## 4 Proposed CB-FBC and Genetic Approach
### Optimization of \(P_{tj_{i}S}(t)\) and \(B_{Sj_{i}}(t)\)  
After obtaining the values of \(x_{j_{i}}^{*}(t)\) and distributing the aerials among the selected \(A(t)\), the values of \(\Psi_{j_{i}}(t)\) become known. Then, in the next step, the values of \(P_{tj_{i}S}(t)\) and \(B_{Sj_{i}}(t)\) should be optimized to accommodate the values of \(\Psi_{j_{i}}(t)\) related to every aerial under constraints C6 and C7 in (7). This optimization problem can be formulated as follows:  
\[\underset{P_{tj_{i}S}(t),B_{Sj_{i}}(t)}{\text{max}} \left(\frac{1}{T}\sum_{t=1}^{T-1}\sum_{j=1}^{N}\min\left(\eta_{Sj_{i}}(t), \ \Psi_{j_{i}}(t)\right)\right.\] s.t \[\left.C6:\sum_{j=1}^{N}B_{Sj_{i}}(t)\leq B_{Smax}\right. \tag{11a}\] \[\left.C7:\right. \left.P_{tj_{i}S}(t)\leq P_{Vmax}\right. \tag{11b}\]  
This optimization problem can be further simplified by reformulating it as minimizing the absolute difference between \(\eta_{Sj_{i}}(t)\) and \(\Psi_{j_{i}}(t)\) via optimizing the control parameters \(P_{tj_{i}S}(t)\) and \(B_{Sj_{i}}(t)\), which can be written as follows:  
\[\underset{P_{tj_{i}S}(t),B_{Sj_{i}}(t)}{\text{min}} \sum_{j=1}^{N}\text{abs}|\eta_{Sj_{i}}(t)-\Psi_{j_{i}}(t)|\] s.t \[\left.C1:\sum_{j=1}^{N}B_{Sj_{i}}(t)\leq B_{Smax}\right. \tag{12a}\] \[\left.C2:\right. P_{tj_{i}S}(t)\leq P_{Vmax} \tag{12b}\]  
This problem is non-linear programming due to the absolute function. Actually, various optimization techniques can handle this optimization problem, including GA, sequential quadratic programming (SQP), particle swarm optimization (PSO), interior point optimization, Bayesian optimization, etc. In this paper, we used the GA because it is well-suited for problems with non-linear objective functions like the one given above. The GA operators (crossover and mutation) allow for effective exploration in complex and non-linear spaces. Also, the GA can handle constraints effectively using penalty functions or specialized constraint-handling mechanisms. This is important in the constrained optimization problem like the one given above where \(P_{ij_{l}|S}(t)\) must not exceed \(P_{Vmax}\) and \(B_{S|i}(t)\) must not exceed \(B_{Smax}\). Thus, the penalty function provided by the GA is best suited to this scenario. Finally, the GA's population-based approach can help prevent premature convergence to suboptimal solutions. It allows for the simultaneous exploration of multiple candidate solutions, increasing the algorithm's robustness. The comparisons with other candidate optimizers to find the best one are out of the scope of this paper, and they will be the subject of our future investigations. In this paper, a \"GA\" algorithmic approach is used to address (11b) within its constraints. The inputs to the \"GA\" algorithm are the values of \(\Psi_{ij_{l}}(t)\), \(B_{Smax}\), \(P_{Vmax}\) in addition to the adjusting parameters of the \"GA\" algorithm such as the population size \(Ps\), the maximum number of generations \(Ge\), the percentage of mutation \(\epsilon_{m}\), and the percentage of crossover \(\epsilon_{c}\). The fitness function will be:  
\[Fit(t)=\sum_{j=1}^{N}\mathrm{abs}\big{|}\eta_{S|i}(t)-\Psi_{ji_{l}}(t)\big{|} \tag{13}\]  
```
Output \(x_{i}^{+}(t)\) Input:\(\mathcal{M}\), \(N\),initiated \(\delta_{i}\) using GPS localization, and \(\Omega\). Initialization: At \(t=0\), \(w_{l}(t)=0\), \(h_{l}(t)=0\), \(\hat{\gamma}_{i}(t)=0\), and \(q_{i}(t)=0\forall i\in\mathcal{M}\). 1 for\(t=1,2,\ldots,T\)do
2 for\(i=1,2,\ldots,M\)do
3 if\(h_{l}(t-1)>0\)then
4\(\hat{\gamma}_{i}(t)=\hat{\gamma}_{i}(t-1)+\sqrt{(3\log t)/(2h_{l}(t-1))}\)
5 end if
6 else
7\(\hat{\gamma}_{i}(t)=1\)
8 end for
9\(q_{i}(t)=\max\left\{q_{i}(t-1)+\delta_{i}-w_{l}(t-1),0\right\}\)
10 choose the super arm \(A(t)\) that achieves: \(A(t)=\arg\max_{A(t)\subset\mathcal{M}}\sum_{i\in A(t)}(q_{i}(t)+\Gamma\hat{ \gamma}_{i}(t))\), \(|A(t)|\leq N\)
11 Select \(d_{i}^{+}(t)\) that minimizes aerial energy consumptions as:
1. Calculate \(\Gamma_{i}(t)\) matrix \(\forall i\in A(t)\) and \(\forall j\in\mathcal{N}\) using (9) assuming \(P_{ij_{l}|S}(t)=P_{Vmax}\)
2.