Intuitionistic Fuzzy Aggregation Techniques - Intuitionistic Fuzzy Aggregation and Clustering - page 128

=

For k 2, we have

μ J 2

κ α ,λ α (α) = μ α + κ α π α + κ α (

1 − λ α v α − μ α − κ α π α )

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

α κ α (

1 − κ α + λ α )

2 − 1

t

1

2

2 − 1 − t

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

0 λ

( 1 − κ α )

α

t =

(1.318)

2
α

v J 2

κ α ,λ α (α) = λ

v (1.319)

α

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

⎛ ⎞

p − 1

− μ α ) 1

p −

⎝

p − 1 − t t

⎠

μ J p

κ α ,λ α (α) = μ α + (

1 − ( 1 − κ α ) v α κ α

0 λ

( 1 − κ α )

α

t =

(1.320)

p
α

v J p

κ α ,λ α (α) = λ

v α (1.321)

then, when n = p + 1, we have

⎛ ⎞

p − 1

t = 0 λ

− μ α ) 1

p −

⎝

p − 1 − t t

⎠

μ J p + 1

κα ,λα ( α ) = μ α + (

1 − ( 1 − κ α ) v

α κ α

( 1 − κ α )

α

⎛

− μ α ) 1

p

⎝ 1

p
α

+ κ α − λ v

α − μ α − (

1 − ( 1 − κ α )

⎛ ⎞ ⎞

p − 1

⎝

p − 1 − t t

⎠ ⎠

+ v α κ α

0 λ

( 1 − κ α )

α

t =

− μ α ) 1 + κ α − κ α 1

p p

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

⎛ ⎛ ⎞ ⎞

p − 1

⎝ ⎝

p − 1 − t t + 1

⎠ + λ

p
α

⎠

− v α κ α

0 λ

( 1 − κ α )

α

t =

p

t

1

1 p + p − t

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

0 λ

( 1 − κ α )

α

t =

(1.322)

p
α

p + 1

v J p + 1

κ α ,λ α (α) = λλ

v

α = λ

v (1.323)

α

α

and hence, (6) holds for n = p + 1. Therefore, (6) holds for all n .

(7) For n = 1, we have