Intuitionistic Fuzzy Aggregation Techniques - Intuitionistic Fuzzy Aggregation and Clustering - page 90

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Theorem 1.34 (Xia et al. 2012c) If r 0, then

ATS − IFWA ( r α 1 , r α 2 ,..., r α n ) = rATS − IFWA (α 1 ,α 2 ,...,α n ) (1.220)

Proof According to Definition 1.21, we have

h − 1

(μ α i )), g − 1 r α = ( rh ( r g( v )) (1.221)

α i

then

ATS − IFWA ( r α 1 , r α 2 ,..., r α n )

h − 1 k + 1

, g − 1 k + 1 h − 1 w i g(g − 1 = w i h ( ( rh (μ α i ))) ( r g( v

α i )))

i = 1 i = 1

h − 1 k + 1

, g − 1 k + 1 = w i ( rh (μ α i )) w i ( r g( v α i )) (1.222)

i = 1 i = 1

and

rATS − IFWA (α 1 ,α 2 ,...,α n )

h − 1 rh h − 1 n

, g − 1 r g g − 1 n

= w i h (μ α i ) w i g( v α i )

i = 1 i = 1

h − 1 r

, g − 1 r

n n

= w i h (μ α i ) w i g( v ) (1.223)

α i

i = 1 i = 1

According to Theorems 1.33 and 1.34, we can get the following result easily:

Theorem 1.35 (Xia et al. 2012c) If r > 0, and β = (μ β , v

β )

is an IFV, then

ATS − IFWA ( r α 1 ⊕ β, r α 2 ⊕ β,..., r α n ⊕ β) = rATS − IFWA (α 1 ,α 2 ,...,α n ) ⊕ β

(1.224)

Theorem 1.36 (Xia et al. 2012c) Let β i = (μ β i , v β i )( i = 1 , 2 ,..., n ) be a

T

collection of IFVs, and w = ( w 1 , w 2 ,..., w n ) their weight vector, such that

i = 1 w i

= 1, then

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

= ATS − IFWA (α 1 ,α 2 ,...,α n ) ⊕ ATS − IFWA (β 1 ,β 2 ,...,β n ) (1.225)

Proof According to Definition 1.21, we have

h − 1

(μ β i )), g − 1 α i ⊕ β i = ( h (μ α i ) + h (g( v ) + g( v )) (1.226)

α β i i