Fig 6. Search space of UCI benchmark function
4.1 Statistical discussion
In each experiment, all algorithms executed 30 timesand each running
performed with 500 iterations for all benchmark functions.From the other
side, this comparison is not enough for reliability. Furthermore, we
applied the Wilcoxon Signed-Rank test, which provide statistically
validate in the results. The test is performed using a pairwise method,
where p and his the significance and logic valuerespectively,that based
on whether the defined hypothesis is rejected or accepted. Test
including the following hypothesis
a = 0.05 (95%)
\(H_{0}\):\(\mu_{1}\)=\(\mu_{2}\)
\(H_{1}\):\(\mu_{1}\neq\mu_{2}\)
Two algorithm with the obtained results are compared and considered
their valuesare near or faraway (\(H_{0}\)and \(H_{1}\)) to each other.
\(H_{0}\)= 0 two algorithm are not different.
\(H_{1}\)= 1 two algorithm are different.
In according with the outcome of Tables 3,4,5,6.We observed our
algorithm surpasses than the others in balancing exploitation and
exploration and finding global minimum. To sum up, our algorithm with
competitive result showed the power on the unimodal and multimodal
benchmark functions in complex problem. One example of divergence for
all algorithmsshowon the Sphere function in Fig 7. As you view, HPO in
less iteration has not proper diversity, but in more dimensionsHPO
algorithm has appropriate diversity and result than the others.
4.1 Real Problem
For further examine, the performance of HPO, a real problem employed in
the function, pressure vessel designs,problem which is constrained
engineering design,are used.HPOalgorithm should have a constraint
handling strategy to optimize the constrained problem. Considering the
equality and non-equality constrain in the problem,algorithms should
control the constraintand object value with considering violence. The
objective of pressure vessel designs is to minimize the total cost with
constraint comprising of welding, forming, and material of a cylindrical
vessel. Four design process factors or four decision variable should be
tuned by algorithms which is mentioned as follows.
_ Thickness of the shell (Ts).
_ Thickness of the head (Th).
_ Inner radius (R).
_ Length of the cylindrical portion without regard to the head
(L).
The objective function and four constraint functions aredefined in
Equation 14.An expression of the Pressure Vessel Design (PVD) is as
follows: both ends of a cylindrical vessel are capped by hemispherical
heads (Fig 8).
Consider\(\overrightarrow{x}\)=\(\left[x_{1}x_{2}x_{3}x_{4}\right]\)=\(\left[T_{s}T_{h}\text{RL}\right]\),
Minimize
f(\(\overrightarrow{x}\))=0.6224\(x_{1}x_{2}x_{3}\)+1.7781\(x_{2}x_{3}^{2}\)+3.1661\(x_{1}^{2}x_{4}\)+19.84\(x_{1}^{2}x_{3}\),
Subject to\(g_{1}\)(\(\overrightarrow{x}\))=-\(x_{1}\)+0.0193\(x_{3}\leq\)0,
\(g_{2}\)(\(\overrightarrow{x}\))=-\(x_{3}\)+0.0954\(x_{3}\leq\)0,
\(g_{3}\)(\(\overrightarrow{x}\))=-\(\pi x_{3}^{2}x_{4}\)-\(\frac{4}{3}\pi x_{3}^{3}\)+1296000\(\leq 0,\)
\({\ g}_{4}\)(\(\overrightarrow{x}\))=-\(x_{4}-240\leq\)0, (14
The upper and lower decision variablesare defined in the following Eq.
15
0\(\leq x_{1}\leq 99,\)
0\(\leq x_{2}\leq 99,\)
10\(\leq x_{3}\leq 200,\)
10\(\leq x_{4}\leq 200\) (15
In Table 7. Comparing all algorithmin vessel pressure design
optimization, which include tuned parameters and cost value. As you
view, the performance of our algorithm take constraint challenge into
consideration outperforms the others.
Table 1. Unimodalbenchmarkfunctions