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rules as a solution candidate for a classification problem. An IF-THEN rule is
represented as follows:
R
i
:
IF
(
V
1
min
≤ x
1
≤ V
1
max
)
∧
(
V
2
min
≤ x
2
≤ V
2
max
)
∧ ..... ∧
(
V
nmin
≤ x
n
≤
V
nmax
)
Then y
=
C
where
R
i
is a rule and each rule represent a chromosome in which each gene
represents an attribute and the consequence gene stands for a class;
n
is the
number of attributes with (
x
1
,x
2
, ...x
n
) are the input attribute set; and
y
is
the output class category assigned with a value of
C
.
V
jmin
and
V
jmax
are the
minimum and maximum bounds of the
j
th feature
x
j
respectively. We encode
rule
R
i
according to Figure1.
Gene1(
A
1
)
Gene
n
(
A
n
)
w
1
V
1
min
V
1
max
˜
w
n
V
nmin
V
nmax
Class
Fig. 1.
An individual representation.
where the weight
w
j
is a real-valued variable taking values in the range [0..1].
This variable indicates whether or not the potential attribute is present in the
corresponding classification rule. More precisely, when
w
j
is smaller than a user-
defined threshold (called Limit) the attribute will not appear in the related rule.
Therefore, the greater the value of the threshold Limit, the smaller the probabil-
ity that the corresponding attribute will be included in the rule. Variable length
rules is implemented by using weights, which simplifies the problem by ignoring
insignificant attributes randomly. The weights are fuzzy indicators representing
attribute significance. The less their significance than a user defined threshold,
the attribute is then ignored.
V
jmin
and
V
jmax
are the limits of the intervals
corresponding to the attribute
A
i
. Note that the above encoding is quite flexible
with respect to the length of the rules. A traditional GA is very limited in this
aspect, since it can only cope with fixed-length rule. In our approach, although
each individual has a fixed length, the genes are interpreted (based on the value
of the weight
w
i
) in such a way that the individual phenotype (the rule) has
a variable length. The start of the first population consists of generating, arbi-
trarily, a fixed number of individuals during the evolution.
2.2 Genetic Operators
For the developed method, the usual one-point crossover operator is stochas-
tically applied with a predefined probability, using two individuals of the se-
lected pool. The crossover point is a percentage of the length of the individual
that defines the starting point from where the crossover breaks the string. We
use arithmetic crossover method in the experiments [22]. The employed method
works as follows:
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