## 4 Conclusions  
In view of the advantages and disadvantages of the AFS algorithm and BAS algorithm, this paper proposes a hybrid algorithm, AFS-MMSBAS, which combines the artificial fish swarm and improved beetle antenna search algorithms. This hybrid algorithm involves the use of a mutation strategy to increase the probability of a beetle escaping the local extreme value and altering its course to a better direction. It also proposes a multi-step detection strategy to improve the convergence speed of the algorithm by adaptively changing the step size. Finally, a simple method is used to combine the AFS algorithm with the improved BAS algorithm. In order to test that the AFS-MMSBAS algorithm is superior to its two constituent algorithms, the AFS-MMSBAS algorithm is compared with the AFS, BAS, and AFS-BAS algorithms at different dimensions. In order to further verify the advantages of  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Fence.**} & \multirow{2}{*}{**Algorithm**} & \multicolumn{3}{c}{**D = 10**} & \multicolumn{3}{c}{**D = 200**} \\ \cline{3-11}  & & & & & & & & & & & \\ \hline \multirow{3}{*}{\(f_{1}\)} & ADSAFS-PSO & 58.5 \(\times\) 10\({}^{-17}\) & 2.1 \(\times\) 10\({}^{-16}\) & 0 & 66.6 \(\times\) 10\({}^{-16}\) & 2862 & 3.14 \(\times\) 10\({}^{3}\) & 7.39 \(\times\) 10\({}^{3}\) & 10.4 & 2.29 \(\times\) 10\({}^{3}\) & 58.4 \\  & AFS-AG & **1.84 \(\times\) 10\({}^{-16}\)** & **3.84 \(\times\) 10\({}^{-16}\)** & 6.65 \(\times\) 10\({}^{-16}\) & 1816 & 1.96 \(\times\) 10\({}^{3}\) & 2.44 \(\times\) 10\({}^{3}\) & 8.5 & 8.31 \(\times\) 10\({}^{3}\) & 42.1 \\  & AFS-AG & 2.73 \(\times\) 10\({}^{-18}\) & 8.25 \(\times\) 10\({}^{-18}\) & 3.27 \(\times\) 10\({}^{-18}\) & 2.62 \(\times\) 10\({}^{-17}\) & **63.97** & **25.3** & **7.64** & 14.2 & **38.57** & **246.7** \\ \hline \multirow{3}{*}{\(f_{1}\)} & ADSAFS-PSO & 9.25 \(\times\) 10\({}^{-15}\) & 1.2 \(\times\) 10\({}^{-14}\) & 4.27 \(\times\) 10\({}^{-16}\) & 4.21 \(\times\) 10\({}^{-16}\) & 3003 & 42.0 & 98.5 & 3.05 & 3.22 \(\times\) 10\({}^{3}\) & 49.4 \\  & AFS-AG & **8.00 \(\times\) 10\({}^{-18}\)** & **1.42 \(\times\) 10\({}^{-18}\)** & **1.84 \(\times\) 10\({}^{-14}\)** & **6.33 \(\times\) 10\({}^{-18}\)** & 20.3 &this algorithm in dealing with high-dimensional problems, the AFS-MMSBAS algorithm is compared with similar algorithms, namely, AAFSA-GE and ADSAFS-PSO. The experimental results show that the AFS-MMSBAS algorithm can solve the problems of the poor optimization and instability of the AFS and BAS algorithms at high dimensions, and that it has a faster convergence speed during the later period. In a word, the AFS-MMSBAS algorithm performs well in high-dimensional problem processing.  
The algorithm proposed in this paper performs well for high-dimensional problem optimization but performs poorly for some function problems, such as step functions. In addition, the algorithm has poor ability to solve low-dimensional problems and has great room for improvement. Therefore, in the future work, the authors plan to improve the algorithm, specifically with respect to its ability to solve high-dimensional partial function optimization and deal with low-dimensional problems. Further, it will be applied to engineering practices.  
[PERSON] and [PERSON] proposed the idea of the paper. [PERSON] and [PERSON] helped manage the annotation group and helped clean the raw annotations. [PERSON] conducted all experiments and wrote the manuscript. [PERSON], [PERSON] and [PERSON] revised and improved the text. [PERSON] and [PERSON] are the people in charge of this project. All authors have read and agreed to the published version of the manuscript.  
This work was supported by the Handan Science and Technology Research and Development Program (19422091008-35) and the Hebei Science and Technology Program (21350101D).  
Not applicable.  
The authors declare no conflict of interest.