## 3 Proposed Method
### Improved SMA
The four variables studied include the width of the shell (\(T\)s), the width of the head (\(T\)h), the inner radius (\(R\)), and the length of the cylindrical section (\(L\)), as shown in Figure 3.9 The formulation of objective functions and four optimization constraints can be described as follows:

Consider:

\[\mathbf{x}=[x_{1}\ x_{2}\ x_{3}\ x_{4}]=[h\ l\ t\ b];\]0

minimize:

\[\mathbf{x}=[x_{1}\ x_{2}\ x_{3}\ x_{4}]=[h\ l\ t\ b];\]1

subject to:

\[\mathbf{x}=[x_{1}\ x_{2}\ x_{3}\ x_{4}]=[h\ l\ t\ b];\]2 \[\mathbf{x}=[x_{1}\ x_{2}\ x_{3}\ x_{4}]=[h\ l\ t\ b];\]3 \[\mathbf{x}=[x_{1}\ x_{2}\ x_{3}\ x_{4}]=[h\ l\ t\ b];\]4 \[\mathbf{x}=[x_{1}\ x_{2}\ x_{3}\ x_{4}]=[h\ l\ t\ b];\]5 \begin{table}
\begin{tabular}{l c c c c} \hline \hline \multirow{2}{*}{**Algorithm**} & \multicolumn{4}{c}{**Optimal Values for Variables**} & \multirow{2}{*}{**Optimal Cost**} \\ \cline{2-2} \cline{4-5}  & \(d\) & \(D\) & \(N\) & \\ \hline DQOBLSMA & 0.050000 & 0.317425 & 14.028013 & 0.012719 \\ OBLSMAL & 0.05000 & 0.317409 & 14.030650 & 0.012721 \\ ESMA & 0.051458 & 0.353086 & 12.050995 & 0.012739 \\ SMA & 0.050000 & 0.317317 & 14.042338 & 0.012726 \\ MEO & 0.057203 & 0.514683 & 7.661607 & 0.014002 \\ JADE & 0.055015 & 0.442128 & 7.613118 & 0.012864 \\ RWGWO & 0.056389 & 0.480684 & 6.712235 & 0.013316 \\ \hline \hline \end{tabular}
\end{table} 
Table 12: Comparison for the tension/compression spring design problem. Figure 4: Tension/compression spring design problem. range of variables:

\[0\leq x_{1}\leq 99,0\leq x_{2}\leq 99,10\leq x_{3}\leq 200,10\leq x_{4}\leq 200\]

Table 13 shows how the DQOBLSMA compares with other competitor algorithms. The results shows the DQOBLSMA is able to find the optimal solution at the lowest cost, obtaining an optimal spend of 5885.623524 when _Ts_ = 0.778246, _Th_ = 0.384708, \(R\) = 40.323469, and \(L\) = 199.950065. ## 6 Conclusions

In this paper, an enhanced SMA (DQOBLSMA) was proposed by introducing two mechanisms, DQRG and OBL, into the original SMA. In the DQOBLSMA, these two strategies further enhance the global search capability of the original SMA: DQRG enhances the exploration capability of the original SMA, and OBL increases the population diversity. The DQOBLSMA overcomes the weaknesses of the original search method and avoids premature convergence. The performance of the proposed DQOBLSMA was analyzed by using 23 classical mathematical benchmark functions. First, the DQOBLSMA and the individual combinations of these two strategies were analyzed and discussed. The results showed that the proposed strategies are effective, and SMA achieved the best performance with the combination of the two mechanisms. Secondly, the results of the DQOBLSMA were compared with five state-of-the-art algorithms ESMA, AEO, JADE, OBLSMAL, and RWGOV. The results show that the DQOBLSMA is competitive with other advanced metaheuristic algorithms. To further validate the superiority of the DQOBLSMA, it was applied to three industrial engineering design problems. The experimental results show that the DQOBLSMA also achieves better results when solving engineering problems and significantly improves the original solutions. As a future perspective, a multi-objective version of the DQOBLSMA will be considered. The proposed algorithm has promising applications in scheduling problems, image segmentation, parameter estimation, multi-objective engineering problems, text clustering, feature selection, test classification, and web applications.