## 14 Applying JADEDO for Solving Engineering Problems
### Speed Reducer Design Problem  
The speed reducer design problem involves optimizing the dimensions and material properties of a speed reducer to minimize its overall weight while satisfying several performance and safety constraints [51]. Figure 17 illustrates the speed reducer design problem, where the objective is to optimize the dimensions of the gearbox components. The design variables include the face width \(x_{1}\), the gear diameters \(x_{2}\) and \(x_{3}\), and the shaft diameters \(x_{6}\) and \(x_{7}\). The overall dimensions \(x_{4}\) and \(x_{5}\) represent the vertical and horizontal sizes of the system. The objective function to be minimized is the weight of the speed reducer, which is expressed as shown in Equation (69):  
\[ \begin{split} f(x)&=0.7854\cdot x_{1}\cdot x_{2}^{2 }\cdot(3.3333\cdot x_{3}^{2}+14.9334\cdot x_{3}-43.0934)\\ &\quad-1.508\cdot x_{1}\cdot(x_{6}^{2}+x_{7}^{2})+7.4777\cdot(x_{ 6}^{3}+x_{7}^{3})\\ &\quad+0.7854\cdot(x_{4}\cdot x_{6}^{2}+x_{5}\cdot x_{7}^{2}). \end{split}  \tag{69}\]  
where \(x_{1}\) through \(x_{7}\) represent the design variables, including gear face width, module, number of teeth, and other related parameters. The optimization process is subject to several constraints that ensure the mechanical performance and reliability of the speed reducer. These constraints are defined by the following inequalities (70)-(80):  
\[c_{1}(x) =\frac{27}{x_{1}\cdot x_{2}^{2}\cdot x_{3}}-1\leq 0, \tag{70}\] \[c_{2}(x) =\frac{397.5}{x_{1}\cdot x_{2}^{2}\cdot x_{3}^{2}}-1\leq 0,\] (71) \[c_{3}(x) =\frac{1.93\cdot x_{4}^{3}}{x_{2}\cdot x_{3}\cdot x_{6}^{4}}-1\leq 0, \tag{72}\]  
Figure 17: Speed reducer design problem. \begin{table}
\begin{tabular}{l l l l l l l l l l l} \hline
**Optimizer** & **Min** & **Mean** & **Max** & **Std** & **Time** & **Rank** & **Best Score** & **X1** & **X2** & **X3** \\ \hline ZOA & 0.012667 & 0.01323 & 0.014017 & 0.00385 & 1.321528 & 4 & 0.012667 & 0.051804 & 0.359452 & 11.13138 \\ RIME & 0.013055 & 0.015576 & 0.018068 & 0.00253 & 0.693365 & 21 & 0.013055 & 0.051 & 0.334085 & 13.02403 \\ SCA & 0.012813 & 0.013086 & 0.013356 & 0.000175 & 0.713861 & 18 & 0.012813 & 0.051 & 0.339724 & 12.50058 \\
**JADEDO** & **0.012665** & **0.012749** & **0.013208** & **0.000185** & **0.990184** & **1** & **0.012665** & **0.051696** & **0.356884** & **11.27924** \\ HHO & 0.012821 & 0.013604 & 0.015683 & 0.000775 & 1.467947 & 19 & 0.012821 & 0.054671 & 0.432778 & 7.911444 \\ SCSO & 0.012674 & 0.013111 & 0.014338 & 0.000687 & 1.03688 & 8 & 0.012674 & 0.051 & 0.340362 & 12.3167 \\ OMA & 0.012693 & 0.012743 & 0.012799 & 3.95E-05 & 1.813852 & 12 & 0.012693 & 0.052914 & 0.386891 & 9.717318 \\ AVOA & 0.012666 & 0.013023 & 0.013517 & 0.00028 & 0.971122 & 3 & 0.012666 & 0.051908 & 0.362019 & 10.98481 \\ \hline \end{tabular}
\end{table} 
Table 9: _Cont._\[c_{4}(x)=\frac{1.93\cdot x_{5}^{3}}{x_{2}\cdot x_{3}\cdot x_{7}^{3}}-1\leq 0, \tag{73}\]  
\[c_{5}(x)=\frac{1}{110\cdot x_{6}^{3}}\cdot\sqrt{\left(\frac{745\cdot x_{4}}{x_{2 }\cdot x_{3}}\right)^{2}+16.9\times 10^{6}}-1\leq 0, \tag{74}\]  
\[c_{6}(x)=\frac{1}{85\cdot x_{7}^{3}}\cdot\sqrt{\left(\frac{745\cdot x_{5}}{x_{2 }\cdot x_{3}}\right)^{2}+157.5\times 10^{6}}-1\leq 0, \tag{75}\]  
\[c_{7}(x)=\frac{x_{2}\cdot x_{3}}{40}-1\leq 0, \tag{76}\]  
\[c_{8}(x)=\frac{5\cdot x_{2}}{x_{1}}-1\leq 0, \tag{77}\]  
\[c_{9}(x)=\frac{x_{1}}{12\cdot x_{2}}-1\leq 0, \tag{78}\]  
\[c_{10}(x)=\frac{1.5\cdot x_{6}+1.9}{x_{4}}-1\leq 0, \tag{79}\]  
\[c_{11}(x)=\frac{1.1\cdot x_{7}+1.9}{x_{5}}-1\leq 0, \tag{80}\]  
To handle any violations of these constraints, a penalty function is incorporated into the objective function, as shown in Equation (81):  
\[f(x)=f(x)+10^{5}\cdot\left(\sum_{i=1}^{11}v_{i}+\sum_{i=1}^{11}\max(0,c_{i}(x) )\right), \tag{81}\]  
where \(v_{i}\) is an indicator function that equals 1 if the corresponding constraint \(c_{i}(x)\) is violated and 0 otherwise.