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α i
= ( μ α i , ˜
v α i , π α i )(
=
,
,...,
)
Let
i
1
2
n
be a collection of IVIFVs, and
T
= (
w 1 ,
w 2 ,...,
w n )
α i
(
=
,
,...,
)
,
w
the weight vector of
i
1
2
n
, where w i
0
n , and i = 1 w i
i
1, then Xu (2011) defined an interval-valued
intuitionistic fuzzy power weighted average (IVIFPWA) operator:
=
1
,
2
,...,
=
I V IFPWA
( α 1 , α 2 ,..., α n )
= (
w 1 (
1
+
T
( α 1 )) α 1 ) (
w 2 (
1
+
T
( α 2 )) α 2 ) ⊕···⊕ (
w n (
1
+
T
( α n )) α n )
i = 1 w i (
1
+
T
( α i ))
(1.70)
which can be transformed into the following form:
( α 1 , α 2 ,..., α n )
IVIFPWA
1
,
n
n
w j ( 1 + T ( α j ))
i = 1 w i (
w j ( 1 + T ( α j ))
i = 1 w i (
μ α j )
μ α j )
=
1 (
1
+
T
( α i ))
,
1 (
1
+
T
( α i ))
1
1
1
j
=
j
=
n
n
w j ( 1 + T ( α j ))
i = 1 w i ( 1 + T ( α i ))
w j ( 1 + T ( α j ))
i = 1 w i ( 1 + T ( α i ))
v α j )
v α j )
,
1 (
,
1 (
j
=
j
=
n
n
w j ( 1 + T ( α j ))
i = 1 w i ( 1 + T ( α i ))
w j ( 1 + T ( α j ))
i = 1 w i ( 1 + T ( α i ))
μ α j )
v α j )
1 (
1
1 (
,
j
=
j
=
n
w j ( 1 + T ( α j ))
n
w j ( 1 + T ( α j ))
n
i =
n
i =
μ α j )
v α j )
1 (
1
1 w i ( 1 + T ( α i ))
1 (
1 w i ( 1 + T ( α i ))
(1.71)
j
=
j
=
where
n
T
( α i ) =
w j Sup
( α i , α j )
(1.72)
j
=
1
j
=
i
and Sup
( α i , α j )
is the support for
α i from
α j , with the following conditions:
(1) Sup
( α i , α j ) ∈[
0
,
1
]
.
(2) Sup
( α i , α j ) =
Sup
( α j , α i )
.
(3) Sup
, where d is a distance
measure, such as the normalized Hamming distance or the normalized Euclidean
distance, where
( α i , α j )
Sup
( α s , α t )
,if d
( α i , α j )<
d
( α s , α t )
 
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