Intuitionistic Fuzzy Aggregation Techniques - Intuitionistic Fuzzy Aggregation and Clustering - page 35

(2) If g 1 ( α 1 ) = g 2 ( α 2 )

, then
(a) If g 2 ( α 1 )<g 2 ( α 2 ) , then α 1 is larger than α 2 , denoted by α 1 > α 2 ;

α 1 = α 2 .
Xu (2011) extended Definition 1.3 to interval-valued intuitionistic fuzzy environ-
ments:

(b) If g 2 ( α 1 ) = g 2 ( α 2 ) , then α 1 is equal to α 2 , denoted by

α i , π α i ) = ( [ μ α i ,μ α i ] , [

v α i , v α i ] , [ π α i , Definition 1.5 (Xu 2011) Let α i = (μ α i , ˜ v

π α i ] )( i = 1 , 2 ) be two IVIFVs, then

α 1 ⊕ α 2 = ( [ μ α 1 + μ α 2 − μ α 1 μ α 2 ,μ α 1 + μ α 2 − μ α 1 μ α 2 ] , [ v α 1 v α 2 , v α 1 v α 2 ] , ( 1 )

− μ α 1 )( − μ α 2 ) − v α 1 v α 2 ,( − μ α 1 )( − μ α 2 ) − v α 1 v α 2 ] ). [ ( 1 1 1 1

α 1 ⊗ α 2 = ( [ μ α 1 μ α 2 ,μ α 1 μ α 2 ] , [

v α 1 + v α 2 − v α 1 v α 2 , v α 1 + v α 2 − v α 1 v α 2 ] ,

( )

2

v α 1 )( v α 2 ) − μ α 1 μ α 2 ,( v α 1 )( v α 2 ) − μ α 1 μ α 2 ] . [ ( 1 − 1 − 1 − 1 −

− μ α 1 ) λ , − μ α 1 ) λ ] , [ ( v α 1 ) λ ,( v α 1 ) λ ] , ( 3 )λα 1 = ( [ 1 − ( 1 1 − ( 1

− μ α 1 ) λ − ( v α 1 ) λ ,( − μ α 1 ) λ − ( v α 1 ) λ ] ),

[ ( 1 1 λ > 0 .

= ( [ (μ α 1 ) λ ,(μ α 1 ) λ ] , [

v α 1 ) λ , v α 1 ) λ ] ,

α 1 ( ) − ( − − ( −

4 1 1 1 1

v α 1 ) λ − (μ α 1 ) λ ,( v α 1 ) λ − (μ α 1 ) λ ] ), [ ( 1 − 1 − λ > 0 .

All the results of the above operations are also IVIFVs, and similar to Theorem 1.2,
the following are all right:

Theorem 1.10 (Xu 2011)
(1) If

1 − λ 1 1 − λ 2

λ 1 >λ 2 , then λ 1 α ≥ λ 2 α, α ≥ α , 0 <λ 1 ,λ 2 ≤ 1.

λ α 1 ≥ λ α 2 , α 1 ≥ α 2 ,

(2) If μ α 1 ≥ μ α 2 , v

α 1 ≤

v

α 2 , then

0 <λ ≤ 1.

(3) If μ α 1 ≥ μ α 3 ,μ α 2 ≥ μ α 4 , v

α 1 ≤

v

α 3 ,

v

α 2 ≤

v

α 4 , then

α 1 ⊕ α 3 ≥ α 2 ⊕ α 4 , α 1 ⊗

α 3 ≥ α 2 ⊗ α 4 .

1.3.2 Power Aggregation Operators for IVIFVs

On the basis of Definitions 1.4 and 1.5, Xu (2011) extended all the operators devel-
oped in Sect. 1.2.3 to aggregate interval-valued intuitionistic fuzzy information.