Intuitionistic Fuzzy Aggregation Techniques - Intuitionistic Fuzzy Aggregation and Clustering - page 74

⎛

r w i w j w k ⎞

1
p + q + r

n

1

⎝ 1

p q

⎠

= 1 − − − ( 1 − v ) ( 1 − v ) ( 1 − v )

β β β i j k

i , j , k = 1

(1.173)

and thus,

⎛
⎝ 1 −

α k w i w j w k ⎞

1
p + q + r

n

1 − μ

p
α

q
α

r

⎠

h α = μ μ

i j

i , j , k = 1

⎛
⎝

r ⎞
⎠

⎛
⎝ 1

r w i w j w k ⎞

1

n

1

p + q +

p q

⎠

+ 1 − − − ( 1 − v

α i )

( 1 − v

α j )

( 1 − v

α k )

i , j , k = 1

⎛

k w i w j w k ⎞

1

n

1

p + q + r

p
β i μ

q
β j μ

r
β

⎝ 1 ⎠

= − − μ

i , j , k = 1

⎛
⎝

p + q + r ⎞
⎠

⎛
⎝ 1 −

r w i w j w k ⎞

1

n

1 − ( 1 − v

p q

⎠

+ 1 −

β i )

( 1 − v

β j )

( 1 − v

β k )

i , j , k = 1

(1.174) = h

β

Then by Xu and Yager (2006)'s ranking method, we get

GIFWBM p , q , r GIFWBM p , q , r (α 1 ,α 2 ,...,α n ) = (β 1 ,β 2 ,...,β n ) (1.175)

which completes the proof.

Theorem 1.19 Let ( α 1 , α 2 ,..., α n ) be any permutation of (α 1 ,α 2 ,...,α n ) , then

GIFWBM p , q , r GIFWBM p , q , r (α 1 ,α 2 ,...,α n ) = ( α 1 , α 2 ,..., α n ) (1.176)

α − and α + be given by Eqs. ( 1.35 ) and ( 1.36 ), then Theorem 1.20 Let

α − ≤

GIFWBM p , q

(α 1 ,α 2 ,...,α n ) ≤ α +

(1.177)

1.5.3 Generalized Intuitionistic Fuzzy Weighted Bonferroni
Geometric Mean

Apparently, the aggregation operators proposed in Sects. 1.5.1 and 1.5.2 are all based
on the arithmetic average, which is one of the basic aggregation techniques and
focuses on the group opinion, and another fundamental one is the geometric mean,