Intuitionistic Fuzzy Clustering Algorithms - Intuitionistic Fuzzy Aggregation and Clustering - page 249

n n

1
n

1
n

μ ik = 1 −

1 |

v ij − v kj |−

1 | π ij − π kj | ,

i , k = 1 , 2 ,..., m (2.192)

j = j =

with n .

BasedonEqs.( 2.186 ) and ( 2.192 ), we have the following concept:

μ ik = 1 if and only ˙ v ij =˙ v kj and π ij = π kj , for all j = 1 , 2 ,...,

Definition 2.34 (Wang et al. 2011) Let y i and y k be two IFSs on X , and Z ( y i , y k ) a

binary relation on X × X ,if

⎧
⎨

( 1 , 0 ), y i = y k ,

1

v ij −

v kj −

π ij − π kj , v ij −

v kj

j = 1

n

j = 1

n

j = 1

n

1
n

1
n

1
n

Z ( y i , y k ) = − ,

⎩

y i = y k ,

(2.193)

then Z is called a closeness degree of y i and y k .

By Eq. ( 2.193 ), we have

( y i , y k )

Theorem 2.20 (Wang et al. 2011) The closeness degree Z ( y i , y k ) of y i and y k is an

intuitionistic fuzzy similarity relation.

Proof (1) Let's first prove that Z ( y i , y k ) is an IFV:

(a) If y i = y k , then Z ( y i , y k ) = ( 1 , 0 ) ;

(b) If y i = y k , then

n n

1
n

1
n

μ ik = 1 −

1 |

v ij − v kj |−

1 | π ij − π kj | j = j =

n

1
n

≤ 1 −

1 |

v ij − v kj + π ij − π kj |

j =

n

1
n

= 1 −

1 | μ ij − μ kj |

(2.194)

j =

Obviously, we have 0 ≤ μ ik ≤ 1, with μ ik = 1 if and only if μ ij = μ kj ,for

all j = 1 , 2 ,..., n , and with μ ik = 0 if and only if μ ij = 1 and μ kj = 0, for all

j = 1 , 2 ,..., n ,or μ ij = 0 and μ kj = 1, for all j = 1 , 2 ,..., n .

v ik = j = 1 | Similarly, we have 0 ≤˙ v ij − v kj | / n ≤ 1, with v ik = ˙ 1 if and only if

v ij = v kj , for all j = 1 , 2 ,..., n , and with ˙ v ik = 0 if and only if v ij = 1 and v kj = 0,

for all j = 1 , 2 ,..., n ,or v ij = 0 and v kj = 1, for all j = 1 , 2 ,..., n .

Also since