Intuitionistic Fuzzy Aggregation Techniques - Intuitionistic Fuzzy Aggregation and Clustering - page 52

⎛
⎝

⎞
⎠

1
p + q

1

α j

n

1
n ( n − 1 )

v p

α i v q

+ 1 − −

i , j = 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

n

1

q

1
n ( n − 1 )

p

(1.130)

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

i , j = 1
i = j

which completes the proof of Theorem 1.13.
Then, in what follows, we introduce some desirable properties of the IFGBM (Xia
et al. 2012a):

(1) (Idempotency) If all α i ( i = 1 , 2 ,..., n ) are equal, i.e., α i = α = (μ α , v

α ,π α )

,

for all i , then

IFGBM p , q (α 1 ,α 2 ,...,α n )

IFGB p , q = (α,α,...,α)

⎛ ⎞ ⎛ ⎞

1 1

n
⊗

n
⊗

⎝ ⎠ = ⎝ ⎠

1
n ( n − 1 )

1
n ( n − 1 )

= ( p α ⊕ q α) (( p + q )α)

p + q p + q

i , j = 1 i , 1

i = j

j =

i = j

1

n )

n ( n − 1 )

( n − 1

=

q ((

p + q )α) = α (1.131)

p +

(2) (Commutativity)

IFGBM p , q IFGBM p , q (α 1 ,α 2 ,...,α n ) = ( α 1 , α 2 ,..., α n ) (1.132)

( α 1 , α 2 ,..., α n ) (α 1 ,α 2 ,...,α n )

where is any permutation of .

Proof Since ( α 1 , α 2 ,..., α n ) is any permutation of (α 1 ,α 2 ,...,α n ) , then