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n
v
α
i
w
i
v
l
1
,
n
=
v
α
i
))
λ
+
(γ
2
(
1
+
(γ
−
1
)(
1
−
−
1
)
(1.288)
i
=
1
n
(
v
α
i
w
i
v
1
,
n
=
α
i
))
λ
−
1
+
(γ
−
1
)(
1
−
v
(1.289)
i
=
1
In the same way, we can prove that the GHIFWG operator also satisfies idem-
potency, monotonicity and boundedness, and some special cases of the GHIFWG
operator can be discussed as below (Xia and Xu 2011):
Case 1
If
λ
=
1, then Eq. (
1.285
) reduces to:
HIFWG
(α
1
,α
2
,...,α
n
)
γ
i
=
1
μ
w
i
α
i
=
i
=
1
(
)
i
=
1
μ
α
i
,
w
i
w
i
1
+
(γ
−
1
)(
1
−
μ
α
i
))
+
(γ
−
1
i
=
1
(
−
i
=
1
(
w
i
w
i
+
(γ
−
)
v
α
i
)
−
v
α
i
)
1
1
1
i
=
1
(
)
i
=
1
(
(1.290)
w
i
w
i
1
+
(γ
−
1
)
v
α
i
)
+
(γ
−
1
1
−
v
α
i
)
which is the Hamacher intuitionistic fuzzy geometric (HIFWG) operator (Xia et al.
2012c).
Case 2
If
γ
=
1 and
λ
=
1, then Eq. (
1.285
) is transformed as:
n
v
α
i
w
i
n
(
w
i
α
IFWG
(α
1
,α
2
,...,α
n
)
=
1
μ
,
1
−
1
−
(1.291)
i
i
=
i
=
1
which is the IFWG operator (Xu and Yager 2006).
Case 3
If
γ
=
1, then by Eq. (
1.285
), we have
GHIFWG
(α
1
,α
2
,...,α
n
)
⎛
λ
⎞
⎠
1
−
μ
α
i
)
λ
w
i
1
v
α
i
w
i
1
λ
1
n
n
(
(
⎝
1
=
−
−
1
−
(
1
,
−
1
−
i
=
1
i
=
1
(1.292)
which is the generalized intuitionistic fuzzy weighted geometric (GIFWG) operator.
Case 4
If
γ
=
2, then Eq. (
1.285
) is transformed as:
GEIFWG
(α
1
,α
2
,...,α
n
)
⎛
⎝
⎞
⎠
n
n
2
v
l
1
,
n
−
v
1
,
n
1
λ
1
λ
1
λ
l
1
l
1
1
1
μ
n
+
3
μ
−
μ
n
−
μ
,
,
,
,
=
,
n
n
v
l
1
,
n
+
3
v
1
,
n
v
l
1
,
n
−
v
1
,
n
1
λ
1
λ
1
λ
1
λ
l
1
l
1
l
1
r
1
μ
n
+
3
μ
+
μ
n
−
μ
+
,
,
,
,
(1.293)
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