Civil Engineering Reference
In-Depth Information
Definition 2.2 (defines the
defuzzification of interval
fuzzy scores
Harloff
(
2011
)).
Let
X
=(
a
,
b
)
be an interval fuzzy number;
c
is the range center; and
l
=
|
b
−
a
|
is the whole distance. The defuzzification value of the interval fuzzy
number is
(
+
|
|
)
ln
1
x
x
f
=
c
+
1
−
(1)
|
x
|
ln
(
1
+
|
x
|
)
1
−
.
(2)
|
x
|
Formula 2.2 is the defuzzification function of the interval length. If
a
→
b
,then
x
f
approaches the range center value
a
+
b
/
2.
2.2
Calculating the Domain of
x
f
Using Formula 2.1
The domain of
x
f
is calculated using Formula 2.1 because Formula 2.2 has a
maximal value of 1 and a minimal value of 0. The domain is calculated using the
following process: lim
x
→
0
ln
(
1
+
|
x
|
)
|
D
x
ln
(
1
+
|
x
|
)
D
x
|
1
|
=
lim
x
→
0
|
=
lim
x
→
0
|
=
1
x
x
1
+
|
x
ln
(
1
+
|
x
|
)
D
x
ln
(
1
+
|
x
|
)
1
lim
x
=
lim
x
=
lim
x
|
=
0
.
|
x
|
D
x
|
x
|
1
+
|
x
→
∞
→
∞
→
∞
Formula 2.2 has a minimal value of 0 and a maximal value of 1. Formula 2.1 has
a minimal value of
c
and a maximal value of
c
+
1. If test scores are expressed as
percentages, then their maximal value is min
and their minimal value
is 0. The domain of
x
f
matches with educational requirements and habits.
(
100
,
c
+
1
)
2.3
Controlled Defuzzification Domain of Interval Fuzzy
Scores within the Smallest Unit
If student scores are expressed as percentages, the maximal score is 100 and the
minimal score is 0. Formula 2.1 was used to transform the defuzzification value of
the interval fuzzy number because the defuzzification value of Formula 2.2 controls
the domain from 0 to 1. Thus, sorting the defuzzification values is the same as
sorting center value
c
.
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