Intuitionistic Fuzzy Aggregation Techniques - Intuitionistic Fuzzy Aggregation and Clustering - page 51

Then by Eq. ( 1.126 ) and the operational law (3) in Definition 1.3, it yields

IFGBM p , q

(α 1 ,α 2 ,...,α n )

p

α j

1

n
⊗
i , j =

1

= α i ⊕ q

n ( n − 1 )

p + q

1

i = j

⎛

⎛
⎝

⎞
⎠

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

j

n

1
n ( n − 1 )

v p
α i v q

1 − − ,

α

i , j = 1

i = j

⎛
⎝

⎞
⎠

1
p + q

n

1

q

1
n ( n − 1 )

p

1 − − ( 1 − μ α i ) ( 1 − μ α j )

i , j = 1

i = j

p + q ⎞
⎠

⎛ ⎞

1

1

α j

n

⎝

1
n ( n − 1 )

⎠

v p

α i v q

− 1 − − (1.127)

i , j = 1

i = j

i.e., Eq. ( 1.119 ) holds. In addition, since

⎛
⎝

⎞
⎠

1
p + q

n

1

q

1
n ( n −

p

0 ≤ 1 − 1 − − ( 1 − μ α i ) ( 1 − μ α j )

)

≤ 1 (1.128)

1

i , j = 1

i = j

⎛
⎝

⎞
⎠

1
p + q

n

1

j

1
n ( n − 1 )

v p
α i v q

0 ≤ 1 − − ≤ 1 (1.129)

α

i , j = 1

i = j

then

⎛
⎝

⎞
⎠

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