# Boosted Reptile Search Algorithm for Engineering and Optimization Problems
## 4 Experimental Results and Discussion
### Series of Analysis 1: Engineering Problems
#### 4.1.5 Speed Reducer Problem  
In order for an airplane's propeller and engine to rotate at an efficient speed, a gearbox between them is needed (see Figure 6[67]). The optimization strategy employed to address the speed reducer problem is intended for minimizing the design's weight while still meeting the requirements for the bending stress of the gear teeth, surface stress, transverse shaft deflections, and stresses in the shafts. The best value for each of the seven design factors must be established to solve this problem. These variables are the face width (\(x_{1}\)), the module of the teeth (\(x_{2}\)), the number of teeth on the pinion (\(x_{3}\)), the lengths of the first and second shafts between the bearings (\(x_{4}\) and \(x_{5}\), respectively), the first shaft's diameter (\(x_{6}\) and \(x_{7}\), respectively), and the module of the teeth (\(x_{2}\)). Given that it contains more limitations than other issues, the speed reducer problem is defined as having a high degree of complexity. Below is a diagram that represents this issue mathematically. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Algorithm**} & \multicolumn{2}{c}{**Estimated Values for Parameters**} & \multirow{2}{*}{**Optimal Weight**} \\ \cline{2-3} \cline{5-4}  & \(x_{1}\) & \(x_{2}\) & \\ \hline RSRFT & 0.78875052 & 0.4080351 & 263.89584 \\ RFO & 0.75356 & 0.55373 & 268.51195 \\ RSA [17] & 0.78873 & 0.40805 & 263.8928 \\ DEDS [61] & 0.78867513 & 0.40824828 & 263.89584 \\ SSA [62] & 0.78866541 & 0.408275784 & 263.89584 \\ MBA [63] & 0.7885650 & 0.4085597 & 263.89585 \\ PSO-DE [58] & 0.7886751 & 0.4082482 & 263.89584 \\ Ray and Saini [54] & 0.795 & 0.395 & 264.3 \\ CS [64] & 0.78867 & 0.40902 & 263.9716 \\ AAA [65] & 0.7887354 & 0.408078 & 263.895880 \\ GOA [66] & 0.78889755557 & 0.40761957011 & 263.89588149 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The value of the estimated parameters using RSRFT to solve the 3-bar truss design problem. Figure 6: The speed reducer problem. \[ \begin{array}{ll}\text{\emph{Assume}}&f(x)=0.7854x_{1}x_{2}^{2}(3.3333x_{3}^{2} +14.9334x_{3}-43.0934)-\\ &1.508x_{1}(x_{6}^{2}+x_{7}^{2})+7.4777x_{6}^{3}+x_{7}^{3}+0.7854x_{4}x_{6}^{2}+ x_{5}x_{7}^{2}\\ \text{\emph{Mintinize}}&f(\vec{u})=(u_{3}+2)u_{2}u_{1}^{2},\\ \text{\emph{Subject to}}&g(1)=\frac{27}{x_{1}x_{2}^{2}x_{3}}-1\leq 0,\\ &g(2)=\frac{397.5}{x_{1}x_{2}^{2}x_{3}^{2}}-1\leq 0,\\ &g(3)=\frac{1.93x_{4}^{3}}{x_{2}x_{3}x_{6}^{4}}-1\leq 0,\\ &g(4)=\frac{1.93x_{5}^{3}}{x_{2}x_{3}x_{7}^{4}}-1\leq 0,\\ &g(5)=\frac{1}{110x_{6}^{3}}\sqrt{(\frac{745x_{4}}{x_{2}x_{3}})^{2}+16.9\times 1 0^{6}}-1\leq 0,\\ &g(6)=\frac{1}{85x_{7}^{3}}\sqrt{(\frac{745x_{5}}{x_{2}x_{3}})^{2}+157.5\times 1 0^{6}}-1\leq 0,\\ &g(7)=\frac{x_{2}x_{3}}{40}-1\leq 0,\\ &g(8)=\frac{5x_{2}}{x_{1}}-1\leq 0,\\ &g(9)=(x_{1}/12x_{2})-1\leq 0,\\ &g(10)=\frac{1.5x_{6}+1.9}{x_{4}}-1\leq 0,\\ &g(11)=\frac{1.1x_{7}+1.9}{x_{5}}-1\leq 0\\ &Variables^{\prime}\text{\emph{range}}&(0.7\leq x_{2}\leq 0.8),(2.6\leq x_{1} \leq 3.6),(17\leq x_{3}\leq 28),(7.3\leq x_{4}\leq 8.3),\\ &(2.9\leq x_{6}\leq 3.9),(7.3\leq x_{5}\leq 8.3),(5\leq x_{7}\leq 5.5)\\ \end{array}  \tag{30}\]  
Table 5 provides the comparative findings between RSRFT and the other MH approaches that have been given in the literature. It is clear that RSRFT surpassed the majority of the techniques that were examined (as given under Optimal Weight), and it was generally assigned the third rank. CS and SBSM were assigned the first and second ranks, respectively [64,68]. Since all three algorithms almost had the same ideal weight, there was little difference between them.