Intuitionistic Fuzzy Aggregation Techniques - Intuitionistic Fuzzy Aggregation and Clustering - page 53

⎛
⎝

⎞
⎠

p

α j

1

n
⊗

1
n ( n − 1 )

IFGBM p , q

(α 1 ,α 2 ,...,α n ) = α i ⊕ q

p + q

i , j = 1

i = j

⎛
⎝

⎞
⎠

p

α j

1

n
⊗

1
n ( n − 1 )

= α i ⊕ q

p + q

i , j = 1

i = j

IFB p , q = ( α 1 , α 2 ,..., α n ) (1.133)

(3) (Monotonicity) Let β i = (μ β i , v β i ,π β i )( i = 1 , 2 ,..., n ) be a collection of

IFVs, if μ α i ≤ μ β i and v ≥ v

β i , for all i , then

α i

IFGBM p , q IFGBM p , q (α 1 ,α 2 ,...,α n ) ≤ (β 1 ,β 2 ,...,β n ) (1.134)

Proof Since μ α i ≤ μ β i and v ≥ v

β i , for all i , then

α i

⎧
⎨
⎩

p q p q

( 1 − μ α i ) ( 1 − μ α j ) ≥ ( 1 − μ β i ) ( 1 − μ β j )

v p α i v q ≥ v p

β i v q

α j

β j

⎧
⎨
⎩

n

1

q

1
n ( n − 1 )

p

− ( 1 − μ α i ) ( 1 − μ α j )

i , j = 1

i = j

n

1

q

1
n ( n − 1 )

p

≤ − ( 1 − μ β i ) ( 1 − μ β j )

⇒

i , j = 1

i = j

1 − v p

β j

1 − v p

α j

1
n ( n − 1 )

n n

1
n ( n − 1 )

α i v q

β i v q

≤

i , j = 1
i = j

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 , 1

i = j

j =

⎛
⎝

⎞
⎠

1
p + q

⇒

n

1

q

1
n ( n − 1 )

p

≥ 1 − − ( 1 − μ β i ) ( 1 − μ β j )

i , j = 1

i = j

1 − v p

β j

1 − v p

α j

1
n ( n − 1 )

n n

1
n ( n − 1 )

α i v q

β i v q

1 − ≥ 1 −

i , j = 1
i = j

i , j = 1
i = j