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⎛
⎝
⎞
⎠
0
.
3044 0
.
3044 0
.
3044 0
.
2977
0
.
3155 0
.
3044 0
.
3055 0
.
3044
2
=
0
.
3055 0
.
2965 0
.
2965 0
.
2950
0
.
3081 0
.
3081 0
.
3044 0
.
2950
⎛
⎝
⎞
⎠
0
.
3044 0
.
3044 0
.
3044 0
.
2977
0
.
2934 0
.
3044 0
.
2945 0
.
3044
3
=
0
.
2945 0
.
3081 0
.
3081 0
.
3248
0
.
2965 0
.
2965 0
.
3044 0
.
3055
Step 2
Utilize the PA operator (
1.17
) to aggregate all the individual fuzzy decision
matrices
F
(
k
)
F
(
k
)
ij
=
(
)
4
×
4
(
k
=
1
,
2
,
3
)
into the collective fuzzy decision matrix
F
=
(
F
ij
)
4
×
4
:
⎛
⎝
⎞
⎠
0
.
4000 0
.
5000 0
.
2000 0
.
2405
.
.
.
.
0
4369 0
6000 0
6094 0
6000
F
=
.
.
.
.
0
6094 0
2308 0
3692 0
6004
0
.
3308 0
.
5308 0
.
5000 0
.
7094
in the
j
th column
of
F
by using the well-known weighted averaging (WA) operator (Harsanyi 1955),
and get the overall preference value
F
j
corresponding to the alternative
y
j
:
Step 3
Aggregate all the preference values
F
ij
(
j
=
1
,
2
,
3
,
4
)
F
1
=
0
.
3431
,
F
2
=
0
.
5534
,
F
3
=
0
.
4529
,
F
4
=
0
.
4988
Since
F
4
>
F
2
>
F
1
>
F
3
, and thus,
y
2
y
4
y
3
y
1
.
It is noted that the rankings of the alternatives are very different in such two
cases, this is because that all the non-membership information and the hesitancy
information are lost in the second case, and thus, the final results obtained in the
second case are obviously less reasonable than those when the experts' evaluations
are expressed in IFVs comprehensively.
In the next section, we shall extend the aggregation operators and the decision
making approaches introduced in this section to interval-valued intuitionistic fuzzy
environments.
1.3 Interval-Valued Intuitionistic Fuzzy Power Aggregation
Operators
1.3.1 Interval-Valued Intuitionistic Fuzzy Values
Atanassov and Gargov (1989) extended IFS to interval-valued intuitionistic fuzzy
environments, and defined an interval-valued intuitionistic fuzzy set (IVIFS),
shown as:
A
={
x
, μ
A
(
x
),
˜
v
A
(
x
)
|
x
∈
X
}
(1.64)
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