# A Multi-Strategy Improved Honey Badger Algorithm for Engineering Design Problems
## 4 Application
### Reducer Design Problems  
In the field of engineering design, the gearbox design problem is a typical optimization challenge that aims to minimize weight and design cost while satisfying a set of key performance constraints. This problem involves seven major design variables: the tooth face width \(b\), the gear module \(m\), the number of pinion teeth \(p\), the lengths of the two shafts \(l_{1}\) and \(l_{2}\), and the diameters of the two shafts \(d_{1}\) and \(d_{2}\). Together, these variables determine  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Algorithm**} & \multicolumn{2}{c}{**Optimum Value**} & \multirow{2}{*}{**Optimal Cost**} \\ \cline{2-3} \cline{5-5}  & \(\boldsymbol{N_{1}}\) & \(\boldsymbol{N_{2}}\) & \\ \hline PSO & \(7.8489\times 10^{-1}\) & \(4.1908\times 10^{-1}\) & \(263.9078\) \\ DBO & \(7.9007\times 10^{-1}\) & \(4.0433\times 10^{-1}\) & \(263.8983\) \\ HBA & \(7.9240\times 10^{-1}\) & \(3.9780\times 10^{-1}\) & \(263.9059\) \\ MIHBA & \(7.8862\times 10^{-1}\) & \(4.0840\times 10^{-1}\) & \(263.8958\) \\ \hline \hline \end{tabular}
\end{table}
Table 11: Comparative experimental results for the three-bar truss design problems.  
Figure 11: Problematic diagram of three-rod truss design.  
the size, performance, and weight of the gearbox. To ensure the safety and reliability of the reducer, the design process must adhere to 11 stringent constraints, including bending stresses on the gear teeth, surface pressures, transverse shaft deflections, and stresses on the shaft. The gearbox design problem is shown in Figure 12, and its mathematical model is given in Equation (22).  
\[\begin{array}{l}\text{{\it Consider}}\overset{\rightarrow}{x}=[x_{1},x_{2},x_{3 },x_{4},x_{5},x_{6},x_{7}]=[b,m,p,l_{1},l_{2},d_{1},d_{2}]\\ \text{{\it Minimize}}\ f(\overset{\rightarrow}{x})=0.7854x_{1}x_{2}^{2}\big{(}3.3333x_{3}^{2}+14.9334x_{3}-43.0934\big{)}\\ \hskip 14.226378 pt-1.508x_{1}\big{(}x_{6}^{2}+x_{7}^{2}\big{)}+7.4777\big{(}x_ {6}^{3}+x_{7}^{3}\big{)}+0.7854\big{(}x_{4}x_{6}^{2}+x_{5}x_{7}^{2}\big{)}\\ \text{{\it Subject to}}\ g_{1}(\overset{\rightarrow}{x})=\frac{27}{x_{1}x_{2}x_{3 }}-1\leq 0\\ \hskip 14.226378 ptg_{2}(\overset{\rightarrow}{x})=\frac{3975}{1.2x_{2}^{2}x_{3 }}-1\leq 0\\ \hskip 14.226378 ptg_{3}(\overset{\rightarrow}{x})=\frac{1.93x_{3}^{2}}{x_{2} x_{4}^{2}x_{3}}-1\leq 0\\ \hskip 14.226378 ptg_{4}(\overset{\rightarrow}{x})=\frac{1.93x_{3}^{2}}{12x_{2} x_{3}}-1\leq 0\\ \hskip 14.226378 ptg_{5}(\overset{\rightarrow}{x})=\frac{\sqrt{(745x_{4}/(x_{3 }x_{3}))^{2}+16.9\times 10^{6}}}{110x_{3}^{2}}-1\leq 0\\ \hskip 14.226378 ptg_{6}(\overset{\rightarrow}{x})=\frac{\sqrt{(745x_{5}/(x_{3 }x_{3}))^{2}+157.5\times 10^{6}}}{85x_{7}^{2}}-1\leq 0\\ \hskip 14.226378 ptg_{7}(\overset{\rightarrow}{x})=\frac{x_{2}x_{3}}{x_{4}}-1 \leq 0\\ \hskip 14.226378 ptg_{8}(\overset{\rightarrow}{x})=\frac{5x_{2}}{x_{1}}-1 \leq 0\\ \hskip 14.226378 ptg_{9}(\overset{\rightarrow}{x})=\frac{x_{1}}{12x_{2}}-1 \leq 0\\ \hskip 14.226378 ptg_{10}(\overset{\rightarrow}{x})=\frac{1.55x_{6}+1.9}{x_{ 1}}-1\leq 0\\ \hskip 14.226378 ptg_{11}(\overset{\rightarrow}{x})=\frac{1.17x+1.9}{x_{5}}-1 \leq 0\\ \hskip 14.226378 pt\textit{Variable range}\ x_{1}\in[2.6,3.6],x_{2}\in[0.7,0.8],x_{3} \in[17,28],\\ \hskip 14.226378 ptx_{4}\in[7.3,8.3],x_{5}\in[7.8,8.3],x_{6}\in[2.9,3.9],x_{7} \in[5.0,5.5]\end{array} \tag{22}\]  
As shown in the experimental results in Table 12, the optimal values obtained by MIHBA are less than the other three algorithms, indicating that MIHBA obtains better values and performs better in the gearbox design problem. In this problem, the optimal cost of MIHBA is 6.69% lower than that of HBA.  
Figure 12: Schemes follow the same formatting.