Intuitionistic Fuzzy Aggregation Techniques - Intuitionistic Fuzzy Aggregation and Clustering - page 101

μ

k
⊕

1 β i

=

i = 1 β i ,

v k
i = 1 β i

k

i =

i = 1 (

)μ β i ) − i = 1 ( 1 + (γ − 1 1 − μ β i )

=

− μ β i ) ,

i = 1 (

) i = 1 ( 1 + (γ − 1 )μ β i ) + (γ − 1 1

γ i = 1 v

β i

(1.250)

i = 1 (

) i = 1 v 1 + (γ − 1 )( 1 − v

β i )) + (γ −

1

β i

Then we prove that Eq. ( 1.249 ) holds for n = k + 1, that is,

k

i = 1 β i

⊕ β k + 1

⎛
⎝

1 + (γ − 1 )μ 1 − μ

1 + (γ − 1 )μ β k + 1 − 1 − μ β k + 1

k
i = 1 β i

k
i = 1 β i

1 + (γ − 1 )μ 1 + (γ − 1 )μ β k + 1 + (γ − 1 ) 1 − μ 1 − μ β k + 1 ,

=

k
i = 1 β i

k
i = 1 β i

⎞
⎠

γ v k

i = 1 β i v β k + 1

1 1 1

) 1 v β k + 1 + (γ − + (γ − 1 ) − v k

i = 1 β i

+ (γ − 1 − 1 ) v k

i = 1 β i

v β k + 1

(1.251)

By the operational laws for IFVs, we have

⎛
⎝ 1 + (γ − 1 )μ

⎞
⎠ 1 + (γ − 1 )μ β k + 1

k
⊕

1 β i i =

1 + (γ − 1 ) 1 + (γ − 1 )μ β k + 1

i = 1 (

)μ β i ) − i = 1 ( − μ β i )

i = 1 ( 1 + (γ − 1 )μ β i ) + (γ − 1 ) i = 1 ( 1 − μ β i )

1 + (γ − 1 1

=

1 − 1 − μ β k + 1

i = 1 (

)μ β i ) − i = 1 (

1 + (γ − 1 1 − μ β i )

−

i = 1 (

) i = 1 (

1 + (γ − 1 )μ β i ) + (γ − 1 1 − μ β i )

)μ β i ) 1 )μ β k + 1 − γ i = 1 ( − μ β i ) 1 − μ β k + 1 γ i = 1 ( 1 + (γ − 1 + (γ − 1 1

=

i = 1 ( 1 + (γ − 1 )μ β i ) + (γ − 1 ) i = 1 ( 1 − μ β i )

γ k + 1
i = 1 ( 1 + (γ − 1 )μ β i ) − γ k + 1 1 ( 1 − μ β i )

i = 1 ( 1 + (γ − 1 )μ β i ) + (γ − 1 ) i = 1 ( 1 − μ β i )

i =

(1.252)

=