# Mutational Chemotaxis Motion Driven MdpI for Engineering Applications
## 5 Engineering Design Problems
### Speed Reducer Design Problem  
The model involves seven parameters, which are represented by \(x_{1}\)-\(x_{7}\), and their meanings are the width of the end face (\(b\)), the tooth module (\(m\)), the number of teeth in the pinion (\(z\)), the length of the first shaft between the bearings (\(l_{1}\)), the length of the second shaft between bearings (\(l_{2}\)), the diameter of the first shaft (\(d_{1}\)), and the second shaft (\(d_{2}\)). The following constraints are carried out within the effective range of satisfying these parameters, and finally, the weight of the reducer is minimized. The specific details in this model are shown below.  
\[\text{Consider}\overset{\rightarrow}{x}=[x_{1}\;\;x_{2}\;x_{3}\;x_{4}\;x_{5 }\;x_{6}\;x_{7}\;]=[b\;m\;p\;l_{1}\;l_{2}\;d_{1}\;d_{2}]\]  
\[\text{Minimize}\;f\!\left(\overset{\rightarrow}{x}\right)=0.7854x_{1}x_{2}^ {2}(3.3333x_{3}^{2}+14.9334x_{3}-43.0934)\]  
\[-1.508x_{1}\!\left(x_{6}^{2}+x_{7}^{2}\right)+7.4777\!\left(x_{6}^{2}+x_{7}^{ 2}\right)+0.7854\!\left(x_{4}x_{6}^{2}+x_{5}x_{7}^{2}\right)\]  
\[\text{Subject to}:\;g_{1}\!\left(\overset{\rightarrow}{x}\right)=\frac{27}{x_ {1}x_{3}x_{2}^{2}}-1\leq 0,\]  
\begin{table}
\begin{tabular}{l c c c c c} \hline \hline \multirow{2}{*}{**Algorithm**} & \multicolumn{4}{c}{**Optimal Values for Variables**} & \multirow{2}{*}{**Optimum Cost**} \\ \cline{2-2} \cline{4-6}  & \(b\) & \(h\) & \(t_{w}\) & \(t_{f}\) & \\ \hline HMCMMFO & 50.0000 & 80.0000 & 0.9000 & 2.321792 & 0.013074 \\ IDARSOA [82] & 50.0000 & 80.0000 & 0.9000 & 2.321769 & 0.013074 \\ SOS [98] & 50.0000 & 80.0000 & 0.9000 & 2.3218 & 0.013074 \\ CS [92] & 50.0000 & 80.0000 & 0.9000 & 2.3217 & 0.013075 \\ AGOA [89] & 43.12663 & 79.91247 & 0.936202 & 2.671865 & 0.013295 \\ ARSM [99] & 37.0500 & 80.0000 & 1.7100 & 2.3100 & 0.015700 \\ IARSM [99] & 48.4200 & 79.9900 & 0.9000 & 2.4000 & 0.131000 \\ \hline \hline \end{tabular}
\end{table}
Table 12: Comparison results of the I-beam design.  
Variable ranges : \(2.6\leq x_{1}\leq 3.6,\ 0.7\leq x_{2}\leq 0.8,\ 17\leq x_{3}\leq 28,7.3\leq x_{4} \leq 28\),  
\[7.3\leq x_{5}\leq 8.3,\ 2.9\leq x_{6}\leq 3.9,\ 5.0\leq x_{7}\leq 5.5\]  
Table 13 below shows the target values obtained by eight algorithms, including HMCMFO, when solving this problem. These algorithms include novel metaheuristics and advanced algorithms. According to the data in the table, the weight of the model obtained by HMCMMFO is 2994.4711, which is smaller than that obtained by other algorithms and satisfies the goal of the model. Compared with SCA and GSA, HMCMMFO has a larger difference in the weight of the reducer. It can be seen that HMCMMFO can achieve good results when solving this problem, and can satisfactorily meet the requirements of industrial problems.  
To sum up, research on the above five engineering examples shows that the HMCMMFO proposed in this paper is an effective method for solving engineering-constraint problems and has an excellent performance in solving practical problems. When it is applied to other fields in the future, such as recommender system [101; 102], power-flow optimization [103], location-based services [104; 105], road-network planning [106], human-activity recognition [107], information-retrieval services [108; 109], image denoising [110], and colorectal-polyp-region extraction [111], it has the potential to perform well.