Intuitionistic Fuzzy Aggregation Techniques - Intuitionistic Fuzzy Aggregation and Clustering - page 134

⎛

⎛

− max κ α j ,μ α j ρ w j ⎞

1
ρ

m

1

⎝

GIFPWAP w (α 1 ,α 2 ,...,α m ) =

⎝ 1 ⎠

( 8 ) − ,

j = 1

ρ ⎞
⎠

⎛

j ρ w j ⎞

1

m

1

− 1 min λ α j ,

⎝ 1 ⎠

1 − − − v

α

j =

1

where κ α j + λ α j ≤ 1, j = 1 , 2 ,..., m .

⎛

⎛

− min κ α j ,μ α j ρ w j ⎞

1
ρ

m

1

⎝

⎝ 1 ⎠

GIFPWAQ w (α 1 ,α 2 ,...,α m ) =

( 9 ) − ,

j = 1

ρ ⎞
⎠

⎛

α j ρ w j ⎞

1

m

1

− 1 max λ α j ,

⎝ 1 ⎠

1 − − − v

j = 1

where κ α j + λ α j ≤ 1, j = 1 , 2 ,..., m .

Proof Now we prove (2) (the others can be proven similarly). We first prove the
following equation by using mathematical induction on m :
w 1 F n

ρ

w 2 F n

ρ

w m F n

ρ

κ α 1 ,λ α 1 (α 1 )

⊕

κ α 2 ,λ α 2 (α 2 )

⊕···⊕

κ α m ,λ α m (α m )

⎛
⎝

⎛

w j ⎞
⎠

1

1
ρ

m

− μ F n

⎝ 1

= − ,

κ α j ,λ α j ( α j

)

j = 1

ρ ⎞
⎠

⎛

ρ w j ⎞
⎠

1

1 1

m

⎝ 1

1 − − − − μ F n (1.337)

κα j ,λα j ( α j

)

j = 1

where

− 1 − κ α j − λ α j n

κ α j + λ α j

1

μ F n

) = μ α j + κ α j π α j

(1.338)

κ α j ,λ α j ( α j

− 1 − κ α j − λ α j n

κ α j + λ α j

1

v F n

) =

v α j + λ α j π α j (1.339)

κ α j ,λ α j ( α j