Information Technology Reference
In-Depth Information
The fuzzy decision matrix for the alternatives ( D) and the criteria ( W) is constructed
as follows:
C
1
C
2
C
n
2
4
3
5
A
1
A
2
...
A
m
~
~
... ~
x
11
x
12
x
1n
D
~
~
... ~
¼
x
21
x
22
x
2n
;
i
¼
1
;
2
; ...;
m
;
j
¼
1
;
2
; ...;
n
ð
4
Þ
...
...
... ...
x
m1
x
m2
...
x
mn
W
¼
ð~
w
1
; ~
w
2
; ...; ~
w
n
Þ
ð
5
Þ
Step 4: Normalize the fuzzy decision matrix.
The raw data are normalized using linear scale transformation to bring the various
criteria scales into a comparable scale. The normalized fuzzy decision matrix R is
given by:
R
¼½
~
r
ij
mxn
;
i
¼
1
;
2
; ...;
m
;
j
¼
1
;
2
; ...;
n
ð
6
Þ
where:
a
ij
c
j
;
b
ij
c
j
;
c
ij
c
j
Þ
c
j
¼
r
ij
¼
ð
and
max
i
c
ij
ð
benefit criteria
Þ
ð
7
Þ
a
j
c
ij
;
a
j
b
ij
;
a
j
a
ij
Þ
a
j
¼
~
r
ij
¼
ð
and
mini
i
a
ij
ð
cost criteria
Þ
ð
8
Þ
Step 5: Compute the weighted normalized matrix.
V for criteria is computed by multiplying the
The weighted normalized matrix
weights (
w
j
) of evaluation criteria with the normalized fuzzy decision matrix
~
~
r
ij
.
V
¼½
~
v
ij
mxn
;
i
¼
1
;
2
; ...;
m
;
j
¼
1
;
2
; ...;
n where
~
v
ij
¼
~
r
ij
ð:Þ~
w
j
ð
9
Þ
Step 6: Compute the fuzzy ideal solution (FPIS) and fuzzy negative ideal
solution (FNIS)
The FPIS and FNIS of the alternatives is computed as follows:
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