Bio-inspired Evolutionary Computing with Context-Awareness and Collective-Effect - Technologies and Applications of Artificial Intelligence - page 110

100 rounds. The smaller the Std. is, the higher the reliability is. The Max. and Min.
are the best and the worst convergent solutions in 100 rounds respectively. A tolera-
ble threshold ( T tor ) is utilized to determine the acceptance of a solution. If the average
error of solution is less than T tor , the global optimal solution is found in an iteration
limitation. The evolution is successful in this round. And, SR. is defined as following:

.

. (23)

The statistics of convergent solution with the single pattern for two benchmark
function with different dimensions are shown in Table 1. Avg. mainly appraises the
quality of convergent solution. The smaller Avg. implies the smaller the average er-
rors of converged solution is and the better the performance of algorithm. The con-
vergent solution is 5.41E-6 occurred in Ackley function with d=128. The errors of
these cases slightly rise in the case of 128-dimension functions. In addition, the Std. is
very small. The average has sufficient reliability. To be stricter than the previous
works [6, 10], the T tor is set to 1.00E-6 in this study, rather than 1.00E-05. If the aver-
age error of convergent solution in an evolution is less than T tor , the bats successfully
finds the global optimal solution in a round. The higher the SR is, the better the ability
of discovering global optimal solution for bats are. The SR is 100% to indicate that
the bats find the global optimal solution in all rounds for a pattern. On the other hand,
the SR is 91% and 73% for the 128-dimension of Griewangk function and 128-
dimension of Ackely function, because the errors of the worst convergent solution are
3.36E-05 and 6.13E-05 respectively. These two errors are greater than tolerable thre-
shold to imply that the bats do not explore the global optimal solution in a small num-
ber of rounds for each pattern. The errors of convergent solution in these unsuccessful
rounds deteriorate the average for a pattern. And, the number of unsuccessful round in
Ackley function is more than that in Griewangk function.

Table 1. The statistics and SR of the GBA with a single pattern in two benchmark function
with two kinds of dimensions

benchmark Griewangk Ackley

Dimension 64 128 64 128

2.90E-16 1.31E-06 3.46E-10 5.41E-06

Avg.

0.00E+00 4.37E-06 1.05E-09 8.52E-06

Std.

1.50E-14 3.36E-05 6.34E-09 6.13E-05

Max.

0.00E+00 3.23E-18 2.37E-15 1.19E-08

Min.

100% 91% 100% 73%

SR.

In order to confirm the solving efficiency of proposed bat algorithm in this study, the
GBA is compared with HBA and BA in 10, 20 and 30 dimensions with an iteration limi-
tation. The simulation results of HBA and BA are provided by [10]. In [10], the iteration
limitations are separately set to 1000, 2000 and 3000 for 10, 20 and 30 dimensions to
verify the quality of global optimal solution. And each function is executed 25 rounds to
derive the statistics in these three kinds of dimensions. The parameters of Griewangk