Intuitionistic and interval-valued intutionistic fuzzy ... - Semantic Scholar

Fuzzy Optim Decis Making (2009) 8:123–139 DOI 10.1007/s10700-009-9056-3

Intuitionistic and interval-valued intutionistic fuzzy preference relations and their measures of similarity for the evaluation of agreement within a group Zeshui Xu · Ronald R. Yager

Published online: 24 March 2009 © Springer Science+Business Media, LLC 2009

Abstract Szmidt and Kacprzyk (Lecture Notes in Artificial Intelligence 3070:388– 393, 2004a) introduced a similarity measure, which takes into account not only a pure distance between intuitionistic fuzzy sets but also examines if the compared values are more similar or more dissimilar to each other. By analyzing this similarity measure, we find it somewhat inconvenient in some cases, and thus we develop a new similarity measure between intuitionistic fuzzy sets. Then we apply the developed similarity measure for consensus analysis in group decision making based on intuitionistic fuzzy preference relations, and finally further extend it to the interval-valued intuitionistic fuzzy set theory. Keywords Group decision making · Intuitionistic fuzzy set · Interval-valued intuitionistic fuzzy set · Similarity measure · Consensus 1 Introduction Since its introduction (Atanassov 1984), the intuitionistic fuzzy set theory has been studied and applied in a variety of fields (Atanassov 1999; Bustince et al. 2007; Xu 2008). Especially, many authors have paid considerable attention to the research on similarity measures and distance measures between intuitionistic fuzzy sets (IFSs)

Z. Xu School of Economics and Management, Southeast University, 210096 Nanjing, Jiangsu, China e-mail: [email protected] R. R. Yager (B) Machine Intelligence Institute, Iona College, New Rochelle, NY 10801, USA e-mail: [email protected]

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(Grzegorzewski et al. 2004; Hung and Yang 2004; Hwang and Yoon 1981; Li and Cheng 2002; Mitchell 2003; Szmidt and Kacprzyk 2000, 2004a,b). Szmidt and Kacprzyk (2000) defined four basic distances between IFSs: the Hamming distance, the normalized Hamming distance, the Euclidean distance, and the normalized Euclidean distance. Li and Cheng (2002) defined the concept of the degree of similarity between IFSs, and introduced several similarity measures between IFSs. They also applied these similarity measures to pattern recognitions. Mitchell (2003) made some modifications to those of Li and Cheng (2002). Grzegorzewski et al. (2004) proposed some distance measures between IFSs and /or interval-valued fuzzy sets based on the Hausdorff metric, which are the generalizations of the well known Hamming distance, the Euclidean distance and their normalized counterparts. Hung and Yang (2004) developed some similarity measures between IFSs based on the Hausdorff from a different point of view. However, all the above similarity measures consider just a distance between the IFSs that are being compared, and can not answer the question if the compared IFSs are more similar or more dissimilar. To overcome this drawback, Szmidt and Kacprzyk (2004a) proposed a similarity measure, which involves both similarity and dissimilarity between IFSs, and showed its usefulness in medical diagnostic reasoning. Szmidt and Kacprzyk (2004b) applied this measure to analyze the extent of agreement in a group of experts. In this paper, we further develop a new similarity measure by analyzing the results of Szmidt and Kacprzyk (2004a), and study its desirable characteristics. Then we apply the developed similarity measure to analyze the consensus of experts’ preferences in group decision making based on intuitionistic fuzzy preference relations, and finally further extend it to deal with the situations where experts’ preferences are expressed in interval-valued intuitionistic fuzzy sets (IVIFSs) or interval-valued intuitionistic fuzzy preference relations. 2 Preliminaries Let a set X be fixed. Zadeh (1965) introduced the concept of fuzzy set, given by F = {
x, µ F (x)|x ∈ X }

(1)

where µ F (x) ∈ [0, 1] is the membership degree of the element x ∈ X . The main characteristic of fuzzy sets is that: the membership function µ assigns to each element x ∈ X a membership degree ranging between zero and one and the nonmembership degree equals one minus the membership degree, i.e., this membership degree combines the evidence for x and the evidence against x. In real life, however, the information of an object corresponding to a fuzzy concept may be incomplete, an expert may assume that the element x belongs to a set A to a certain degree, but it is possible that he/she is not so sure about it, i.e., there may be a hesitation or uncertainty about the membership degree of x in A (Deschrijver et al. 2004), in such cases, the sum of the membership degree and the non-membership degree of an element in a universe may be less than one. To solve this issue, Atanassov (1984) introduced the concept of intuitionistic fuzzy set (IFS), which is the generalization of fuzzy set, shown as follows:

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An IFS A in X is an object having the form: A = {
x, µ A (x), v A (x)|x ∈ X }

(2)

where the functions µ A : X → [0, 1] and v A : X → [0, 1] determine the degree of membership and the degree of non-membership of the element x ∈ X , such that 0 ≤ µ A (x) + v A (x) ≤ 1, for all x ∈ X

(3)

π A (x) = 1 − µ A (x) − v A (x), for all x ∈ X

(4)

and

π A (x) is called the degree of indeterminacy of x to A, or called the degree of hesitancy of x to A (Szmidt and Kacprzyk 2000). Especially, if π A (x) = 0, for all x ∈ X , then the IFS A is reduced to a fuzzy set. The complement of A is given as (Szmidt and Kacprzyk 2004a): Ac = {
x, v A (x), µ A (x)|x ∈ X }. For convenience, here, we call α = (µα , vα , πα ) the intuitionistic fuzzy value (IFV), where µα ∈ [0, 1], vα ∈ [0, 1], µα + vα ≤ 1, πα = 1 − µα − vα

(5)

and the complement of α is denoted by α c = (vα , µα , πα ). Let X = {x1 , x2 , . . . , xn } be a finite universe of discourse, A and B be two IFSs in X, then Szmidt and Kacprzyk (2000) defined the normalized Hamming distance between A and B as below: d(A, B) =

n 1
(|µ A (x j ) − µ B (x j )| + |v A (x j ) − v B (x j )| + |π A (x j ) − π B (x j )|) 2n j=1

(6) By (4), we have: |π A (x j ) − π B (x j )| = |1 − µ A (xi ) − v A (xi ) − (1 − µ B (xi ) − v B (xi ))| = |(µ A (xi ) − µ B (xi )) − (v A (xi ) − v B (xi ))|

(7)

thus, (6) can be rewritten as: d(A, B) =

n 1
(|µ A (x j ) − µ B (x j )| + |v A (x j ) − v B (x j )| 2n j=1

+ |(µ A (xi ) − µ B (xi )) − (v A (xi ) − v B (xi ))|)

(8)

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Let α1 = (µα1 , vα1 , πα1 ) and α2 = (µα2 , vα2 , πα2 ) be two IFVs, then d(α1 , α2 ) =

1 (|µα1 − µα2 | + |vα1 − vα2 | + |πα1 − πα2 |) 2

(9)

is called the normalized Hamming distance between α1 and α2 . By (5), we have: |πα1 − πα2 | = |(µα1 − µα2 ) − (vα1 − vα2 )|

(10)

thus, (9) can be rewritten as: d(α1 , α2 ) =

1 (|µα1 − µα2 | + |vα1 − vα2 | + |(µα1 − µα2 ) − (vα1 − vα2 )|) (11) 2

Grzegorzewski et al. (2004) suggested a distance measure based on the Hausdorff metric, shown as follows: d1 (α1 , α2 ) = max{|µα1 − µα2 |, |vα1 − vα2 |}

(12)

Below we show the relation between the distance measures (9) and (12). It is well known that the following relation between any two real numbers x and y holds: max{x, y} =

1 (x + y + |x − y|) 2

(13)

then, from (12) and (13), it follows that: d1 (α1 , α2 ) =

1 (|µα1 − µα2 | + |vα1 − vα2 | + ||µα1 − µα2 | − |vα1 − vα2 ||) (14) 2

Since |(µα1 − µα2 ) − (vα1 − vα2 )| ≥ ||µα1 − µα2 | − |vα1 − vα2 ||

(15)

d(α1 , α2 ) ≥ d1 (α1 , α2 )

(16)

then

Obviously, the distance measure (12) only involves the first two parameters describing IFSs, while the distance measure (9) takes into account all three parameters of IFSs. Szmidt and Kacprzyk (2000) showed that the third parameter can not be omitted when calculating distance between two IFSs (or IFVs). As a result, in the remainder of this paper, we shall consider only the distance measure (9) together with the normalized Hamming distance (6), and use them as a base to develop some similarity measures between IFSs (or IVIFSs) for consensus analysis in fuzzy group decision making.

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Let α1 and α2 be two IFVs belonging to an IFS, and α2c be the complement of α2 , then Szmidt and Kacprzyk (2004a) defined the degree of similarity between α1 and α2 as: s(α1 , α2 ) =

d(α1 , α2 ) d(α1 , α2c )

(17)

From (17), we have 0 ≤ s(α1 , α2 ) ≤ ∞. s(α1 , α2 ) has the following properties: (1) s(α1 , α2 ) = 0 means the identity of α1 and α2 ; (2) s(α1 , α2 ) = 1 means that α1 is to the same extent similar to α2 and α2c ; (3) α1 = α2c (or α1c = α2 ), i.e., d(α1 , α2c ) = d(α1c , α2 ) = 0 means the complete dissimilarity of α1 and α2 , and s(α1 , α2 ) = ∞; (4) α1 = α2 = α2c means the highest “fuzziness”—not too constructive a case when looking for compatibility (both similarity and dissimilarity). The prominent characteristic of the similarity measure (17) is that: it takes into account not only a pure distance between two IFVs but also examines if the compared IFVs are more similar or more dissimilar to each other. However, in the practical applications, it is generally expected that the degree of similarity would describe to what extent the IFVs are similar, so the most similar (identical) IFVs should have the largest degree of similarity, which can not be reflected by (17). Moreover, we find that if d(α1 , α2c ) → 0, then s(α1 , α2 ) → ∞, which is inconvenient in the practical applications because it is somewhat difficult to express these enormous numbers. In the next section, we shall extend the idea of the TOPSIS of Hwang and Yoon (1981) to develop a new similarity measure between two IFSs, which can not only overcome the disadvantages of the similarity measure (17), but also preserve the advantage of (17), i.e. still will examine if the compared values are more similar or more dissimilar to each other. 3 Similarity measure between IFSs Definition 1 Let α1 and α2 be two IFVs, and α2c be the complement of α2 , then  s˙ (α1 , α2 ) =

0.5,

d(α1 ,α2c ) , d(α1 ,α2 )+d(α1 ,α2c )

if α1 = α2 = α2c ; otherwise

(18)

is called the degree of similarity between α1 and α2 . From (18), we know that s˙ (α1 , α2 ) have the following desirable properties: (1) (2) (3) (4) (5)

0 ≤ s˙ (α1 , α2 ) ≤ 1; s˙ (α1 , α2 ) = s˙ (α2 , α1 ) = s˙ (α1c , α2c ); s˙ (α1 , α2c ) = s˙ (α1c , α2 ); s˙ (α1 , α2 ) = 1 ⇔ α1 = α2 which means the identity of α1 and α2 ; s˙ (α1 , α2 ) > 0.5 ⇔ d(α1 , α2 ) < d(α1 , α2c ) which means that α1 is more similar to α2 than α2c ;

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(6) s˙ (α1 , α2 ) = 0.5 ⇔ d(α1 , α2 ) = d(α1 , α2c ) which means that α1 is to the same extent similar to α2 and α2c ; (7) s˙ (α1 , α2 ) < 0.5 ⇔ d(α1 , α2 ) > d(α1 , α2c ) which means that α1 is more similar to α2c than α2 ; (8) s˙ (α1 , α2 ) = 0 ⇔ α1 = α2c which means the complete dissimilarity of α1 and α2 . Based on Definition 1, in the following, we define the degree of similarity between two IFSs: Definition 2 Let A and B be two IFSs in X, and B c be the complement of B, then n n d(α j , β cj ) 1
1
s˙ (A, B) = s˙ (α j , β j ) = n n d(α j , β j ) + d(α j , β cj ) j=1

(19)

j=1

is called the degree of similarity between A and B, where α j and β j are j-th IFVs of A and B respectively. Especially,˙s (A, B) = 1, i.e., A = B means the identity of A and B; s˙ (A, B) > 0.5 indicates that A is more similar to B than B c ; s˙ (A, B) = 0.5 indicates that A is to the same extent similar to B and B c ; s˙ (A, B) < 0.5 means that A is more similar to B c than B; and s˙ (A, B) = 0, i.e., A = B c means the complete dissimilarity of A and B. From (19), we have: (1) 0 ≤ s˙ (A, B) ≤ 1; (2) s˙ (A, B) = s˙ (B, A) = s˙ (B c , Ac ); (3) s˙ (A, B c ) = s˙ (Ac , B). 4 Consensus analysis in group decision making based on intuitionistic fuzzy preference relations In this section, we shall apply the developed similarity measure between two IFSs (or IFVs) to group decision making based on intuitionistic fuzzy preference relations. The group decision making problem considered in this paper can be described as follows: Let X = {x1 , x2 , . . . , xn } be a discrete set of alternatives, and let E = {e1 , e2 , . . . , em } be the set of experts. Let λ =(λ1 , λ2 , . . . , λm )T be the weight vector of experts, where λk > 0, k = 1, 2, . . . , m, m k=1 λk = 1. In the process of decision making, an expert usually needs to provide his/her preference information over alternatives. Especially, in some real-life situations, such as negotiation processes, the high technology project investment of venture capital firms, supply chain management, etc., the expert may provide his/her preferences over alternatives to a certain degree, but it is possible that he/she is not so sure about it (Deschrijver and Kerre 2003). Thus, it is very suitable to express the expert’s preference values with intuitionistic fuzzy values rather than exact numerical values or linguistic variables (Atanassov et al. 2005; Herrera et al. 2005; Szmidt and Kacprzyk 2002; Xu and Yager 2006; Xu 2007). Szmidt and Kacprzyk (2002) first generalized the fuzzy preference relation to the intuitionistic

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fuzzy preference relation. Later, Xu (2007) introduced the concepts of intuitionistic fuzzy preference relation as below: Definition 3 An intuitionistic fuzzy preference relation R on the set X is represented by a matrix R = (ri j )n×n ⊂ X × X with ri j = (µi j , vi j , πi j ), for all i, j = 1, 2, . . . , n, where ri j is an IFV, composed by the certainty degree µi j to which xi is preferred to x j , the certainty degree vi j to which xi is non-preferred to x j , and the hesitancy degree πi j of xi is preferred to x j . Furthermore, µi j ,vi j and πi j satisfy the following characteristics: 0 ≤ µi j + vi j ≤ 1, µ ji = vi j , v ji = µi j , πi j = π ji = 1 − µi j − vi j = 1 − µ ji − v ji µii = vii = 0.5, πii = 0, for all i, j = 1, 2, . . . , n

(20)

Suppose that the experts ek (k = 1, 2, . . . , m) provide intuitionistic fuzzy preferences for each pair of alternatives, and construct the intuitionistic fuzzy preference relations  (k)  R (k) = ri j n×n (k = 1, 2, . . . , m), respectively. Then, we have:  (k)  Theorem 1 Let R (k) = ri j n×n (k = 1, 2, . . . , m) be m intuitionistic fuzzy preference relations given by the experts ek (k = 1, 2, . . . , m) respectively, and λ =  (k) (k) (k)  (k) (λ1 , λ2 , . . . , λm )T be the weight vector of experts, where ri j = µi j , vi j , πi j , m λk > 0, k = 1, 2, . . . , m, k=1 λk = 1, then the aggregation R = (ri j )n×n of  (k)  (k) R = ri j n×n (k = 1, 2, . . . , m) is also an intuitionistic fuzzy preference relation, where ri j = (µi j , vi j , πi j ), µi j =

m


λk µi(k) j , vi j =

k=1

=

m


m


λk vi(k) j , πi j

k=1

(k)

λk πi j , µii = vii = 0.5, πii = 0, i, j = 1, 2, . . . , n

(21)

k=1

 (k)  Definition 4 Let R (k) = ri j n×n (k = 1, 2, . . . , m) be m intuitionistic fuzzy preference relations, and R = (ri j )n×n be their aggregated (or collective) intuitionistic fuzzy preference relation, then n n    1

 (k) s˙ R (k) , R = 2 s˙ ri j , ri j n

(22)

i=1 j=1

 (k)  is called the degree of similarity between R (k) and R, where s ri j , ri j is the degree (k)

of similarity between ri j and ri j , calculated by (18). By Definition 4, we have the following properties:

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  0 ≤ s˙ R (k) , R ≤ 1;   c     (k)  s˙ R , R = s˙ R, R (k) = s˙ R c , R (k) ;   c   s˙ R (k) , R c = s˙ R (k) , R ;  (k)  s˙ R , R = 1 if and only if R (k) and R is identical (or completely similar);   s˙ R (k) , R = 0 if and only if R (k) and R is completely dissimilar.

  Definition 5 Let R (k) = ri(k) j n×n (k = 1, 2, . . . , m) be m intuitionistic fuzzy preference relations, and R = (ri j )n×n be their collective intuitionistic fuzzy preference relation, then R (k) and R are of acceptable similarity, if   s˙ R (k) , R > α

(23)

where α is the threshold of acceptable similarity, which can be determined by the experts in advance in practical applications. In general, we take α = 0.5.   In the process of group decision making, if s˙ R (k) , R ≤ α, then we shall return R (k) together with R to the expert ek , and at the same time, inform him/her of some elements of R (k) with small degrees of similarity, which are needed to be revaluated. We repeat this procedure until R (k) and R are of acceptable similarity. Example 1 For a group decision making problem, there are four alternatives xi (i = 1, 2, 3, 4), and three experts ek (k = 1, 2, 3) (whose weight vector is λ = (0.5, 0.3, 0.2)T ). The experts ek (k = 1, 2, 3) compare these four alternatives and construct the  (k)  intuitionistic fuzzy preference relations R (k) = ri j 4×4 (k = 1, 2, 3), respectively, shown as follows: ⎡

R (1)

(0.5, 0.5, 0) ⎢ (0.4, 0.2, 0.4) =⎢ ⎣ (0.4, 0.5, 0.1) (0.1, 0.7, 0.2)

⎡

R (2)

(0.5, 0.5, 0) ⎢ (0.4, 0.3, 0.3) =⎢ ⎣ (0.5, 0.4, 0.1) (0.3, 0.6, 0.1)

⎡

R (3)

123

(0.5, 0.5, 0) ⎢ (0.1, 0.8, 0.1) =⎢ ⎣ (0.4, 0.3, 0.3) (0.4, 0.6, 0)

(0.2, 0.4, 0.4) (0.5, 0.5, 0) (0.5, 0.3, 0.2) (0.5, 0.4, 0.1)

(0.5, 0.4, 0.1) (0.3, 0.5, 0.2) (0.5, 0.5, 0) (0.2, 0.8, 0)

⎤ (0.7, 0.1, 0.2) (0.4, 0.5, 0.1) ⎥ ⎥ (0.8, 0.2, 0) ⎦ (0.5, 0.5, 0)

(0.3, 0.4, 0.3) (0.5, 0.5, 0) (0.4, 0.4, 0.2) (0.3, 0.5, 0.2)

(0.4, 0.5, 0.1) (0.4, 0.4, 0.2) (0.5, 0.5, 0) (0.2, 0.7, 0.1)

⎤ (0.6, 0.3, 0.1) (0.5, 0.3, 0.2) ⎥ ⎥ (0.7, 0.2, 0.1) ⎦ (0.5, 0.5, 0)

(0.8, 0.1, 0.1) (0.5, 0.5, 0) (0.3, 0.5, 0.2) (0.5, 0.4, 0.1)

(0.3, 0.4, 0.3) (0.5, 0.3, 0.2) (0.5, 0.5, 0) (0.7, 0.3, 0)

⎤ (0.6, 0.4, 0) (0.4, 0.5, 0.1) ⎥ ⎥ (0.3, 0.7, 0) ⎦ (0.5, 0.5, 0)

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By (15), we get the collective intuitionistic fuzzy preference relation R = (ri j )4×4 : ⎡

(0.5, 0.5, 0) ⎢ (0.34, 0.25, 0.31) R =⎢ ⎣ (0.43, 0.43, 0.14) (0.22, 0.65, 0.13)

(0.25, 0.34, 0.31) (0.5, 0.5, 0) (0.43, 0.37, 0.20) (0.44, 0.43, 0.13)

(0.43, 0.43, 0.14) (0.37, 0.43, 0.20) (0.5, 0.5, 0) (0.30, 0.67, 0.03)

⎤ (0.65, 0.22, 0.13) (0.43, 0.44, 0.13) ⎥ ⎥ (0.67, 0.30, 0.03) ⎦ (0.5, 0.5, 0)

then by (18), we have:   (1)   (1)   (1)   (1) s˙ r11 , r11 = s˙ r22 , r22 = s˙ r33 , r33 = s˙ r44 , r44 = 0.50  (1)   (1)   (1)   (1)  s˙ r12 , r12 = s˙ r21 , r21 = 0.66, s˙ r13 , r13 = s˙ r31 , r31 = 0.50  (1)   (1)   (1)   (1)  s˙ r14 , r14 = s˙ r41 , r41 = 0.82, s˙ r23 , r23 = s˙ r32 , r32 = 0.75  (1)   (1)   (1)   (1)  s˙ r24 , r24 = s˙ r42 , r42 = 0.54, s˙ r34 , r34 = s˙ r43 , r43 = 0.79  (2)   (2)   (2)   (2)  s˙ r11 , r11 = s˙ r22 , r22 = s˙ r33 , r33 = s˙ r44 , r44 = 0.50  (2)   (2)   (2)   (2)  s˙ r12 , r12 = s˙ r21 , r21 = 0.63, s˙ r13 , r13 = s˙ r31 , r31 = 0.50  (2)   (2)   (2)   (2)  s˙ r14 , r14 = s˙ r41 , r41 = 0.83, s˙ r23 , r23 = s˙ r32 , r32 = 0.50  (2)   (2)   (2)   (2)  s˙ r24 , r24 = s˙ r42 , r42 = 0.48, s˙ r34 , r34 = s˙ r43 , r43 = 0.82  (3)   (3)   (3)   (3)  s˙ r11 , r11 = s˙ r22 , r22 = s˙ r33 , r33 = s˙ r44 , r44 = 0.50  (3)   (3)   (3)   (3)  s˙ r12 , r12 = s˙ r21 , r21 = 0.45, s˙ r13 , r13 = s˙ r31 , r31 = 0.50  (3)   (3)   (3)   (3)  s˙ r14 , r14 = s˙ r41 , r41 = 0.68, s˙ r23 , r23 = s˙ r32 , r32 = 0.35  (3)   (3)   (3)   (3)  s˙ r24 , r24 = s˙ r42 , r42 = 0.54, s˙ r34 , r34 = s˙ r43 , r43 = 0.07

Thus, from (22), it follows that       s˙ R (1) , R = 0.63, s˙ R (2) , R = 0.60, s˙ R (3) , R = 0.45   Without loss of generality, here, we suppose that α = 0.5. Since s˙ R (1) , R > 0.5,    s˙ R (2) , R > 0.5 and s˙ R (3) , R < 0.5, then R (1) and R, R (2) and R are of acceptable similarity, but the degree of similarity between R (3) and R is unacceptable. Con  (3)   (3)   (3)   (3) sider that the similarity degrees s˙ r12 , r12 , s˙ r21 , r21 , s˙ r23 , r23 , s˙ r32 , r32 ,  (3)   (3)   (3)   (3)  s˙ r34 , r34 , and s˙ r43 , r43 are less than α, especially, s˙ r34 , r34 and s˙ r43 , r43 have the smallest similarity degrees. Thus, we need to return R (3) together with R to the (3) (3) (3) expert e3 , and suggest him/her to revaluate the elements r12 , r23 and r34 . Suppose that  (3)  the revaluated intuitionistic fuzzy preference relation (denoted by R¯ (3) = r¯i j 4×4 ) is as follows: 

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⎡

R¯ (3)

(0.5, 0.5, 0) ⎢ (0.5, 0.4, 0.1) =⎢ ⎣ (0.4, 0.3, 0.3) (0.4, 0.6, 0)

(0.4, 0.5, 0.1) (0.5, 0.5, 0) (0.5, 0.3, 0.2) (0.5, 0.4, 0.1)

(0.3, 0.4, 0.3) (0.3, 0.5, 0.2) (0.5, 0.5, 0) (0.3, 0.6, 0)

⎤ (0.6, 0.4, 0) (0.4, 0.5, 0.1) ⎥ ⎥ (0.6, 0.3, 0.1) ⎦ (0.5, 0.5, 0)

Then by (21), we aggregate the individual intuitionistic fuzzy preference relations R (1) , R (2) and R¯ (3) into the collective intuitionistic fuzzy preference relation R¯ = (¯ri j )4×4 : ⎡

(0.5, 0.5, 0) ⎢ (0.42, 0.27, 0.31) R¯ = ⎢ ⎣ (0.43, 0.43, 0.14) (0.22, 0.65, 0.13)

(0.27, 0.42, 0.31) (0.5, 0.5, 0) (0.47, 0.33, 0.20) (0.44, 0.43, 0.13)

(0.43, 0.43, 0.14) (0.33, 0.47, 0.20) (0.5, 0.5, 0) (0.22, 0.73, 0.05)

⎤ (0.65, 0.22, 0.13) (0.43, 0.44, 0.13) ⎥ ⎥ (0.73, 0.22, 0.05) ⎦ (0.5, 0.5, 0)

then by (18), we have:   (1)   (1) s˙ r12 , r¯12 = s˙ r21 , r¯21 = 0.71,  (1)   (1)  s˙ r34 , r¯34 = s˙ r43 , r¯43 = 0.89,  (2)   (2)  s˙ r34 , r¯34 = s˙ r43 , r¯43 = 0.91,  (3)   (3)  s˙ r¯23 , r¯23 = s˙ r¯32 , r¯32 = 0.85,

 (1)   (1)  s˙ r23 , r¯23 = s˙ r32 , r¯32 = 0.85  (2)   (2)  s˙ r12 , r¯12 = s˙ r21 , r¯21 = 0.81  (3)   (3)  s˙ r¯12 , r¯12 = s˙ r¯21 , r¯21 = 0.52  (3)   (3)  s˙ r¯34 , r¯34 = s˙ r¯43 , r¯43 = 0.77

and the other elements keep constant. Thus, from (22), it follows that       s˙ R (1) , R¯ = 0.66, s˙ R (2) , R¯ = 0.63, s˙ R¯ (3) , R¯ = 0.61 hence, each individual intuitionistic fuzzy preference relation and the collective intuitionistic fuzzy preference relation are of acceptable similarity. In the next section, we shall extend the developed similarity measure to the interval-valued intuitionistic fuzzy set theory. 5 Similarity measure between IVIFSs Atanassov and Gargov (1989) introduced the notion of interval-valued intuitionistic fuzzy set (IVIFS), which is characterized by a membership function and a non-membership function, whose values are intervals rather than exact numbers: Let a set X be fixed, an IVIFS A˜ over X is an object having the form: A˜ = {
x, µ˜ A˜ (x), v˜ A˜ (x)|x ∈ X }

(24)





where µ˜ A˜ (x) = µ˜ L˜ (x), µ˜ U˜ (x) ⊂ [0, 1] and v˜ A˜ (x) = v˜ L˜ (x), v˜ U˜ (x) ⊂ [0, 1] are A

A

A

A

intervals, µ˜ L˜ (x) = inf µ˜ A˜ (x), µ˜ U˜ (x) = sup µ˜ A˜ (x), v˜ L˜ (x) = inf v˜ A˜ (x), v˜ U˜ (x) = A A A A sup v˜ A˜ (x), and

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133

µ˜ UA˜ (x) + v˜ U (x) ≤ 1, for all x ∈ X A˜

(25)

and π˜ A˜ (x) = [π˜ L˜ (x), π˜ U˜ (x)], where A

A

(x), π AU˜ (x) = 1 − µ LA˜ (xi ) − v AL˜ (x), for all x ∈ X π AL˜ (x) = 1 − µUA˜ (xi ) − vU A˜ (26) Especially, if µ A˜ (x) = µ˜ L˜ (x) = µ˜ U˜ (x) and v A˜ (x) = v˜ L˜ (x) = v˜ U˜ (x), then, A˜ is A A A A ˜ where reduced to an IFS. A˜ c is denoted as the complement of A, A˜ c = {
x, v˜ A (x), µ˜ A (x)|x ∈ X }

(27)

Here, we call α˜ = (µ˜ α˜ , v˜α˜ , π˜ α˜ ) the interval-valued intuitionistic fuzzy value (IVIFV), where L U

U ˜U µ˜ α˜ = µ˜ αL˜ , µ˜ U α˜ ⊂ [0, 1], v˜ α˜ = v˜ α˜ , v˜ α˜ ⊂ [0, 1], µ α˜ + v˜ α˜ ≤ 1,

L U U π˜ α˜ = π˜ α˜ , π˜ α˜ = 1 − µ˜ U ˜ αL˜ − v˜αL˜ α˜ − v˜ α˜ , 1 − µ

(28)

˜ where Similar to (27), α˜ c is denoted as the complement of α,   α˜ c = v˜α˜ , µ˜ α˜ , π˜ α˜

(29)



 , Let α˜ i = µ˜ α˜ i , v˜α˜ i , π˜ α˜ i (i = 1, 2) be two IVIFVs, where µ˜ α˜ i = µ˜ αL˜ i , µ˜ U α ˜ i L U L U v˜α˜ i = v˜α˜ i , v˜α˜ i , π˜ α˜ i = π˜ α˜ i , π˜ α˜ i , i = 1, 2, then α˜ 1 = α˜ 2 , if and only if L L U U ˜U ˜ α˜L1 = π˜ α˜L2 , π˜ αU ˜ αU µ˜ αL˜ 1 = µ˜ αL˜ 2 , µ˜ U α˜ 1 = µ α˜ 2 , v˜ α˜ 1 = v˜ α˜ 2 , v˜ α˜ 1 = v˜ α˜ 2 , π ˜1 = π ˜2 (30)

Similar to (9), the normalized Hamming distance between α˜ 1 and α˜ 2 can be defined as: d(α˜ 1 , α˜ 2 ) =

 L L |µ˜ αL˜ 1 − µ˜ αL˜ 2 | + |µ˜ U ˜U α˜ 1 − µ α˜ 2 | + |v˜ α˜ 1 − v˜ α˜ 2 |  + |v˜αU˜ 1 − v˜αU˜ 2 | + |π˜ α˜L1 − π˜ α˜L2 | + |π˜ αU ˜ αU ˜1 − π ˜2 | 1 4

(31)

Let X = {x1 , x2 , . . . , xn } be a finite universe of discourse, A˜ = {< x j , µ˜ A˜ (x j ), v˜ A˜ (x j ) > |x j ∈ X } and B˜ = {< x j , µ˜ B˜ (x j ), v˜ B˜ (x j ) > |x j ∈ X } be two



IVIFSs, where µ˜ A˜ (x j ) = µ˜ L˜ (x j ), µ˜ U˜ (x j ) , v˜ A˜ (x j ) = v˜ L˜ (x j ), v˜ U˜ (x j ) , µ˜ B˜ (x j ) = A A A A



L µ˜ ˜ (x j ), µ˜ U˜ (x j ) , v˜ B˜ (x j ) = v˜ L˜ (x j ), v˜ U˜ (x j ) , x j ∈ X , then below we introduce the B B B B ˜ normalized Hamming distance between A˜ and B:

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Z. Xu, R. R. Yager n 
˜ B) ˜ = 1 |µ˜ LA˜ (x j ) − µ˜ LB˜ (x j )| d( A, 4n j=1

U + |µ˜ UA˜ (x j ) − µ˜ UB˜ (x j )| + |v˜ AL˜ (x j ) − v˜ BL˜ (x j )| + |v˜ U ˜ (x j ) − v˜ B˜ (x j )|  A + |π˜ AL˜ (x j ) − π˜ BL˜ (x j )| + |π˜ AU˜ (x j ) − π˜ BU˜ (x j )| (32)

Definition 6 Let α˜ 1 and α˜ 2 be two IVIFVs, and α˜ 2c be the complement of α˜ 2 , then  s(α˜ 1 , α˜ 2 ) =

0.5,

d(α˜ 1 ,α˜ 2c ) d(α˜ 1 ,α˜ 2 )+d(α˜ 1 ,α˜ 2c )

if α˜ 1 = α˜ 2 = α˜ 2c ; , otherwise

(33)

is called the degree of similarity between α˜ 1 and α˜ 2 . From (33), we have: 0 ≤ s(α˜ 1 , α˜ 2 ) ≤ 1; s(α˜ 1 , α˜ 2 ) = s(α˜ 2 , α˜ 1 ) = s(α˜ 1c , α˜ 2c ); s(α˜ 1 , α˜ 2c ) = s(α˜ 1c , α˜ 2 ); s(α˜ 1 , α˜ 2 ) = 1 ⇔ α˜ 1 = α˜ 2 which means the identity of α˜ 1 and α˜ 2 ; s(α˜ 1 , α˜ 2 ) > 0.5 ⇔ d(α˜ 1 , α˜ 2 ) < d(α˜ 1 , α˜ 2c ) which means that α˜ 1 is more similar to α˜ 2 than α˜ 2c ; (6) s(α˜ 1 , α2 ) = 0.5 ⇔ d(α˜ 1 , α˜ 2 ) = d(α˜ 1 , α˜ 2c ) which means that α˜ 1 is to the same extent similar to α˜ 2 and α˜ 2c ; (7) s(α˜ 1 , α˜ 2 ) < 0.5 ⇔ d(α˜ 1 , α˜ 2 ) > d(α˜ 1 , α˜ 2c ) which means that α˜ 1 is more similar to α˜ 2c than α˜ 2 ; (8) s(α˜ 1 , α˜ 2 ) = 0 ⇔ α˜ 1 = α˜ 2c which means the complete dissimilarity of α˜ 1 and α˜ 2 . (1) (2) (3) (4) (5)

Based on Definition 6, we define the degree of similarity between two IVIFSs as follows: ˜ then Definition 7 Let A˜ and B˜ be two IVIFSs in X, and B˜ c be the complement of B, n n d(α˜ j , β˜ cj ) 1
1
˜ ˜ ˜ s(α˜ j , β j ) = s( A, B) = n n d(α˜ j , β˜ j ) + d(α˜ j , β˜ cj ) j=1 j=1

(34)

˜ where α˜ j and β˜ j are j-th IVIFVs is called the degree of similarity between A˜ and B, ˜ ˜ of A and B, respectively. From (34), we have: ˜ B) ˜ ≤ 1; (1) 0 ≤ s( A, ˜ B) ˜ = s( B, ˜ A) ˜ = s( B˜ c , A˜ c ); (2) s( A, ˜ B˜ c ) = s( A˜ c , B). ˜ (3) s( A, ˜ B) ˜ = 1, i.e., A˜ = B˜ means the identity of A˜ and B; ˜ s( A, ˜ B) ˜ > 0.5 Especially, s( A, c ˜ ˜ ˜ ˜ ˜ indicates that A is more similar to B than B ; s( A, B) = 0.5 indicates that A is to the ˜ B) ˜ < 0.5 indicates that A˜ is more similar to same extent similar to B˜ and B˜ c ; s( A, c c ˜ ˜ ˜ ˜ ˜ ˜ ˜ B than B; s( A, B) = 0, i.e., A = B means the complete dissimilarity of A˜ and B.

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135

Example 2 Let A˜ and B˜ be two IVIFSs, α˜ j and β˜ j are j-th IVIFVs of A˜ and B˜ respectively, j = 1, 2, 3, where α˜ 1 = ([0.3, 0.4], [0.2, 0.3], [0.3, 0.5]), α˜ 2 = ([0.4, 0.5], [0.1, 0.2], [0.3, 0.5]) α˜ 3 = ([0.3, 0.6], [0.2, 0.3], [0.1, 0.5]), β˜1 = ([0.5, 0.6], [0.1, 0.2], [0.2, 0.4]) β˜2 = ([0.2, 0.4], [0.4, 0.5], [0.1, 0.4]), β˜3 = ([0.4, 0.5], [0.3, 0.5], [0, 0.3]) ˜ then the IVIFVs β˜ c ( j = 1, 2, 3) in B˜ c are as follows: Let B˜ c be the complement of B, j β˜1c = ([0.1, 0.2], [0.5, 0.6], [0.2, 0.4]), β˜2c = ([0.4, 0.5], [0.2, 0.4], [0.1, 0.4]), β˜3c = ([0.3, 0.5], [0.4, 0.5], [0, 0.3]) ˜ then by (33), If we use (34) to calculate the degree of similarity between A˜ and B, we have: s(α˜ 1 , β˜1 ) = 0.6, s(α˜ 2 , β˜2 ) = 0.33, s(α˜ 3 , β˜3 ) = 0.5 hence s˙ (A, B) =

1 × (0.6 + 0.33 + 0.5) = 0.48 < 0.5 3

˜ In addition, s(α˜ 1 , β˜1 ) = 0.6 which indicates that A˜ is more similar to B˜ c than B. indicates that α˜ 1 is more similar to β˜1 than β˜1c ; s(α˜ 2 , β˜2 ) = 0.33 indicates that α˜ 2 is more similar to β2c than β2 ; s(α˜ 3 , β˜3 ) = 0.5 indicates that α˜ 3 is to the same extent similar to β˜3 and β˜3c . In what follows, we apply the above similarity measure between two IVIFSs (or IVIFVs) to group decision making with interval-valued intuitionistic fuzzy preference information. Based on IVIFSs, we first define the concept of interval-valued intuitionistic preference relation: Definition 8 An interval-valued intuitionistic preference relation R˜ on the set X is represented by a matrix R˜ = (˜ri j )n×n ⊂ X × X with r˜i j = (µ˜ i j , v˜i j , π˜ i j ), for all i, j = 1, 2, . . . , n, where r˜i j is an IVIFV, composed by the certainty degree µ˜ i j to which xi is preferred to x j , the certainty degree v˜i j to which xi is non-preferred to x j , and the hesitancy degree π˜ i j of xi is preferred to x j . Furthermore,µ˜ i j ,v˜i j and π˜ i j satisfy the following characteristics:

L U µ˜ i j = µ˜ iLj , µ˜ U ˜ ji = v˜i j , v˜ ji = µ˜ i j , i j ⊂ [0, 1], v˜i j = v˜i j , v˜i j ⊂ [0, 1], µ U π˜ i j = π˜ ji = [1 − µ˜ U ˜ iLj − v˜iLj ] i j − v˜i j , 1 − µ U µ˜ ii = v˜ii = [0.5, 0.5], π˜ ii = [0, 0], µ˜ U i j + v˜i j ≤ 1, for all i, j = 1,2,…, n (35)

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Suppose that the experts ek (k = 1, 2, . . . , m) provide interval-valued intuitionistic fuzzy preferences for each pair of alternatives, and construct the interval-valued intui  tionistic fuzzy preference relations R˜ (k) = r˜i(k) j n×n (k = 1, 2, . . . , m), respectively. Then, we have the following result:   Theorem 2 Let R˜ (k) = r˜i(k) j n×n (k = 1, 2, . . . , m) be m interval-valued intuitionistic fuzzy preference relations given by the experts dk (k = 1, 2, . . . , m) respectively,  (k) (k) and λ = (λ1 , λ2 , . . . , λm )T be the weight vector of experts, where r˜i j = µ˜ i j ,  (k)  L  (k) U (k)  (k)  L  (k) U (k) (k)  (k) , v˜i j = v˜i j , v˜i j , λk > 0, k = 1, 2, v˜i j , π˜ i j , µ˜ i j = µ˜ i j , µ˜ i j  (k)  m (k) . . . , m, k=1 λk = 1, then the aggregation R˜ = (˜ri j )n×n of R˜ = r˜i j n×n (k = 1, 2, . . . , m) is also an interval-valued intuitionistic fuzzy preference relation, where       L U ˜ i j = π˜ iLj , π˜ iUj , r˜i j = (µ˜ i j , v˜i j , π˜ i j ), µ˜ i j = µ˜ iLj , µ˜ U i j , v˜i j = v˜i j , v˜i j , π µ˜ iLj

=

m


m m

 (k)  L  (k) U  (k)  L U L λk µ˜ i j µ˜ i j = λk µ˜ i j , v˜i j = λk v˜i j ,

k=1

v˜iUj =

m


k=1

k=1

m m
 (k) U L
 (k)  L  (k) U λk v˜i j π˜ i j = λk π˜ i j , π˜ iUj = λk π˜ i j ,

k=1

k=1

k=1

i, j = 1, 2, . . . , n

(36)

 (k)  Definition 9 Let R˜ (k) = r˜i j n×n (k = 1, 2, . . . , m) be m interval-valued intuitionistic fuzzy preference relations, R˜ = (˜ri j )n×n be their aggregated (or collective) interval-valued intuitionistic fuzzy preference relation, then n n    1

 (k) s R˜ (k) , R˜ = 2 s r˜i j , r˜i j n

(37)

i=1 j=1

  ˜ where s r˜ (k) , r˜i j is the degree is called the degree of similarity between R˜ (k) and R, ij (k)

of similarity between r˜i j and r˜i j , calculated by (32). By Definition 9, we have:   (1) 0 ≤ s R˜ (k) , R˜ ≤ 1;        (k)  ˜ R˜ (k) = s R˜ c , R˜ (k) c ; (2) s R˜ , R˜ = s R,   c   (3) s R˜ (k) , R˜ c = s R˜ (k) , R˜ ;  (k)  (4) s R˜ , R˜ = 1 if and only if R˜ (k) and R˜ is identical (or completely similar);   (5) s R˜ (k) , R˜ = 0 if and only if R˜ (k) and R˜ is completely dissimilar.  (k)  Definition 10 Let R˜ (k) = r˜i j n×n (k = 1, 2, . . . , m) be m interval-valued intuitionistic fuzzy preference relations, and R˜ = (˜ri j )n×n be the collective interval-valued intuitionistic fuzzy preference relation, then R˜ (k) and R˜ are of acceptable similarity, if

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 R˜ (k) , R˜ > α

137

(38)

where α is the threshold of acceptable similarity, which can be determined by the experts in advance in practical applications. In general, we take α = 0.5.   In the process of group decision making, if R˜ (k) , R˜ ≤ α, then we shall return R˜ (k) together with R˜ to the expert ek , and inform him/ her of some elements of R (k) with small degrees of similarity, which are needed to be revaluated. We repeat this procedure until R˜ (k) and R˜ are of acceptable similarity. Example 3 Suppose that three experts ek (k = 1, 2, 3) (whose weight vector is λ = (0.4, 0.3, 0.3)T ) compare three alternatives xi (i = 1, 2, 3) and construct the fol (k)  lowing interval-valued intuitionistic fuzzy preference relations R˜ (k) = r˜i j 3×3 (k = 1, 2, 3), respectively: ⎤ ⎡ ([0.5, 0.5], [0.5, 0.5], [0, 0]) ([0.1, 0.2], [0.7, 0.8], [0, 0.2]) ([0.6, 0.9], [0, 0.1], [0, 0.4]) (1) ˜ ⎦ ⎣ [0, 0.2]) ([0.5, 0.5], [0.5, 0.5], [0, 0]) ([0, 0.1], [0.8, 0.9], [0, 0.2]) ([0.7, 0.8], [0.1, 0.2], R = ([0, 0.1], [0.6, 0.9], [0, 0.4]) ([0.8, 0.9], [0, 0.1], [0, 0.2]) ([0.5, 0.5], [0.5, 0.5], [0, 0]) ⎡

⎤ ([0, 0.1], [0.7, 0.9], [0, 0.3]) ([0.7, 0.8], [0.1, 0.2], [0, 0.2]) ([0.5, 0.5], [0.5, 0.5], [0, 0]) ([0.2, 0.3], [0.5, 0.7], [0, 0.3]) ⎦ ([0.1, 0.2], [0.7, 0.8], [0, 0.2]) ([0.5, 0.7], [0.2, 0.3], [0, 0.3]) ([0.5, 0.5], [0.5, 0.5], [0, 0]) ([0.5, 0.5], [0.5, 0.5], [0, 0])

R˜ (2) = ⎣ ([0.7, 0.9], [0, 0.1], [0, 0.3])

R˜ (3) ⎡

⎤ ([0.5, 0.5], [0.5, 0.5], [0, 0]) ([0.2, 0.3], [0.4, 0.6], [0.1, 0.4]) ([0.6, 0.8], [0, 0.1], [0.1, 0.4]) ⎣ ⎦ ([0.4, 0.6], [0.2, 0.3], [0.1, 0.3]) ([0.5, 0.5], [0.5, 0.5], [0, 0]) ([0.1, 0.2], [0.7, 0.8], [0, 0.2]) = ([0, 0.1], [0.6, 0.8], [0.1, 0.4]) ([0.7, 0.8], [0.1, 0.2], [0, 0.2]) ([0.5, 0.5], [0.5, 0.5], [0, 0])

By (36), we get the collective interval-valued intuitionistic fuzzy preference relation R˜ = (˜ri j )4×4 : R˜ = ⎡ ⎤ ([0.5, 0.5], [0.5, 0.5], [0, 0]) ([0.10, 0.20], [0.61, 0.77], [0.03, 0.29]) ([0.63, 0.84], [0.03, 0.13], [0.03, 0.34]) ⎢ ⎥ ([0.09, 0.19], [0.68, 0.81], [0, 0.23]) ⎦ ⎣ ([0.61, 0.77], [0.10, 0.20], [0.03, 0.29]) ([0.5, 0.5], [0.5, 0.5], [0, 0]) ([0.03, 0.13], [0.63, 0.84], [0.03, 0.34]) ([0.68, 0.81], [0.09, 0.19], [0, 0.23]) ([0.5, 0.5], [0.5, 0.5], [0, 0])

then by (33), we have:  (1)   (1)   (1)   (1)   (1)  s r˜11 , r˜11 = s r˜22 , r˜22 =s r˜33 , r˜33 =0.50 s r˜12 , r˜12 =s r˜21 , r˜21 = 0.91  (1)   (1)   (1)   (1)  s r˜13 , r˜13 = s r˜31 , r˜31 = 0.92, s r˜23 , r˜23 = s r˜32 , r˜32 = 0.87  (2)   (2)   (2)  s r˜11 , r˜11 = s r˜22 , r˜22 = s r˜33 , r˜33 = 0.50  (2)   (2)   (2)   (2)  s r˜12 , r˜12 = s r˜21 , r˜21 = 0.85, s r˜13 , r˜13 = s r˜31 , r˜31 = 0.86  (2)   (2)  s r˜23 , r˜23 = s r˜32 , r˜32 = 0.77  (3)   (3)   (3)   (3)   (3)  s r˜11 , r˜11 = s r˜22 , r˜22 =s r˜33 , r˜33 =0.50, s r˜12 , r˜12 =s r˜21 , r˜21 = 0.70  (3)   (3)   (3)   (3)  s r˜13 , r˜13 = s r˜31 , r˜31 = 0.91, s r˜23 , r˜23 = s r˜32 , r˜32 = 0.97

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Thus, from (37), it follows that       s R˜ (1) , R˜ = 0.77, s R˜ (2) , R˜ = 0.72, s R˜ (3) , R˜ = 0.74 then each individual intuitionistic fuzzy preference relation and the collective intuitionistic fuzzy preference relation are of acceptable similarity, and thus we need not returning them to the experts for revaluation. 6 Concluding remarks In this paper, we have developed a new similarity measure between IFSs, which can not only overcome the disadvantages of Szmidt and Kacprzyk’s similarity measure, but also examine if the compared values are more similar or more dissimilar to each other so as to avoid drawing conclusions about strong similarity between two IFSs on the basis of the small distances between these sets. We have applied the similarity measure for consensus analysis in group decision making based on intuitionistic fuzzy preference relations. Furthermore, we have extended the similarity measure to interval-valued intuitionistic fuzzy situations, and used the extended similarity measure to assess the consensus of experts’ preferences in group decision making with interval-valued intuitionistic fuzzy information. Acknowledgment The work was supported by the National Science Fund for Distinguished Young Scholars of China (No.70625005).

References Atanassov, K. (1984). Intuitionistic fuzzy sets. In V. Sgurev (Ed.), VII ITKR’s Session, Sofia (June 1983). Central Sci. and Tech. Library, Bulg. Academy of Sciences Atanassov, K. (1999). Intuitionistic fuzzy sets: Theory and applications. Heidelberg: Physica-Verlag. Atanassov, K., & Gargov, G. (1989). Interval-valued intuitionistic fuzzy sets. Fuzzy Sets and Systems, 31, 343–349. Atanassov, K., Pasi G., & Yager R. R. (2005). Intuitionistic fuzzy interpretations of multi-criteria multiperson and multi-measurement tool decision making. International Journal of Systems Science, 36, 859–868. Bustince, H., Herrera, F, & Montero, J. (2007). Fuzzy sets and their extensions: Representation, aggregation and models. Heidelberg: Physica-Verlag. Deschrijver, G., Cornelis, C., & Kerre, E. E. (2004). On the representation of intuitionistic fuzzy t-norms and t-conorms. IEEE Transactions on Fuzzy Systems, 12, 45–61. Deschrijver, G., & Kerre, E. E. (2003). On the composition of intuitionistic fuzzy relations. Fuzzy Sets and Systems, 136, 333–361. Grzegorzewski, P. (2004). Distances between intuitionistic fuzzy sets and/or interval-valued fuzzy sets based on the Hausdorff metric. Fuzzy Sets and Systems, 148, 319–328. Herrera, F., Martínez, L., & Sánchez, P. J. (2005). Manahging non-homogeneous information in group decision making. European Journal of Operational Research, 166, 115–132. Hung, W. L., & Yang, M. S. (2004). Similarity measures of intuitionistic fuzzy sets based on Hausdorff distance. Pattern Recognition letters, 25, 1603–1611. Hwang, C. L., & Yoon, K. (1981). Multiple attribute decision making: Methods and applications. Berlin: Springer.

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