## 3 The Proposed Model
### Design of WMAO Algorithm
while (The ending criteria are not satisfied) do  Calculate the fitness value for every \(X_{i}\). Find the optimal individual \(X_{b}(t)\) for \((i=1,2\ldots,N)\) do  If \(t\leq\left(\frac{2}{3}\right)*T\)  Upgrade the \(X_{i}\) based on Equation (7). If the FF \((Fit)(X_{1}(t+1))<Fit(X(t))\) then \(X_{b}(t)=(X_{1}(t+1))\)In the WMAO algorithm, the mutation was considered a significant technique to support the technique's jump from local optimum [20]. In this phase, the wavelet mutating approach for enhancing the execution of the AO method is presented. Once the mutation probability \(P\) is set, every individual after the exploitation and exploration phase of the method will be getting a mutation chance via wavelet mutation strategy. Where \(rand<P\), an individual executes Morlet wavelet mutation. The formula of mutation is:

\[X_{i}^{new}(t)= \begin{cases}X_{i}(t)+\sigma(UB-X_{i}(t)),&rand<0.5\\ X_{i}(t)+\sigma(X_{i}(t)-LB),&rand\geq 0.5\end{cases}  \tag{18}\]

where \(X_{i}(t)(i=1,2,\ \cdots,\ N)\) denotes the \(i\)-th individual location in \(t\)-th generation, and \(LB\) and \(UB\) are the lower and upper bounds of the present search space. Correspondingly, \(\sigma\) represents the wavelet mutating coefficient. Its formula is [21]:

\[\sigma=\frac{1}{\sqrt{\alpha}}\psi\Big{(}\frac{\varphi}{\alpha}\Big{)} \tag{19}\]

where \(\psi(\varphi/\alpha)=e^{-\left(\varphi/\alpha\right)^{2}/2}\). \(\cos\left(5\varphi/\alpha\right)\) is the Morlet wavelet function, and 99% of its energy can be concentrated within \(-2.5\) and \(2.5\), so \(\varphi\) signifies an arbitrary number within \(-2.5\alpha\) and \(2.5\alpha\). The a is the scaling parameter and its expression is [22]:

\[\alpha=s\cdot(\frac{1}{s})^{(1-\frac{t}{r\max})} \tag{20}\]where \(s\) indicates a given constant. The mutant individual \(X_{i}^{new}\) is gained after the wavelet mutation function is completed, and greedy choice can be made among the original individual \(X_{i}\) and mutant individual \(X_{i}^{new}\), i.e.,

\[X_{i}(t+1)= \begin{cases}X_{i}^{new}(t),&f(X_{i}^{new}(t))\leq f(X_{i}(t))\\ X_{i}(t),&f(X_{i}^{new}(t))>f(X_{i}(t))\end{cases}  \tag{21}\]

This procedure assures that an individual with superior fitness will be entering the next iteration, thereby enhancing the convergence speed and optimization capability of the algorithm. ### Process Involved in the WMAO-EAR Technique

In the presented WMAO-EAR technique, an FF is derived using distinct constraints, such as delay, energy, distance, and security. The main concept of the proposed algorithm is to decrease the communication distance between the selected nodes and CH [23]. It also focuses on decreasing the delay to transmit the information from one node to another. In contrast, network energy has to be higher, i.e., during data transmission, it needs to employ a small amount of energy. Finally, the node must tolerate the risk attained in the network. The objective function of the adapted CH demonstrated in Equation (22), while the value of \(\eta\) needs to depend on \(0<\eta<1\). Now, \(v_{m}\) and \(v_{n}\) indicate the operation, as demonstrated in Equations (23) and (24). The constraints on delay, energy, distance, and security are indicated as \(\sigma_{1}\), \(\sigma_{2}\), \(\sigma_{3}\), and \(\sigma_{4}\). The condition of this constraint is indicated as \(\sigma_{1}+\sigma_{2}+\sigma_{3}+\sigma_{4}=1\). In Equation (24), \(Y^{Z}-S_{s}\) denotes the distance between the normal and sink nodes. \[N_{n}=\eta v_{n}+(1-\eta)v_{m} \tag{22}\]

\[v_{m}=\sigma_{1}*v_{i}^{dis}+\sigma_{2}*v_{mu}+\sigma_{3}*v_{\mu}{}^{\prime \prime}+\sigma_{4}*v_{\mu}{}^{\prime\prime\prime} \tag{23}\]

\[v_{n}=\frac{1}{b}\sum_{z=1}^{b}\|Y^{K}-S_{I}\| \tag{24}\]

Equation (25) illustrates the FF for distance, where \(di_{j}(m)\) is related to packet transmission from the normal nodes to CH and CH to \(BS\). Generally, \(v_{\mu}\) lies amongst \([0,\ 1]\) and the value goes high when the distance between the normal node and CH is higher.