Environmental Engineering Reference
In-Depth Information
Table 8
Criteria weight using AHP
w
Financial (
w
1
)
Services (
w
2
)
Qualitative (
w
3
)
EMS (
w
4
)
Value
0.07
0.28
0.53
0.12
Potential suppliers are ranked based on high ranked criteria using fuzzy GRA
technique. Fuzzy set theory (Zadeh
1965
) has been extensively used for modeling
decision making processes based on imprecise and vague information such as
judgment of decision makers. Also Grey System Theory (GST) is a mathematical
method that is applied to imprecise information in the form of interval values and
developed by Deng (
1989
).
General description in concept of intuitionistic fuzzy set (IFS) has been
explained in following. If
X
be a fixed set, an IFS
A
in
X
is given by Atanassov
(
1986
) as follows:
A
= {(
X
,
µ
A
(
X
)
,
ν
A
(
X
))|
X
∈
X
}
(14)
where the functions
µ
A
(
x
) :
X
→[
0, 1
]
,
x
∈
X
→ µ
A
(
x
) ∈[
0, 1
]
and
ν
A
(
x
) :
X
→[
0, 1
]
,
x
∈
X
→ ν
A
(
x
) ∈[
0, 1
]
satisfy the condition
0
≤
µ
A
(
X
)
+ ν
A
(
X
) ≤
1
for all
x
∈
X
.
The numbers
µ
A
(
x
)
and
ν
A
(
X
)
define the degree of membership and non-member-
ship for the element
x
∈
X
to the set A, respectively.
Ratings of potential alternatives with respect to selected criteria could be expressed
using linguistic variables presented in Table
9
, then linguistic variables can convert to
intuitionistic fuzzy numbers (IFN) (Junior et al.
2014
; Zhang and Liu
2011
).
Afterward the intuitionistic fuzzy decision matrices of each decision maker for
selection of two kind suppliers a (steel sheet) and b (PET granule) constructed,
and the linguistic evaluation converted into IFNs. Finally the intuitionistic fuzzy
decision matrices (R) of each decision maker for each supplier formed. For exam-
ple matrix
R
a
2
(3 4; three potential suppliers of steel sheet and four high ranked
criteria) is related to opinion of second decision maker (
d
2
) for supplier a:
(
0.85, 0.1, 0.05
)(
0.65, 0.25, 0.1
)(
0.85, 0.1, 0.05
)(
0.65, 0.25, 0.1
)
(
0.35, 0.55, 0.1
)(
0.25, 0.65, 0.1
)
0.05, 0.95, 0
)
0.65, 0.25, 0.1
)
(
0.5, 0.4, 0.1
)
0.95, 0.05, 0
)
0.85, 0.1, 0.05
)(
0.65, 0.25, 0.1
)
R
a
2
=
(15)
Table 9
Conversion between
linguistic variables and IFNs
Number
Linguistic variables (Importance)
IFNs
1
Extreme low (EL)
(0.05, 0.95, 0.00)
Very low (VL)
(0.15, 0.80, 0.05)
2
Low (L)
(0.25, 0.65, 0.10)
3
4
Medium low (ML)
(0.35, 0.55, 0.10)
Medium (M)
(0.50, 0.40, 0.10)
5
Medium high (MH)
(0.65, 0.25, 0.10)
6
7
High (H)
(0.75, 0.15, 0.10)
Very high (VH)
(0.85, 0.10, 0.05)
8
Extreme high (EH)
(0.95, 0.05, 0.00)
9
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