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In-Depth Information
2.
Conversion
a)
Discrete to Floating Conversion
: This conversion schema transforms the parent
solution into the required continuous solution.
b)
DE Strategy
: The DE strategy transforms the parent solution into the child
solution using its inbuilt crossover and mutation schemas.
c)
Floating to Discrete Conversion
: This conversion schema transforms the con-
tinuous child solution into a discrete solution.
3.
Selection
a)
Validation
: If the child solution is feasible, then it is evaluated and accepted in
the next population, if it improves on the parent solution.
3.3.1
Permutative Population
The first part of the heuristic generates the permutative population. A permutative so-
lution is one, where each value within the solution is unique and systematic. A basic
description is given in Equation 3.1.
P
G
=
{
x
1
,
G
,
x
2
,
G
,...,
x
NP
,
G
}
,
x
i
,
G
=
x
j
,
i
,
G
x
j
,
i
,
G
=0
=(int)
rand
j
[0
,
1]
x
(
hi
)
j
+
x
(
lo
)
j
x
(
lo
)
j
•
−
+ 1
∈
x
0
,
i
,
x
1
,
i
,...,
x
j
−
1
,
i
if x
j
,
i
/
i
=
{
1
,
2
,
3
,...,
NP
} ,
j
=
{
1
,
2
,
3
,..,
D
}
(3.1)
where
P
G
represents the population,
x
j
,
i
,
G
=0
represents each solution within the popu-
lation and
x
(
lo
)
j
and
x
(
hi
)
j
represents the bounds. The index
i
references the solution from
1to
NP
,and
j
which references the values in the solution.
3.3.2
Forward Transformation
The transformation schema represents the most integral part of the code. [23] developed
an effective routine for the conversion.
Let a set of integer numbers be represented as in Equation 3.2:
x
i
∈
x
i
,
G
(3.2)
which belong to solution
x
j
,
i
,
G
=0
. The equivalent continuous value for
x
i
is given as
1
10
2
.
The domain of the variable
x
i
has length = 5 as shown in 5
10
2
<
5
10
2
•
•
≤
10
2
. The precision of the
value to be generated is set to two decimal places (2 d.p.) as given by the superscript
two (2) in 10
2
. The range of the variable
x
i
is between 1 and 10
3
. The lower bound
is 1 whereas the upper bound of 10
3
•
was obtained after extensive experimentation.
The upper bound 10
3
provides optimal filtering of values which are generated close
together [27].
The formulation of the forward transformation is given as:
1 +
x
i
•
f
•
5
x
i
=
−
(3.3)
10
3
−
1
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