Information Technology Reference
In-Depth Information
α
j
=
α
Theorem 1.8
(Idempotency) If
, for all
j
, then
IFPWG
(α
1
,α
2
,...,α
n
)
=
α
(1.41)
α
−
and
α
+
be given by Eqs. (
1.35
) and (
1.36
),
Theorem 1.9
(Boundedness) Let
then
α
−
≤
(α
1
,α
2
,...,α
n
)
≤
α
+
IFPWG
(1.42)
The fundamental characteristic of both the IFPWA and IFPWG operators is that
they weight all the given IFVs themselves, and the weighting vectors depend upon
the input arguments and allow the values being aggregated to support and reinforce
each other. However, in many group decision making problems, such as person-
nel evaluation, diving games, etc., we need to rearrange all the given arguments in
descending (or ascending) order, and then weight the ordered positions of the input
arguments so as to relieve the influence of unfair arguments on the decision result by
assigning low weights to those “false” or “biased” ones. As a result, motivated by
the idea of Yager (1988, 2001)'s ordered weighted average, Xu (2011) introduced an
intuitionistic fuzzy power ordered weighted average (IFPOWA) operator:
IFPOWA
(α
1
,α
2
,...,α
n
)
=
ω
1
α
index
(
1
)
⊕
ω
2
α
index
(
2
)
⊕···⊕
ω
n
α
index
(
n
)
(1.43)
which can be further expressed as:
IFPOWA
(α
1
,α
2
,...,α
n
)
⎛
n
n
n
⎝
1
−
μ
α
index
(
j
)
)
ω
j
v
α
index
(
j
)
)
ω
j
−
μ
α
index
(
j
)
)
ω
j
=
−
1
(
1
,
1
(
,
1
(
1
j
=
j
=
j
=
α
index
(
j
)
)
ω
j
⎞
n
⎠
−
1
(
v
(1.44)
j
=
(
)
where
index
is an indexing function such that
index
i
is the index of the
i
th largest of
α
j
=
(μ
α
j
,
v
α
j
,π
α
j
)(
=
,
...,
)
α
index
(
i
)
the IFVs
j
1
2
n
, and thus,
is the
i
th largest
of the IFVs
α
j
(
j
=
1
,
2
,...,
n
)
.
ω
i
(
i
=
1
,
2
,...,
n
)
are a collection of weights
such that
D
i
TV
D
i
−
1
TV
i
n
ω
i
=
g
−
g
,
D
i
=
V
index
(
j
)
TV
=
V
index
(
i
)
j
=
1
i
=
1
V
index
(
j
)
=
1
+
T
(α
index
(
j
)
)
(1.45)
and
T
(α
index
(
j
)
)
denotes the support of the
j
th largest IFV
α
index
(
j
)
by all the other
IFVs, i.e.,
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