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where
n
n
(
μ
α
i
w
i
(
−
μ
α
i
))
λ
−
μ
α
i
w
i
l
1
−
μ
α
i
))
λ
+
1
μ
n
=
1
+
(
1
3
,μ
,
n
=
1
+
(
1
,
i
=
1
i
=
1
(1.294)
n
n
(
3
v
α
i
w
i
(
v
α
i
w
i
v
l
1
,
n
=
α
i
))
λ
+
v
1
,
n
=
α
i
))
λ
−
1
+
(
1
−
v
,
1
+
(
1
−
v
i
=
1
i
=
1
(1.295)
which is a generalized Einstein intuitionistic fuzzy weighted geometric (GEIFWG)
operator.
Case 5
If
γ
=
2 and
λ
=
1, then Eq. (
1.285
) is transformed as:
EIFWG
(α
1
,α
2
,...,α
n
)
2
i
=
1
μ
i
=
1
(
−
i
=
1
(
w
i
α
w
i
w
i
1
+
v
)
1
−
v
)
α
α
i
i
i
=
i
=
1
(
+
i
=
1
μ
α
i
,
i
=
1
(
+
i
=
1
(
w
i
w
i
w
i
2
−
μ
α
i
))
w
i
1
+
v
α
i
)
1
−
v
α
i
)
(1.296)
which is the EIFWG operator (Xia et al. 2012c).
In what follows, we apply the GHIFWA and GHIFWG operators to decision
making (Xia and Xu 2011):
For a multi-attribute decision making problem, let
Y
,
G
and
w
be as defined pre-
viously. The expert provides the performance of the alternative
y
i
under the attribute
G
j
denoted by the IFVs
α
ij
=
(μ
ij
,
v
ij
)(
i
=
1
,
2
,...,
m
;
j
=
1
,
2
,...,
n
)
.All
the IFVs
α
ij
(
i
=
1
,
2
,...,
m
;
j
=
1
,
2
,...,
n
)
construct the intuitionistic fuzzy
decision matrix
B
b
ij
)
n
×
n
.
To obtain the alternative(s), the following steps are given (Xia and Xu 2011):
=
(
=
(
b
ij
)
n
×
n
into the
Step 1
Transform the intuitionistic fuzzy decision matrix
B
normalized intuitionistic fuzzy decision matrix
R
=
(
r
ij
)
n
×
n
, where
b
ij
,
f or benef it attribute G
i
r
ij
=
,
i
=
1
,
2
,...,
m
;
j
=
1
,
2
,...,
n
b
ij
,
f or cost attribute G
i
(1.297)
Step 2
Aggregate the intuitionistic fuzzy values
r
i
of the alternative
y
i
by the
GAIFWA or GHIFWG operator:
n
j
=
1
w
j
r
ij
1
λ
r
i
=
GHIFWA
(
r
i
1
,
r
i
2
,...,
r
in
)
=
(1.298)
or
n
⊗
1
λ
r
w
j
ij
r
i
=
GHIFWG
(
r
i
1
,
r
i
2
,...,
r
in
)
=
1
λ
(1.299)
j
=
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