Intuitionistic Fuzzy Aggregation Techniques - Intuitionistic Fuzzy Aggregation and Clustering - page 55

IFGBM p , q

(α 1 ,α 2 ,...,α n )

⎛ ⎞

n
⊗

n

p

α j

1 1

p

⎝

n
⊗

⎠ =

1

1

= α i ⊕

1 (

α i )

q

n ( n − 1 )

p

p + q

i , j = 1 i =

i = j

⎛

1

n

1

n

p p

n n

1

p

1

α i

1 1

⎝ 1

v p = − − − ( 1 − μ α i ) , − − ,

i = 1 i = 1

p ⎞
⎠

1

n

1

n

1
p

1

n n

1

p

1

α i

1 1

v p − − ( 1 − μ α i ) − − −

i = 1 i = 1

IFGBM p , 0 = (α 1 ,α 2 ,...,α n ) (1.137)

which is called a generalized intuitionistic fuzzy geometric mean (GIFGM).
Case 2 If p = 2 and q → 0, then Eq. ( 1.119 ) is transformed as:

n
⊗

n

1
2

1

IFGBM 2 , 0 (α 1 ,α 2 ,...,α n ) =

1 (

2 α i )

i =

⎛

1

n

1

n

1
2

1
2

1

2

1

α i

n n 1 1

⎝ 1

v 2 = − − − ( 1 − μ α i ) , − − ,

i = 1 i = 1

2 ⎞
⎠

1

n

1

n

1
2

1

1

2

1

α i

n n 1 1

v 2 − − ( 1 − μ α i ) − − − (1.138)

i = 1 i = 1

which is called an intuitionistic fuzzy square geometric mean (IFSGM).
Case 3 If p 0, then Eq. ( 1.119 ) reduces to the intuitionistic fuzzy

geometric mean (IFGM) (Xu and Yager 2006):

= 1 and q →

IFGBM 1 , 0

(α 1 ,α 2 ,...,α n )

n
⊗

1
n
i

=

1 α i =

n

n

n n n

μ α i 1

v α i

1

v α i

μ α i 1

n

1
n

1
n

1

= , 1 − − , − − (1.139)

i = 1 i = 1 i = 1 i = 1

Case 4 If p = q = 1, then Eq. ( 1.119 ) reduces to the following:

IFGBM 1 , 1 (α 1 ,α 2 ,...,α n )

⎛
⎝

⎞
⎠

α i ⊕ α j

1
2

n
⊗
i , j =

1

=

n ( n − 1 )

1

i = j