Intuitionistic Fuzzy Aggregation Techniques - Intuitionistic Fuzzy Aggregation and Clustering - page 76

To aggregate the intuitionistic fuzzy information, we further introduce the
following:

T be the weight vector

Definition 1.16 (Xia et al. 2012b) Let w = ( w 1 , w 2 ,..., w n )

n and i = 1 w i

of α i ( i = 1 , 2 ,..., n ) such that w i > 0, i = 1 , 2 ,..., = 1. If

GIFWBGM p , q , r (α 1 ,α 2 ,...,α n )

α k w i w j w k

p

1

n
⊗ = α i ⊕ q α j ⊕ r (1.181)

p + q + r

i , j , k = 1

then GIFWBGM p , q , r is called a generalized intuitionistic fuzzy weighted Bonferroni
geometric mean (GIFWBGM).

Based on the operational laws of the IFVs, and similar to Theorem 1.16, we can
derive the following theorem:

Theorem 1.22 (Xia et al. 2012b) The aggregated value by using the GIFWBGM is
also an IFV, and

GIFWBGM p , q , r (α 1 ,α 2 ,...,α n )

⎛
⎝

⎛

r w i w j w k ⎞

1
p + q + r

n

1

⎝ 1

p q

⎠

= 1 − − − ( 1 − μ α i ) ( 1 − μ α j ) ( 1 − μ α k ) ,

i , j , k = 1

p + q + r ⎞
⎠

⎛

k w i w j w k ⎞

1

n

1

⎝ 1

v p
α i v q j v r

⎠

− − (1.182)

α α

i , j , k = 1

Now let us discuss some desirable properties of the GIFGBM (Xia et al. 2012b):

α i ( = , ,..., ) α i = α = (μ α ,

v α ,π α ) Theorem 1.23 If all i 1 2 n are equal, i.e., ,

for all i , then

GIFWBGM p , q (α 1 ,α 2 ,...,α n ) = α (1.183)

Theorem 1.24 Let β i = (μ β i , v

β i ,π β i )(

i = 1 , 2 ,..., n ) be a collection of IFVs,

if μ α i ≤ μ β i and v ≥ v

β i , for all i , then

α i

GIFWBGM p , q GIFWBGM p , q (α 1 ,α 2 ,...,α n ) ≤ (β 1 ,β 2 ,...,β n ) (1.184)

Theorem 1.25 Let ( α 1 , α 2 ,..., α n ) be any permutation of (α 1 ,α 2 ,...,α n ) , then

GIFWBGM p , q GIFWBGM p , q (α 1 ,α 2 ,...,α n ) = ( α 1 , α 2 ,..., α n ) (1.185)