# Mutational Chemotaxis Motion Driven MdpI for Engineering Applications
## 5 Engineering Design Problems
### Tension-Compression String Problem  
The engineering model is a typical engineering-constraint problem, which minimizes the weight of the spring under the constraints of model parameters. The parameters involved are wire diameter (\(d\)), average coil diameter (\(D\)), and the finite number of coils (\(N\)). Specific model details are shown below. \[\text{Consider: }\overset{\rightarrow}{x}=[x_{1}\ x_{2}\ x_{3}\ ]=[d\ D\ N]\]  
\[\text{Minimize: }\ f\binom{\rightarrow}{x}=x_{1}^{2}x_{2}(x_{3}+2)\]  
\[\text{Subject to: }\ g_{1}\binom{\rightarrow}{x}=1-\frac{4x_{2}^{3}x_{3}}{71,7 85x_{1}^{4}}\leq 0,\]  
\[ \begin{array}{l}g_{2}\binom{\rightarrow}{x}=\frac{4x_{2}^{2}-x_{3}x_{2}}{12,566\left(x_{2}x_{3}^{2}-x_{4}^{4}\right)}+\frac{1}{1508x_{1}^{2}}\leq 0, \\ g_{3}\binom{\rightarrow}{x}=1-\frac{140.45x_{1}}{x_{2}^{2}x_{3}}\leq 0, \\ g_{4}\binom{\rightarrow}{x}=\frac{x_{1}+x_{2}}{1.5}-1\leq 0.\end{array} \]  
Figure 6: Convergence effect of advanced algorithms. \[\text{Variable range}:\;0.05\leq x_{1}\leq 2,\;0.25\leq x_{2}\leq 1.3,\;2\leq x_{3} \leq 150\]  
Table 9 below shows the parameters and minimum weight obtained by the 11 methods for solving this problem. Through comparison, it is found that the improved algorithm proposed in this paper obtains the smallest spring weight of 0.012665, compared to the traditional mathematical methods and well-known metaheuristic algorithms. Compared with WEMFO, which is also an improved MFO algorithm, the minimum cost of HMCMMFO is much better than WEMFO. It can be seen that HMCMMFO can solve this problem effectively and with significant results. ### Tension-Compression String Problem  
This engineering model contains three constraint functions and two parameters. The effective parameters are applied to the constraints to find the optimal cost. The specific details of this model are described below. \[\text{Objective function}:\;f(x)=\left(2\sqrt{2}x_{1}+x_{2}\right)\times I\]  
\[\text{Subject to}:\;g_{1}(x)=\frac{\sqrt{2}x_{1}+x_{2}}{\sqrt{2}x_{1}^{2}+2x_ {1}x_{2}}P-\sigma\leq 0,\]  
\[g_{2}(x)=\frac{x_{2}}{\sqrt{2}x_{1}^{2}+2x_{1}x_{2}}P-\sigma\leq 0,\]  
\[g_{3}(x)=\frac{1}{\sqrt{2}x_{2}+x_{1}}P-\sigma\leq 0.\]  
\[\text{Variable range}:\;0\leq x_{i}\leq 1\;(\;i=1,\;2),\,l=100\text{cm},P=2 \;\text{kN}/\text{cm}^{2},\sigma=2\;\text{kN}/\text{cm}^{2}\]  
As shown in Table 10 below, the cost of HMCMMFO for this problem is 263.895843, with \(x_{1}=0.788673\) and \(x_{2}=0.408253\). Compared with other mature algorithms, HMCMMFO ranks first with the smallest optimum cost. It shows that the method proposed in this paper has advantages in solving the optimization problem, which can effectively reduce engineering consumption and help to solve practical problems in the real world. \begin{table}
\begin{tabular}{l c c c} \hline \hline \multirow{2}{*}{**Algorithm**} & \multicolumn{3}{c}{**Optimal Values for Variables**} & \multirow{2}{*}{**Optimum Cost**} \\ \cline{2-3} \cline{5-5}  & \(\mathbf{x_{1}}\) & \(\mathbf{x_{2}}\) & \\ \hline HMCMMFO & 0.788673 & 0.408253 & 263.895843 \\ AGOA [89] & 0.788584 & 0.4085050 & 263.895849 \\ MBA [90] & 0.788565 & 0.4085597 & 263.895852 \\ CLSCA [91] & 0.78806 & 0.40998 & 263.8964 \\ CS [92] & 0.78867 & 0.40902 & 263.9716 \\ GWO [73] & 0.79477 & 0.39192 & 263.987 \\ FOA [93] & 0.768316 & 0.4691685 & 264.229309 \\ CBA [94] & 0.00871 & 0.98083 & 265.5966 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Comparison results of the three-bar design problem.