## 3 Proposed Method
### Improved SMA
The proposed DQOBLSMA adds two different mechanisms to the original SMA. To verify their respective effects, they are compared when separated. Different combinations between SMA and two mechanisms are listed below:

* SMA combined with DQRG and OBL (DQOBLSMA);
* SMA combined with DQRG (DQSMA);
* SMA combined with OBL(OBLSMA);
* Original SMA;

Table 4 gives the comparison results between the original SMA and the improved algorithm after adding the mechanism. The ranking of the four algorithms is given at the end of the table, and it can be seen that the first-ranked algorithm is the DQOBLSMA. This ranking was obtained using the [PERSON] ranking test [53] and reveals the overall performance rankings of the compared algorithms against the tested functions. In these cases, the ranking from best to worst was roughly as follows: DQOBLSMA \(>\) OBLSMA \(>\) SMA \(>\) DQSMA. With the addition of both mechanisms, the performance of the DQOBLSMA is more stable, and the global search capability is much improved. When comparing DQSMA with OBLSMA, we can see that OBLSMA is much stronger than DQSMA, indicating that the contribution of OBL to the performance of SMA is more significant than the contribution of DQRG to the performance of SMA. When comparing DQSMA with SMA, we can see that DQSMA becomes worse on unimodal functions but stronger on most multimodal and fixed-dimensional multimodal functions than the original SMA in terms of optimization. [PERSON]'s rank-sum test was used to verify the significance of the DQOBLSMA against the original SMA and SMA with the addition of one mechanism. The results are shown in Table 5. Based on these results and those in Table 4, the DQOBLSMA outperformed SMA on 13 benchmark functions, DQSMA on 17 benchmark functions, and OBLSMA on 8 benchmark functions. Thus, the DQOBLSMA algorithm proposed in this paper combines DQRG with OBL. Although DQSMA and OBLSMA can both find the solutions, there are more benefits to be gained by combining the two strategies. In conclusion, the DQOBLSMA offers better optimization performance and is significantly better than SMA, DQSMA, and OBLSMA. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multicolumn{1}{c}{**DQORISMA**} & \multicolumn{2}{c}{**DQSMA**} & \multicolumn{2}{c}{**OBLSMA**} & \multicolumn{2}{c}{**SMA**} \\ \cline{2-10} \multicolumn{1}{c}{**Function**} & \multic



### Benchmark Function Experiments

As seen from Table 6, on unimodal benchmark functions (F1-F7), the DQOBLSMA can achieve better results than other optimization algorithms. For F1, F3, and F6, the DQOBLSMA could find the theoretical optimal value. For all unimodal functions, the DQOBLSMA obtained the smallest mean values and standard deviations compared to other algorithms, showing the best accuracy and stability. From the results shown in Tables 7 and 8, the DQOBLSMA outperformed the other algorithms for most of the multimodal and fixed-dimensional multimodal functions. For the multimodal functions F8-F13, the DQOBLSMA obtained almost all the best mean and standard deviation values, and obtainedthe global optimal solution for four functions (F8-F11). As shown in Table 8, the DQOBLSMA obtained theoretically optimal values in 8 of the 10 fixed-dimensional multimodal functions (F14-F23). Although the DQOBLSMA did not outperform JADE in F14-F23, it exceeded ESMA and OBLSMAL in overall performance. These results show that the DQOBLSMA also provides powerful and robust exploitation capabilities. In addition, Table 9 presents [PERSON]'s rank-sum test results to verify the significant differences between the DQOBLSMA and the other five algorithms. It is worth noting that \(p\)-values less than 0.05 mean significant differences between the respective pairs of compared algorithms. The DQOBLSMA outperformed all other algorithms to varying degrees, and outperformed OBLSMAL, ESMA, MEO, JADE, and RWGWO, on 14, 15, 16, 15, and 18 benchmark functions, respectively.