Information Technology Reference
In-Depth Information
Forward Transformation
The transformation scheme represents the most integral part of the code. [6] developed
an effective routine for the conversion of permutative-based indices into the continuous
domain. Let a set of integer numbers be represented as
x
i
∈
x
i
,
G
which belong to solution
x
j
,
i
,
G
=0
. The formulation of the forward transformation is given as:
x
i
=
−
1 +
α
x
i
(2.15)
where the value
α
is a small number.
Backward Transformation
The reverse operation to forward transformation, converts the real value back into inte-
ger as given in 2.16 assuming
x
to be the real value obtained from 2.15.
int [
x
i
]=
1 +
x
i
/
α
(2.16)
The value
x
i
is rounded to the nearest integer. [9], [2, 3] have applied this method
to an enhanced DE for floor shop problems.
2.3.5
Smallest Position Value Approach
The smallest position value (SPV) approach is the idea of [20] in which a unique solu-
tion representation of a continuous DE problem formulation is presented and the SPV
rule is used to determine the permutations. Applying this concept to the GTSP, in which
a tour is required, integer parts of the parameter values (
s
j
) in a continuous DE problem
formulation represent the nodes (
v
j
). Then the random key values (
s
j
) are determined
by simply subtracting the integer part of the parameter
x
j
from its current value consid-
ering the negative signs, i.e.,
s
j
=
x
j
−
int (
x
j
). Finally, with respect to the random key
values (
s
j
) , the smallest position value (SPV) rule of [20] is applied to the random key
vector to determine the tour
. They adapted the encoding concept of [1] for solving the
GTSP using GA approach, where each set
V
j
has a gene consisting of an integer part
π
between
1
,
V
j
and a fractional part between [0
,
1]. The integer part indicates which
node from the cluster is included in the tour, and the nodes are sorted by their fractional
part to indicate the order. The objective function value implied by a solution
x
with
m
nodes is the total tour length, which is given by
m
1
j
=1
d
π
j
π
j
+1
+
d
π
m
π
1
−
F
(
π
)=
(2.17)
{
1
,..,
20
}
{
1
,..,
5
}
{
6
,,..,
10
}
{
11
,..,
15
}
V
=
and
V
1
=
,
V
2
=
,
V
3
=
and
V
4
=
{
16
,..,
20
}
. Table 2.4 shows the solution representation of the DE for the GTSP.
In Table 2.4, noting that
1
,
V
j
, the integer parts of the parameter values (
s
j
) are
respectively decoded as
. These decoded values are used to extract the nodes
from the clusters
V
1
,
V
2
,
V
3
,
V
4
. The first node occupies the fourth position in
V
1
,the
second node occupies the third position in
V
2
, the third node occupies the first position in
{
4
,
3
,
1
,
3
}
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