## 3 Optimization Methodology
### Archimedes Optimization Algorithm
It reduces with increasing time according to (51):  
\[\mathbf{d^{t+1}}=exp(\frac{t_{max}-t}{t_{max}})-(\frac{t}{t_{max}}) \tag{51}\]where \(\mathbf{d^{t+1}}\) reduces with increasing time that provides the capability to converge in the previously specified promising zone. To guarantee a balance between the exploration and the exploitation in the proposed AOA, this variable must be handled appropriately. The text continues here. * Exploration step (colliding among objects happens). If \(\mathbf{TF}\leq 0.5\), colliding among objects happens, an arbitrary material (\(\mathbf{mr}\)) must be nominated and the acceleration of for repetition \(t+1\) according to (52) must be modernized: \[\mathbf{acc_{i}^{t+1}=\frac{den_{mr}+vol_{mr}\times acc_{mr}}{den_{i}^{t+1}\times vol _{i}^{t+1}}}\] (52) where \(\mathbf{acc_{i}}\), \(\mathbf{den_{i}}\), and \(\mathbf{vol_{i}}\) are the acceleration, the density, and the volume of the object \(\mathbf{I}\), whereas \(\mathbf{acc_{mr}}\), \(\mathbf{vol_{mr}}\) and \(\mathbf{den_{mr}}\) are the acceleration, the volume, and the density of arbitrary material. It is significant to indicate that \(\mathbf{TF}\leq 0.5\) guarantees exploration through one third of repetitions. Using a value other than 0.5 will affect the behavior of changing from exploration to exploitation steps. * Exploitation step (no colliding among objects). If \(\mathbf{TF}>0.5\), there is no colliding among objects, modernize the acceleration of the object for repetition (\(\mathbf{t+1}\)) according to (53): \[den_{i}=rand(i,Dim) \tag{45}\]0 (53) where \(\mathbf{acc_{best}}\) refers to the best object acceleration. * Normalize the object acceleration. Normalize the object acceleration to compute the percentage of variation according to (54): \[den_{i}=rand(i,Dim) \tag{45}\]1 (54) where \(\mathbf{l}\) and \(\mathbf{u}\) represent the scope of normalization and put it at 0.1 and 0.9, respectively. The \(\mathbf{acc_{i}^{t+1}-norm}\) calculates the percentage of the period that every agent will alteration. The value of acceleration will be great when the object is away from the global optimum, which means that the object will be in the exploration stage; other than that, it will be in the exploitation stage. This clarifies how the inspection modifies from the exploration stage to the exploitation stage. In an ordinary case, the factor of acceleration initiates with high value and reduces with increasing time. This aids search agents to move away from local solutions and at the same time transfer towards the global best solution. However, it is significant to state that there may still a small number of search agents that require extra time to stay in the exploration stage than in the normal case. Therefore, the proposed AOA attains the equilibrium between the exploration stage and the exploitation stage. * Modernize location If \(\mathbf{TF}\leq 0.5\) (exploration stage), the \(\mathbf{ith}\) object's location for following repetition \(t+1\) according to (55) \[den_{i}=rand(i,Dim) \tag{45}\]2 (55) where \(C_{1}\) referes to a constant that equals 2. Other than that, when \(\mathbf{TF}>0.5\)(exploitation stage), the objects modernize their locations according to (56). \[den_{i}=rand(i,Dim) \tag{45}\]3 (56) where \(C_{2}\) referes to a constant that equals 6. T rises with increasing time and it is proportional to transfer factor and it is determined according to \(\mathbf{T=C_{3}\times TF}\). Additionally, it rises with increasing time through the scope [\(\mathbf{C_{3}\times 0.3}\), 1] and it possesses a particular percentage from the best location, at first. It begins with small percentage which causes a huge difference between the best location and the present location; consequently, the random walk step-size will be big. As the search continues, this percentage will rise progressively to reduce the difference between the best location and the present location. This results in an appropriate equilibrium between the exploration and the exploitation.