szmidt and Kacprzyk [20] for properties and its connections with the Jaccard coefficient) Definition 4 Sim(F,E)= d(f,e) d(f,e C ) (29) where: d(f, E) is a distance from F(µ F,ν F,π F )toe(µ E,ν E,π E ), d(f,e C ) is the distance from F(µ F,ν F,π F ) toe C (ν E,µ E,π E ), E C isacomplementofe. Distancesd(.,.)in(28)and(29)areanyproperly calculated distances (cf. Szmidt and Kacprzyk[22]), e.g. (7) or(8). We can notice that there is not a direct connection between the entropy (28) and its counterpart similarity (29) for any distance d(.,.). The general conclusion for intuitionistic fuzzy setsisthatthereisnotabijectiveconnection between the similarity and entropy. This fact isobviousinviewofthemeaningofentropy and similarity. The entropy tells us if it is possible to say if an element(s) belongs to a set considered or not. The similarity, especially in a case of intuitionistic fuzzy sets, is not just the same that can be illustrated by the following example. We wish to rent one ofthetwoaccessibleflatsconsideringasetof criteria. We know absolutely nothing about the flats. So the entropy of elements representing each of them is equal to 1 (we are not abletosayiftheflatsbelongtoaset of flats fulfilling our criteria, not to speak about the extent to which they fulfill the considered criteria). But canwe saythat bothflats are similar? Oneofthemcanbegreat,theother onejustterrible. Thefactthatweknownothing(entropyequalto1)doesnotmeanatall those elements orsets are similaror that we can conclude about their similarity. Conclusion: in general(!) knowing the entropy we are not able to conclude about the similarity of intuitionistic fuzzy sets. 5 Concluding remarks We revised the axioms for entropy and proposed two different formulas for calculations. Both formulas compare the distances to the nearer and to the farer crisp elements. The first formula (19) compares the distances by subtracting them, the second formula (28) compares the distances by their ratio. We also presented the counterpart similarity measures (25) and (29) for both measuresofentropy. Theideaofthequalityconnections between the similarity measures and their counterpart entropy measures is clarified by Figure 2 and Figure 3, respectively. We discussed in detail the entropy formula (19) for two types of distances: the normalized Hamming and the normalized Euclidean distance. For the former the measure is equivalent to the considering of distances between elements and their complements(as given by Hung[10]). Unfortunately, for the Euclidean distance we can not show that the entropy (19) can be equivalently expressed in terms of the distances between elements and their complements(as for the Hamming distance). Next, we discussed connections between the entropy(19) and similarity(25) for intuitionisticfuzzysets. Weshowedthat(19)isclosely related to the similarity measure(25) we proposed in(szmidt and Kacprzyk[21]). But the

8 relationship between the two measures occurs for the Hamming distance, and not for the Euclidean distance. We examined relationships between the measure of entropy(28) and similarity(29). But for these formulas any relationships occur. References [1] Atanassov K. (1983), Intuitionistic Fuzzy Sets. VII ITKR Session. Sofia (Centr. Sci.- Techn. Libr. of Bulg. Acad. of Sci., 1697/84) (in Bulgarian). [2] Atanassov K. (1999), Intuitionistic Fuzzy Sets: Theory and Applications. Springer- Verlag. [3] Burillo P. and Bustince H. (1996). Entropy of intuitionistic fuzzy sets and on intervalvalued fuzzy sets. Fuzzy Sets and Systems, 78, [4] HigashiM.andKlirG.(1982).Onmeasures of fuzziness and fuzzy complements. Int. J. Gen. Syst., 8, [5] De Luca A. and Termini S. (1972). A definition of a non-probabilistic entropy in the setting of fuzzy sets theory. Inform. and Control, 20, [6] FanJ.andW.Xie(1999).Distancemeasure and induced fuzzy entropy. Fuzzy Sets and Systems, 104, [7] FanJ-L.,MaY-L.andXieW-X.(2001).On some properties of distance measures. Fuzzy Sets and Systems, 117, [8] Fan J-L. and Ma Y-L. (2002). Some new fuzzy entropy formulas. Fuzzy Sets and Systems, 128, [9] HuQ.andYuD.(2004).Entropiesoffuzzy indiscernibility relation and its operations. Int. J. of Uncertainty, Knowledge-Based Systems, 12(5), [10] HungW-L.(2003).Anoteonentropyofintuitionistic fuzzy sets. Int. J. of Uncertainty, Fuzziness and Knowledge-Based Systems, 11 (5), [11] Kosko B. (1986) Fuzzy entropy and conditioning. Inform. Sciences, 40, [12] Pal N.R. and Bezdek J.C. (1994). Measuring fuzzy uncertainty. IEEE Trans. on Fuzzy Systems, 2(2), [13] Pal N.R. and Pal S.K. (1991). Entropy: a new definition and its applications. IEEE Trans. on Systems, Man, and Cybernetics, 21(5), [14] Szmidt E. (2000) Applications of Intuitionistic Fuzzy Sets in Decision Making. (D.Sc. dissertation) Techn. Univ., Sofia, [15] Szmidt E. and Baldwin J. (2003) New Similarity Measure for Intuitionistic Fuzzy Set Theory and Mass Assignment Theory. Notes onifss,9(3), [16] SzmidtE.andBaldwinJ.(2004)Entropyfor Intuitionistic Fuzzy Set Theory and Mass Assignment Theory. Notes on IFSs, 10, 3, [17] Szmidt E. and Kacprzyk J.(2000) Distances between intuitionistic fuzzy sets, Fuzzy Sets and Systems, 114(3), [18] Szmidt E. and Kacprzyk J. (2001) Entropy for intuitionistic fuzzy sets. Fuzzy Sets and Systems, 118(3), [19] SzmidtE.andKacprzykJ.(2002).AnIntuitionistic Fuzzy Set Base Approach to Intelligent Data Analysis(an application to medical diagnosis). In A. Abraham, L. Jain, J. Kacprzyk(Eds.): Recent Advances in Intelligent Paradigms and Applications. Springer- Verlag, [20] Szmidt E. and Kacprzyk J.(2004) Similarity of intuitionistic fuzzy sets and the Jaccard coefficient. IPMU 2004, [21] Szmidt E. and Kacprzyk J. (2005) A New Concept of a Similarity Measure for Intuitionistic Fuzzy Sets and its Use in Group Decision Making. In V. Torra, Y. Narukawa, S. Miyamoto(Eds.): Modelling Decisions for AI. LNAI 3558, Springer 2005, [22] Szmidt E. and Kacprzyk J. Distances Between Intuitionistic Fuzzy Sets: Straightforward Approaches may not work.(accepted.) [23] Yager R.R.(1997). On measures of fuzziness and negation. Part I: Membership in the unit interval. Int. J. Gen. Syst., 5, [24] Zadeh L.A. (1965). Fuzzy sets. Information and Control, 8,