Intuitionistic Fuzzy Aggregation Techniques - Intuitionistic Fuzzy Aggregation and Clustering - page 89

Based on Theorem 1.31, the following property can be obtained:

α − α + be given by Eqs. ( 1.179 ) and Theorem 1.32 (Xia et al. 2012c) Let and

( 1.180 ), then

α − ≤ (α 1 ,α 2 ,...,α n ) ≤ α + ATS − IFWA (1.215)

T be the weight vector

Theorem 1.33 (Xia et al. 2012c) Let w = ( w 1 , w 2 ,..., w n )

, such that i = 1 w i of the IFVs α i ( i = 1 , 2 ,..., n ) = 1. If β = (μ β , v

β )

is an IFV,

then

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

(1.216)

Proof Since

h − 1

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

β ))

(1.217)

α i

then

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

⎛

⎝ h − 1 ⎛

⎞

⎠ , g − 1 ⎛

⎞
⎠

⎞
⎠

n n

w i h ( h − 1 w i g(g − 1

⎝ ⎝

= ( h (μ α i ) + h (μ β ))) (g( v α i ) + g( v β )))

i = 1 i = 1

⎛

⎝ h − 1 ⎛

⎞

⎠ , g − 1 ⎛

⎞
⎠

⎞
⎠

n n

⎝ ⎝

(1.218) = w i ( h (μ α i ) + h (μ β )) w i (g( v α i ) + g( v

β ))

i = 1 i = 1

and

ATS − IFWA (α 1 ,α 1 ,...,α n ) ⊕ β

h − 1 n

, g − 1 n = w i h (μ α i ) w i g( v

α i )

⊕ (μ β , v

β )

h − 1 h h − 1 n

i = 1 i = 1

= w i h (μ α i ) + h (μ β ) ,

g − 1 g − 1 n

i = 1

g w i g( v

α i )

+ g( v

β )

h − 1 n

, g − 1 n

i = 1

= w i h (μ α i ) + h (μ β ) w i g( v

α i ) + g(

v

β )

h − 1 n

, g − 1 n

(1.219)

i = 1 i = 1

= w i ( h (μ α i ) + h (μ β )) w i (g( v

α i ) + g(

v

β ))

i = 1 i = 1

which completes the proof.