# Improvement Technique for Group Search Optimization Using Experimental Design Method
## 2 Related Work
### Gso  
The GSO algorithm mainly simulates visual search mechanisms in a group of animals, e.g., the regional replication strategy and global search of sparrows. Assumedly sparrows are an example [40] to constitute an SIO algorithm. In the GSO algorithm, each individual in the group is called a member. The concept assumes that a group of members is searching for the location that contains the most food in the search space, i.e., the global solution. Initially, all members of the group are randomly scattered throughout the entire search space, but no members of the group know the best solution. Then, the members who are located in a good foraging area are used as producers and most of the members located in the good foraging area search for a better foraging area according to the PS model. To avoid being guided to the optimal local solution by most of the members, a small number of members are used to conduct random searches in the entire search space and to evaluate whether there is a better foraging area than the current producer, thereby guiding members to find the global solution in the search space. In an \(N\)-dimension search space, the \(i\)th member in the position of the \(k\)th iteration is denoted \(x_{i}^{k}\in R^{N}\). The angle is denoted \(\phi_{i}^{k}=\left(\phi_{i1}^{k},\cdots,\,\phi_{i(N-1)}^{k}\right)\in R^{N-1}\), and the search direction is defined as \(D_{i}^{k}\left(\varphi_{i}^{k}\right)=\left(d_{i1}^{k},\cdots,d^{k}\right)\in R^ {N}\), where \(\varphi_{i}^{k}\) can converted to \(d_{i}^{k}\) by Equations (1)-(3) [44]:  
\[d_{i1}^{k}=\prod\
olimits_{q=1}^{N-1}cos\left(\varphi_{iq}^{k}\right) \tag{1}\]  
\[d_{ij}^{k}=sin\left(\varphi_{i(j-1)}^{k}\right)\prod\
olimits_{q=1}^{N-1}cos \left(\varphi_{iq}^{k}\right),j=2,\ldots,N-1 \tag{2}\]  
\[d^{k}=sin\left(\varphi_{i(N-1)}^{k}\right) \tag{3}\]  
In each iteration, the member located at the position with the best fitness value is called the producer, which is denoted \(X_{p}\). According to a fixed ratio, the other members are distributed randomly to scroungers and rangers. Generally, 80% of members are scroungers, and the remaining 20% are rangers [45; 46]. Scroungers and rangers are, denoted \(X_{s}\) and \(X_{r}\), respectively. Figure 1 shows that, based on the current location, the producer randomly selects three angles to scan, and the three new positions can be obtained by Equations (4)-(6):  
1. Zero direction: \[X_{pz}=X_{p}^{k}+r_{1}l_{max}D_{p}^{k}\left(\varphi^{k}\right)\] (4)
2. Right-half side: \[X_{pr}=X_{p}^{k}+r_{1}l_{max}D_{p}^{k}\left(\varphi^{k}+r_{2}\theta_{max}/2\right)\] (5)
3. Left-half side: \[X_{pl}=X_{p}^{k}+r_{1}l_{max}D_{p}^{k}\left(\varphi^{k}-r_{2}\theta_{max}/2\right)\] (6)  
where \(\theta_{\max}\) and \(l_{\max}\) are the maximum pursuit angle and the maximum pursuit distance of the producer, respectively; \(\theta_{\max}\) and \(l_{\max}\in R^{1}\); \(\theta_{\max}\) is calculated from \(\pi/a^{2}\) where \(a=round\left(\sqrt{n+1}\right);l_{\max}\) is calculated from \(\sqrt{\sum_{i=1}^{n}\left(UB_{i}-LB_{i}\right)^{2}}\), where \(UB_{i}\) and \(LB_{i}\) are the upper and lower bounds of the \(i\)th dimension, respectively; \(r_{1}\) is a random variable of normal distribution between \(-1\) and \(1\), and \(r_{1}\in R^{1}\); and \(r_{2}\) denotes a random variable uniformly distributed between \(0\) and \(1\), and \(r_{2}\in R^{n-1}\). If the above search strategy finds a fitness value better than the current one, it replaces the current position of the producer as the best individual position. Otherwise, it remains at the current position and the following strategy uses the following Equation (7) to update its pursuit angle:  
\[\varphi^{k+1}=\varphi^{k}+r_{2}\alpha_{max} \tag{7}\]  
where \(\alpha_{\max}\) is the maximum rotation angle of the producer, \(r_{2}\) is a random variable of uniform distribution between \(0\) and \(1\), and \(r_{2}\in R^{n-1}\).