INTUITIONISTIC FUZZY CLUSTERING ALGORITHM BASED ON ...

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International Journal of Information Technology & Decision Making Vol. 12, No. 1 (2013) 95118 c World Scienti¯c Publishing Company ° DOI: 10.1142/S0219622013500053

INTUITIONISTIC FUZZY CLUSTERING ALGORITHM BASED ON BOOLE MATRIX AND ASSOCIATION MEASURE

HUA ZHAO College of Sciences PLA University of Science and Technology Nanjing, Jiangsu 210007, China zhaohua [email protected] ZESHUI XU* and ZHONG WANG† College of Sciences, PLA University of Science and Technology Nanjing, Jiangsu 210007, China *[email protected] † [email protected]

In this paper we develop a measure for calculating the association coe±cient between Atanassov's intuitionistic fuzzy sets (A-IFSs), and show its desirable axiomatic properties. Then we present an algorithm for clustering A-IFSs. The algorithm ¯rst utilizes the association coe±cient of A-IFSs to construct an association matrix, and then calculates the -cutting matrix of the association matrix no matter whether it is an equivalent matrix or not. After that, the -cutting matrix is used to cluster A-IFSs (if the -cutting matrix is just only a similarity matrix, then we can easily transform it into an equivalent matrix). Three examples are used to show the e®ectiveness of the association coe±cient and the algorithm for clustering A-IFSs. Furthermore, we extend the algorithm to cluster interval-valued intuitionistic fuzzy sets (IVIFSs), and ¯nally, we use another numerical example to illustrate the latter algorithm. Keywords: Atanassov's intuitionistic fuzzy set; interval-valued intuitionistic fuzzy set; association coe±cients; association matrix; -cutting matrix; clustering algorithm. MSC 2000: 90B50, 91B06

1. Introduction Clustering is the process of grouping data into clusters so that objects in the same cluster have high similarity in comparison to each other, but are very dissimilar to objects in other clusters.1 As a principal technique, cluster algorithms have been widely used in data mining,2,3 information retrieval,4 image retrieval,5 and bioinformatics,6 etc. Further information on clustering algorithms can be found in the literature,79 etc.

*Corresponding

author. 95

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H. Zhao, Z. S. Xu & Z. Wang

Atanassov10,11 extended fuzzy set12 to intuitionistic fuzzy set (A-IFS), each element of which is assigned a membership degree and a nonmembership degree. A-IFS seems to be more useful in dealing with fuzziness and uncertainty than fuzzy set. Gau and Buehrer13 introduced the concept of vague set, which is equivalent to A-IFS.14 Atanassov and Gargov15 further generalized A-IFS into the notion of interval-valued intuitionistic fuzzy set (IVIFS). Up to now, the A-IFS theory has been applied to various ¯elds, including decision making,1621 medical diagnosis,22 and pattern recognition,23,24 etc. In recent years, cluster analysis has been in vigorous development. However, there still exist many problems to be solved. One important problem concerns the data type of clustering algorithms. For example, how to cluster samples with intuitionistic fuzzy information? To solve this problem, some scholars have attempted to propose e±cient and convenient algorithms. Zhang et al.25 constructed a kind of intuitionistic fuzzy similarity matrix by de¯ning the concept of intuitionistic fuzzy similarity degree, then changed it into an intuitionistic fuzzy equivalent matrix, and ¯nally presented a clustering method based on the -cutting matrix. Xu et al.26 pointed out that this clustering technique has some shortcomings for requiring too much computational e®ort, producing the loss of too much information in the process of calculation and not being able to be extended to cluster IVIFSs. Chen et al.27 put forward a di®erent clustering algorithm based on intuitionistic fuzzy equivalent relations by giving new composition operations based on modular operations between A-IFSs, which solve the problem of the loss of too much information in some degree. Afterwards, Xu et al.26 proposed a new clustering technique by transforming an association matrix into an equivalent association matrix, the -cutting matrix is used to cluster the given A-IFSs, and then, they extended the algorithm to IVIFSs. Cai et al.28 presented a clustering method based on the intuitionistic fuzzy equivalent dissimilarity matrix and ð; Þ-cutting matrices. A common characteristic of the above clustering techniques is that all these intuitionistic fuzzy clustering methods are on the basis of the intuitionistic fuzzy equivalent matrices and the transitive closure technique of transformating a similarity matrix into an equivalent one. However, when the quantity of the samples to be clustered becomes large, the amount of computational e®ort will be very large. Can we directly cluster the given A-IFSs by the -cutting matrix of a similarity matrix without complicated transformation? It is a pity that there is less work on this topic. To overcome the shortcoming, in this paper, we propose a new intuitionistic fuzzy clustering procedure to deal with intuitionistic fuzzy data by -cutting matrix of a similarity matrix, and extend the algorithm to cluster IVIFSs. To do that, the rest of the paper is organized as follows. Section 2 reviews the concept of A-IFSs and introduces the intuitionistic fuzzy association coe±cient. Section 3 introduces the concepts and properties of equivalent association matrix and Boole matrix, and then gives a clustering algorithm for A-IFSs called intuitionistic fuzzy Boole clustering algorithm. Section 4 uses several examples to illustrate the e®ectiveness of the developed association coe±cient and the intuitionistic fuzzy Boole clustering

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algorithm. In Sec. 5, we extend the algorithm to cluster IVIFSs and then use a numerical example to illustrate our algorithm, and conclude the paper in Sec. 6. 2. Intuitionistic Fuzzy Association Measures 2.1. The basic concepts about A-IFS Let a set X be a universe of discourse. A fuzzy set12 is de¯ned as F ¼ fhx; F ðxÞij x 2 Xg, where the function F : X ! ½0; 1 de¯nes the degree of membership of the element x to the set F. Atanassov11 generalized the concept of fuzzy set by adding a hesitation degree, and de¯ned the concept of intuitionistic fuzzy set (A-IFS) as A ¼ fhx; A ðxÞ; A ðxÞijx 2 Xg, where the function A : X ! ½0; 1 de¯nes the degree of membership, and A : X ! ½0; 1 de¯nes the degree of nonmembership of x to A, respectively, with the condition: 0 A ðxÞ þ A ðxÞ 1. For each A-IFS A in X, we call A ðxÞ ¼ 1 A ðxÞ A ðxÞ the degree of indeterminacy of x to A, for all x 2 X. Obviously, if A ðxÞ ¼ 0, then the A-IFS A reduces to a fuzzy set. 2.2. The association coe±cient of A-IFSs Since clustering is the grouping of similar objects, we usually need to ¯nd some sort of measure that can determine the degree of the relationship between two objects. Usually, there are three main types of measures which can estimate this relation: distance measures, similarity measures and association measures. The choice of a good measure will directly in°uence the clustering e®ect. Next we shall seek for some association measures to be prepared for clustering analysis. An association measure is an important tool for determining the degree of the relationship between two objects. Many scholars have given various association measures. We ¯rst review an axiomatic de¯nition of association measure of A-IFSs, which has been widely employed in the literature on intuitionistic fuzzy theory. For better description, let ðXÞ be the set of all A-IFSs, and A; B 2 ðXÞ. De¯nition 1.29 Let be a mapping : ððxÞÞ2 ! ½0; 1, then the association coe±cient between two A-IFSs A and B is de¯ned as ðA; BÞ, which has the following properties: (1) 0 ðA; BÞ 1; (2) ðA; BÞ ¼ 1 if A ¼ B; and (3) ðA; BÞ ¼ ðB; AÞ. Moreover, Gerstenkorn and Ma¯ko29 proposed a method to calculate the association coe±cient of A-IFSs, which was formulated in the following way: Pn j¼1 A ðxj Þ B ðxj Þ þ A ðxj Þ B ðxj Þ ﬃ: 1 ðA; BÞ ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð1Þ Pn Pn 2 2 2 2 j¼1 ð A ðxj Þ þ A ðxj ÞÞ j¼1 ð B ðxj Þ þ B ðxj ÞÞ Hong and Hwang30 further considered the case where the set X is in¯nite and de¯ned another association coe±cient of A-IFSs as follows: R ðA ðxÞ B ðxÞ þ A ðxÞ B ðxÞÞdx X ﬃ: 2 ðA; BÞ ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð2Þ R R 2 ðxÞ þ 2 ðxÞÞdx 2 ðxÞ þ 2 ðxÞÞdx ð ð B B A A X X

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1 ðA; BÞ and 2 ðA; BÞ satisfy the three conditions in De¯nition 1, yet they cannot guarantee the necessity in the condition (2) of De¯nition 1. Hong and Hwang30 and Mitchell31 pointed out that if association coe±cients do not guarantee the necessity in the condition (2), then some situations where the obtained results are counter-intuitive will appear, although in most cases the association coe±cient may give reasonable result. For this reason, Xu et al.26 proposed another axiomatic de¯nition for the association measure of A-IFSs, which is an improved version of De¯nition 1. De¯nition 2.26 Let be a mapping : ððxÞÞ2 ! ½0; 1, then the association coe±cient between two A-IFSs A and B is de¯ned as ðA; BÞ, which has the following properties: (1) 0 ðA; BÞ 1; (2) ðA; BÞ ¼ 1 if and only if A ¼ B; and (3) ðA; BÞ ¼ ðB; AÞ. Furthermore, Szmidt and Kacprzyk32 pointed out that omitting any one of the three parameters may lead to incorrect results, and therefore, we should take the three parameters into account when computing the association coe±cients between A-IFSs. Based on the two ideas above when constructing an association coe±cient between A-IFSs, we will improve Eq. (1) to a new form, satisfying all the conditions proposed by Hong and Hwang,30 Mitchell,31 and Szmidt and Kacprzyk32: Pn j¼1 ðA ðxj Þ B ðxj Þ þ A ðxj Þ B ðxj Þ þ A ðxj Þ B ðxj ÞÞ ﬃ: 3 ðA; BÞ ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Pn Pn 2 2 2 2 2 2 j¼1 ð A ðxj Þ þ A ðxj Þ þ A ðxj ÞÞ j¼1 ð B ðxj Þ þ B ðxj Þ þ B ðxj ÞÞ ð3Þ It is clear that 3 ðA; BÞ takes the third parameter of A-IFS (the hesitation degree) into consideration, moreover, we will prove that it also satis¯es all the three conditions of De¯nition 2. Proof. Because A; B 2 ðXÞ, then from the concept of A-IFS and Eq. (3), we know that 3 ðA; BÞ 0. To prove the inequality 3 ðA; BÞ 1, we can use the famous CauchySchwarz inequality: vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ! ! u n n n X X u X t 2 2 ð4Þ ab a b i i

i¼1

i

i¼1

i

i¼1

with equality if and only if the two vectors a ¼ ða1 ; a2 ; . . . ; an Þ and b ¼ ðb1 ; b2 ; . . . ; bn Þ are linearly dependent, that is, there is a nonzero real number such that a ¼ b. From Eq. (4), we know that 3 ðA; BÞ 1 with equality if and only if there is a nonzero real number such that A ðxi Þ ¼ B ðxi Þ;

A ðxi Þ ¼ B ðxi Þ;

A ðxi Þ ¼ B ðxi Þ;

for all xi 2 X

ð5Þ

while because A ðxi Þ ¼ 1 A ðxi Þ A ðxi Þ;

B ðxi Þ ¼ 1 B ðxi Þ B ðxi Þ;

for all xi 2 X ð6Þ

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then by Eq. (5), we know that ¼ 1, and thus, 3 ðA; BÞ ¼ 1 if and only if A ¼ B. Hence we complete the proof of the conditions (1) and (2) in De¯nition 2. In addition, by Eq. (3) we know that Pn j¼1 ðA ðxj Þ B ðxj Þ þ A ðxj Þ B ðxj Þ þ A ðxj Þ B ðxj ÞÞ ﬃ 3 ðA; BÞ ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Pn Pn 2 2 2 2 2 2 j¼1 ð A ðxj Þ þ A ðxj Þ þ A ðxj ÞÞ j¼1 ð B ðxj Þ þ B ðxj Þ þ B ðxj ÞÞ Pn j¼1 ðB ðxj Þ A ðxj Þ þ B ðxj Þ A ðxj Þ þ B ðxj Þ A ðxj ÞÞ ﬃ ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Pn Pn 2 2 2 2 2 2 j¼1 ð B ðxj Þ þ B ðxj Þ þ B ðxj ÞÞ j¼1 ð A ðxj Þ þ A ðxj Þ þ A ðxj ÞÞ ¼ 3 ðB; AÞ:

ð7Þ

Thus, the condition (3) in De¯nition 2 also holds. It is very interesting that when we add the third parameter, i.e., the indeterminacy degree of A-IFSs, to 1 ðA; BÞ, we get a good association coe±cient 3 ðA; BÞ, which not only takes the third parameter of A-IFS (the hesitation degree) into consideration, but also satis¯es all the three conditions of De¯nition 2. In many cases, for instance, in cluster analysis, the weights of the attributes are always di®erent, so we should take them into account, and thus extend 3 ðA; BÞ to the following form: Pn j¼1 wj ðA ðxj Þ B ðxj Þ þ A ðxj Þ B ðxj Þ þ A ðxj Þ B ðxj ÞÞ ﬃ; 4 ðA; BÞ ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Pn Pn 2 2 2 2 2 2 j¼1 wj ð A ðxj Þ þ A ðxj Þ þ A ðxj ÞÞ j¼1 wj ð B ðxj Þ þ B ðxj Þ þ B ðxj ÞÞ ð8Þ where w ¼ ðw1 ; w2 ; . . . ; wn ÞT is the weight vector of xj ðj ¼ 1; 2; . . . ; nÞ with wj 0, Pn j ¼ 1; 2; . . . ; n and j¼1 wj ¼ 1. Similar to Eq. (3), Eq. (8) also satis¯es all the conditions of De¯nition 2. If the universe of discourse, X, is continuous and the weight of the element R x 2 X ¼ ½a; b is wðxÞ, where wðxÞ 0 and ab wðxÞdx ¼ 1, then Eq. (8) is transformed into the following form: Rb wðxÞðA ðxÞB ðxÞ þ A ðxÞ B ðxÞ þ A ðxÞB ðxÞÞdx a ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ: q 5 ðA; BÞ ¼ R Rb b 2 2 2 2 2 2 wðxÞð ðxÞ þ ðxÞ þ ðxÞÞdx wðxÞð ðxÞ þ ðxÞ þ ðxÞÞdx B B B A A A a a ð9Þ 1 2 ½0; 1 (in this case, If all the elements have the same importance, i.e., wðxÞ ¼ ba ðb aÞ 1Þ, for any x 2 ½a; b, then Eq. (9) is replaced by Rb ðA ðxÞB ðxÞ þ A ðxÞ B ðxÞ þ A ðxÞB ðxÞÞdx a ﬃ : ð10Þ 6 ðA; BÞ ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Rb 2 Rb 2 2 2 2 2 ð ðxÞ þ ðxÞ þ ðxÞÞdx ð ðxÞ þ ðxÞ þ ðxÞÞdx B B B A A A a a

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3. Intuitionistic Fuzzy Clustering Algorithm In this section, we will focus on investigating the algorithm for clustering intuitionistic fuzzy information. 3.1. Intuitionistic fuzzy association matrix Let X be a discrete universe of discourse, X ¼ fx1 ; x2 ; . . . ; xn g, and w ¼ ðw1 ; w2 ; . . . ; wn ÞT be the weight vector of xi ði ¼ 1; 2; . . . ; nÞ, with wi 0, i ¼ 1; 2; . . . ; n, and Pn i¼1 wi ¼ 1. Let Aj ðj ¼ 1; 2; . . . ; mÞ be a collection of m A-IFSs, where Aj ¼ fhxi ; Aj ðxi Þ; Aj ðxi Þijxi 2 Xg;

i ¼ 1; 2; . . . ; n; j ¼ 1; 2; . . . ; m

ð11Þ

Aj ðxi Þ ¼ 1 Aj ðxi Þ Aj ðxi Þ;

i ¼ 1; 2; . . . ; n; j ¼ 1; 2; . . . ; m

ð12Þ

and

is the degree of indeterminacy of xi to Aj . In what follows, we will introduce the intuitionistic fuzzy association matrix, which was ¯rst proposed by Xu et al.26 De¯nition 3.26 Let Aj ðj ¼ 1; 2; . . . ; mÞ be m A-IFSs, then P ¼ ðij Þmm is called an association matrix, where ij ¼ ðAi ; Aj Þ is the intuitionistic fuzzy association coe±cient between Ai and Aj , which has the following properties: (1) 0 ij 1 for all i; j ¼ 1; 2; . . . ; m; (2) ij ¼ 1 if and only if Ai ¼ Aj ; and (3) ij ¼ ji , for all i; j ¼ 1; 2; . . . ; m. 3.2. Equivalent association matrix De¯nition 4. Let P ¼ ðij Þmm be an association matrix, if P satis¯es: ð1Þ P is reflexive; i:e:; I P;

ð13Þ

ð2Þ P is symmetric; i:e:; P ¼ P; ð3Þ P is transitive; i:e:; P 2 P:

ð14Þ ð15Þ

T

Then P ¼ ðij Þmm is called an equivalent association matrix, where P 2 ¼ P P ¼ ðij Þmm is a composition matrix26 of P, whose element ij satis¯es ij ¼ maxf minfik ; kj gg; k

i; j ¼ 1; 2; . . . ; m:

ð16Þ

De¯nition 5.26 Let P ¼ ðij Þmm be an equivalent association matrix, then we call P ¼ ð ij Þmm the -cutting matrix of P, where 0 if ij < ; ij ¼ i; j ¼ 1; 2; . . . ; m ð17Þ 1 if ij ; and is the con¯dence level with 2 ½0; 1.

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Theorem 1. P ¼ ðij Þmm is an equivalent association matrix if and only if P ¼ ð ij Þmm is an equivalent Boole matrix, for all 2 ½0; 1, where P ¼ ð ij Þmm is a -cutting matrix of P. Proof. Based on the properties of -cutting matrix, we have (1) P is re°exive, i.e., I P if and only if I P , i.e., I P ; (2) P is symmetric, i.e., P T ¼ P if and only if ðP T Þ ¼ P , i.e., ðP ÞT ¼ P ; (3) P is transitive, i.e., P 2 P if and only if P P P . From Theorem 1, we can see that if the association matrix is equivalent, then its -cutting matrix is an equivalent Boole matrix, and then we can use the equivalent Boole matrix to do clustering directly. But if the association matrix does not satisfy the transitivity, then we know that the -cutting matrix of P is just only a similar Boole matrix, and thus we cannot do clustering. In this situation, we can transform the similar Boole matrix into an equivalent matrix for clustering. Let us see the following theorem: Theorem 2.33 Let R be a similar Boole matrix over a discrete universe of discourse X ¼ fx1 ; x2 ; . . . ; xn g, then R is transitive if and only if R has not the following special submatrices: 1 1 1 1 1 0 0 1 ; ; ; ð18Þ 1 0 0 1 1 1 1 1 no matter how the matrix R is arranged. We can judge from Theorem 2 whether or not a similar Boole matrix is an equivalent one. 3.3. Algorithm for clustering A-IFSs Let X ¼ fx1 ; x2 ; . . . ; xn g be an attribute space, and w ¼ ðw1 ; w2 ; . . . ; wn ÞT be the P weight vector of xi ði ¼ 1; 2; . . . ; nÞ, with wi 0, i ¼ 1; 2; . . . ; n, and ni¼1 wi ¼ 1. Let Aj (j ¼ 1; 2; . . . ; m) be a collection of m A-IFSs expressing m samples to be clustered, having the following form: Aj ¼ fhxi ; Aj ðxi Þ; Aj ðxi Þijxi 2 Xg;

i ¼ 1; 2; . . . ; n; j ¼ 1; 2; . . . ; m

ð19Þ

Aj ðxi Þ ¼ 1 Aj ðxi Þ Aj ðxi Þ;

i ¼ 1; 2; . . . ; n; j ¼ 1; 2; . . . ; m

ð20Þ

and

is the degree of indeterminacy of xi to Aj . Based on Theorems 1 and 2, in what follows, we develop an intuitionistic fuzzy clustering algorithm based on Boole matrix and association measure. Algorithm I Step 1. Use Eq. (3) or Eq. (8) (if the weights of the attributes are the same, we use Eq. (3); otherwise, we use Eq. (8)) to compute the association coe±cients of the IFSs

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Aj ðj ¼ 1; 2; . . . ; mÞ, and then construct an association matrix P ¼ ðij Þmm , where ij ¼ 3 ðAi ; Aj Þ or ij ¼ 4 ðAi ; Aj Þ, i; j ¼ 1; 2; . . . ; m. Step 2. Construct a -cutting matrix P ¼ ð ij Þmm of P by using Eq. (17). Step 3. If P is an equivalent Boole matrix, then we can cluster the m samples as follows: If all the elements of the ith column are the same as the corresponding elements of the jth column in P , then the A-IFSs Ai and Aj are in the same cluster. By this principle, we can cluster all these m samples Aj ðj ¼ 1; 2; . . . ; mÞ. If P is not an equivalent Boole matrix, then by Theorem 2, we know that no matter how the matrix P is arranged, it must have some of the special submatrices in Eq. (18). In such cases, we can transform the elements 0 into 1 in such special submatrices until P has not any special submatrix, and thus we get a new equivalent matrix P . Step 4. Employ the equivalent matrix P to classify all the given A-IFSs Aj ðj ¼ 1; 2; . . . ; mÞ by the procedure in Step 3. Step 5. End. The principal of choosing ¸: Based on the idea of constructing the association matrix whose elements are association coe±cients between every two alternatives (samples) in this paper, we balance the similarity degree between two alternatives mainly through the association coe±cient (that is, the con¯dence level) of them. We choose the con¯dence level from the biggest one to the smallest one in the association matrix. After that, in terms of the chosen con¯dence level , we construct the corresponding -cutting matrix. With this principle, the clustering results come into being, the smaller the con¯dence level is, the more detailed the clustering will be.

4. Numerical Example Example 1. A military equipment development team needs to cluster ¯ve combat aircrafts according to their operational e®ectiveness. In order to group these combat aircrafts Ai ði ¼ 1; 2; . . . ; 5Þ with respect to their comprehensive functions, a team of military experts have been set up to provide their assessment information on Ai ði ¼ 1; 2; . . . ; 5Þ. The attributes which are considered here in assessment of Ai ði ¼ 1; 2; . . . ; 5Þ are: (1) x1 is the aircraft power; (2) x2 is the ¯re power (a military capability to direct force at an enemy); (3) x3 is the capacity for target detection; (4) x4 is the controlling ability; (5) x5 is the survivability; (6) x6 is the range of voyage; and (7) x7 is the electronic countermeasure e®ect. The military experts evaluate the performances of the combat aircrafts Ai ði ¼ 1; 2; . . . ; 5Þ according to the attributes xj ðj ¼ 1; 2; . . . ; 7Þ, and gives the data as follows: A1 ¼ fhx1 ; 0:5; 0:3i; hx2 ; 0:6; 0:3i; hx3 ; 0:4; 0:3i; hx4 ; 0:8; 0:1i; hx5 ; 0:7; 0:2i; hx6 ; 0:5; 0:2i; hx7 ; 0:4; 0:3ig

Intuitionistic Fuzzy Clustering Algorithm

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A2 ¼ fhx1 ; 0:6; 0:2i; hx2 ; 0:5; 0:3i; hx3 ; 0:5; 0:2i; hx4 ; 0:6; 0:2i; hx5 ; 0:6; 0:3i; hx6 ; 0:6; 0:3i; hx7 ; 0:5; 0:2ig A3 ¼ fhx1 ; 0:7; 0:1i; hx2 ; 0:6; 0:3i; hx3 ; 0:7; 0:2i; hx4 ; 0:5; 0:3i; hx5 ; 0:5; 0:2i; hx6 ; 0:5; 0:2i; hx7 ; 0:6; 0:3ig A4 ¼ fhx1 ; 0:4; 0:3i; hx2 ; 0:7; 0:2i; hx3 ; 0:5; 0:3i; hx4 ; 0:6; 0:2i; hx5 ; 0:7; 0:1i; hx6 ; 0:4; 0:3i; hx7 ; 0:7; 0:2ig A5 ¼ fhx1 ; 0:6; 0:2i; hx2 ; 0:6; 0:3i; hx3 ; 0:6; 0:2i; hx4 ; 0:5; 0:3i; hx5 ; 0:8; 0:1i; hx6 ; 0:6; 0:1i; hx7 ; 0:6; 0:1ig: Suppose that the weights of the attributes xj ðj ¼ 1; 2; . . . ; 7Þ are equal, now we utilize Algorithm I to group these combat aircrafts Ai ði ¼ 1; 2; . . . ; 5Þ: Step 1. Use Eq. (3) to compute Ai ði ¼ 1; 2; . . . ; 5Þ, and then construct ij ¼ 3 ðAi ; Aj Þ, i; j ¼ 1; 2; . . . ; 5: 0 1:000 0:964 B B 0:964 1:000 B B P ¼ B 0:917 0:948 B B 0:952 0:941 @ 0:947 0:963

the association coe±cients of the IFSs an association matrix P ¼ ðij Þmm , where 0:917 0:952 0:947

1

C 0:948 0:941 0:963 C C C 1:000 0:946 0:957 C: C 0:946 1:000 0:957 C A 0:957 0:957 1:000

Step 2. By Eq. (17) we give a detailed analysis with respect to the threshold , and then we get all the possible clusters of the combat aircrafts Ai ði ¼ 1; 2; . . . ; 5Þ: (1) If ¼ 1, then Ai ði ¼ 1; 2; . . . ; 5Þ are grouped into the following ¯ve types: fA1 g; fA2 g; fA3 g; fA4 g; fA5 g: (2) If ¼ 0:964, then by Eq. (17), the -cutting matrix P ¼ ð ij Þmm of P is: 1 0 1 1 0 0 0 B1 1 0 0 0C C B C B P ¼ B 0 0 1 0 0 C: C B @0 0 0 1 0A 0 0 0 0 1 According to Theorem 2, we know that P is an equivalent Boole matrix, we can use P to cluster the combat aircrafts Ai ði ¼ 1; 2; . . . ; 5Þ directly, and then Ai ði ¼ 1; 2; . . . ; 5Þ are grouped into the following four types: fA1 ; A2 g; fA3 g; fA4 g; fA5 g:

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(3) If ¼ 0:963, then the -cutting matrix P ¼ ð ij Þmm of P is: 0

1 B1 B B P ¼ B 0 B @0 0

1 1 0 0 1

0 0 1 0 0

0 0 0 1 0

1 0 1C C C 0 C: C 0A 1

From Theorem 2, we know that P is not an equivalent Boole matrix, we should ¯rst transform P into an equivalent Boole matrix by changing the element \0" in the special submatrices into \1" and get 0

1 B1 B B P ¼ B 0 B @0 1

1 1 0 0 1

0 0 1 0 0

0 0 0 1 0

1 1 1C C C 0 C: C 0A 1

and thus, Ai ði ¼ 1; 2; . . . ; 5Þ are grouped into the following three type: fA1 ; A2 ; A5 g; fA3 g; fA4 g: (4) If ¼ 0:957, then the -cutting matrix P ¼ ð ij Þmm of P is: 1 0 1 1 0 0 0 B1 1 0 0 1C C B C B P ¼ B 0 0 1 0 1 C : C B @0 0 0 1 1A 0 1 1 1 1 Similarly, P is not an equivalent Boole matrix, we should ¯rst transform P into an equivalent Boole matrix by changing the element \0" in the special submatrices into \1" and get 1 0 1 1 1 1 1 B1 1 1 1 1C C B C B P ¼ B 1 1 1 1 1 C C B @1 1 1 1 1A 1 1 1 1 1 and thus, Ai ði ¼ 1; 2; . . . ; 5Þ are grouped into the following one type: fA1 ; A2 ; A3 ; A4 ; A5 g: In the following, a simple comparison is made among the method proposed in this paper, Xu et al.'s method26 which may be regarded as a generalization of Yang and Shih's method34 and Pelekis et al.'s method35 in Table 1:

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Table 1. Comparisons of the derived results.

Classes 5

The results derived by intuitionistic fuzzy Boole method

The results developed by Xu et al.'s method

The results developed by Pelekis et al.'s method

fA1 g; fA2 g; fA3 g; fA4 g; fA5 g fA1 g; fA2 g; fA3 g; fA4 g; fA5 g fA1 g; fA2 g; fA3 g; fA4 g; fA5 g

4

fA1 ; A2 g; fA3 g; fA4 g; fA5 g

fA1 ; A2 g; fA3 g; fA4 g; fA5 g

fA2 ; A5 g; fA1 g; fA3 g; fA4 g

3

fA1 ; A2 ; A5 g; fA3 g; fA4 g

fA1 ; A2 ; A5 g; fA3 g; fA4 g

fA2 ; A4 ; A5 g; fA1 g; fA3 g fA1 ; A2 g; fA3 ; A4 ; A5 g

2 1

fA1 ; A2 ; A3 ; A4 ; A5 g

fA1 ; A2 ; A3 ; A4 ; A5 g

fA1 ; A2 ; A3 ; A4 ; A5 g

Through Table 1, we know that the method proposed in this paper has the same clustering results with those of Xu et al.'s method26 and Pelekis et al.'s method35 can make more detailed clustering results. In order to demonstrate the e®ectiveness of the proposed clustering algorithm, intuitionistic fuzzy Boole clustering method, we further conduct an experiment with more samples to compare these methods. Example 2. Below we ¯rst introduce the experimental data sets, and then make a comparison among these methods. Experimental data sets: Suppose the military experts evaluate the performance of another group of combat aircrafts Ai ði ¼ 1; 2; . . . ; 10Þ according to the attributes xj ðj ¼ 1; 2; . . . ; 7Þ, and give the data as: A1 ¼ fhx1 ; 0:5; 0:3i; hx2 ; 0:6; 0:3i; hx3 ; 0:4; 0:3i; hx4 ; 0:8; 0:1i; hx5 ; 0:7; 0:2i; hx6 ; 0:5; 0:2i; hx7 ; 0:4; 0:3ig A2 ¼ fhx1 ; 0:6; 0:2i; hx2 ; 0:5; 0:3i; hx3 ; 0:5; 0:2i; hx4 ; 0:6; 0:2i; hx5 ; 0:6; 0:3i; hx6 ; 0:6; 0:3i; hx7 ; 0:5; 0:2ig A3 ¼ fhx1 ; 0:7; 0:1i; hx2 ; 0:6; 0:3i; hx3 ; 0:7; 0:2i; hx4 ; 0:5; 0:3i; hx5 ; 0:5; 0:2i; hx6 ; 0:5; 0:2i; hx7 ; 0:6; 0:3ig A4 ¼ fhx1 ; 0:4; 0:3i; hx2 ; 0:7; 0:2i; hx3 ; 0:5; 0:3i; hx4 ; 0:6; 0:2i; hx5 ; 0:7; 0:1i; hx6 ; 0:4; 0:3i; hx7 ; 0:7; 0:2ig A5 ¼ fhx1 ; 0:6; 0:2i; hx2 ; 0:6; 0:3i; hx3 ; 0:6; 0:2i; hx4 ; 0:5; 0:3i; hx5 ; 0:8; 0:1i; hx6 ; 0:6; 0:1i; hx7 ; 0:6; 0:1ig A6 ¼ fhx1 ; 0:8; 0:1i; hx2 ; 0:5; 0:2i; hx3 ; 0:7; 0:1i; hx4 ; 0:7; 0:1i; hx5 ; 0:7; 0:2i; hx6 ; 0:8; 0:1i; hx7 ; 0:7; 0:2ig A7 ¼ fhx1 ; 0:7; 0:2i; hx2 ; 0:6; 0:3i; hx3 ; 0:8; 0:1i; hx4 ; 0:8; 0:1i; hx5 ; 0:6; 0:3i; hx6 ; 0:5; 0:4i; hx7 ; 0:8; 0:1ig A8 ¼ fhx1 ; 0:5; 0:2i; hx2 ; 0:7; 0:2i; hx3 ; 0:7; 0:2i; hx4 ; 0:6; 0:2i; hx5 ; 0:5; 0:3i; hx6 ; 0:7; 0:1i; hx7 ; 0:6; 0:2ig

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A9 ¼ fhx1 ; 0:6; 0:2i; hx2 ; 0:5; 0:3i; hx3 ; 0:6; 0:3i; hx4 ; 0:5; 0:2i; hx5 ; 0:8; 0:1i; hx6 ; 0:8; 0:1i; hx7 ; 0:5; 0:2ig A10 ¼ fhx1 ; 0:9; 0:0i; hx2 ; 0:9; 0:1i; hx3 ; 0:8; 0:1i; hx4 ; 0:7; 0:2i; hx5 ; 0:5; 0:15i; hx6 ; 0:3; 0:65i; hx7 ; 0:15; 0:75ig: Comparison results among these methods are listed in Table 2. Again we can see from Table 2 that the method in this paper has the same clustering result with that of Xu et al.'s method26 and Pelekis et al.'s method35 can make more detailed clustering results. It is worthy of pointing out that the clustering results of the intuitionistic fuzzy Boole clustering method proposed in this paper are exactly the same with those of Xu et al.'s method,26 but our method does not need to use the transitive closure technique to calculate the equivalent matrix of the association matrix, and thus requires much less computational e®ort than Xu et al.'s method.26 Let us examine into the computing process of the two methods: whether in Xu et al.26 or in this work, the clustering processes are all based on -cutting matrix. Before getting the -cutting matrix, Xu et al.26 ¯rst transformed the intuitionistic fuzzy association matrix into an intuitionistic fuzzy equivalent association matrix by transitive closure technique, which needs lots of computational e®ort. In this work, we get the -cutting matrix directly from the intuitionistic fuzzy association matrix. Furthermore, Let m and n represent the amount of alternatives and attributes, respectively. Then the computational complexity of our method is Oðnm 2 Þ, Xu et al.'s method is Oðð1 þ kÞnm 2 Þ where k (usually, k 2Þ represents the transfer times until we get the equivalent matrix, and Pelekis et al.'s method is Oðnm 2 þ jcmÞ where c is the number of the clusters, j is the times of judgement if jjU jþ1 U j jjF > " is valid. In summary, Xu et al.'s method26 and Pelekis et al.'s method35 have relatively high computational complexity, which indeed motivates the intuitionistic fuzzy Boole clustering method proposed in this paper. Furthermore, from Examples 1 and 2 we can see that the clustering results have much to do with the threshold , the smaller the con¯dence level is, the more detailed the clustering will be. Either in Example 1 or in Example 2, we all use the association coe±cient Eq. (3) but not Eq. (1), the reason is that Eq. (1) cannot guarantee the necessity in the condition (2) of De¯nition 1 and omits the hesitation degree, which may lead to incorrect results. The following example shows these ideas. Example 3. Suppose the military experts evaluate the performance of another group of combat aircrafts Ai ði ¼ 1; 2; . . . ; 9Þ according to the attributes xj ðj ¼ 1; 2; . . . ; 7Þ,

fA1 g; fA2 g; fA3 g; fA4 g; fA5 g; fA6 g; fA7 g; fA8 g; fA9 g; fA10 g fA5 ; A9 g; fA1 g; fA2 g; fA3 g; fA4 g; fA6 g; fA7 g; fA8 g; fA10 g fA3 ; A8 g; fA5 ; A9 g; fA1 g; fA2 g; fA4 g; fA6 g; fA7 g; fA10 g

fA1 g; fA2 g; fA3 g; fA4 g; fA5 g; fA6 g; fA7 g; fA8 g; fA9 g; fA10 g

fA5 ; A9 g; fA1 g; fA2 g; fA3 g; fA4 g; fA6 g; fA7 g; fA8 g; fA10 g

fA3 ; A8 g; fA5 ; A9 g; fA1 g; fA2 g; fA4 g; fA6 g; fA7 g; fA10 g

10

9

8

fA1 ; A2 ; A5 ; A9 g; fA3 ; A8 g; fA4 g; fA6 g; fA7 g; fA10 g fA1 ; A2 ; A5 ; A6 ; A9 g; fA3 ; A8 g; fA4 g; fA7 g; fA10 g fA1 ; A2 ; A3 ; A5 ; A6 ; A8 ; A9 g; fA4 g; fA7 g; fA10 g fA1 ; A2 ; A3 ; A4 ; A5 ; A6 ; A8 ; A9 g; fA7 g; fA10 g fA1 ; A2 ; A3 ; A4 ; A5 ; A6 ; A7 ; A8 ; A9 g; fA10 g fA1 ; A2 ; A3 ; A4 ; A5 ; A6 ; A7 ; A8 ; A9 ; A10 g

fA1 ; A2 ; A5 ; A9 g; fA3 ; A8 g; fA4 g; fA6 g; fA7 g; fA10 g

fA1 ; A2 ; A5 ; A6 ; A9 g; fA3 ; A8 g; fA4 g; fA7 g; fA10 g

fA1 ; A2 ; A3 ; A5 ; A6 ; A8 ; A9 g; fA4 g; fA7 g; fA10 g

fA1 ; A2 ; A3 ; A4 ; A5 ; A6 ; A8 ; A9 g; fA7 g; fA10 g

fA1 ; A2 ; A3 ; A4 ; A5 ; A6 ; A7 ; A8 ; A9 g; fA10 g

fA1 ; A2 ; A3 ; A4 ; A5 ; A6 ; A7 ; A8 ; A9 ; A10 g

6

5

4

3

2

1

7

Classes

The results developed by Xu et al.'s method

Comparisons of the clustering results.

The results derived by IF Boole method

Table 2.

fA1 ; A2 ; A3 ; A4 ; A5 ; A6 ; A7 ; A8 ; A9 ; A10 g

fA1 ; A2 ; A3 ; A4 ; A5 ; A6 ; A7 ; A8 ; A9 g; fA10 g

fA1 ; A2 ; A5 ; A6 ; A9 g; fA3 ; A4 ; A7 ; A8 g; fA10 g

fA1 ; A4 g; fA3 ; A7 ; A8 g; fA2 ; A5 ; A6 ; A9 g; fA10 g

fA1 ; A4 g; fA2 ; A3 ; A8 g; fA5 ; A6 ; A9 g; fA7 g; fA10 g

fA1 ; A4 g; fA3 ; A8 g; fA5 ; A6 ; A9 g; fA2 gfA7 g; fA10 g

fA1 ; A4 g; fA2 ; A8 g; fA5 ; A9 g; fA3 g; fA6 g; fA7 g; fA10 g

fA1 ; A4 g; fA5 ; A9 g; fA2 g; fA3 g; fA6 g; fA7 g; fA8 g; fA10 g

fA5 ; A9 g; fA1 g; fA2 g; fA3 g; fA4 g; fA6 g; fA7 g; fA8 g; fA10 g

fA1 g; fA2 g; fA3 g; fA4 g; fA5 g; fA6 g; fA7 g; fA8 g; fA9 g; fA10 g

The results developed by Pelekis et al.'s method

Intuitionistic Fuzzy Clustering Algorithm 107

108

H. Zhao, Z. S. Xu & Z. Wang

and give the data as: A1 ¼ fhx1 ; 0:5; 0:3i; hx2 ; 0:6; 0:3i; hx3 ; 0:4; 0:3i; hx4 ; 0:8; 0:1i; hx5 ; 0:7; 0:2i; hx6 ; 0:5; 0:2i; hx7 ; 0:4; 0:3ig A2 ¼ fhx1 ; 0:6; 0:2i; hx2 ; 0:5; 0:3i; hx3 ; 0:5; 0:2i; hx4 ; 0:6; 0:2i; hx5 ; 0:6; 0:3i; hx6 ; 0:6; 0:3i; hx7 ; 0:5; 0:2ig A3 ¼ fhx1 ; 0:7; 0:1i; hx2 ; 0:6; 0:3i; hx3 ; 0:7; 0:2i; hx4 ; 0:5; 0:3i; hx5 ; 0:5; 0:2i; hx6 ; 0:5; 0:2i; hx7 ; 0:6; 0:3ig A4 ¼ fhx1 ; 0:4; 0:3i; hx2 ; 0:7; 0:2i; hx3 ; 0:5; 0:3i; hx4 ; 0:6; 0:2i; hx5 ; 0:7; 0:1i; hx6 ; 0:4; 0:3i; hx7 ; 0:7; 0:2ig A5 ¼ fhx1 ; 0:6; 0:2i; hx2 ; 0:6; 0:3i; hx3 ; 0:6; 0:2i; hx4 ; 0:5; 0:3i; hx5 ; 0:8; 0:1i; hx6 ; 0:6; 0:1i; hx7 ; 0:6; 0:1ig A6 ¼ fhx1 ; 0:8; 0:1i; hx2 ; 0:5; 0:2i; hx3 ; 0:7; 0:1i; hx4 ; 0:7; 0:1i; hx5 ; 0:7; 0:2i; hx6 ; 0:8; 0:1i; hx7 ; 0:7; 0:2ig A7 ¼ fhx1 ; 0:7; 0:2i; hx2 ; 0:6; 0:3i; hx3 ; 0:8; 0:1i; hx4 ; 0:8; 0:1i; hx5 ; 0:6; 0:3i; hx6 ; 0:5; 0:4i; hx7 ; 0:8; 0:1ig A8 ¼ fhx1 ; 0:5; 0:2i; hx2 ; 0:7; 0:2i; hx3 ; 0:7; 0:2i; hx4 ; 0:6; 0:2i; hx5 ; 0:5; 0:3i; hx6 ; 0:7; 0:1i; hx7 ; 0:6; 0:2ig A9 ¼ fhx1 ; 0:6; 0:2i; hx2 ; 0:5; 0:3i; hx3 ; 0:6; 0:3i; hx4 ; 0:5; 0:2i; hx5 ; 0:8; 0:1i; hx6 ; 0:8; 0:1i; hx7 ; 0:5; 0:2ig: If we use Eq. (1) to compute the association coe±cients of the IFSs Ai ði ¼ 1; 2; . . . ; 9Þ, then the association matrix P ¼ ðij Þmm , where ij ¼ 1 ðAi ; Aj Þ , i; j ¼ 1; 2; . . . ; 9 will be: 1 0 1:000 0:971 0:931 0:960 0:945 0:933 0:934 0:943 0:948 B 0:971 1:000 0:973 0:956 0:970 0:970 0:971 0:972 0:970 C C B C B B 0:931 0:973 1:000 0:945 0:968 0:964 0:965 0:973 0:953 C C B C B B 0:960 0:956 0:945 1:000 0:962 0:923 0:952 0:950 0:938 C C B C B P ¼ B 0:945 0:970 0:968 0:962 1:000 0:967 0:946 0:965 0:985 C: C B B 0:933 0:970 0:964 0:923 0:967 1:000 0:963 0:969 0:971 C C B C B B 0:934 0:971 0:965 0:952 0:946 0:963 1:000 0:960 0:923 C C B C B @ 0:943 0:972 0:973 0:950 0:965 0:969 0:960 1:000 0:960 A 0:948 0:970 0:953 0:938 0:985 0:971 0:923 0:960 1:000 If we use Eq. (3) to compute the association coe±cients of the IFSs Ai ði ¼ 1; 2; . . . ; 9Þ, then the association matrix P ¼ ðij Þmm , where ij ¼ 3 ðAi ; Aj Þ,

Intuitionistic Fuzzy Clustering Algorithm

109

i; j ¼ 1; 2; . . . ; 9 will be: 0

1:000 0:964 0:917 0:952 0:947 0:914 0:914 0:934 0:933

B B 0:964 B B B 0:917 B B B 0:952 B B P ¼ B 0:947 B B B 0:914 B B 0:914 B B B 0:934 @

1

C 1:000 0:948 0:941 0:963 0:959 0:950 0:959 0:964 C C C 0:948 1:000 0:946 0:957 0:945 0:948 0:969 0:936 C C C 0:941 0:946 1:000 0:957 0:908 0:934 0:950 0:923 C C C 0:963 0:957 0:957 1:000 0:950 0:930 0:960 0:976 C: C C 0:959 0:945 0:908 0:950 1:000 0:956 0:953 0:961 C C 0:950 0:948 0:934 0:930 0:956 1:000 0:947 0:911 C C C 0:959 0:969 0:950 0:960 0:953 0:947 1:000 0:955 C A

0:933 0:964 0:936 0:923 0:976 0:961 0:911 0:955 1:000 Based on the above two association matrices, using the intuitionistic fuzzy Boole clustering method proposed in this paper, we can make a comparison between the clustering results of the two association coe±cients: We can see from Table 3 that Eq. (3) can derive more detailed clustering results than Eq. (1). Since Eq. (1) cannot guarantee the necessity in the condition (2) of De¯nition 1 and omits the hesitation degree, some information may be missing. Namely, Eq. (1) can not re°ect all the information that the intuitionistic fuzzy data contains. Considering the stated reasons above, it is not hard for us to comprehend why Eq. (3) can get more detailed classes than Eq. (1). Therefore, compared to Eq. (1), Eq. (3) has much more potential for practical applications.

Table 3. Comparisons of the clustering results of Eqs. (1) and (3). Classes

The clustering result using Eq. (1)

The clustering result using Eq. (3)

9

fA1 g; fA2 g; fA3 g; fA4 g; fA5 g; fA6 g; fA7 g; fA8 g; fA9 g fA5 ; A9 g; fA1 g; fA2 g; fA3 g; fA4 g; fA6 g; fA7 g; fA8 g

fA1 g; fA2 g; fA3 g; fA4 g; fA5 g; fA6 g; fA7 g; fA8 g; fA9 g fA5 ; A9 g; fA1 g; fA2 g; fA3 g; fA4 g; fA6 g; fA7 g; fA8 g fA3 ; A8 g; fA5 ; A9 g; fA1 g; fA2 g; fA4 g; fA6 g; fA7 g

8 7 6

fA2 ; A3 ; A8 g; fA5 ; A9 g; fA1 g; fA4 g; fA6 g; fA7 g

5 4 3 2 1

fA1 ; A2 ; A3 ; A7 ; A8 g; fA5 ; A6 ; A9 g; fA4 g fA1 ; A2 ; A3 ; A5 ; A6 ; A7 ; A8 ; A9 g; fA4 g fA1 ; A2 ; A3 ; A4 ; A5 ; A6 ; A7 ; A8 ; A9 g

fA1 ; A2 ; A5 ; A9 g; fA3 ; A8 g; fA4 g; fA6 g; fA7 g fA1 ; A2 ; A5 ; A6 ; A9 g; fA3 ; A8 g; fA4 g; fA7 g fA1 ; A2 ; A3 ; A5 ; A6 ; A8 ; A9 g; fA4 g; fA7 g fA1 ; A2 ; A3 ; A4 ; A5 ; A6 ; A8 ; A9 g; fA7 g fA1 ; A2 ; A3 ; A4 ; A5 ; A6 ; A7 ; A8 ; A9 g

110

H. Zhao, Z. S. Xu & Z. Wang

5. Interval-Valued Intuitionistic Fuzzy Clustering Algorithm 5.1. The basic concept related to IVIFS Atanassov and Gargov15 considered that sometimes using exactly real numbers to characterize the membership degrees and the nonmembership degrees for certain elements of A is unsuitable, but fuzzy ranges should be given. Hence, they de¯ned the notion of IVIFS as follows. An IVIFS A~ over X is an object having the form: A~ ¼ fhx; ~A~ ðxÞ; ~A~ ðxÞijx 2 Xg, where ~A~ ðxÞ ½0; 1 and ~A~ ðxÞ ½0; 1 are intervals, and sup ~A~ ðxÞ þ sup ~A~ ðxÞ 1, for every x 2 X. Especially, if each of the intervals ~A~ ðxÞ and ~A~ ðxÞ contains exactly one element, i.e., if A~ ðxÞ ¼ inf ~A~ ðxÞ ¼ sup ~A~ ðxÞ, A~ ðxÞ ¼ inf ~A~ ðxÞ ¼ sup ~A~ ðxÞ, for every x 2 X, then, the given IVIFS A~ reduces to an ordinary A-IFS. 5.2. The association coe±cient of IVIFSs ~ Let ðXÞ be the set of all IVIFSs over X, Xu et al.26 de¯ned the concept of association coe±cient between two IVIFS as follows: :

:

2 ~ De¯nition 6.26 Let be a mapping : ððxÞÞ ! ½0; 1, then the association : ~ ~ ~ ~ coe±cient between two IVIFSs A and B is de¯ned as ðA; BÞ, which satis¯es : ~ ~ : ~ ~ ~ the following conditions: (1) 0 ðA; BÞ 1; (2) ðA; BÞ ¼ 1 if and only if A~ ¼ B; : ~ ~ : ~ ~ and (3) ðA; BÞ ¼ ðB; AÞ.

In the case where X ¼ fx1 ; x2 ; . . . ; xn g is a discrete universe of discourse, we extend 3 ðA; BÞ to IVIFSs to calculate the association coe±cient between two IVIFSs A~ and B~ as below: Pn ~ B~ ðxj Þ j¼1 fA; : ~ ~ p ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 7 ðA; BÞ ¼ P n ; ð21Þ Pn j¼1 gA~ ðxj Þ j¼1 gB~ ðxj Þ where LA~ ðxj ÞÞ2 þ ð~ LA~ ðxj ÞÞ2 þ ð~ U ðx ÞÞ2 þ ð~ U ðx ÞÞ2 gA~ ðxj Þ ¼ ð~ LA~ ðxj ÞÞ2 þ ð~ A~ j A~ j þð~ U ðx ÞÞ2 ; A~ j

ð22Þ

gB~ ðxj Þ ¼ ð~ LB~ ðxj ÞÞ2 þ ð~ LB~ ðxj ÞÞ2 þ ð~ LB~ ðxj ÞÞ2 þ ð~ U ðx ÞÞ2 þ ð~ U ðx ÞÞ2 B~ j B~ j þ ð~ U ðx ÞÞ2 ; B~ j

ð23Þ

~ LA~ ðxj Þ~ LB~ ðxj Þ þ ~ LA~ ðxj Þ~ fA; LB~ ðxj Þ þ ~ LA~ ðxj Þ~ LB~ ðxj Þ ~ B~ ðxj Þ ¼ ðx Þ~ U ðx Þ þ ~ U ðx Þ~ U ðx Þ þ ~U ðx Þ~ U ðx Þ: þ ~ U B~ j B~ j B~ j A~ j A~ j A~ j

ð24Þ

If we need to consider the weights of the element xi 2 X, then Eq. (21) can be extended to its weighted counterpart: Pn ~ B~ ðxj Þ j¼1 wj fA; : ~ ~ 8 ðA; BÞ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ; ð25Þ Pn Pn j¼1 wj gA~ ðxj Þ j¼1 wj gB~ ðxj Þ

Intuitionistic Fuzzy Clustering Algorithm

111

where w ¼ ðw1 ; w2 ; . . . ; wn ÞT is the weight vector of xi ði ¼ 1; 2; . . . ; nÞ, with wj 0, P i ¼ 1; 2; . . . ; n, and nj¼1 wj ¼ 1. If w1 ¼ w2 ¼ ¼ wn ¼ n1 , then Eq. (25) reduces to Eq. (21). In the following, we prove that Eq. (25) satis¯es all the conditions of De¯nition 6. ~ B~ 2 ðXÞ, ~ Proof. Since A; then 0 ~ LA~ ðxj Þ ~ U ðx Þ 1; A~ j

0 ~ LA~ ðxj Þ ~ U ðx Þ 1; A~ j

0 ~ LA~ ðxj Þ ~U ðx Þ 1; A~ j

for all xj 2 X

ðx Þ 1; 0 ~ LB~ ðxj Þ ~ U B~ j

0 ~ LB~ ðxj Þ ~ U ðx Þ 1; B~ j

~U ðx Þ 1; 0 ~ LB~ ðxj Þ B~ j

for all xj 2 X

ð26Þ

ð27Þ

: ~ ~ and thus, by Eq. (25), we get 8 ðA; BÞ 0. According to the famous CauchySchwarz inequality Eq. (4), we have vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ! !ﬃ u n n n X X u X t wj fA; wj gA~ ðxj Þ wj gB~ ðxj Þ ð28Þ ~ B~ j¼1

j¼1

j¼1

:

~ BÞ ~ 1 with equality if and only if there exists a nonzero real number and thus, 8 ðA; , such that ~ LA~ ðxj Þ ¼ ~ LB~ ðxj Þ;

~ U ðx Þ ¼ ~ U ðx Þ; B~ j A~ j

~ LA~ ðxj Þ ¼ ~ LB~ ðxj Þ;

~ U ðx Þ ¼ ~ U ðx Þ; B~ j A~ j

~ LA~ ðxj Þ ¼ ~ LB~ ðxj Þ;

~U ðx Þ ¼ ~ U ðx Þ B~ j A~ j

for all xj 2 X

ð29Þ

while because ~ LA~ ðxj Þ ¼ 1 ~ U ðx Þ ~ U ðx Þ; A~ j A~ j

~U ðx Þ ¼ 1 ~ LA~ ðxj Þ ~ LA~ ðxj Þ; A~ j

for all xj 2 X; ð30Þ

~ LB~ ðxj Þ ¼ 1 ~ U ðx Þ ~ U ðx Þ; B~ j B~ j

~U ðx Þ ¼ 1 ~ LB~ ðxj Þ ~ LB~ ðxj Þ; B~ j

for all xj 2 X: ð31Þ

~ which completes the proofs of the Then by Eq. (29), we have ¼ 1, i.e., A~ ¼ B, conditions (1) and (2) in De¯nition 6. Furthermore, by Eq. (25), we have Pn ~ B~ ðxj Þ j¼1 wj fA; : ~ ~ 8 ðA; BÞ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Pn Pn j¼1 wj gA~ ðxj Þ j¼1 wj gB~ ðxj Þ Pn

j¼1 wj fB~ ;A~ ðxj Þ : ~ ~ ¼ 8 ðB; AÞ: ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Pn Pn j¼1 wj gB~ ðxj Þ j¼1 wj gA~ ðxj Þ

ð32Þ

: ~ ~ BÞ also satis¯es the condition (3) of De¯nition 6. Thus, we prove that 8 ðA;

If the universe of discourse, X, is continuous and the weight of the element R x 2 X ¼ ½a; b is wðxÞ, where wðxÞ 0 and ab wðxÞdx ¼ 1, then we get the continuous

112

H. Zhao, Z. S. Xu & Z. Wang

form of Eq. (25):

Rb wðxÞfA; ~ B~ ðxÞdx a ~ ~ 9 ðA; BÞ ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ; Rb Rb wðxÞg ðxÞdx wðxÞg ðxÞdx ~ ~ B A a a :

ð33Þ

where LA~ ðxÞÞ2 þ ð~ LA~ ðxÞÞ2 þ ð~ U ðxÞÞ2 þ ð~ U ðxÞÞ2 þ ð~ U ðxÞÞ2 ; ð34Þ gA~ ðxÞ ¼ ð~ LA~ ðxÞÞ2 þ ð~ A~ A~ A~ gB~ ðxÞ ¼ ð~ LB~ ðxÞÞ2 þ ð~ LB~ ðxÞÞ2 þ ð~ LB~ ðxÞÞ2 þ ð~ U ðxÞÞ2 þ ð~ U ðxÞÞ2 þ ð~ U ðxÞÞ2 ; ð35Þ B~ B~ B~ ~ LA~ ðxj Þ~ LB~ ðxj Þ þ ~ LA~ ðxj Þ~ fA; LB~ ðxj Þ þ ~ LA~ ðxj Þ~ LB~ ðxj Þ ~ B~ ðxj Þ ¼ þ ~ U ðx Þ~ U ðx Þ þ ~ U ðx Þ~ U ðx Þ þ ~U ðx Þ~ U ðx Þ: B~ j B~ j B~ j A~ j A~ j A~ j If all the elements have the same importance, then Eq. (33) reduces to Rb f ~ B~ ðxÞdx : a A; ~ BÞ ~ ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ: 10 ðA; Rb Rb g ðxÞdx g ðxÞdx a A~ a B~

ð36Þ

ð37Þ

For convenience, we introduce the concept of interval-valued intuitionistic fuzzy association matrix: :

: De¯nition 7.26 Let A~j ðj ¼ 1; 2; . . . ; mÞ be m IVIFSs, then P ¼ ðij Þmm is called : : an association matrix, where ij ¼ ðAi ; Aj Þ is the interval-valued intuitionistic fuzzy association coe±cient of A~i and A~j , which has the following properties: (1) 0 : : : : ij 1 for all i; j ¼ 1; 2; . . . ; m; (2) ij ¼ 1 if and only if A~i ¼ A~j ; and (3) ij ¼ ji , for all i; j ¼ 1; 2; . . . ; m.

Based on the de¯nition above, in the next section we will develop an algorithm for clustering IVIFSs. 5.3. Algorithm for clustering IVIFSs Let X ¼ fx1 ; x2 ; . . . ; xn g be an attribute space, and w ¼ ðw1 ; w2 ; . . . ; wn ÞT be the weight vector of the elements xi ði ¼ 1; 2; . . . ; nÞ, with wj 0, j ¼ 1; 2; . . . ; n, and Pn ~ j¼1 wj ¼ 1. Let A j ðj ¼ 1; 2; . . . ; mÞ be a collection of m IVIFSs expressing m samples to be clustered, with A~j ¼ fhxi ; ~A~j ðxi Þ; ~A~j ðxi Þijxi 2 Xg; i ¼ 1; 2; . . . ; n; j ¼ 1; 2; . . . ; m; ð38Þ h i where ~A~j ðxi Þ ¼ ~ LA~ ðxi Þ; ~ U ðx Þ ½0; 1 and ~A~j ðxi Þ ¼ ½~ LA~ ðxi Þ; ~ U ðx Þ ½0; 1 A~j i A~j i j j L U are intervals, and ~ A~j ðxi Þ ¼ ½~ A~ ðxi Þ; ~ A~ ðxi Þ, where ~ LA~ ðxi Þ j ðx Þ ~U A~j i

j

j

¼1

~ U ðx Þ A~j i

~ U ðxi Þ; A~

¼1

~ LA~ ðxi Þ j

~ LA~ ðxi Þ; j

is the degree of hesitation of xi to A~j .

j

for all xi 2 X

ð39Þ

Intuitionistic Fuzzy Clustering Algorithm

113

Drawing support from the association coe±cients of IVIFSs, we can extend Algorithm I to interval-valued intuitionistic fuzzy environment: Algorithm II Step 1. Use Eq. (21) or Eq. (25) (if the weights of the attributes are the same, we use Eq. (21); otherwise, we use Eq. (25)) to calculate the association coe±cients of the : : IVIFSs A~j ðj ¼ 1; 2; . . . ; mÞ, and then construct an association matrix P ¼ ðij Þmm , : : : : where ij ¼ 7 ðA~i ; A~j Þ or ij ¼ 8 ðA~i ; A~j Þ, i; j ¼ 1; 2; . . . ; m. :

:

Step 2. Construct a -cutting matrix P ¼ ð ij Þmm of P by using Eq. (17). Step 3. See Algorithm I. Step 4. See Algorithm I. Step 5. End. Example 4. Suppose that there are six samples A~i ði ¼ 1; 2; . . . ; 6Þ to be classi¯ed. According to the attributes xj ðj ¼ 1; 2Þ, their attribute values are expressed by IVIFSs as follows: A~1 ¼ fhx1 ; ½0:60; 0:80; ½0:10; 0:20i; hx2 ; ½0:50; 0:70; ½0:10; 0:30ig; A~2 ¼ fhx1 ; ½0:30; 0:50; ½0:25; 0:45i; hx2 ; ½0:70; 0:85; ½0:00; 0:15ig; A~3 ¼ fhx1 ; ½0:45; 0:65; ½0:15; 0:35i; hx2 ; ½0:60; 0:80; ½0:05; 0:20ig; A~4 ¼ fhx1 ; ½0:34; 0:54; ½0:25; 0:45i; hx2 ; ½0:50; 0:70; ½0:10; 0:30ig; A~5 ¼ fhx1 ; ½0:40; 0:60; ½0:25; 0:40i; hx2 ; ½0:65; 0:80; ½0:10; 0:20ig; A~6 ¼ fhx1 ; ½0:45; 0:65; ½0:15; 0:35i; hx2 ; ½0:47; 0:67; ½0:05; 0:25ig: Suppose that the weights of the attributes xj ðj ¼ 1; 2Þ are equal, now we utilize Algorithm 2 to group these samples A~i ði ¼ 1; 2; . . . ; 6Þ: Step 1. Use Eq. (21) to compute the association coe±cients of the IFSs : : A~i ði ¼ 1; 2; . . . ; 6Þ, and then construct an association matrix P ¼ ðij Þmm , where : : ~ ~ ij ¼ 7 ðA i ; A j Þ, i; j ¼ 1; 2; . . . ; 6: 1 0 1:000 0:908 0:973 0:944 0:950 0:977 C B B 0:908 1:000 0:979 0:975 0:987 0:950 C C B B 0:973 0:979 1:000 0:982 0:992 0:986 C : C B P¼B C: B 0:944 0:975 0:982 1:000 0:981 0:983 C C B C B @ 0:950 0:987 0:992 0:981 1:000 0:967 A 0:977 0:950 0:986 0:983 0:967 1:000 Step 2. By Eq. (17) we give a detailed analysis with respect to the threshold , and then we get all the possible clusters of the samples A~i ði ¼ 1; 2; . . . ; 6Þ:

114

H. Zhao, Z. S. Xu & Z. Wang

(1) If ¼ 1, then A~i ði ¼ 1; 2; . . . ; 6Þ are grouped into the following six types: fA~1 g; fA~2 g; fA~3 g; fA~4 g; fA~5 g; fA~6 g: (2) If ¼ 0:992, then by Eq. (17), the -cutting 0 1 0 0 0 B B0 1 0 0 B B B0 0 1 0 P ¼ B B0 0 0 1 B B B0 0 1 0 @ 0 0 0 0

:

:

matrix P ¼ ð ij Þmm of P is: 1 0 0 C 0 0C C C 1 0C C: 0 0C C C 1 0C A 0 1

According to Theorem 2, we know that P is an equivalent Boole matrix, we can use P to cluster the samples A~i ði ¼ 1; 2; . . . ; 6Þ directly, and then A~i ði ¼ 1; 2; . . . ; 6Þ are grouped into the following ¯ve types: fA~1 g; fA~2 g; fA~3 ; A~5 g; fA~4 g; fA~6 g: (3) If ¼ 0:987, then the -cutting matrix 0 1 0 B B0 1 B B B0 0 P ¼ B B0 0 B B B0 1 @

:

:

P ¼ ð ij Þmm of P is: 1 0 0 0 0 C 0 0 1 0C C C 1 0 1 0C C: 0 1 0 0C C C 1 0 1 0C A

0 0 0 0 0 1 Similar to (2), A~i ði ¼ 1; 2; . . . ; 6Þ are grouped into the following four types: fA~1 g; fA~2 ; A~3 ; A~5 g; fA~4 g; fA~6 g: (4) If ¼ 0:986, then the -cutting matrix 0 1 0 B B0 1 B B B0 0 P ¼ B B0 0 B B B0 1 @

:

:

P ¼ ð ij Þmm of P is: 1 0 0 0 0 C 0 0 1 0C C C 1 0 1 1C C: 0 1 0 0C C C 1 0 1 0C A

0 0 1 0 0 1 By Theorem 2, we know that P is not an equivalent Boole matrix, to transform P into an equivalent Boole matrix, we should change the element \0" in the

Intuitionistic Fuzzy Clustering Algorithm

special submatrices into \1" and then we get 0 1 0 0 0 B0 1 1 0 B B0 1 1 0 B P ¼ B B0 0 0 1 B @0 1 1 0 0 1 1 0

115

1 0 1C C 1C C C: 0C C 1A

0 1 1 0 1 1

1

Obviously, is an equivalent Boole matrix, by which we can group A~i ði ¼ 1; 2; . . . ; 6Þ into the following three types: P

fA~1 g; fA~2 ; A~3 ; A~5 ; A~6 g; fA~4 g: (5) If ¼ 0:982, then the -cutting matrix 0 1 0 B0 1 B B0 0 B P ¼ B B0 0 B @0 1 0 1

:

:

P ¼ ð ij Þmm of P is 1 0 0 0 0 0 0 1 1C C 1 1 1 1C C C: 1 1 0 0C C 1 0 1 0A 1 0 0 1

Similar to (4), A~i ði ¼ 1; 2; . . . ; 6Þ are grouped into the following two types: fA~1 g; fA~2 ; A~3 ; A~4 ; A~5 ; A~6 g: (6) If ¼ 0:977, then the -cutting matrix 0 1 0 B0 1 B B0 1 B P ¼ B B0 1 B @0 1 1 1

:

:

P ¼ ð ij Þmm of P is: 1 0 0 0 1 1 1 1 1C C 1 1 1 1C C C: 1 1 1 1C C 1 1 1 1A 1 1 1 1

Similar to (4), A~i ði ¼ 1; 2; . . . ; 6Þ are grouped into the following one types: fA~1 ; A~2 ; A~3 ; A~4 ; A~5 ; A~6 g: 6. Concluding Remarks A-IFS is a powerful tool to deal with vagueness and uncertainty in our real world. In the last few years, the theory of A-IFSs has been applied in a variety of ¯elds, such as decision making, medical diagnosis, and pattern recognition, etc. Recently, some researchers have made their e®orts on clustering intuitionistic fuzzy data. Several of the existing clustering algorithms construct equivalent matrices and use -cutting matrices to cluster samples with intuitionistic fuzzy information. Considering that

116

H. Zhao, Z. S. Xu & Z. Wang

there is little study on the clustering method by using the -cutting matrix of a similarity matrix, in this paper, we have focused on this issue. We have proposed some methods for calculating the association coe±cients of A-IFSs, and developed an easy and feasible method to cluster intuitionistic fuzzy data. We have also utilized the association coe±cient of A-IFSs to construct an association matrix, and then calculated the -cutting matrix of the association matrix no matter whether or not it is an equivalent matrix. We have further used the -cutting matrix to cluster A-IFSs, and then extended the algorithm to cluster IVIFSs. Several practical examples have been used to show the feasibility and e®ectiveness of the developed association coe±cients and algorithms. This paper is a preliminary study on intuitionistic fuzzy clustering with Boole matrix and association measure. The latent applications of our algorithm in the ¯elds of data mining and information retrieval, etc., may be the directions for future research.

Acknowledgments The work was supported by the National Natural Science Foundation of China (No. 71071161 and No. 61273209), and the Pre-Research Foundation of PLA University of Science and Technology (No. 20110511). The authors are very grateful to the anonymous reviewers for their constructive comments and suggestions that have led to an improved version of this paper.

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Some issues on intuitionistic fuzzy aggregation operators based on ...