Information Technology Reference
In-Depth Information
where
Sup
α
index
(
j
)
, α
index
(
i
)
indicates the support of
i
th largest IVIFV
α
index
(
i
)
for
α
index
(
j
)
, and
g
:[
,
]→[
,
]
the
j
th largest IVIFV
0
1
0
1
is a BUM function.
Especially, if
g(
x
)
=
x
, then the IVIFPOWAoperator (
1.80
) reduces to the IVIFPA
operator (
1.75
).
Then based on the IVIFPOWA operator (
1.80
) and the geometric mean, Xu (2011)
defined an interval-valued intuitionistic fuzzy power ordered weighted geometric
(IVIFPOWG) operator:
IVIFPOWG
( α
1
, α
2
,..., α
n
)
1
−
ω
1
n
1
−
ω
2
n
1
−
ω
n
n
=
( α
index
(
1
)
)
⊗
( α
index
(
2
)
)
⊗···⊗
( α
index
(
n
)
)
(1.84)
−
1
−
1
−
1
which can be further expressed as:
IVIFPOWG
( α
1
, α
2
,..., α
n
)
⎛
⎡
⎤
n
n
1
−
ω
j
n
−
1
1
−
ω
j
n
−
1
⎝
⎣
1
(μ
α
index
(
j
)
)
1
(μ
α
index
(
j
)
)
⎦
,
=
,
j
=
j
=
⎡
⎤
n
n
1
−
ω
j
n
−
1
1
−
ω
j
n
−
1
⎣
1
v
α
index
(
j
)
)
v
α
index
(
j
)
)
⎦
,
−
1
(
1
−
,
1
−
1
(
1
−
j
=
j
=
⎡
n
n
n
1
−
ω
j
n
−
1
1
−
ω
j
n
−
1
1
−
ω
j
n
−
1
⎣
v
α
index
(
j
)
)
1
(μ
α
index
(
j
)
)
v
α
index
(
j
)
)
1
(
1
−
−
,
1
(
1
−
j
=
j
=
j
=
⎤
⎦
⎞
⎠
n
1
−
ω
j
n
−
1
1
(μ
α
index
(
j
)
)
−
(1.85)
j
=
where
are a collection of weights satisfying the conditions (
1.82
)
and (
1.83
). Especially, if
g(
ω
i
(
i
=
1
,
2
,...,
n
)
x
)
=
x
, then the IVIFPOWG operator (
1.84
) reduces to
the IVIFPG operator (
1.79
).
Let
be a vector of
n
IVIFVs, then all the IVIFPWA, IVIFPWG,
IVIFPOWA and IVIFPOWG operators have the following properties (Xu 2011):
( α
1
, α
2
,..., α
n
)
( α
1
, α
2
,..., α
n
)
Theorem 1.11
(Commutativity) Assume that
is any permutation
of
( α
1
, α
2
,..., α
n
)
, then
( α
1
, α
2
,..., α
n
)
IVIFPWA
( α
1
, α
2
,..., α
n
)
=
IVIFPWA
(1.86)
( α
1
, α
2
,..., α
n
)
IVIFPWG
( α
1
, α
2
,..., α
n
)
=
IVIFPWG
(1.87)
( α
1
, α
2
,..., α
n
)
IVIFPOWA
( α
1
, α
2
,..., α
n
)
=
IVIFPOWA
(1.88)
( α
1
, α
2
,..., α
n
)
IVIFPOWG
( α
1
, α
2
,..., α
n
)
=
IVIFPOWG
(1.89)
Search WWH ::
Custom Search