## 3 The Proposed Model
### Design of WMAO Algorithm  
[PERSON] et al. [19] introduced the fundamental formula of the Aquila optimizer (AO). Usually, the AO algorithm mimics the natural behavior of Aquila for catching the prey. Like other metaheuristic (MH) models, AO was a population-based optimization algorithm that takes place by creating an early population \(X\) that has \(N\) agents. In Equation (6), the mathematical modeling of the algorithm is given. \[X_{ij}=r_{1}\times\left(UB_{j}-LB_{j}\right)+LB_{j},\ i=1,2,\ldots Nj=1,2, \ldots,Dim \tag{6}\]  
\(UB_{j}\) and \(LB_{j}\) indicate parameters of the searching space, \(r_{1}\in[0,\ 1]\) denotes a random number, and \(Dim\) shows the dimension of the agent. Next, exploration and exploitation are performed until the optimal solution is found. To perform exploration and exploitation, two strategies are employed. The initial strategy performs the exploration based on the average of agents \((X_{M})\) and the finest agent \(X_{b}\), as follows:  
\[X_{i}(t+1)=X_{b}(t)\times\left(1-\frac{t}{T}\right)+(X_{M}(t)-X_{b}(t)\times rand), \tag{7}\]  
\[X_{M}(t)=\frac{1}{N}\sum_{i=1}^{N}X(t),\forall j=1,2,\ldots,Dim \tag{8}\]  
where \(\left(1-\frac{t}{T}\right)\) controls the search in the exploration stage, \(T\) indicates the maximal amount of iterations, and \(rand\) shows a random number within [0, 1]. Figure 2 showcases the flowchart of the AO method. The next strategy used to perform the Levy flight \((Levy(D))\) distribution and \(X_{b}\) for updating the exploration capability of the solution is:  
\[X_{i}(t+1)=X_{b}(t)\times Levy(D)+X_{R}(t)+(y-x)\times rand \tag{9}\]  
Figure 2: Flowchart of AO. \[Levy\left(D\right)=s\times\frac{u\times\sigma}{|v|^{\frac{1}{\beta}}},\sigma= \left(\frac{\Gamma(1+\beta)\times sine\left(\frac{\pi\beta}{2}\right)}{\Gamma \left(\frac{1+\beta}{2}\right)\times\beta\times 2\left(\frac{\beta-1}{2}\right)}\right) \tag{10}\]  
where \(s=0.01\) and \(\beta=1.5\), \(u\) and \(v\) denote random numbers, and \(\Gamma\) indicates a constant number. In Equation (9), \(X_{R}\) indicates an arbitrarily selected agent. Furthermore, \(y\) and \(x\) imitate the spiral shape, as given below:  
\[y=r\times\ \cos\theta,\ x=r\times\ \sin\theta \tag{11}\]  
\[r=r_{2}+U\times D_{1},\ \theta=-\omega\times D_{1}+\theta_{1},\ \theta=\frac{3 \times\pi}{2} \tag{12}\]  
where \(\omega=0.005\), \(U=0.00565\), \(r_{2}\in[0,20]\) represents a random number, and \(D_{l}\) indicates integer numbers from \(1\) to the length of the search domain. An initial strategy is used to upgrade agents inside the exploitation stage based on \(X_{b}\) and \(X_{M}\), like exploration as follows:  
\[X_{i}(t+1)=\left(X_{b}(t)-X_{M}(t)\right)\times\alpha-rand+\left(\left(UB-LB \right)\times rand+LB\right)\times\delta \tag{13}\]  
where \(\alpha\) and \(\delta\) signify the exploitation adjustment parameter and \(rand\in[0,\ 1]\) denotes a random number. In the next strategy of exploitation, the agent gets upgraded based on \(X_{b}\), _Levy_, and the quality function \(QF\):  
\[X_{i}(t+1)=QF\times X_{b}(t)-\left(G_{1}\times X(t)\times rand\right)-G_{2} \times Levy(D)+rand\times G_{1} \tag{14}\]  
\[X_{i}(t+1)=X_{b}(t)\times\left(1-\frac{t}{T}\right)+(X_{M}(t)-X_{b}(t)\times rand), \tag{7}\]0  
where \(rand\ ()\) denotes a function that produces a random number and \(G_{1}\) specifies distinct motions that are used to track the best individual solution, as follows:  
\[X_{i}(t+1)=X_{b}(t)\times\left(1-\frac{t}{T}\right)+(X_{M}(t)-X_{b}(t)\times rand), \tag{7}\]1  
where \(rand\) shows a random number. Given that, \(G_{2}\) specifies reducing value from \(2\) to \(0\), and it is evaluated by Equation (17):  
\[X_{i}(t+1)=X_{b}(t)\times\left(1-\frac{t}{T}\right)+(X_{M}(t)-X_{b}(t)\times rand), \tag{7}\]2  
Algorithm 1 illustrates the fundamental steps of the AO. ``` Input :  Describe the number of solutions \(N\), the overall number of iterations \(T\), and the dimension of every solution \(Dim\). Set the initial value for the parameter of \(AO\). Produce early population \(X\).