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compute the fuzzy weighted decision matrix for the alternatives. For example, for
alternative A1, the fuzzy weight for criteria C1 is given by:
~
v
ij
¼
ð
0
:
111
;
0
:
629
;
1
Þð:Þð
3
;
7
;
9
Þ
¼
ð
0
:
333
;
4
:
407
;
9
Þ
Likewise, the fuzzy weights of the three alternatives for the 17 criteria are
computed (Table
9
). Then, the fuzzy positive ideal solution (A*) and the fuzzy
negative ideal solutions (A
) are computed using Eqs. (
10
) and (
11
) for the
−
for criteria C1, A
¼
ð
alternatives. For example,
0
:
333
;
0
:
333
;
0
:
333
Þ
and
Table 9 Weighted normalized alternatives, FPIS and FNIS
Criteria
A1
A2
A3
Min
Max
A
A*
−
C1
(0.33,
4.40, 9)
(1,
5.963, 9)
(0.33,
4.407, 9)
0.33
9
(0.333,
0.333, 0.333)
(9,
9, 9)
C2
(0.11,
2.08, 9)
(0.11,
1.12, 7)
(0.111,
2.407, 9)
0.11
9
(0.111,
0.111, 0.111)
(9,
9, 9)
C3
(0.11,
2.78, 9)
(0.33,
3.14, 9)
(0.11,
1.667, 9)
0.11
9
(0.11, 0.111,
0.111)
(9,
9, 9)
C4
(0.11,
4.40, 9)
(0.33,
3.14, 7)
(0.11,
3.148, 9)
0.111
9
(0.111,
0.111, 0.111)
(9,
9, 9)
C5
(0.11,
2.27, 7)
(0.11,
2.72, 9)
(0.33,
2.407, 7)
0.111
9
(0.111,
0.111, 0.111)
(9,
9, 9)
C6
(1,
5.395, 9)
(0.33,
2.58, 7)
(0.33,
3.519, 9)
0.333
9
(0.333,
0.333, 0.333)
(9,
9, 9)
C7
(0.56,
3.89, 9)
(0.56,
2.33, 9)
(0.556,
4.407, 9)
0.556
9
(0.556,
0.556, 0.556)
(9,
9, 9)
C8
(0.33,
1.0, 3)
(0.33,
.895, 9)
(0.333,
1.133, 9)
0.333
9
(0.333,
0.333, 0.333)
(9,
9, 9)
C9
(0.56,
4.82, 9)
(0.56,
5.39, 9)
(0.556,
3.691, 9)
0.556
9
(0.556,
0.556, 0.556)
(9,
9, 9)
C10
(0.33,
3.51, 9)
(0.33,
.704, 3)
(0.333,
3.049, 9)
0.333
9
(0.333,
0.333, 0.333)
(9,
9, 9)
C11
(0.56,
4.29, 9)
(1.67,
6.53, 9)
(0.556,
5.963, 9)
0.556
9
(0.556,
0.556, 0.556)
(9,
9, 9)
C12
(0.56,
1.98, 5)
(0.56,
1.98, 7)
(1.67,
5.395, 9)
0.556
9
(0.556,
0.556, 0.556)
(9,
9, 9)
C13
(0.33,
4.45, 9)
(0.33,
3.04, 9)
(0.333,
4.457, 9)
0.333
9
(0.333,
0.333, 0.333)
(9,
9, 9)
C14
(2.78, 7,
9)
(0.56,
3.37, 7)
(0.556,
2.852, 7)
0.556
9
(0.556,
0.556, 0.556)
(9,
9, 9)
C15
(0.11,
1.76, 9)
(0.11,
1.76, 9)
(0.111,
2.037, 9)
0.111
9
(0.111,
0.111, 0.111)
(9,
9, 9)
C16
(0.11,
1.67, 9)
(0.11,
1.67, 7)
(0.111,
2.037, 9)
0.111
9
(0.111,
0.111, 0.111)
(9,
9, 9)
C17
(0.11,
2.78, 9)
(0.33,
4.25, 9)
(0.556,
4.259, 9)
0.111
9
(0.111,
0.111, 0.111)
(9,
9, 9)
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