Environmental Engineering Reference
In-Depth Information
The importance of decision makers in the group decision making process are
shown in Table
10
. These linguistic variables used can be converted into IFNs
(Zhang and Liu
2011
) and weight of decision makers are calculated using
Eq. (
16
).
(
k
+
µ
k
(
µ
k
+
ν
k
))
µ
k
k
=
(16)
t
µ
k
k
=
1
(µ
k
+
π
k
(
µ
k
+
ν
k
))
where
K
=
1
K
=
1
(
K
=
NUMBER OF DECISION MAKERS
)
.
In group decision making process, all the individual decision opinions need to
be fused into a group opinion to construct an aggregated intuitionistic fuzzy deci-
sion matrix. In order to do that, the IFWA operator given by Xu (
2007
) is utilized:
T
r
(
1
)
ij
,
r
(
2
)
ij
,
...
,
r
(
t
)
ij
=
1
r
(
1
)
⊕
2
r
(
2
)
⊕···⊕
t
r
(
t
)
r
ij
=
IFWA
ij
ij
ij
t
t
t
t
(17)
k
k
− µ
(
k
)
ij
ν
(
k
)
ij
− µ
(
k
)
ij
)
k
−
((
ν
(
k
)
ij
)
k
)
=
1
−
1
,
,
(
1
k
=
1
k
=
1
k
=
1
k
=
1
Therefore according to Eq.
17
all the intuitionistic fuzzy decision matrices
(
R
a
1
,
...
,
R
a
5
) aggregated into a complex intuitionistic fuzzy decision matrix
R
a
.
(
0.15, 0.11, 0.04
)(
0.21, 0.15, 05
)
0.22, 0.16, 0.06
)(
0.33, 0.24, 0.09
)
R
A
=
(
0.58, 0.48, 0.1
)
0.65, 0.55, 0.1
)
0.72, 0.65, 0.07
)(
0.53, 0.43, 0.1
)
(
0.39, 0.31, 0.07
)(
0.34, 0.28, 0.05
)(
0.23, 0.19, 0.04
)(
0.38, 0.31, 0.07
)
(18)
Next point is forming of reference sequence which is the optimal sequence of the
criteria values. In the intuitionistic fuzzy decision matrix, the maximum value
α
+
=
(
1, 0, 0
)
can be used as the reference value. That, in this case the reference
sequence
r
0
is:
r
0
= (
r
0
j
)
1
×
n
=
α
+
α
+
...α
+
(
n
=
number of criteria
)
(19)
Equation (
19
) is used in order to calculate the distance
δ
ij
between
r
ij
and
r
0
j
(Table
11
). Distance measure between two IFNs is defined as follows:
Let
α
1
=
(µ
α
1
,
ν
α
1
,
π
α
1
)
and
α
2
=
(µ
α
2
,
ν
α
2
,
π
α
2
)
be two IFNs, then
d
(∝
1
,
∝
2
) =
1
2
(µ
α
1
− µ
α
2
+ ν
α
1
− ν
α
2
+ π
α
1
− π
α
2
)
(20)
Table 10
The importance of decision makers
Acronym
Decision maker type
Linguistic variables
ʻ
Environmental expert
Important
0.22
d
1
d
2
Purchase manager
Very important
0.26
Research and development staff
Important
0.22
d
3
Purchase staff
Medium
0.15
d
4
d
5
Purchase staff
Medium
0.15
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