Intuitionistic Fuzzy Aggregation Techniques - Intuitionistic Fuzzy Aggregation and Clustering - page 88

h − 1 n h − 1 n

= w i h (μ α i ) + 1 − w i h (μ α i ) = 1 (1.208)

i = i = 1 1

which completes the proof of Theorem 1.29.

Then we can investigate some desirable properties of the ATS-IFWA operator:

Theorem 1.30 (Xia et al. 2012c) If all α i ( i = 1 , 2 ,..., n ) are equal, i.e., α i =

α = (μ α , v α ) , for all i , then

− (α 1 ,α 2 ,...,α n ) = α

ATS IFWA (1.209)

Proof Let α i = α = (μ α , v α ) ,wehave

n
⊕

ATS − IFWA (α 1 ,α 2 ,...,α n ) = ATS − IFWA (α,α,...,α) = w i α

i = 1

h − 1 n

, g − 1 n = w i h (μ α ) w i g( v

α )

i = 1 i = 1

h − 1

(μ α )) , g − 1 = ( h (g( v

α ))

= α (1.210)

Theorem 1.31 (Xia et al. 2012c) Let β i = (μ β i , v β i )( i = 1 , 2 ,..., n ) be a collec-

tion of IFVs, if μ α i ≤ μ β i and v ≥ v

β i , for all i , then α i

ATS − IFWA (α 1 ,α 2 ,...,α n ) ≤ ATS − IFWA (β 1 ,β 2 ,...,β n ) (1.211)

Proof We have known that h ( t ) = g( 1 − t ) , and g : [ 0 , 1 ]→ 0 , 1 ] is a strictly

decreasing function, then h ( t ) is a strictly increasing function. Since μ α i ≤ μ β i and

v ≥ v

β i , then we have
h − 1 n

α i

h − 1 n

(μ α i ) ≤ (μ β i )

w i h w i h (1.212)

i = 1 i = 1

g − 1 n ≥ g − 1 n

w i g( v α i ) w i g( v β i )

(1.213)

i = 1 i = 1

and thus,

S ( ATS-IFWA (α 1 ,α 2 ,...,α n )) ≤ S ( ATS-IFWA (β 1 ,β 2 ,...,β n )) (1.214)

which completes the proof.