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Table 3.
Comparison among different values of parameter ʻ in tournament selection strategy
with benchmark functions (
f
8
(
x
) -
f
13
(
x
)
)
ʻ
f
8
(
x
)
f
9
(
x
)
f
10
(
x
)
f
11
(
x
)
f
12
(
x
)
f
13
(
x
)
avg
SD
-12569.5
2.48E-12
0
0
7.55E-15
0
0
0
1.57E-32
8.35E-48
1.35E-32
5.57E-48
0.02
avg
SD
-12569.5
2.19E-12
0
0
1.39E-14
1.45E-15
0
0
1.57E-32
8.35E-48
1.35E-32
5.57E-48
0.1
avg
SD
-12569.5
2.34E-12
0
0
2.21E-14
3.89E-15
0
0
1.57E-32
8.35E-48
3.53E-32
8.81E-32
0.2
avg
SD
-12569.5
2.55E-12
0
0
3.27E-14
5.10E-15
0
0
1.79E-30
6.58E-30
9.14E-29
4.91E-28
0.4
avg
SD
-12569.5
2.68E-12
0
0
4.64E-14
1.29E-14
0
0
2.67E-30
1.15E-29
2.43E-28
6.84E-28
0.6
avg
SD
-12569.5
3.04E-12
0
0
7.16E-14
2.58E-14
0
0
1.61E-27
5.76E-27
4.39E-25
2.39E-24
0.8
avg
SD
-12569.5
2.74E-12
0
0
8.27E-14
2.85E-14
0
0
6.48E-28
1.97E-27
1.69E-24
7.44E-24
0.9
In the comparative experiment of RWS and TS, the Eq. (3) has to be switched
because the benchmark functions are all the minimum optimization problems. In this
paper, the switch function is as follows:
1
,
(4)
f
=
i
fit
−
OPT
+
1
i
where
fit
i
is the fitness value of the solution
i
, and
OPT
presents the minimum value
of the problem. In this work, the minimum value of the benchmark functions have
already been given. If the minimum value is unknown, we can set the optimal value
that has been found so far as
OPT
. Then the selection Eq. (3) becomes:
f
.
(5)
p
=
∑
i
i
SN
f
n
n
=
1
It can be seen that roulette wheel selection based ABC algorithm is inconvenient
when solving minimum optimization problems since the switch function has to be
designed under different situations. Within Eqs. (4) and (5), the optimizations of
benchmark functions by ABC algorithm are showed in Table 4, which also displays
the experiment results of TSABC algorithm with
= 0.05.
According to the data in Table 4, we can notice that the optimization results by
ABC are almost same with that of TSABC under
ʻ
= 0.05. That is to say, roulette
wheel selection strategy and tournament selection with
ʻ
= 0.05 produce the same
level of selection pressure. This selection pressure may have good performances in
solving some optimization problems, but there are still many other problems which
need less or more selection pressure to reach the optimal result. However, roulette
wheel selection based ABC algorithm has no adjustable parameters to control it.
ʻ
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