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f
and crowding operators to keep genetic drift from
the genetic algorithm application. The role of the
crowding operators is to identify how many indi-
viduals dominate the environmental niche. Then,
the competition for the next generation in selection
step increases rapidly. The individuals have the
lower possibilities to survive in next generation.
The percentage of the population that is allowed to
reproduce is called generation gap. The number of
individuals that are initially selected as candidates
to be replaced by a particular offspring is called
the crowding factors (Shrinivas and Deb 1994;
Coello et al. 2001).
A sharing function that is achieved by perform-
ing the selection is suggested by Goldberg and
Richardson (1987). The sharing function defines
the degraded fitness values obtained by dividing
the original fitness function value of an individ-
ual by a quantity proportional to the number of
individuals around it (Shrinivas and Deb 1994).
Goldberg and Richardson (1987) defined a shar-
ing function sh d i ( ), and the sharing function can
be expressed as different functions by using the
power factor α which is generally 1, but it will be
dependent on the optimization problem charac-
teristics. The general format of the sharing func-
tion is (Goldberg and Richardson 1987)
f
=
i
,
(2)
s
= (
M
i
sh d
)
ij
j
1
where M is the number of individuals located in
vicinity of the i th individual and d ij is the p -di-
mensional Euclidean distance (Coello et al. 2001)
p
(
2
d
=
x
x
) ,
(3)
ij
k i
,
k j
,
k
=
1
where p refers to the number of variables en-
coded in the evolutionary algorithms. As σ share
value is generally selected between 1 to 2, Deb
et al (1989) suggested an equation to determine
the value of the sharing parameters (Coello et al.
2001)
(
p
2
x
x
)
r
q
k
,max
k
,min
k
1
σ share
=
=
,
p
p
2
q
(4)
where r is the volume of a p -dimensional hyper-
sphere of the radius of σ share and q is the number
of Pareto-solutions that GAs need to find. These
non-dominated Pareto ranking and sharing func-
tions will be the backbone of the proposed multi-
objective genetic algorithms (MOGAs).
α
d
ij
1
,
if
d
<
σ
sh d
(
)
=
,
ij
share
σ
ij
share
0
,
otherwise
Multi-Objective Genetic
Algorithms (MOGA)
(1)
Fonseca and Fleming (1993) proposed a modi-
fication of the simple genetic algorithm (SGA)
at the selection level. The basic concepts of the
proposed MOGA are the ranking based on the
Pareto dominance and sharing function. The
Pareto dominance-based rank is the same as one
plus the number that certain individual dominates
where d ij is the metric distance between the in-
dividual string i , j , and σ share is the sharing
parameter or radius to control the range of the
sharing. From the sharing function, the modified
fitness is defined as (Goldberg and Richardson
1987)
 
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