# Boosted Reptile Search Algorithm for Engineering and Optimization Problems
## 4 Experimental Results and Discussion
### Series of Analysis 1: Engineering Problems
#### 4.1.6 Multiple Disc Clutch Brake Problem  
Finding the values of five design factors to reduce the mass of an MDCB is the primary goal of research into the multiple disc clutch brake (MDCB) problem, which was cited in [63]. The design parameters are shown in Figure 7 as the inner radius \(x_{1}\), outer radius \(x_{2}\), disc thickness \(x_{3}\), actuation force \(x_{4}\), and number of friction surfaces \(x_{5}\). Below is a diagram that represents this issue mathematically.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multicolumn{6}{c}{**Estimated Values for Parameters**} & \multirow{2}{*}{**Optimal Weight**} \\ \cline{2-2} \cline{4-7}  & \(x_{1}\) & \(x_{2}\) & \(x_{3}\) & \(x_{4}\) & \(x_{5}\) & \(x_{6}\) & \(x_{7}\) \\ \hline SES [71] & 3.506163 & 0.700831 & 17 & 7.460181 & 7.962143 & 3.362900 & 5.308949 & 3025.005127 \\ MDA [72] & 3.5 & 0.7 & 17 & 7.3 & 7.670396 & 3.542421 & 5.245814 & 3019.583365 \\ SBSM [68] & 3.506122 & 0.700006 & 17 & 7.549126 & 7.859330 & 3.365576 & 5.289773 & 3008.08 \\ SCA [11] & 3.508755 & 0.7 & 17 & 7.3 & 7.8 & 3.461020 & 5.289213 & 3030.563 \\ CS [64] & 3.5015 & 0.7000 & 17 & 7.6050 & 7.8181 & 3.3520 & 5.2875 & 3000.9810 \\ PSO [73] & 3.5001 & 0.7000 & 17.0002 & 7.5177 & 7.7832 & 3.3508 & 5.2867 & 3145.922 \\ FA [74] & 3.507495 & 0.7001 & 17 & 7.719674 & 8.080854 & 3.351512 & 5.287051 & 3010.137492 \\ hHHO-SCA [75] & 3.506119 & 0.7 & 17 & 7.3 & 7.99141 & 3.452569 & 5.286749 & 3029.873076 \\ \hline \hline \end{tabular}
\end{table}
Table 5: _Cont._  
Figure 7: The multiple disc clutch brake problem.  
\[\begin{array}{ll}\text{\emph{Assume}}&f(x)=\Pi(r_{o}^{2}-r_{i}^{2})t(Z+1) \rho\\ &1.508x_{1}(x_{6}^{2}+x_{7}^{2})+7.4777x_{6}^{3}+x_{7}^{3}+0.7854x_{4}x_{6}^{2}+x _{5}x_{7}^{2}\\ \text{\emph{Minimize}}&f(\vec{u})=(u_{3}+2)u_{2}u_{1}^{2},\\ \text{\emph{Subject to}}&g_{1}(x)=l_{max}-(Z+1)(t+\delta)\geq 0,g_{2}(x)=r_{o}-r_{i}- \Delta r\geq 0\\ &g_{3}(x)=P_{max}\;v_{sr\ max}-P_{rz}\;v_{sr}\geq 0\,g_{4}(x)=Pmax-Prz\geq 0,\\ &g_{5}(x)=v_{sr\ max}-v_{sr}\geq 0,\;g_{6}=T_{max}-T\geq 0\\ &g_{7}(x)=M_{h}-sM_{5}\geq 0,g_{8}(x)=T\geq 0\\ \text{\emph{where}}&M_{h}=\frac{2}{3}\mu FZ\frac{r_{o}^{3}-r_{i}^{2}}{r_{o}^{2 }-r_{i}^{3}},\;\;\;P_{rz}=\frac{F}{\Pi(r_{o}^{2}-r_{i}^{2})},\
u_{rz}=\frac{2 \Pi(r_{o}^{3}-r_{i}^{3})}{90(r_{o}^{2}-r_{i}^{2})},\;\;\;\;T=\frac{l_{z}\;\Pi \;n}{30(M_{h}+M_{f})}\\ &\Delta r=20\;\text{mm},\;l_{z}=55\;\text{kgmm}^{2},\;P_{max}=1\;\text{MPa}, \;F_{max}=1000\;\text{N},\\ &v_{sr\ max}=10\;\text{m/s},\;l_{max}=30\;\text{mm},\;r_{i\ min}=60,\\ &T_{max}=15\;\text{s},\;\mu=0.5,\;s=1.5,\;M_{s}=40\;\text{Nm},\;M_{f}=3\;\text {Nm},\;n=250\;\text{rpm},\\ &F_{max}=1000,\;Z_{min}=2,\;Z_{max}=9,r_{i\ max}=80,r_{o\ min}=90,\\ &r_{o\ max}=110,\;t_{min}=1.5,t_{max}=3,\;F_{min}=600\\ \end{array} \tag{31}\]  
Table 6 lists the outcomes of RSRFT and other MH approaches collected from the literature. From this table, it is clear that RSRFT obtained the ideal cost (0.31176) (as given under Optimal Weight), which was then followed by CMVO, MFO, MVO, and the WCA, which had the same efficiency.