## 3 Utilized Meta-Heuristic Algorithms
### Standard Genetic Algorithm
If \(x_{1}\), \(x_{2}\), \(x_{3}\), and \(x_{4}\) represent the numbers of teeth on gears A, B, D, and F, respectively, the goal is to minimize Equation (12):

\[f(x)=\left(\frac{1}{6.931}-\frac{x_{3}x_{2}}{x_{1}x_{4}}\right)^{2} \tag{12}\]

where \(x_{1}\), \(x_{2}\), \(x_{3}\) and \(x_{4}\) must be in the interval of [12; 60] and gear ratio is:

\[f(x)=0.6224x_{1}x_{3}x_{4}+1.7781x_{3}^{2}x_{2}+3.1611x_{1}^{2}x_{4}+19.8621x_{ 1}^{2}x_{3} \tag{5}\]0

If the performances of the algorithms given in Table 13 are examined, it is obvious that the performances of the PSO, CS, and BIRA algorithms are better than the other algorithms. Though the gear ratio values, obtained from Equation (13), of all three algorithms are the same (the first six digits of the decimal part of the results), it can be observed that PSO is more successful than BIRA and CS (albeit by a small margin). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Values** & **SGA** & **DE** & **PSO** & **CS** & **GWO** & **BIRA** \\ \hline x\({}_{1}\) & 51.6703 & 35.71 & 44.9135 & 60 & 45.5971 & 39.5025 \\ x\({}_{2}\) & 15.1419 & 13.3157 & 14.0528 & 30.2716 & 31.1309 & 12 \\ x\({}_{3}\) & 15.7607 & 22.4754 & 18.2749 & 14.0054 & 12 & 16.3961 \\ x\({}_{4}\) & 31.8489 & 56.8036 & 39.6311 & 48.9752 & 56.7843 & 34.5217 \\ Gear ratio & 0.1450175 & 0.1475386 & **0.1442795** & 0.1442791 & 0.1442804 & 0.1442796 \\ Cost & 5.4429 \(\times 10^{-7}\) & 1.0627 \(\times 10^{-5}\) & **0** & 1.1755 \(\times 10^{-15}\) & 1.216 \(\times 10^{-12}\) & 2.3559 \(\times 10^{-28}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 13: The results of the algorithms for designing the gear train. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Values** & **SGA** & **DE** & **PSO** & **CS** & **GWO** & **BIRA** \\ \hline x\({}_{1}\) & 1.2526 & 4.4106 & 1.267 & 1.2211 & 1.2614 & 1.1 \\ x\({}_{2}\) & 2.5953 & 10.6419 & 0.6263 & 0.6228 & 0.624 & 0.6 \\ x\({}_{3}\) & 55.492 & 52.8848 & 65.6414 & 56.729 & 64.9616 & 55.4458 \\ x\({}_{4}\) & 75.5108 & 145.4679 & 10 & 56.6558 & 11.1642 & 60.5331 \\ g\({}_{1}\)(x) & \(-0.1816\) & \(-3.39\) & \(-1.3294\times 10^{-4}\) & \(-0.1262\) & \(-0.0076\) & \(-0.0299\) \\ g\({}_{2}\)(x) & \(-2.0659\) & \(-10.1374\) & \(-5.6956\times 10^{-5}\) & \(-0.0816\) & \(-0.0043\) & \(-0.071\) \\ g\({}_{3}\)(x) & \(-1.6963\times 10^{5}\) & \(-6.017\times 10^{5}\) & \(-2.4103\times 10^{4}\) & \(-4.1523\times 10^{4}\) & \(-320.3189\) & \(-2.6226\times 10^{3}\) \\ g\({}_{4}\)(x) & \(-162.4892\) & \(-9.54321\) & \(-230\) & \(-183.3442\) & \(-228.8358\) & \(-179.4669\) \\ g\({}_{5}\)(x) & \(-0.1526\) & \(-3.3106\) & \(-0.167\) & \(-0.1211\) & \(-0.1614\) & 0 \\ g\({}_{6}\)(x) & \(-1.9953\) & \(-10.0419\) & \(-0.0263\) & \(-0.0228\) & \(-0.024\) & 0 \\ Cost (S) & 19676.01 & 103412.12 & 7457.3 & 7952.03 & 7358.68 & **7140.59** \\ \hline \hline \end{tabular}
\end{table} 
Table 12: The results of algorithms the for designing the pressure vessel. Figure 6: The set of four gears of the train (taken from [25]). #### 4.4.3 Tension/Compression Spring Design Problem

Firstly, [PERSON] [48] presented the tension/compression spring design problem, shown in Figure 7. The main purpose of the problem is to minimize the cost of three parameters: wire diameter d (\(x_{1}\)), mean coil diameter D (\(x_{2}\)), and the number of active coils P (\(x_{3}\)). In addition to minimizing Equation (14), it must not violate any constraints from Equations (15)-(18). \[f(x)=(x_{3}+2)x_{2}x_{1}^{2} \tag{14}\]

\[g_{1}(x)=1-\frac{x_{2}^{2}x_{3}}{71785x_{1}^{4}}\leq 0 \tag{15}\]

\[g_{2}(x)=\frac{4x_{2}^{2}-x_{1}x_{2}}{12566(x_{2}x_{1}^{3}-x_{1}^{4})}+\frac{1 }{5108x_{1}^{2}}-1\leq 0 \tag{16}\]

\[g_{3}(x)=1-\frac{140.45x_{1}}{x_{2}^{2}x_{3}}\leq 0 \tag{17}\]

\[g_{4}(x)=\frac{x_{2}+x_{1}}{1.5}-1\leq 0 \tag{18}\]

where \(0.05\leq x_{1}\leq 2\), \(0.25\leq x_{2}\leq 1.30\) and \(2\leq x_{3}\leq 15\). When the performances of the algorithms in Table 14 are observed, it can be seen that SGA and DE are stuck at the same local value.