Intuitionistic Fuzzy Aggregation Techniques - Intuitionistic Fuzzy Aggregation and Clustering - page 125

− μ α ) 1

n

J κ α ,λ α (α) =

( 7 ) μ α + ( 1 − ( 1 − κ α )

n − 1

t n − 1 − t n

α

− v

α κ α λ α

0 (

1 − κ α ) λ ,λ v .

α

α

t =

P n ( 8 )

κ α ,λ α (α) = (

max (κ α ,μ α ) , min (λ α , v α )) , where κ α + λ α ≤ 1 .

( 9 ) Q

κ α ,λ α (α) = (

min (κ α ,μ α ) , max (λ α , v

α )) ,

where κ α + λ α ≤ 1 ,

which translate one IFV to another IFV.

Proof (1), (3), (8) and (9) are obvious. By the idea of Liu and Wang (2007), we can
also easily prove (2). Next, motivated also by the idea of Liu and Wang (2007), we
prove (4), (5), (6) and (7) using mathematical induction on n :
(4) For n

= 1, we have

H 1

κ α ,λ α (α) =

μ H 1

κ α ,λ α ( α ) ,

v H 1 = (κ α μ α , v

α + λ α π α )

κ α ,λ α ( α )

1

1 1

= κ

α μ α ,

v α + (

− ν α ) − ( − λ α )

1 1

1 − 1

t 1 − 1 − t

− μ α λ α

0 κ

( 1 − λ α ) (1.304)

α

t =

For n = 2, we have

2

μ H 2

κ α ,λ α (α) = κ

α μ α

(1.305)

v H 2

κ α ,λ α (α) = v

α + λ α π α + λ α ( 1 − κ α μ α − v α − λ α π α )

= v α + ( 1 − v α )(λ α + λ α ( 1 − λ α )) − μ α λ α ( 1 − λ α + κ α )

⎛

t ⎞

1

2

2 − 1

2 − 1 − t

⎝ ⎠

= v

α + (

1 − v

α )

− ( 1 − λ α ) − μ α λ α

0 κ

( 1 − λ α )

(1.306)

α

t =

Suppose that it is true for n = p , that is,

p
α μ α

(1.307)

μ H p

κ,λ (α) = κ

⎛
⎝

⎞
⎠ (1.308)

p − 1

κ,λ (α) = v α + ( 1 − v α ) 1 − ( 1 − λ α )

p − μ α λ α

p − 1 − t

t

v H p

0 κ

( 1 − λ α )

α

t =

then, when n = p + 1, we have

p p + 1

μ H p + 1

κ α ,λ α (α) = κκ

α μ α = κ

μ α

α