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where the numerators
kk k
in the right-hand side expressions are
some constants of the (respective) variations.
From the above expressions it is evident that all three probabilities do not
depend on the fitness of any particular solution and have the same value for all
solutions of the populations. Consequently, solutions both with high and with low
fitness values are subjected to the same level of reproduction, mutation, and
crossover. Also, when the population converges to an optimal global or local
solution, the increase of
p
(
c
) and
p
(
m
) may eventually cause disruption of the near-
optimal solutions. Therefore, the population will neither converge to a local
optimum nor converge to the global optimum. Therefore, though we may prevent
the GA from getting stuck at a local optimum solution, the performance of the GA
- in terms of generations required for convergence will be very large - will
certainly deteriorate.
To overcome this problem, we need to preserve the “good” solutions of the
population by using some higher value of
p
(
r
) and lower values of
p
(
c
) and
p
(
m
)
for higher fitness solutions and some higher values of
p
(
c
) and
p
(
m
) for lower
fitness solutions. This is because high fitness values support the convergence speed
of the GA, whereas low fitness solutions prevent the GA from getting stuck at local
optima.
Thereby, the value of
p
(
m
) should not only depend on
,
a d
rep
mu
cross
f f
but also on
the fitness value of the solution. Similarly, the
p
(
c
) value should not only depend
on the difference
(
)
max
avg
f f
but also on the fitness of the two parent solutions.
Furthermore, if the value of the difference
(
)
max
avg
(
f
f
)
can identify whether the
max
avg
GA is converging or not, then the difference
f f
will possibly also
identify the convergence of the GA because, in our experiment (Palit and Popovic,
2000), all the populations for subsequent generations are selected from the mating
pool that consists of the best 50% populations of the current generation. Therefore,
by using both of them as a measure of GA convergence, the adaptive values of the
control parameter of the GAs are set as follows. For
(
)
avg
min
f
t
f
selrep
avg
it is
§
f
f
·
selrep
avg
pr
()
k
k
(9.4)
¨
¸
¨
¸
1rep
1repbias
f
f
©
¹
max
avg
and for
f
f
selrep
avg
it is
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