## 3 Proposed MISAO
### Random Follower Search Strategy  
The original SAO method relied heavily on the initial member to determine the relocation of population members, resulting in a subsequent dependency. However, this reliance can ensnare the algorithm to a solution only optimal within a limited scope, impeding its ability to break free. This predicament emerges from the strong inclination of people in the population to solely adhere to the most superior individual at present without engaging in adequate diversity and investigation. In order to tackle this issue, the random follower search approach [36] has been implemented to improve the variety of individuals in population intelligence algorithms and facilitate a more comprehensive exploration of the solution space. The utilization of the random follower search approach facilitates the promotion of communication and collaboration among populations. By reducing the dependence on the current most successful individual to update the positions of the population members, the significance of exchanging knowledge and sharing experiences between individuals becomes vital. This compels individuals within the population to interact with one another more proactively, thereby promoting the exchange of knowledge and experience between them. Consequently, the algorithm's experience speeds up convergence, enhancing efficiency and problem-solving capability.  
The detailed mathematical formulation is given below:  
\[V_{j}=\begin{cases}\overset{-j}{s}_{i}^{j}+c_{1}\cdot(lb+c_{2}\cdot(ub-lb)),j \leq k\text{ and }rand>0.5\\ \overset{-j}{s}_{i}^{j}-c_{1}\cdot(lb+c_{2}\cdot(ub-lb)),j\leq k\text{ and }rand\leq 0.5\\ \left(\text{Temp}+\overset{-j}{s}_{i}^{j}\right)/2,\text{ otherwise}\end{cases} \tag{13}\]  
\[\overline{s}_{i}=s_{i}(pick) \tag{14}\]  
\[pick=random(dim) \tag{15}\]  
where the parameters \(ub\) and \(lb\) denote the upper and lower bounds of the exploration space of the ant colony search agent, respectively; \(\overset{-j}{s}_{i}^{j}\) denotes the \(j-th\) dimension of the \(i-th\) population member \(s_{i}\) after random combination; \(c_{1}\) is an adaptive step control parameter, which determines the path of the ant colony search agent's evolution to a certain extent; \(c_{2}\) is a random number in the range of \([-1,\,1]\); \(dim\) denotes the dimension of the exploration space of the feasible domain; the parameter \(k\) is a random integer with the value range of \([1,\,\text{dim}]\); and Temp denotes the current member of search agent exchange, and its corresponding mathematical model is as follows:  
\[\text{Temp}=\begin{cases}S_{best}^{j},i=1\\ \overset{j}{s}_{i-1}^{j},\text{otherwise}\end{cases} \tag{16}\]  
where \(S_{best}^{j}\) denotes the location of the \(j-th\) dimension currently evaluated as the best quality and \(\overline{s}_{i-1}^{j}\) denotes the \(j-th\) dimension of the \(i-1 st\) population member \(s_{i-1}\) after the random combination process. The adaptive step control parameter \(c_{1}\) is defined as:  
\[c_{1}=2\times e^{-\left(\frac{\overset{+}{\mathit{des}}}{\mathit{Max}_{ \mathit{des}}}\right)^{2}} \tag{17}\]  
The flowchart of the suggested MSAO is presented in Figure 3.