## 2 Related Works
### Archimedes Optimization Algorithm for Optimizing VME
#### 2.2.2 Archimedes Optimization Algorithm  
The AOA, a new heuristic algorithm proposed by [PERSON] et al. [45], is designed for complex problems prone to local optimal solutions. In this algorithm, each individual is represented as an immersed object, and its acceleration is updated based on collisions with neighboring objects. The individual's new position is determined by considering factors such as density, volume, and acceleration. The initial location of an individual is defined as follows:  
\[\mathrm{O_{i}}=\mathrm{lb_{i}}+\mathrm{r}\times(\mathrm{ub_{i}}-\mathrm{lb_{i}}) \tag{13}\]  
In Equation (13), \(\mathrm{lb_{i}}\) and \(\mathrm{ub_{i}}\) denote the lower and upper bounds of the search range, and r denotes a random number in the range [0,1]. Acceleration, density, and volume are initialized as follows:  
\[\left\{ \begin{array}{l}\mathrm{acc_{i}}=\mathrm{lb_{i}}+\mathrm{r}\times( \mathrm{ub_{i}}-\mathrm{lb_{i}})\\ \mathrm{den_{i}}=\mathrm{r}\\ \mathrm{vol_{i}}=\mathrm{r}\end{array} \right. \tag{14}\]  
In this step, the initial population is evaluated, and the individual with the best fitness is selected and assigned \(\mathrm{acc_{best}}\), \(\mathrm{den_{best}}\), and \(\mathrm{vol_{best}}\). The updated density and volume are as follows:  
\[\left\{ \begin{array}{l}\mathrm{den_{i}^{t+1}}=\mathrm{den_{i}^{t}+r\times \left(\mathrm{den_{best}}-\mathrm{den_{i}^{t}}\right)}\\ \mathrm{vol_{i}^{t+1}}=\mathrm{vol_{i}^{t}+r\times\left(\mathrm{vol_{best}}- \mathrm{vol_{i}^{t}}\right)}\end{array} \right. \tag{15}\]  
In Equation (15), t represents the current iteration, \(\mathrm{den_{best}}\) and \(\mathrm{vol_{best}}\) represent the best individual density and volume found so far, and \(\mathrm{acc_{best}}\) denotes the current individual optimal acceleration. The transition from collision to balance between individuals is controlled by the balance factor TF. This signifies the transition from the exploration phase to the exploitation phase, as follows:  
\[\mathrm{TF}=\mathrm{exp}\bigg{(}\frac{\mathrm{t}-\mathrm{t_{max}}}{\mathrm{t_ {max}}}\bigg{)} \tag{16}\]  
In Equation (16), \(\mathrm{t_{max}}\) represents the maximum number of iterations. The density factor d determines the position state of the individual, which helps the AOA search from global to local optima:  
\[\text{d}^{t+1}=\text{exp}(\frac{t_{\text{max}}-t}{t_{\text{max}}})-(\frac{t}{t_{ \text{max}}}) \tag{17}\]  
The individual acceleration at iteration \(t+1\) is updated as follows:  
\[ \begin{cases}\text{acc}_{i}^{t+1}=\frac{\text{den}_{\text{m}r}+\text{vol}_{ \text{m}r}\times\text{acc}_{\text{out}}}{\text{den}_{i}^{t+1}\times\text{vol} _{i}^{t+1}}&\text{TF}\leq 0.5\\ \text{acc}_{i}^{t+1}=\frac{\text{den}_{\text{m}r}+\text{vol}_{\text{m}r}\times \text{acc}_{\text{out}}}{\text{den}_{i}^{t+1}\times\text{vol}_{i}^{t+1}}&\text{ TF}>0.5\end{cases}  \tag{18}\]  
In Equation (18), \(\text{den}_{\text{m}r}\), \(\text{vol}_{\text{m}r}\), and \(\text{acc}_{\text{m}r}\), denote the density, volume, and acceleration, respectively, of a random individual. Acceleration within the algorithm is normalized to facilitate optimal search behavior. When the target is distant from the global optimum, the acceleration is set at a high value, indicative of the exploration stage. Conversely, when the target nears the global optimum, the acceleration is reduced, signifying the transition to the exploitation stage:  
\[\text{acc}_{i-\text{norm}}^{t+1}=\text{u}\times\frac{\text{acc}_{1}^{t+1}- \text{min}(\text{acc})}{\text{max}(\text{acc})-\text{min}(\text{acc})}+1 \tag{19}\]  
In Equation (19), u and l are the standardization parameters and are set to 0.9 and 0.1, respectively. The individual position at iteration \(t+1\) is updated as follows:  
\[