Intuitionistic Fuzzy Aggregation Techniques - Intuitionistic Fuzzy Aggregation and Clustering - page 127

p

μ H ∗ , p

κ α ,λ α (α) = κ

α μ α

(1.313)

⎛
⎝

⎞
⎠

p − 1

v α ) 1

p − μ α κ α λ α

p − 1 − t t

v H ∗ , p

κ α ,λ α (α) =

v α + ( 1 − − ( 1 − λ α )

0 κ

( 1 − λ α )

α

t =

(1.314)

= +

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 (1.315)

α

⎛ ⎞

p − 1

v α ) 1

p − μ α κ α λ α

⎝

p − 1 − t t

⎠

v H ∗ , p + 1

κ α ,λ α (α) =

v α + ( 1 − − ( 1 − λ α )

0 κ

( 1 − λ α )

α

t =

⎛

v α ) 1

p

⎝ 1

p + 1
α

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

⎛ ⎞ ⎞

p − 1

⎝

p − 1 − t t

⎠ ⎠

+ μ α κ α λ α

0 κ

( 1 − λ α )

α

t =

α ) 1

− (

p p

= v

α + (

1 − v 1 − λ α ) − λ α ( 1 − λ α )

⎛ ⎛ ⎞ ⎞

p − 1

⎝ ⎝

p − 1 − t t + 1

⎠ + κ

p
α

⎠

− μ α κ α λ α

0 κ

( 1 − λ α )

α

t =

p

t

1

1 p + p − t

= v

α + (

1 − v

α )

− ( 1 − λ α ) − μ α κ α λ α

0 κ

( 1 − λ α )

α

t =

(1.316)

= +

and thus, (5) holds for n p 1. Therefore, (5) holds for all n .

(6) For k = 1, we have

J 1

κ α ,λ α (α) = (μ α + κ α π α ,λ α

v

α )

1

1

= μ α + ( 1 − μ α ) − ( 1 − κ α )

1 − 1

t

t = 0 λ

1 − 1 − t 1

α

− v

α κ α

( 1 − κ α ) ,λ v (1.317)

α

α