Intuitionistic Fuzzy Aggregation Techniques - Intuitionistic Fuzzy Aggregation and Clustering - page 137

= (

w n ) F n

κ α ,λ α ( α ) ρ

1
ρ

w 1 + w 2 +···+

= F n

κ α ,λ α ( α ) ρ

1
ρ

F n

=

κ α ,λ α ( α ) .

Similarly, we can prove the others.

Theorem 1.46 (Xia and Xu 2010)

α D n GIFPWAD w (α 1 ,α 2 ,...,α m ) ≤ α D n . (1) ≤

α F n α F n , where

GIFPWAF w (α 1 ,α 2 ,...,α m )

(2) ≤ ≤ κ α j + λ α j ≤ 1,

j = 1 , 2 ,..., m .

α G n GIFPWAG w (α 1 ,α 2 ,...,α m ) ≤ α G n . (3) ≤

α H n GIFPWAH w (α 1 ,α 2 ,...,α m ) ≤ α H n . (4) ≤

α H n (α 1 ,α 2 ,...,α m ) ≤ α H n .

GIFPWAH ∗ , n
w

(5) ≤

α J n

GIFPWAJ w (α 1 ,α 2 ,...,α m ) ≤ α J n .

≤

(6)

α J n (α 1 ,α 2 ,...,α m ) ≤ α J n .

GIFPWAJ ∗ , n
w

(7) ≤

α P n α P n , where GIFPWAP w (α 1 ,α 2 ,...,α m )

(8) ≤ ≤ κ α j + λ α j ≤ 1,

j = 1 , 2 ,..., m .

α Q n α Q n , where

GIFPWAQ w (α 1 ,α 2 ,...,α m )

≤ ≤ κ α j + λ α j ≤

(9) 1,

j = 1 , 2 ,..., m , and

mi j (μ D n

α D n =

κ α j ,λ α j (α j ) ), ma j ( v D n κ α j ,λ α j (α j ) )

ma j (μ D n

α D n =

κ α j ,λ α j (α j ) ), mi j ( v D n κ α j ,λ α j (α j ) )

mi j (μ F n

α F n

=

) ), ma j ( v F n ) )

κα j ,λα j (α κα j ,λα j (α

j j

ma j (μ F n

α F n =

κ α j ,λ α j (α j ) ), mi j ( v F n κ α j ,λ α j (α j ) )

mi j (μ G n

α G n =

κ α j ,λ α j (α j ) ),

ma j ( v G n

κ α j ,λ α j (α j ) )

ma j (μ G n

α G n

=

κ α j ,λ α j (α j ) ), mi j ( v G n κ α j ,λ α j (α j ) )

mi j (μ H n

α H n =

κ α j ,λ α j (α j ) ), ma j ( v H n κ α j ,λ α j (α j ) )

ma j (μ H n

α H n =

κ α j ,λ α j (α j ) ), mi j ( v H n κ α j ,λ α j (α j ) )

mi j (μ H ∗ , n

α H n =

κ α j ,λ α j (α j ) ), ma j ( v H ∗ , n κ α j ,λ α j (α j ) )