## 3 The Proposed Model
### Parameter Optimization  
Finally, the AO technique is utilized to fine-tune the parameters related to the BNN method so that the classification performance will be improved [24]. The AO technique simulates AO's social activity for catching their prey. AO is a population-based optimized approach, related to other meta-heuristic systems, which start with establishing a primary population \(X\) with \(N\) agents. The subsequent formula is utilized for executing this process. Figure 2 demonstrates the flowchart of the AO technique.  
\[X_{ij}=r_{1}\times\left(UB_{j}-LB_{j}\right)+LB_{j},\ i=1,2,\ Nj=1,2,\ \ldots,\ Dim \tag{9}\]  
In Equation (9), \(UB_{j}\) and \(LB_{j}\) demonstrate the restrictions of searching space. \(r_{1}\in[0,1]\) implies the arbitrary value and \(Dim\) denotes the dimensional of agents. In the AO approach, the following step is to perform exploration and exploitation until the optimal solution is established [25]. Two stages exist in exploitation and exploration. The \(X_{b}\) optimal agent and (\(X\)) the average of agents are applied in the exploration, and it is mathematically expressed in the following:  
\[X_{i}(t+1)=X_{b}(t)\times\left(\frac{1-t}{T}\right)+\left(X_{M}(t)-X_{b}(t)* rand\right), \tag{10}\]  
\[X_{M}(t)=\frac{1}{N}\sum_{i=1}^{N}X(t),\ \forall j=1,2,\ \ldots,\ Dim \tag{11}\]  
The exploration stage can be controlled using \((\frac{1-t}{T})\) in Equation (10). The maximal amount of generations is represented as \(T\). The exploration stage applies the Levy flight (\(Levy(D)\)) distribution as well as \(X_{b}\) to upgrade the solution as follows:  
\[X_{i}(t+1)=X_{b}(t)\times Levy(D)+X_{R}(t)+(y-x)*rand, \tag{12}\]  
\[Levy\ (D)=s\times\frac{u\times\sigma}{|v|^{\frac{1}{\beta}}},\sigma=\left( \frac{\Gamma(1+\beta)\times sine\left(\frac{\pi\beta}{2}\right)}{\Gamma\left( \frac{1+\beta}{2}\right)\times\beta\times 2^{\frac{\delta-1}{2}}\right)} \tag{13}\]  
In Equation (13), \(s=0.01\) and \(\beta=1.5\). \(u\) and \(v\) indicate the arbitrary values. \(X_{R}\) refers to arbitrarily selected agents. Additionally, \(y\) and \(x\) refer to two variables utilized for stimulating the spiral shape:  
\[y=r\times cos(\theta),\ x=r\times sin(\theta) \tag{14}\]  
\[r=r_{1}+U\times D_{1},\ \theta=-\omega\times D_{1}+\theta_{1},\ \theta 1=\frac{3 \times\pi}{2} \tag{15}\]  
In Equation (15), \(\omega=0.005\) and \(U=0.00565\). \(r_{1}\in[0,\ 20]\) stands for an arbitrary value. The initial method utilized to improve the agent in the exploitation stage depends on \(X_{b}\) and \(X_{M}\), comparable to exploration as follows:  
\[X_{i}(t+1)=\left(X_{b}(t)-X_{M}(t)\right)\times\alpha-rnd+\left(UB\times rnd+ LB\right)\times\delta \tag{16}\]  
In Equation (16), \(UB=\left(UB-LB\right)\), \(\alpha\), and \(\delta\) refer to the exploitation adjustment parameter. \(rnd\in[0,\ 1]\) denotes a random number. The agent is upgraded by \(X_{b}\), Levy, and the quality function \(QF\) in the next exploitation phase:  
\[X_{i}(t+1)=QP\times X_{b}(t)-GX-G_{2}\times Levy(D)+rnd\times G_{1} \tag{17}\]  
\[GX=\left(G_{1}\times X(t)\times rnd\right)\]\[QP(t)=t^{\frac{2\times rnd(\;(\;)-1\;)-1}{(1-T)^{2}}} \tag{18}\]  
Moreover, \(G_{1}\) denotes the motion applied to track the optimum individual solution, as follows:  
\[G_{1}=2\times md(\;)-1,\;G_{2}=2\times\left(1-\frac{t}{T}\right) \tag{19}\]  
In Equation (19), \(rnd\) refers to a random number. Furthermore, \(G_{2}\) indicates a variable decreased from two to zero as follow:  
\[G_{2}=2\times\left(1-\frac{t}{T}\right) \tag{20}\]  
The AO method derives a fitness function (FF) to exhibit the better performance of the classification. It defines a positive integer to characterize the improved performance of the solution candidate. In this study, the reduction of classification error rate was regarded as the FF, as follows.  
\[\begin{array}{rl}fitness(x_{i})&=ClassifierErrorRate(x_{i})\\ &=\frac{number\;of\;misclassified\;samples}{Total\;number\;of\;samples}*100\end{array} \tag{21}\]