## 3 Proposed Algorithm
### Optimized Transfer Energy Distribution
Taking the first derivative of (6), i.e., \(\frac{\partial\mathcal{L}\{:\}}{\partial E_{t}^{rn}}\), the optimal decision is obtained for the prediction of energy transfer in a relay node after arriving at RTS and CTS packets. Then, the predicted best transfer energy at the relay nodes is computed as follows:  
\[E_{t}^{*rn}=\left(\psi-\frac{2^{2R}N_{0}}{|h_{r_{n},D}|^{2}d_{r_{n},D}^{-\alpha }}\right)^{+} \tag{7}\]  
where \(\psi=\frac{\mu}{2\ln 2(\lambda+\rho+T(1+\mu))}\). After this, this predicted \(E_{t}^{*\tau_{n}}\) is piggybacked in the SRT frame, forward through the engaging optimum node which has its backoff timer die earlier during the contention relay choice task. Additionally, the optimum-elected relay distributed energy should fulfill the energy criteria if the primary term is assigned to the range higher than the second term and \(\left(\cdot\right)^{+}=\min\bigl{(}E_{t}^{\tau_{n}},E_{t-max}\bigr{)}\), or else the criteria are not fulfilled. Subsequently, the origin determines its best transfer of energy \(E_{t}^{*S}\) as a factor of \(E_{t}^{*\tau_{n}}\). By considering the first-order derivative of (6) with respect to \(E_{t}^{S}\) i.e., \(\frac{\partial\mathcal{L}\{\cdot\}}{\partial E_{t}^{*}}\), get  
\[E_{t}^{*S}=\left(\zeta-\frac{E_{t}^{*\tau_{n}}|h_{\tau_{n},D}|^{2}d_{S,\tau_{ n}}^{-\alpha}}{|h_{S,\tau_{n}}|^{2}d_{\tau_{n},D}^{-\alpha}}\right)^{+} \tag{8}\]  
where \(\zeta=\frac{\mu}{2\ln^{2}(\lambda+\sigma+T(1+z))}\)  
The optimum distributed energy at the origin has to fulfill the criteria with \(\left(\cdot\right)^{+}=\min\bigl{(}E_{t}^{S},E_{t-max}\bigr{)}\). It creates the reliable and independent distribution of the direct transfer energy, so power efficacy is increased. The best decision of the asymmetric strategy in (7) and (8) are achieved iteratively through modifying the Lagrangian multipliers in (9) based on the Karush Kuhn-Tucker (KKT) criteria from (6) with \(\varepsilon\) decided to be a limited step size. Every possible supporter executes this process to acquire their optimum ranges in (7) and (8), which is applied to deciding the optimum supporter. \[\text{s.t}\ E_{t}^{S}+E_{t}^{rn}\leq E_{\text{overall}},\forall n\in N \tag{5a}\]0 \[\text{s.t}\ E_{t}^{S}+E_{t}^{rn}\leq E_{\text{overall}},\forall n\in N \tag{5a}\]1 \[\text{s.t}\ E_{t}^{S}+E_{t}^{rn}\leq E_{\text{overall}},\forall n\in N \tag{5a}\]2 \[\text{s.t}\ E_{t}^{S}+E_{t}^{rn}\leq E_{\text{overall}},\forall n\in N \tag{5a}\]3 \[\text{s.t}\ E_{t}^{S}+E_{t}^{rn}\leq E_{\text{overall}},\forall n\in N \tag{5a}\]4 \[\text{s.t}\ E_{t}^{S}+E_{t}^{rn}\leq E_{\text{overall}},\forall n\in N \tag{5a}\]5  
```
Begin  Initialize \(N,E_{t-max},\alpha,T,R,N_{0},E_{0}\), \(I\) (maximum iteration) and the Lagrange multipliers such that \(\lambda,\mu,z,u,\sigma,\rho\geq 0\); for(each node, N)  Create \(d_{S,\tau_{n}}\) and \(d_{\tau_{n},D}\) such that \(r_{n}\) is situated in the collaborative area from origin to the target such that \(d_{S,\tau_{n}}\
eq d_{\tau_{n}D}\) and \(E_{\tau_{n}}\);  Obtain \(E_{min}=\min\{E_{S},E_{t_{1}},\ldots,E_{\tau_{n}}\}\); for(eachiteration,1), Create \(h_{S,D},h_{S,\tau_{n}}\) and \(h_{\tau_{n},D}\) in a random manner; if\(\bigl{(}|h_{S,\tau_{n}}|^{2}\geq|h_{\tau_{n},D}|^{2},|h_{\tau_{n},D}|^{2} \geq|h_{S,D}|^{2},d_{S,\tau_{n}}\
eq d_{\tau_{n},D}\bigr{)}\)  Calculate and modify (7) & (8) by their suitable Lagrange multipliers in (9); Else  No possible supporters that fulfil (7); End if End for  Find the best \(E_{t}^{*\tau_{n}}\) and \(E_{t}^{*S}\); endfor; End
```  
**Algorithm 1** Refs. [10; 11]  
Thus, this ELQSSM-ML algorithm can reduce the energy use of each node in the network by optimizing the transmitted energy, and thus, the network lifespan is increased efficiently.