A New Method for Measuring Similarity between Intuitionistic Fuzzy Sets Based on Normal Distribution Functions Zehua Lv, Chuanbo Chen, Wenhai Li School of Computer Sci. & Tech.,Huazhong University of Science and Technology
[email protected] Abstract This paper puts forward a new kind of similarity measure between Intuitionistic Fuzzy Sets (IFSs) based on normal distribution functions. At first, we propose a method to express an intuitionistic fuzzy set by a series of normal distribution functions. Then, we use these normal distribution functions to calculate the degree of similarity between IFSs. The properties of the proposed similarity measure are proved and several numerical examples are taken to validate it. Compared with the existing methods, the proposed similarity measure is more reasonable and more suitable for any special situation. Moreover, by comparing the proposed similarity measure with the existing measures, our method shows that it is much more reliable than the existing measures to linguistic variables. Though having a little difficulty for calculation, the similarity measure presents a brandnew method to deal with fuzzy information.
1. Introduction The notion of Intuitionistic Fuzzy Sets (IFSs) for fuzzy sets generalization, proposed by Atanassov [1], has gained successful applications in various fields. The concept of vague sets introduced by Gau and Buehere[2] is another generalization of fuzzy sets. Bustince and Burillo[3] pointed out that the notion of vague sets is the same as that of intuitionistic fuzzy sets. Since this fuzzy set generalization can present the degrees of membership and non-membership with a degree of hesitancy, the knowledge and semantic representation become more meaningful and applicable [4,5]. Afterwards the theory of IFSs has been widely studied and applied in variety of areas such as logic programming [6,7], decision making problems [8,9,24], medical diagnostics [10,19,22], pattern recognitions [21,25] and so on.
In many applications, the similarity between fuzzy sets is very important. Hyung [11] and S.M. Chen [12] put forward the definition of similarity between fuzzy sets and between elements. S.M. Chen [13] proposed a set of methods for measuring the degree of similarity between vague sets and between elements. Hong and Kim [14] showed by examples that the similarity measure proposed by Chen do not fit well in some cases and proposed a set of modified measures. Recently, Li and Cheng [15] discussed similarity measures on IFSs and showed how these measures may be used in pattern recognition problems. However, Li and Cheng’s similarity measure may not be effective in some special cases. In order to overcome these drawbacks of Li and Cheng’s methods, Liang and Shi [16] proposed several new similarity measures and also discussed the relationships between these measures. Examples showed that Liang and Shi’s similarity measures are more reasonable than Li and Cheng’s. On the other hand, Mitchell [17] interpreted IFSs as ensembles of ordered fuzzy sets from a statistical viewpoint to modify Li and Cheng’s methods. Wen-Liang Hung and Miin-Shen Yang [18] presented a new method for similarity measures between IFSs based on Hausdorff distance, some examples showed that this kind of similarity measures is much simpler and well suited to be used with linguistic variables. In this paper, we firstly present a method to express an intuitionistic fuzzy set by a series of normal distribution functions, and then we calculate the similarity measure by the normal distribution functions. Examples show that this kind of similarity measure is much reasonable and suits any special situation. The remaining part of this paper is organized as follows: In section 2, we give a brief overview of some existing typical similarity measures of IFSs. In section 3, we present a method to express an intuitionistic fuzzy set by a series of normal distribution functions, and then use these functions to generate a new kind of similarity measure to calculate
the degree of similarity. In section 4, some examples are proposed to compare the proposed similarity measure with the existing measures. Finally, conclusion is made in section 5.
2. Preliminaries Measuring the similarity between IFSs is important in pattern recognition research. Some methods have previously been advanced to calculate the degree of similarity. In the study of the similarity between IFSs, Li and Cheng [15] introduced the following definition. Definition1. A mapping S : IFSs( X ) × IFSs( X ) → [0,1]. S ( A, B ) is said to be the degree of similarity between A ∈ IFSs( X ) and B ∈ IFSs( X ), if S ( A, B ) satisfies the properties (P1-P4): (P1) 0 ≤ S ( A, B ) ≤ 1 (P2) If A = B , S ( A, B) = 1 (P3) S ( A, B) = S ( B, A) (P4) If A⊆ B ⊆C, A, B, C ∈ IFSs ( X ) then: S ( A, C) ≤ S( A, B) , S ( A, C ) ≤ S ( B, C ) . Next, we will recall some kind of existing typical similarity measures. Assume that there are two IFSs A and B in, X = {x1 , x2 ,", xn } , the degree of similarity between A and B can be calculated as follows [14]: 1
n
p ∑ m A (i) − m B (i) n i =1 Where: m A (i) = (t A ( xi ) + 1 − f A ( xi )) 2 ,
S dp ( A, B ) = 1 −
p
p
(1)
mB (i) = (t B ( xi ) + 1 − f B ( xi )) 2 , 1 ≤ p ≤ ∞ .
From S dp ( A, B) , we note that m A (i ) is the median value of the interval [t A ( x i ),1 − f A ( x i )] . Then, if the median values of subinterval are equal respectively, the similarity measures between these two IFSs are equal to 1. In order to overcome the drawback of S dp ( A, B) , Liang and Shi [16] proposed the following similarity measures between IFSs. Let ϕ tAB (i ) = t A ( x i ) − t B ( x i ) 2 ϕ fAB (i ) = (1 − f A ( x i )) − (1 − f B ( x i )) 2
And the degree of similarity between the two IFSs, A and B , can be calculated as follows: n 1 (2) S ep ( A, B) = 1 − p p ∑ (ϕ tAB (i) + ϕ fAB (i )) p n i= 2 In order to get more information on IFSs, Liang and Shi considered another definition of similarity measures between A and B [16]. Let: mA1 (i) = (µ A ( xi ) + mA (i)) 2 , mA2 (i) = (mA (i) + 1 − f A ( xi )) 2 , mB1 (i) = (µ B ( xi ) + mB (i)) 2 , mB 2 (i) = (mB (i) + 1 − f B ( xi )) 2 . Then, we can calculate the degree of similarity
between the IFSs, A and B , as follows:
S sp = 1 −
n
1 p
p
n
∑ (ϕ
s1
( i ) + ϕ s 2 ( i ))
p
(3)
i =1
Where ϕs1 = mA1 (i) − mB1 (i) 2 , ϕs2 = mA2 (i) − mB2 (i) 2 . Finally, they considered the following similarity measure to obtain all available information on IFSs. Let ϕ1 (i ) = ϕ s1 (i ) + ϕ s 2 (i) , ϕ 2 (i) = mA (i) − mB (i) , l A (i) = (1 − f A ( xi ) − µ A ( xi )) 2 , lB (i) = (1− f B (xi ) − µB (xi )) 2 . Let ϕ 3 (i ) = max{ l A ( i ), l B (i )} − min{ l A (i ), l B (i )} . The degree of similarity between the two IFSs, A and B , can be calculated as follows: S hp ( A , B ) = 1 − Where 0 ≤ ω m ≤ 1 ,
3
∑
1 p
n
3
n
p
∑ (∑ ω i =1
m
ϕ m (i )) p
(4)
m =1
ωm =1.
m =1
Mitchell [17] adopted a statistical approach and interpreted IFSs as ensembles of ordered fuzzy sets to modify Li and Cheng’s similarity measures. Let ρ µ ( A, B) and ρ f ( A, B ) denote the similarity measures between the “low” membership function µ A and µ B and between the “high” membership function 1− fA and 1− fB, respectively, as follows: 1
ρ µ ( A, B) = S (µ A , µ B ) = 1 − p
n
n
p
∑ µ A ( xi ) − µB ( xi )
p
i =1
p
1 p n ∑ f A (xi ) − f B (xi ) n i=1 Then, they defined the modified similarity measure between A and B with 1 (5) S mod ( A, B) = ( ρ µ ( A, B) + ρ f ( A, B )) 2 Hung and Yang proposed a new kind of similarity measure based on Hausdorff distance [18]. Let A and B be two IFSs in X = { x1 , x 2 , " , x n } and let I A ( xi ) and I B ( xi ) be subintervals on [0, 1] denoted by the following: I A ( xi ) = [ µ A ( xi ),1 − f A ( xi )] ,
ρµ ( A, B) = S(1− f A ,1− fB ) =1− p
I B ( xi ) = [ µ B ( xi ),1 − f B ( xi )] , i = 1 , 2 , " , n . Let H ( I A ( x i ), I B ( x i )) be the Hausdorff distance
between I A ( xi ) and I B ( xi ) . In [18], they defined the distance d H ( A, B ) between A and B as follows: 1 n ∑ H ( I A ( x i ), I B ( x i )) n i =1 Then, the similarity measures between A and B is defined as follows: d H ( A, B ) =
S l ( A, B ) = 1 − d H ( A, B ) S e ( A, B ) =
e
− d H ( A, B )
−e −1
1− e 1 − d H ( A, B) S c ( A, B) = 1 + d H ( A, B)
−1
(6) (7) (8)
Examples show that this kind of similarity measure is much simpler than others and is well suited to be used with linguistic variables.
3. Similarity measure between intuitionistic fuzzy sets In this section, we firstly present a method to express an intuitionistic fuzzy set by a series of normal distribution functions. Then, the definition of similarity between intuitionistic fuzzy sets based on normal distribution function is given. Lastly, in order to coincide with the intuition, this similarity measure is normalized at the end of this section.
3.1. A method to express intuitionistic fuzzy sets by normal distribution functions The difference between fuzzy set and classical crisp set is that the fuzzy set has not clear boundaries, we can not distinguish whether an object subjecting to a fuzzy set or not, but the membership degree is precise. Intuitionistic fuzzy set is further vague, its characteristic is that not only an object belonging to a set is not known, but also the membership degree is not precise. We only know that the membership degree is in a subinterval of [0,1]. For example, assume an intuitionistic fuzzy set B = {( x1 , 0.2, 0.6)} , we merely know that the degree of x1 belonging to B is in the subinterval [0.2,0.4]; in other words, the inevitability of x1 belonging to B is 0.2 and the possibility is 0.4. Therefore, we will naturally consider the following two questions: 1. whether all the values in the subinterval have the same probability as the membership degree of x1 ? 2. If the values’ probabilities as membership in the subinterval are not the same, which kind of distribution do they accord with? Herein we take a vote model for an example, assume an intuitionistic fuzzy set A = {( x ,0.3,0.3)}, it can be interpreted as “the vote for a resolution is 3 in favor 3 against and 4 abstentions ” , then how to draw a conclusion to this vote? The general thinking is that the final result is 0.5, or takes an average of 0.3 and 0.7; therefore， we draw the conclusion that the median value, [t A ( xi ), 1 − f A ( xi )] , has the biggest probability as the membership degree, with the distance away from
µ , the probability becomes small. Therefore, we can use a normal distribution function to substitute [t A ( xi ),1 − f A ( xi )] to express the membership degree of xi . In the first instance, we suppose that there is only one element in the universe of X , X = {x} . Let A be an intuitionistic fuzzy set in X and A = [t A ( x), 1 − f A ( x)] x , || I A (x) ||= 1− f A ( x) − t A (x) . If || I A ( x ) ||= 0 , t A (x) = 1− f A (x) , the intuitionistic fuzzy set degenerates into a fuzzy set. At this time, the corresponding normal distribution function is ϕ ( z ) = exp(−π ( z − µ ) 2 ) . This normal distribution function satisfies E(ϕ(z)) = µ = t A (x) , σ = D(ϕ(z)) =1 2π and ϕ ( µ ) = 1 . If || I A ( x) ||≠ 0 , the corresponding normal distribution function satisfies µ = E (ϕ ( z )) = (1− f A(x) +tA (x)) 2,
(
)
σ = D(ϕ ( z )) = 1 2π (1 − || I A ( x ) || 2) as the value of || I A ( x) || becomes bigger, the value of σ becomes bigger.
3.2. Similarity measure between intuitionistic fuzzy sets Let A and B be two intuitionistic fuzzy sets in the universe X = {x1} A =[t A (x1),1− f B (x1)] x1 , B = [t B ( x1),1− f B (x1)] x1 , the corresponding normal distribution functions are ϕ xA1 ( z) and ϕxB1 (z) respectively. ϕ xB (z )
ϕ xB (z )
ϕ (z) A x
ϕ xA (z
z t t z 1 1 (a) The expectation of ϕ xA (z ) (b) The expectation of ϕ xA (z ) is is smaller than that of ϕ xB (z ) bigger than that of ϕ xB (z ) Fig.1 The similarity between two intuitionistic fuzzy
When (t A (x1 ) +1 − f A ( x1 )) 2 < (t B (x1) +1− f B (x1 )) 2 , as shown in Fig.1 (a), the similarity S N ( A, B) between A and B is as follows: S N ( A, B ) = S 1
∫ = ∫
t
−∞ t
−∞
+∞
ϕ xB1 ( z ) dz + ∫ ϕ xA1 ( z ) dz t +∞
(9)
ϕ ( z ) dz + ∫ ϕ ( z ) dz A x1
t
B x1
Here t is the x-coordinate of the intersection point of these two functions. When (tB (x1 ) +1− f B (x1 )) 2 < (t A (x1 ) +1− f A (x1 )) 2 , as shown in Fig.1 (b), then the degree of similarity between A and B is as follows:
S N ( A , B ) = S1 =
∫ ∫
t
−∞ t
−∞
∫ ( z ) dz + ∫
ϕ xA1 ( z ) dz + ϕ
B x1
+∞
t +∞
t
ϕ xB1 ( z ) dz (10) ϕ xA1 ( z ) dz
Here t is the same as before. When (t B ( x1 ) +1 − f B ( x1 )) 2 = (t A ( x1 ) + 1 − f A ( x1 )) 2 , as shown in Fig.2 (a), if || I A ( x1 ) || > || I B ( x1 ) || , the xcoordinates of the two intersection points are t1 and t 2 respectively. Then, the degree of similarity between A and B is as follows: t1
SN ( A, B) = S1
∫ = ∫
−∞ t1
−∞
t2
+∞
t1 t2
t2 +∞
ϕxB1 (z)dz + ∫ ϕxA1(z)dz + ∫ ϕxB1 (z)dz
(11)
ϕ ( z)dz + ∫ ϕ ( z)dz + ∫ ϕ ( z)dz A x1
t1
B x1
t2
A x1
If || I A ( x1 ) || < || I B ( x1 ) || , as shown in Fig.2 (b), the degree of similarity between A and B is as follows: t1
SN ( A, B) = S1
∫ = ∫
−∞ t1
−∞
t2
+∞
t1 t2
t2 +∞
ϕxA1 (z)dz + ∫ ϕxB1(z)dz + ∫ ϕxA1 (z)dz
(12)
ϕxB1( z)dz + ∫ ϕxA1( z)dz + ∫ ϕxB1( z)dz t1
t2
ϕ xA ( z )
ϕ xB (z )
ϕ xA (z )
ϕ xB ( z )
1 z 0 t1 t 2 t1 t 2 1 z 0 (a) The variance of ϕ xA (z ) is (b) The variance of ϕ xA (z ) is bigger than that of ϕ xB (z ) smaller than that of Fig.2 Similarity between 2 IFSs which have the same expectations
We can also explain this kind of similarity measure in geometry, the degree of similarity between A and B is the ratio of ∆1 to ∆ 2 , here ∆1 denotes the common area surrounded by the two normal distribution curves and the rectangular coordinates, ∆ 2 denotes the whole area surrounded by the two normal distribution curves and the rectangular coordinate, which will be shown in Fig.3 ∆1 ∆ (13) S ( A, B ) = S = 1 = N
0
1
∆2
∆1 + ∆ 01 + ∆ 02
ϕ xA ( z )
ϕ xB ( z )
∆ 01
∆ 02 ∆1
follows: S N ( A, B ) =
1 n ∑ Si n i =1
(14)
Where S i represents the degree of similarity between the two intuitionistic fuzzy sets Ai and Bi . Ai = [t A (xi ),1 − f A (xi )] xi , B i = [t B ( x i ),1 − f B ( x i )] x i Proposition 1. The defined degree of similarity S N ( A, B) between the two IFSs A and B in X = {x1 , x2 ,", xn } satisfies the following properties (P1-P4). (P1) 0 ≤ S N ( A, B) ≤ 1 (P2) A = B if and only if S N ( A, B) = 1 (P3) S N ( A, B) = S N ( B, A) (P4) If A⊆ B ⊆C , A, B, C ∈ IFSs( X ) then S N ( A, C ) ≤ S N ( A, B) , SN ( A, C) ≤ SN (B, C) . Proof: Omitted. Next, we will discuss the similarity measure proposed in definition 1. If an element belongs fully to a set when it has an attribute(s) we are interested in, we represent this situation by the IFSs A = {(x, 1, 0)} . If an element does not belong to a set when it has not an attribute(s) we are interested in, we represent it by the IFSs B = {(x, 0, 1)} , A and B represent quite “opposite” situation in terms of IFSs, the similarity should be equal to zero intuitively. We use the method proposed in definition 1 to calculate the similarity between A and B , the corresponding normal distribution functions are: ϕ A ( x ) = exp( −π ( x − 1) 2 ) ， ϕ B ( x ) = exp( −π x 2 ) . The similarity measure degree
between A and B is S N ( A, B) = 0.12 , this result does not equal zero, it is counterintuitive. Therefore, we modify definition 2 as follows. Definition 3: Assume that there are two IFSs A and B in X = {x1 , x2 ,", xn } , the degree of similarity between the two IFSs, A and B , can be calculated as follows: S N ( A, B ) =
n 1 ( S i − 0.12) ∑ n(1 − 0.12) i =1
(15)
Where S i represents the degree of similarity between the two intuitionistic fuzzy sets Ai and Bi . Ai = [t A ( xi ),1 − f A ( xi )] xi , Bi = [t B ( xi ),1 − f B ( xi )] xi . 1
Fig.3 Geometry explain of similarity between two IFSs
Definition 2. Assume that there are two IFSs A and B in X = {x1 , x2 ,", xn } , the degree of similarity between the two IFSs, A and B , can be calculated as
4.
Numerical measures
examples
of
similarity
In this section, we present several examples to show that the proposed similarity can be better used to measure the similarity between intuitionistic fuzzy sets
than any other existing similarity measures. For convenience, we consider p =1 and ωi =1 3 , where i = 1,2,3 , in similarity measure S dp [15], S ep , S sp , S hp [16], S mod [17] and S l , S e , S c [18]. We now give an example to show that the proposed method in this paper can well measure the similarity between two intuitionistic fuzzy sets in some special situations. Example 1: In [16], Liang and Shi assumed that there are two patterns denoted with IFSs in X = {x1 , x 2 , x3 } . The two patterns are denoted as follows: A1 = {( x1 ,0.2,0.2), ( x2 ,0.2,0.2), ( x3 ,0.2,0.2)} A2 = {( x1 ,0.4,0.4), ( x2 ,0.4,0.4), ( x3 ,0.4,0.4)} Assume that a sample B ={(x1,0.3,0.3),(x2 ,0.3,0.3),(x3,0.1,0.3)} is given. By (1)-(8), we have： S d1 ( A1 , B) = S d1 ( A2 , B ) = 0.967 S e1 ( A1 , B ) = 0.900 , S e1 ( A2 , B) = 0.867 S s1 ( A1 , B ) = S s1 ( A2 , B) = 0.933 S h1 ( A1 , B ) = 0.933 , S h1 ( A2 , B ) = 0.900 1 1 ( A1 , B ) = 0.900 , S mod S mod ( A2 , B ) = 0.867 S l ( A1 , B) = 0.900 , S l ( A2 , B) = 0.833
S e ( A1 , B) = 0.849 ， S e ( A2 , B) = 0.757 S c ( A1 , B) = 0.818 ， S c ( A2 , B) = 0.714 On the other hand, by (14) we have: S N ( A1 , B) = 0.814 , S N ( A2 , B) = 0.706 The similarity can also be normalized by (15) S N ( A1 , B) = 0.788 , S N ( A, B) = 0.666 Based on the results above, we can see that the sample B belongs to the pattern A1 according to the principle of the maximum degree of similarity between IFSs. But the similarity measures S dp and S sp can not classify the patterns. For querying a database friendlier with fuzzy queries, it is necessary to define the degree of similarity between these fuzzy sets. The similarity measure of intuitionistic fuzzy sets based on Hausdorff distance which was proposed by Wen-Liang Hung and Miin-Shen Yang [18] is suitable to be used with linguistic variables, but we find that this method is likely to produce mistakes in some situations. Let F = {( x, µ F ( x), v F ( x)) | x ∈ X } be an IFS in X . For any positive real number n , the authors in [23] defined IFS F n as follows: n n F n = {( x, [µ F ( x) ] ,1 − [1 − v F ( x) ] ) | x ∈ X } (16) Using the above operation, we define the concentration and dilation of F as follows: Concentration: CON(F) = F 2
Dilation: DIL(F) = F1 2 . Similar to fuzzy set, CON(F ) and DIL(F ) may be treated as “Very ( F )” and “More or Less ( F )”, respectively. We first review the example presented by WenLiang Hung and Miin-Shen Yang [18]. Example 2. Hung and Yang in [18] let an IFS F in X = {6,7,8,9,10} F = {(6,0.1,0.8),(7,0.3,0.5),(8,0.6,0.2),(9,0.9,0.0),(10,1.0,0.0)}.
By taking into account the characterization of linguistic variables, they regarded F as “Large” in X . Using the operations of concentration and dilation, F 1 2 May be treated as “More or Less Large”, F 2 May be treated as “Very Large”, F 4 May be treated as “Very Very Large” In [18], they used the similarity measure based on Hausdorff distance to calculate the degree of similarity between these IFSs, and then drew the conclusion that (17) S ( L., M .L .L .) > S ( L., V . L.) Here L. denotes “Large”, V .L. denotes “Very Large”, and M.L.L. denotes “More or Less Large”. Next, we give another example to show that the inequation S ( L., M .L.L.) > S (L.,V.L.) is not always right. Example 3 Let F1 ={(6,0.2,0.7)}, then F11 2 ={(6,0.45,0.45)}, 2 12 F1 = {(6,0.04,0.91)} .We regard F1 as “Large”, F1 as
“More or less Large” and F1 2 as “Very Large”, respectively. From (6), we get that: (18) S (M .L.L., L.) = 0.75 < S (V .L., L.) = 0.79 On the other hand, if we let F2 = {(8,0.6,0.2)} , from (10), we get 12 2 F2 = {(8,0.77,0.11)} , F2 = {(8,0.36 ,0.36)} from (6), we get that (19) S (M .L.L., L.) = 0.825 > S (V .L., L.) = 0.76 2 From (18), the sample F belongs to the pattern F , but from (19), the sample F 1 2 belongs to the pattern F . From the example proposed above, we find that this kind of similarity measure [18] is not always reliable in application to compound linguistic variables. On the other hand, by (14) we have: 12 S N ( F1 , F1 ) = 0.57 , S N ( F1 , F12 ) = 0.36 , 12
S N ( F2 , F2 ) = 0.76, S N ( F2 , F22 ) = 0.66 It can be seen that the inequation S ( M .L.L., L.) > S (V .L., L.) is always right, we then can draw the conclusion that the sample “More or Less Large” belongs to the sample “Large”. Compared with the existing similarity measure, the method proposed in this paper has a few difficulties in computation. The computation difficulties mainly exist on two sides: On the one hand, we should calculate the
point of intersection between the two normal distribution functions; On the other hand, we should use definite integral to calculate the areas surrounded by the normal distribution curves and the rectangle coordinates.
5. Conclusion and future work In this article, we propose a method to express an intuitionistic fuzzy set by a series of normal distribution functions through the investigation of vote model. Then, a new definition of measuring similarity between IFSs is presented. The properties of the similarity measure were proved and several examples were used to make comparisons between the proposed similarity measure and the existing methods. The results showed that this similarity measure is superior to any other existing methods. This method not only suits any special situation, but is also more reliable than the one proposed by Wen-Liang Hung in application to linguistic variables [18]. However, the only disadvantage of this method is that the calculation complexity is high. Therefore, in our future work we will find a method to simplify the calculation complexity or using another approximate method to substitute it. References [1] Atanassov, K., “Intuitionistic fuzzy sets”, Fuzzy Sets Systems, vol.20, pp. 87–96, 1986. [2] Gau W.L., Buehere D.J., “Vague sets”, IEEE Trans. Systems Man Cybernet, vol.23, pp. 610-614, 1993. [3] H.Bustince, P.Burillo, “Vague sets are intuitionistic fuzzy sets”, Fuzzy Sets and Systems, vol. 79, pp. 403-405, 1996. [4] Atanassov, K., “New operations defined over the intuitionistic fuzzy sets”, Fuzzy Sets Systems, vol. 61, pp. 137-142, 1994. [5] Atanassov K., Intuitionistic Fuzzy Sets: Theory and Applications, Physica-Verlag Heidelberg., New York, 1999. [6] Atanassov, K., Gargov, G., “Intuitionistic fuzzy logic”, C.R. Acad. Bulg. Soc., vol. 43, pp. 9–12, 1990. [7] Atanassov, K., Georgeiv, G., “Intuitionistic fuzzy prolog”, Fuzzy Sets Systems, vol. 53, pp. 121–128, 1993. [8] Szmidt, E., Kacprzyk, J., “Intuitionistic fuzzy sets in group decision making”, Notes on IFS, vol. 2, pp. 11–14, 1996. [9] Deng-Feng Li, “Multiattribute decision making models and methods using intuitionistic fuzzy sets”, Journal of Computer and System Science, vol. 70, pp. 73-85, 2005. [10] De S.K., Biswas, R., Roy A.R., “An application of intuitionistic fuzzy sets in medical diagnosis”, Fuzzy Sets Systems, vol. 117, pp. 209–213, 2001.
[11] Hyung L.K., Song Y.S., Lee K.M., “Similarity measures between fuzzy sets and between elements”, Fuzzy Sets Systems, vol. 62, no. 3, pp. 291–293, 1994. [12] S.M. Chen, “Measures of similarity between vague sets”, Fuzzy Sets and Systems, vol. 74, pp. 217-223, 1995. [13] S.M. Chen, “Similarity measures between vague sets and between elements”, IEEE Trans. Systems Man Cybernet, vol. 27, pp. 153-158, 1997. [14] D.H. Hong, C. Kim, “A note on similarity measures between vague sets and between elements”, Information Science, vol. 115, pp. 83-96, 1999. [15] Li, D., Cheng, C., “New similarity measures of intuitionistic fuzzy sets and application to pattern recognition”, Pattern Recognition Letters., vol. 23, pp. 221–225, 2002. [16] Liang Z., Shi P., “Similarity measures on intuitionistic fuzzy sets”, Pattern Recognition Letters, vol. 24, pp. 2687 – 2693, 2003. [17] Mitchell, H.B., “On the Dengfeng–Chuntian similarity measure and its application to pattern recognition”, Pattern Recognition Letters, vol. 24, pp. 3101–3104, 2003. [18] Wen-Liang Hung, Miin-She Yang, “Similarity measures of intuitionistic fuzzy sets based on Hausdorff distance”, Pattern Recognition Letters, vol. 25, pp. 1603-1611, 2004. [19] Eulalia Szmidt, Janusz Kacprzyk, “A similarity measure for intuitionistic fuzzy sets and its application in supporting medical diagnostic reasoning”, ICAIAC, Lnai 3070, pp. 388-393, 2004. [20] Li Dengfeng, Cheng Chuntian, “New similarity measure of intuitionistic fuzzy sets and application to pattern recognitions”, Pattern Recognition Letters, vol. 23, pp. 221-225, 2002. [21] H.B. Mitchell, “On the Dengfeng-Chuntian similarity measure and its application to pattern recognition”, Pattern Recognition Letters, vol. 24, pp. 3101-3104, 2003. [22] Supriya Kumar De, Ranjit Biswas, Akhil Ranjan Roy, “An application of intuitionistic fuzzy sets in medical diagnosis”, Fuzzy Sets and Systems, vol. 117, pp. 209213, 2001. [23] De, S.K., Biswas R., Roy A.R., “Some operations on intuitionistic fuzzy sets”, Fuzzy Sets Systems, vol. 114, pp. 477–484, 2000. [24] Eulalia Szmidt, Janusz Kacprzyk, “A new concept of similarity measure for intuitionistic fuzzy sets and its use in group decision making”, MDAI, LNAI 3558, pp. 272-282, 2005. [25] Hua-Wen Liu, “New similarity measures between intuitionistic fuzzy sets and between elements”, Mathematical and Computer Modeling, vol. 42, pp. 6170, 2005.