## 3 Utilized Meta-Heuristic Algorithms
### Standard Genetic Algorithm
\begin{table}
\begin{tabular}{c c} \hline
**Methods** & **Mean Rank** \\ \hline SGA & 4.87500 \\ DE & 5.17708 \\ PSO & 2.67708 \\ CS & 5.84375 \\ GWO & 3.41666 \\ RA & 3.61458 \\ BIRA & **2.39583** \\ \hline \end{tabular}
\end{table} 
Table 10: Statistical values for the Friedman test. \begin{table}
\begin{tabular}{c c} \hline
**Statistical Variant** & **Value** \\ \hline Number of functions & 48 \\ \(X^{2}\) & 116.99142 \\ df & 6 \\ Probability value & 6.97748 \\ \hline \end{tabular}
\end{table} 
Table 10: Statistical values for the Friedman test. \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Methods** & **SGA** & **DE** & **PSO** & **CS** & **GWO** & **RA** & **BIRA** \\ \hline SGA & 1 & 0.005677 & 0.000357 & 3.545623 \(\times 10^{-5}\) & 0.000474 & 0.000370 & 4.121701 \(\times 10^{-6}\) \\ DE & & 1 & 3.050312 \(\times 10^{-6}\) & 1.338213 \(\times 10^{-5}\) & 1.338213 \(\times 10^{-5}\) & 0.00012 & 3.463021 \(\times 10^{-7}\) \\ PSO & & 1 & 1.068801 \(\times 10^{-8}\) & 0.000706 & 0.000239 & 0.269221 \\ CS & & & & 1 & 1.013714 \(\times 10^{-6}\) & 4.547581 \(\times 10^{-8}\) & 6.118679 \(\times 10^{-7}\) \\ GWO & & & & 1 & 0.059862 & 0.000991 \\ RA & & & & & 1 & 0.000236 \\ BIRA & & & & & 1 \\ \hline \end{tabular}
\end{table} 
Table 11: The pairwise results of the Wilcoxon Signed Rank Test. \begin{table}
\begin{tabular}{c c} \hline
**Methods** & **Mean Rank** \\ \hline SGA & 4.87500 \\ DE & 5.17708 \\ PSO & 2.67708 \\ CS & 5.84375 \\ GWO & 3.41666 \\ RA & 3.61458 \\ BIRA & **2.39583** \\ \hline \end{tabular}
\end{table} 
Table 9: The Friedman test mean rank values. may not comply with these constraints. That is why, for each iteration, if individuals in the population violate the constraints, these individuals are sentenced to death and the new generation is recreated from variants of individuals that did not violate the rules, and the process continues. #### 4.4.1 Pressure Vessel Design Problem

The pressure vessel, a containment unit for various gas or liquid pressures of different sizes, is susceptible to critical faults during manufacturing or operation, which could lead to severe damage or injury. Hence, it is crucial to design the pressure vessel with the correct structure [46]. The design of the vessel, shown in Figure 5, necessitates the adjustment of several parameters (materials): shell thickness (\(T_{s}=x_{1}\)), spherical head thickness (\(T_{h}=x_{2}\)), radius of cylindrical shell (\(R=x_{3}\)), and shell length (\(L=x_{4}\)). The aim of the problem is to minimize the total cost of the materials, given in Equation (5), without violating any constraints (given in Equations (6)-(11)). \[f(x)=0.6224x_{1}x_{3}x_{4}+1.7781x_{3}^{2}x_{2}+3.1611x_{1}^{2}x_{4}+19.8621x_{ 1}^{2}x_{3} \tag{5}\]

\[g_{1}(x)=0.0193x_{3}-x_{1}\leq 0 \tag{6}\]

\[g_{2}(x)=0.00954x_{3}-x_{2}\leq 0 \tag{7}\]

\[g_{3}(x)=750X1728-\pi x_{3}^{2}x_{4}-\frac{4}{3}\pi x_{3}^{3}\leq 0 \tag{8}\]

\[g_{4}(x)=x_{4}-240\leq 0 \tag{9}\]

\[g_{5}(x)=1.1-x_{1}\leq 0 \tag{10}\]

\[g_{6}(x)=0.6-x_{2}\leq 0 \tag{11}\]

where \(0.0625\leq x_{1},x_{2}\leq 6.1875\) and \(x_{1}\) and \(x_{2}\) are integer multipliers of \(0.0625\) in, and \(x_{3}\) and \(x_{4}\) are continuous integer values in the respectively following interval: \(40\) in \(\leq x_{3}\leq 80\) in and \(20\) in \(\leq x_{4}\leq 60\) in. In Table 12, the best results where the algorithms did not violate the constraints (given in Equations (6)-(11)) are shared. In the pressure vessel design problem, PSO could not show the performance it showed in the test functions and fell behind GWO and BIRA. According to Table 12, BIRA presented the most successful result compared to all tested algorithms by rea

Figure 5: The structure of a pressure vessel (taken from [25]). #### 4.4.2. Gear Train Design Problem

In the gear train design problem, the objective is to approach a gear ratio value of 1/6.931, or 0.1442793, as closely as possible [47]. As shown in Figure 6, a set of four gears constitutes the train. The tuning process involves determining the numbers of teeth on these gears.