Intuitionistic Fuzzy Aggregation Techniques - Intuitionistic Fuzzy Aggregation and Clustering - page 143

⎛
⎝

ρ ⎞
⎠

⎛
⎝ 1

w j ⎞
⎠

⎛
⎝ 1

w j ⎞
⎠

1

1
ρ

1

1

m m

F n

κ α j ,λ α j (α j ) ) ρ

≤ − − μ − 1 − − − ( 1 − v F n

κ α j ,λ α j (α j )

j = 1 j = 1

(1.382)

GIFPWAF w (α 1 ,α 2 ,...,α m ) α ∗ GIFPWAF w (α 1 ,α 2 ,...,α m )

Let α = and = ,

then by Eq. ( 1.382 ), we have

(α F n ) S (α F n ) ≤ S (1.383)

(α F n )

If S (α F n )< S , then by using Xu and Yager (2006)'s ranking method, we

have

GIFPWAF w (α 1 ,α 2 ,...,α m )< GIFPWAF w (α 1 ,α 2 ,...,α m ) (1.384)

(α F n ) If S (α F n ) = S , then

⎛
⎝

ρ ⎞
⎠

⎛
⎝ 1

w j ⎞
⎠

⎛
⎝ 1

) ) ρ w j ⎞

1
ρ

1

1 1

m m

− μ F n

⎠

− − 1 − − − ( 1 − v F n

κ α j ,λ α j (α

κα j ,λα j (α j )

j

j = 1 j = 1

⎛
⎝

ρ ⎞
⎠

⎛
⎝ 1

w j ⎞
⎠

⎛
⎝ 1

w j ⎞
⎠

1

1
ρ
−

1

1

m m

− μ F n

κ α j ,λ α j (α j ) ) ρ

= − 1 − − − ( 1 − v F n

κ α j ,λ α j (α j )

j = 1 j = 1

(1.385)

Since μ F n

κ α j ,λ α j (α j ) ≤ μ F n

and v F n

κ α j ,λ α j (α j ) ≥

v F n , for all j , then

κ α j ,λ α j (α j ) κ α j ,λ α j (α j )

⎛

w j ⎞
⎠

⎛

w j ⎞
⎠

1
ρ

1

1
ρ

1

m m

− μ F n − μ F n

⎝ 1 ⎝ 1

− = −

κ α j ,λ α j (α j )

κ α j ,λ α j (α j )

j = 1 j = 1

(1.386)

⎛

κ α j ,λ α j (α j ) ) ρ w j ⎞

1
ρ

1

m

⎝ 1 ⎠

1 − − − ( 1 − v F n

j = 1

⎛

w j ⎞
⎠

1

1
ρ

m

⎝ 1

κ α j ,λ α j (α j ) ) ρ

= 1 − − − ( 1 − v F n (1.387)

j = 1