## 3 Optimization Methodology
### Archimedes Optimization Algorithm  
Steps of AOA Algorithm  
The mathematical construction for the algorithm of AOA is presented in this sub-section. Theoretically, AOA is deemed as a universal algorithm where it involves both exploitation and exploration procedures. The pseudocode of the AOA is indicated in Algorithm 1; it includes the preliminary population of objects, population assessment, and modernizing parameters. Mathematically, the stages of the suggested AOA are indicated as follows:  
1 **Preparation** Set the locations of overall objects using (44):  
\[O_{i}=lb_{i}+rand(i,Dim)\times(ub_{i}-lb_{i}),i=1,2,\ldots,N \tag{44}\]  
where \(O_{i}\) refers to the ith object from the population that have \(N\) (search agents) objects. \(ub_{i}\) and \(lb_{i}\) are the higher and lower limits of the search scope, respectively. Dim represents the dimension variables. Set the initial value of density (_den_) and volume (\(\
u o\)) for every ith object according to Equations (45) and (46). \[den_{i}=rand(i,Dim) \tag{45}\]  
\[vol_{i}=rand(i,Dim) \tag{46}\]  
where rand refers to a random number within [0,1]. Finally, set the initial value of ith object acceleration (_acc_) using (47):  
\[acc_{i}=lb_{i}+rand(i,Dim)\times(ub_{i}-lb_{i}) \tag{47}\]  
Figure 6: Some objects dipped in the same liquid [45]. In this step, assess preliminary population and nominate the object that has the best fitness value. Specify, the best location (\(\mathbf{x_{best}}\)), the best density (\(\mathbf{den_{best}}\)), the best volume (\(\mathbf{v_{best}}\))), and the best acceleration (\(\mathbf{acc_{best}}\)). **Modernize volumes and densities** The volume and the density for every object \(\mathbf{i}\) at the repetition (\(\mathbf{t+1}\)) is modernized according to (48) and (49):  
\[\mathbf{vol_{i}^{t+1}}=\mathbf{vol_{i}^{t}+rand\times(\mathbf{vol_{best}-vol_{i}^{t}})} \tag{48}\]  
\[\mathbf{den_{i}^{t+1}}=\mathbf{den_{i}^{t}+rand\times(\mathbf{den_{best}-den_{i}^{t}})} \tag{49}\]  
where \(\mathbf{vol_{best}}\) is the volume correlated to the best object that has been obtained so far, and \(rand\) is a random number that is distributed uniformly. **Modernize volume and density for every object according to (49)**  
Modernize the factors of transfer and decreasing of density \(\mathbf{TF}\) and \(\mathbf{d}\) according to (50) and (51), respectively. **If \(\mathbf{TF\leq 0.5}\) then Exploration stage**  
Modernize the object acceleration according to (52) and normalize this acceleration according to (54)  
Modernize the object location according to (55)  
**else**  
Exploitation stage  
Modernize the object acceleration according to (53) and normalize this acceleration according to (54)  
Modernize direction flag \(\mathbf{F}\) according to (57)  
Modernize the object location according to (56)  
**end if**  
**end for**  
Assess every object and nominate one of them that has best fitness significance. Set \(\mathbf{t=t+1}\)  
**End while**  
**return** object that has best fitness significance  
**end Procedure**  
In the AOA algorithm, the population objects (search agents) are searching for the best promising area in all of the search space by the exploration phase and then searching for the best location (best object) in this promising area by the exploitation phase. \(\mathbf{TF}\) is a factor that is changing with iteration to transfer the algorithm from the exploration phase to the exploitation phase through the simulation time, and can be evaluated as follows:  
\[\mathbf{TF=exp(\frac{t-t_{max}}{t_{max}})} \tag{50}\]  
where the \(\mathbf{TF}\) factor rises progressively with increasing time till up to 1; \(t_{max}\) and \(\mathbf{t}\) are the maximum repetitions number and repetition number, respectively. Likewise, density decreasing factor d also supports the proposed AOA on universal to local inspection.