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0.5
0
−0.5
1
1
0.5
0.5
0
0
Δ
e
k
−0.5
−0.5
e
k
−1
−1
Fig. 4 View of the fuzzy rule-base for the standard FLC
For the software implementation of the proposed metaheuristics, the control
parameters of each algorithm are set as follows:
DSA: random numbers Stopover site research p
1
¼
p
2
¼
0
:
3rand 0
ðÞ
;
1
;
GSA: initial value of gravitational constant G
0
¼
75, parameter
g ¼
20, initial
value of the Kbest agents K
0
¼
N
¼
30 which is decreased linearly to 1;
ABC: Limit of abandonment L
¼
60;
PSO: cognitive and social coef
cients equal
to c
1
¼
c
2
¼
2,
inertia factor
decreasing linearly from 0.9 to 0.4;
GAO: Stochastic Uniform selection and Gaussian mutation methods, Elite
Count equal to 2 and Crossover Fraction equal to 0.8.
In order to get statistical data on the quality of results and so to validate the
proposed approaches, we run all implemented algorithms 20 times. Feasible solu-
tions are usually found within an acceptable CPU computation time. The obtained
optimization results are summarized in Tables
3
and
4
.
4.3 Results Analysis and Discussion
According to the statistical analysis of Tables
3
and
4
, as well as the numerical
simulations in Figs.
5
,
6
,
7
and
8
, we observe that the proposed approaches produce
near results in comparison with each other and with the standard GAO-based
method. Globally, the algorithms convergences always take place in the same
region of the design space whatever is the initial population. This result indicates
that the algorithms succeed in
finding a region of the interesting research space to
explore.
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