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