Intuitionistic Fuzzy Aggregation Techniques - Intuitionistic Fuzzy Aggregation and Clustering - page 73

GIFWBM p , q , r GIFWBM p , q , r

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

Let and

...,β n ) (α) (β) α β

, and let S and S be the scores of and , then Eq. ( 1.169 ) is equal to

S (α) ≤ . Now we discuss the following cases:

Case 1 If S

S (β)

(α) < S (β) , then by Xu and Yager (2006)'s ranking method, it can be

obtained that

GIFWBM p , q GIFWBM p , q (α 1 ,α 2 ,...,α n )< (β 1 ,β 2 ,...,β n ) (1.170)

Case 2 If S (α) = S (β) , then

⎛

α k w i w j w k ⎞

1
p + q + r

n

1

⎝ 1

p q r

⎠

− − μ

α i μ α j μ

i , j , k = 1

⎛
⎝

r ⎞
⎠

⎛
⎝ 1

r w i w j w k ⎞

1

p + q +

n

1

⎠

p q

− − − − ( −

v α i )

( −

v α j )

( −

v α k ) 1 1 1 1

i , j , k = 1

⎛

β k w i w j w k ⎞

1

1

p + q + r

n

⎝ 1

p q

⎠

r

= − − μ

β i μ β j μ

i , j , k = 1

⎛

r ⎞
⎠

⎛

r w i w j w k ⎞

1

p + q +

n

1

⎝

⎝ 1 ⎠

p q

− 1 − − − ( 1 − v β i ) ( 1 − v β j ) ( 1 − v β k )

i , j , k = 1

(1.171)

Since μ α i ≤ μ β i and v α i ≥ v β i , for all i ,wehave

⎛

α k w i w j w k ⎞

1
p + q + r

1

n

⎝ 1

p q r

⎠

− − μ

α i μ α j μ

i , j , k = 1

⎛

k w i w j w k ⎞

1
p + q + r

n

1

p
β i μ

q
β j μ

⎝ 1

r
β

⎠

= − − μ (1.172)

i , j , k = 1

and

⎛

r w i w j w k ⎞

1

p + q + r

n

1

⎝ 1

p q

⎠

1 − − − ( 1 − v

α i )

( 1 − v

α j )

( 1 − v

α k )

i , j , k = 1