Intuitionistic Fuzzy Aggregation Techniques - Intuitionistic Fuzzy Aggregation and Clustering - page 135

=

(1) For m 2: Since

ρ

F n ρ 1

μ F n

κ α 1 ,λ α 1 (α 1 )

=

) ,

1 − − v F n (1.340)

κ α 1 ,λ α 1 (α

)

κ α 1 ,λ α 1 (α

1

1

ρ F n ρ 1

μ F n

κ α 2 ,λ α 2 (α 1 )

=

κ α 2 ,λ α 2 (α 2 ) ,

1 − − v F n (1.341)

κ α 2 ,λ α 2 (α

)

2

then

w 1 F n

ρ

w 2 F n

ρ

κ α 1 ,λ α 1 (α 1 )

⊕

κ α 2 ,λ α 2 (α 2 )

⎛

ρ w j ⎞
⎠

1 w j

1 1

2 2

− μ F n

⎝ 1

= − , − − v F n

κ α 1 ,λ α 1 (α 1 )

κ α 1 ,λ α 1 (α 1 )

j = 1 j = 1

(1.342)

(2)IfEq.( 1.332 ) holds for m = p , that is,

w 1 F n

ρ

w 2 F n

ρ

w p F n

κ α p ,λ α p α p ρ κ α 1 ,λ α 1 (α 1 )

⊕

κ α 2 ,λ α 2 (α 2 )

⊕···⊕

⎛
⎝

⎛

w j ⎞
⎠

1

1
ρ

p

− μ F n

⎝ 1

= − ,

κ α j ,λ α j ( α j

)

j = 1

ρ ⎞
⎠

⎛
⎝ 1

ρ w j ⎞
⎠

1

1 1

p

1 − − − − v F n (1.343)

κ α j ,λ α j ( α j

)

j = 1

= +

then, when m p 1, by the operational laws given in Sect. 1.8 ,wehave

ρ ⊕···⊕ w p + 1 F n

κ α p + 1 ,λ α p + 1 α p + 1 ρ

w 1 F n

ρ ⊕ w 2 F n

κ α 1 ,λ α 1 (α 1 ) κ α 2 ,λ α 2 (α 2 )

⎛
⎝ 1 −

ρ w j ⎞
⎠

1 − μ F n w j

1 − 1 − v F n

p p

κ α j ,λ α j α j

=

κ α j ,λ α j α j

,

j = 1 j = 1

1 − 1 − μ F n

κ α p

w p + 1

ρ w p + 1

1 − 1 − v F n

κ α p +

⊕ ,

1 ( α p + 1 )

1 ( α p + 1 )

1 ,λ α p +

1 ,λ α p + +

⎛
⎝ 1 −

ρ w j ⎞
⎠ (1.344)

1 − μ F n w j

1 − 1 − v F n

p + 1 p + 1

κ α j ,λ α j α j

=

κ α j ,λ α j α

j

,

j = 1 j = 1

i.e. Eq. ( 1.337 ) holds for m = p + 1. Thus, Eq. ( 1.337 ) holds for all m . Therefore,