## 3 Proposed ATMALS-GSK Algorithm
### Gsk  
The GSK algorithm begins with a randomly generated population where individuals, seen as \"juniors,\" lack prior experience. As the optimization evolves, these individuals interact with their environment. Through these interactions, they gain knowledge and eventually reach a \"senior\" stage, where they can share what they have learned with others, enhancing the overall intelligence of the population. This learning process is divided into the junior phase (beginner-intermediate) and the senior phase (intermediate-expert). During the search process, individuals (i.e., solutions) continuously interact and influence one another through cooperation and competition. In the early stage of life, knowledge is mostly acquired from close, familiar networks, while the ability to judge others is still underdeveloped. As individuals engage with broader social and professional networks, they gain diverse experiences and develop the capacity to assess, categorize, and share knowledge with others. This transition enhances both personal learning and the collective performance of the population. The mathematical formulation of this process is detailed below. The overall pseudocode of GSK can be seen in Algorithm 1. Let \(x_{j},\forall j=1,2,3,\ldots N\) represent feasible solutions (i.e., individuals), where \(N\) is the population size of the algorithm. Each solution \(x_{j}\) is characterized using a vector \(x_{ij}=\left(x_{j1},x_{j2},\ldots,x_{jD}\right)\), where \(D\) denotes the number of knowledge domains or dimensions assigned to that person, i.e., decision variables. The corresponding fitness value for each individual \(j\) is given by \(f_{j}\)\(\forall j=1,2,3,\ldots N\). The vector representation of an individual \(j\) during the junior phase (early-middle stage) is illustrated in Figure 1(a), while its representation during the senior phase (middle-later stage) is depicted in Figure 1(b). Figure 2 illustrates that in the junior gaining and sharing phase (early-middle stage), more dimensions of each vector are updated compared to the senior phase. In contrast, during the senior phase (middle-later stage), the senior scheme updates more dimensions than the junior scheme. The number of dimensions updated in both phases depends on the knowledge rate, which controls how much knowledge is transferred between generations. Figure 2: Solution representation of the GSK algorithm during junior and senior phases. The specific number of updated dimensions in each phase is determined using a nonlinear formula that adjusts over time, which can be formulated as follows:  
\[D(Junior)=ProbSize\times\left(1-\frac{G}{MaxGen}\right)^{K} \tag{1}\]  
\[D(Senior)=ProbSize-D(Junior)=ProbSize\left(1-\left(1-\frac{G}{MaxGen}\right)^{K}\right) \tag{2}\]  
where \(K\) (\(K\) > 0) is a real number denoting the knowledge rate, \(ProbSize\) is the problem size (dimensions), \(G\) is the iteration number, and \(MaxGen\) is the maximum number of iterations. ```
Inputs: - \(N\): Population size - \(D\): Number of dimensions (decision variables) - \(K\): Knowledge rate (controls the volume of knowledge exchange) - \(K_{f}\): Knowledge factor (controls intensity of learning) - \(K_{f}\): Knowledge ratio (probability of applying knowledge update) - \(MaxGen\): Maximum number of generations - \(P\): Proportion for grouping in senior phase Output: Global best solution (\(X_{best}\))
1 Initialize generation counter G \(\gets\) 0
2 Generate an initial population of N individuals:
3 for1 each \(i\) = 1:\(N\)
4 Initialize individual \(x_{i}\) randomly in \(D\)-dimensional space
5 Evaluate fitness \(f\) (\(x_{i}\)) for each individual \(i\)
6 for2 each generation \(G\) = 1 to \(MaxGen\):
7 Compute knowledge-sharing dimensions for both phases
8 Determine \(D(Junior)\) and \(D(Senior)\) using Equations (1) and (2).