## 3 Results and Analysis
### Computational Complexity Analysis of FAS-MMSBAS Algorithm  
The main improvement to the algorithms in this paper is that after the convergence of the AFS algorithm becomes slow, the improved BAS algorithm is used to continue to search for the optimal position. In the later stage, the optimization is changed from population optimization to individual optimization, which reduces the complexity of the algorithm. When the optimization problem dimension is D and the population number is N, the  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Function Name** & **Expression** \\ \hline Sphere & \(f_{1}=\sum\limits_{i=1}^{D}x_{i}^{2}\) \\ Rosenbrock & \(f_{2}=\sum\limits_{i=1}^{D-1}\left[100(x_{i+1}-x_{i}^{2})^{2}+(x_{i}-1)^{2}\right]\) \\ SchwefelP222 & \(f_{3}=\sum\limits_{i=1}^{D}|x_{i}|+\prod\limits_{i=1}^{D}|x_{i}|\) \\ Quartic & \(f_{4}=\sum\limits_{i=1}^{D}\left(ix_{i}^{4}\right)+random[0,1)\) \\ Rastrigin & \(f_{5}=\sum\limits_{i=1}^{D}\left[x_{i}^{2}-10\text{cos}(2\pi x_{i})+10\right]\) \\ Griewank & \(f_{6}=\frac{1}{400}\sum\limits_{i=1}^{D}x_{i}^{2}-\prod\limits_{i=1}^{D}\left[ cos\left(\frac{x_{i}}{\sqrt{i}}\right)\right]+1\) \\ Ackley & \(f_{7}=-20 exp\left(-0.2\sqrt{\frac{1}{\sum\limits_{i=1}^{D}x_{i}^{2}}}\right)- exp\left(\frac{1}{\sum\limits_{i=1}^{D}cos(2\pi x_{i})}\right)+20+e\) \\ Levy and Montalvo 2 & \(f_{8}=0.1\left\{sin^{2}(3\pi x_{1})+\sum\limits_{i=1}^{D}(x_{i}-1)^{2}[1+sin^{ 2}(3\pi x_{i}+1)]+(x_{D}-1)^{2}[1+sin^{2}(2\pi x_{D})]\right\}\) \\ [PERSON] & \(f_{9}=-\sum\limits_{i=1}^{D}x_{i}sin\left(\sqrt{|x_{i}|}\right)\) \\ [PERSON] & \(f_{11}=-\sum\limits_{i=1}^{D}\left(x_{i}^{2}+x_{i+1}^{2}\right)^{0.25}\left[ sin^{2}\left(50\left(x_{i}^{2}+x_{i+1}^{2}\right)^{0.1}\right)+1\right]\) \\ [PERSON] & \(f_{10}=\sum\limits_{i=1}^{D}\left(floor(x_{i}+0.5)\right)^{2}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Expressions of benchmark test function; this table is a supplement to Table 1.  
complexity analysis of the FAS-MMSBAS algorithm is as follows: the number of iterations of the AFS algorithm is t\({}_{1}\), the number of iterations of the MMSBAS algorithm is t\({}_{2}\), and the total number of iterations is T = t\({}_{1}\) + t\({}_{2}\). In the optimization stage of the AFS algorithm, the computational complexity of initializing the artificial fish group is O(DN). In the iteration process, the computational complexity of clustering behavior is O(DN\({}^{2}\)), and that of chasing behavior is O(DN\({}^{2}\)); thus, the total complexity of this stage is O(DN) + O(2 DN\({}^{2}\))t\({}_{1}\). In the iteration process of the MMSBAS algorithm, the number of multi-directional detection is c\({}_{1}\), the number of multi-step detection is 3, the computational complexity of the multi-directional multistep detection strategy is O(3c\({}_{1}\)D), the computational complexity of the mutation strategy is O(D), and the total computational complexity is O(c\({}_{2}\)D)t2, c1 \(<\) c2. Combined with the two stages, the total computational complexity of the algorithm can be approximated as O(DN\({}^{2}\))t\({}_{1}\) + O(c\({}_{2}\)D)t\({}_{2}\).  
It can be seen from Table 3 that when the number of iterations is the same, the order of algorithm complexity from low to high is BAS \(<\) AFS-DBAS \(<\) AFS-MMBAS \(<\) AFS.