Intuitionistic Fuzzy Aggregation Techniques - Intuitionistic Fuzzy Aggregation and Clustering - page 92

log γ + ( 1 − γ) t
t

,

Case 3 If g( t ) = γ ∈ ( 0 , +∞ ) , then the ATS-IFWA operator

reduces to the following:

i = 1 (

− i = 1 (

w i w i

1 + (γ − 1 )μ α i ) 1 − μ α i )

i = 1 (

) i = 1 (

HIFWA (α 1 ,α 2 ,...,α n ) =

w i ,

1 + (γ − 1 )μ α i )

w i

+ (γ − 1 1 − μ α i )

γ i = 1 v w i α i

i = 1 ( 1 + (γ − 1 )( 1 − v α i ))

+ (γ − 1 ) i = 1 v w i

w i

α i

(1.231)

which is called a Hammer intuitionistic fuzzyweighted averaging (HIFWA) operator.
Especially, if γ = 1, then the HIFWA operator reduces to the IFWA operator; if

γ = 2, then the HIFWA operator reduces to the EIFWA operator.

Case 4 If g( log γ − 1

γ 1 , t

t ) = ∈ ( 1 , +∞ ) , then the ATS-IFWA operator reduces

t

−

to the following:

FIFWA (α 1 ,α 1 ,...,α n )

1 1

i = 1 (γ 1 i = 1 (γ

1 − μ α i v α i

w i w i

− 1 ) − 1 )

= − log + , log +

γ γ

γ − 1 γ − 1

(1.232)

which is called a Frank intuitionistic fuzzy weighted averaging (FIFWA) operator.
Especially, if 1, then the FIFWA operator reduces to the IFWA operator.

Motivated by the geometric mean, the following definition is given:

γ →

Definition 1.23 (Xia et al. 2012c) If

n
⊗

w i
i

ATS − IFWG (α 1 ,α 2 ,...,α n ) =

1 α

(1.233)

i =

then ATS-IFWG is called an Archimedean t-cornorm and t-norm based intuitionistic
fuzzy geometric (ATS-IFWG) operator.

Based on the operational laws of the IFVs given in Definition 1.21, we can derive
the following theorem:

Theorem 1.37 (Xia et al. 2012c) The aggregated value by using the ATS-IFWG
operator is also an IFV, and

n
⊗ − (α 1 ,α 2 ,...,α n ) =

w i α i

ATS IFWG

i = 1

g − 1 n h − 1 n = w i g(μ α i ) , w i h ( v α i )

i = 1 i = 1

(1.234)