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In-Depth Information
Equation 3.3 when broken down, shows the value
x
i
multiplied by the length 5 and
a scaling factor
f
. This is then divided by the upper bound minus one (1). The value
computed is then decrement by one (1). The value for the scaling factor
f
was estab-
lished after extensive experimentation. It was found that when
f
was set to 100, there
was a tight grouping of the value, with the retention of optimal filtration
sofvalues.
The subsequent formulation is given as:
1 +
x
i
•
f
•
5
1 +
x
i
•
f
•
5
x
i
=
−
=
−
(3.4)
10
3
10
3
−
1
−
1
Illustration
:
Take a integer value 15 for example. Applying Equation 3.3, we get:
1 +
15
500
999
•
x
i
=
−
= 6
.
50751
This value is used in the DE internal representation of the population solution pa-
rameters so that mutation and crossover can take place.
3.3.3
Backward Transformation
The reverse operation to forward transformation, backward transformation converts the
real value back into integer as given in Equation 3.5 assuming
x
i
to be the real value
obtained from Equation 3.4.
•
10
3
1
•
10
3
1
int [
x
i
]=
(1 +
x
i
)
−
=
(1 +
x
i
)
−
(3.5)
5
•
f
500
The value
x
i
is rounded to the nearest integer.
Illustration
:
Take a continuous value -0.17. Applying equation Equation 3.5:
10
3
1
int [
x
i
]=
(1 +
−
0
.
17)
•
−
=
|
3
.
3367
|
= 3
500
The obtained value is 3, which is the rounded value after transformation.
These two procedures effectively allow DE to optimise permutative solutions.
3.3.4
Recursive Mutation
Once the solution is obtained after transformation, it is checked for feasibility. Feasibil-
ity refers to whether the solutions are within the bounds and unique in the solution.
⎧
⎨
u
j
,
i
,
G
+1
=
u
1
,
i
,
G
+1
,...,
u
j
−
1
,
i
,
G
+1
u
i
,
G
+1
if
x
(
lo
)
x
(
lo
)
x
i
,
G
+1
=
≤
≤
(3.6)
u
j
,
i
,
G
+1
⎩
x
i
,
G
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