Intuitionistic Fuzzy Aggregation Techniques - Intuitionistic Fuzzy Aggregation and Clustering - page 84

γ =

1, then the operational laws (9)-(12) reduce to the correspond-
ing operational laws (1)-(4) of Definition 1.3; if

Especially, if

γ =

2, then the operational laws

(9)-(12) above reduce to the operational laws (5)-(8).

log γ − 1
γ 1 ,

If g( t ) = γ ∈ ( 1 , +∞ ) , then

t

−

1 1

1 − μ α 1 1 − μ α 2

+ (γ

− 1 )(γ − 1 )

( 13 )α 1 ⊕ α 2 = − log ,

γ

γ − 1

1

v α 1 v α 2

+ (γ

− 1 )(γ − 1 )

log ,γ> 1 .

γ

γ − 1

log 1

+ (γ μ α 1

)(γ μ α 2 − 1 − 1 )

( 14 )α 1 ⊗ α 2 = ,

γ

γ − 1

1

1 − v α 1 1 − v α 2

+ (γ

− )(γ − )

1 1

− ,γ> .

1 log 1

γ

γ − 1

1 1 1

− μ α − 1 ) λ
(γ − 1 ) λ − 1

1 v α − 1 ) λ

(γ − 1 ) λ − 1

+ (γ

(γ

( 15 ) λα =

− , + ,λ> ,γ> .

log log 0 1

γ γ

log 1 1

+ (γ μ α − 1 ) λ
(γ − 1 ) λ − 1

v α − 1 ) λ
(γ − 1 ) λ − 1

1 −

+ (γ

) α λ =

( 16

, 1 − log ,λ > 0 ,γ> 1 ,

γ γ

which are the ones defined based on Frank t-conorm and t-norm. Especially, if 1,

then the operational laws (13)-(16) reduce to the corresponding operational laws (1)-
(4) of Definition 1.3.

γ =

Moreover, in what follows, we discuss some relationships of the above operational
laws of the IFVs:

Theorem 1.27 (Xia et al. 2012c)

α 1 ⊕ α 2 = α 2 ⊕ α 1 .

(1)

α 1 ⊗ α 2 = α 2 ⊗ α 1 .

(2)

(3) λ(α 1 ⊕ α 2 ) = λα 1 ⊕ λα 2 .

(α 1 ⊗ α 2 ) λ = α 1 ⊗ α 2 .

(4)

(5) λ 1 α ⊕ λ 2 α = (λ 1 + λ 2 )α .

α λ 1 ⊗ α λ 2 = α λ 1 + λ 2 .

(6)