Intuitionistic Fuzzy Aggregation Techniques - Intuitionistic Fuzzy Aggregation and Clustering - page 49

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Definition 1.8 (Xia et al. 2012a) For any p q 0, if

⎛ ⎞

p

α j

1

n
⊗

⎝

1
n ( n − 1 )

⎠

IFGBM p , q

(α 1 ,α 2 ,...,α n ) = α i ⊕ q (1.118)

p + q

i , j = 1

i = j

then IFGBM p , q is called an intuitionistic fuzzy geometricBonferronimean (IFGBM).

By using the operations and the relations of IFVs given in Sect. 1.2.2 , we can
derive the following theorem:

Theorem 1.13 (Xia et al. 2012a) Let p , q > 0, then the aggregated value by using

the IFGBM is also an IFV, and

IFGBM p , q (α 1 ,α 2 ,...,α n )

⎛
⎝

⎛
⎝

⎞
⎠

1

p + q

n

1

q

1
n ( n − 1 )

p

= 1 − 1 − − ( 1 − μ α i ) ( 1 − μ α j ) ,

, =

i j 1

i = j

⎛
⎝

⎞
⎠

⎛
⎝

⎞
⎠

1
p + q

1
p + q

1

j

n n 1

n ( n − 1 )

1

q

1

v p
α i v q

p

1 − − , 1 − − ( 1 − μ α i ) ( 1 − μ α j )

n ( n − 1 )

α

i , j = 1 i , j = 1

i = j i = j

q ⎞
⎠

⎛
⎝

⎞
⎠

1

p +

1

j

n 1

v p
α i v q

n ( n − 1 )

(1.119) − 1 − −

α

i , j = 1

i = j

Proof By the operational laws (1) and (3) described in Definition 1.3, we have

p

v p

p

v p

α i = ( − ( − μ α i ) ,

α i ,(

− μ α i ) −

α i )

p 1 1 1 (1.120)

q

v q
α

q

v q
α

q α j = ( 1 − ( 1 − μ α j ) , ,( 1 − μ α j ) − ) (1.121)

j j

and then

p q

v p

α i v q

p q

v p

α i v q

p α i ⊕ q α j = ( 1 − ( 1 − μ α i ) ( 1 − μ α j ) ,

α j ,(

1 − μ α i ) ( 1 − μ α j ) −

α j )
(1.122)