## 4 Proposed CB-FBC and Genetic Approach
### Optimization of \(x_{j_{i}}(t)\)  
Herein, the assignment vector \(x_{j_{i}}(t)\) will be optimized under constraints C1 to C5. This sub-problem is a non-linear dynamic time sequential combinatorial optimization problem, where a group of clusters should be selected at each time \(t\) under the aforementioned constraints. Additionally, the M&C center has no information about this catastrophic area, including the number of survivors, their distributions, their traffic needs, etc. Such kinds of optimization problems cannot be solved using traditional optimization techniques as there is a lack of full information about the problem. More specifically, the values of \(\Psi_{j_{i}}(t)\) are unknown for the optimizer itself. In these kinds of optimization problems, artificial intelligence by means of online learning provides a sufficient solution. This is due to the exploration and exploitation mechanism of online learning that can arrive at sufficient solutions without knowing the details of the problem. Thus, in this paper, this problem is modeled as a budget constraint combinatorial bandit with arms' fairness [27]. Generally, the MAB hypothesis is an efficient online learning methodology where a player intends to maximize his long-term reward via attempting the bandit's arms [19]. The player has no prior knowledge about arms except his observations about the played arms and their associated rewards. Within the bandit game, the player attempts to compromise between consistently exploiting the arm associated with the most considerable noticeable reward until now or exploring new ones [19]. In some MAB games, a cost should be paid for the played arms, which is subject to the player's limited budget. These types of MAB games are called budget-constrained MAB games [28]. Conversely, combinatorial bandits with arms' fairness given in [29] are bandits where the player chooses a group of arms named the super arm in each time step while guaranteeing long-term fairness in the arms' selection over time horizon. In this section, we will propose the \"CB-FBC\" algorithm to address the problem of \(x_{j_{i}}(t)\) optimization. In this regard, the player will be the nearest survival GBS, the arms are the post-disaster clusters, and the reward is the sum of the aerials' data rates. The aerials' remaining battery capacity limits this game. Also, fairness should be maintained in post-disaster area clusters coverage based on their users' densities. Algorithm 1 summarizes the envisioned \"CB-FBC\" algorithm, which will be implemented in the survival GBS as it is the bandit player. The algorithm inputs are the sets of all available post-disaster clusters and aerials, i.e., \(\mathcal{M}\) and \(\mathcal{N}\), the initial values of users' densities \(\delta_{i}\), which are pre-estimated using GPS localization, and the design parameter \(\Omega\). The algorithm's output is the aerials-clusters assignment vector \(x_{j_{i}}^{*}(t)\) at time \(t\). For initialization, at \(t=0,w_{i}(t)\) which is the selection vector, \(h_{i}(t)\) which is the number of times cluster \(i\) was drawn up to time \(t\), \(\gamma_{i}(t)\) which is the mean data rate of cluster \(i\) up to time \(t\), and \(q_{i}(t)\) which is the queue of cluster \(i\), are all initiated by zero \(\forall i\in\mathcal{M}\). \(q_{i}(t)\) assures fairness among the clusters, as discussed later. For \(t=1\) to \(T\), the upper confidence bound (UCB) for \(\forall i\in\mathcal{M}\) are set to \(\tilde{\gamma}_{i}(t)=\gamma_{i}(t-1)+\sqrt{(3\log t)/(2h_{i}(t-1))}\) if \(h_{i}(t-1)>0\) or \(\tilde{\gamma}_{i}(t)=1,ifh_{i}(t-1)\leq 0\). Then, \(q_{i}(t)\) is estimated for \(\forall i\in\mathcal{M}\) as illustrated in Algorithm 1. At each trial \(t\), the value of \(q_{i}(t)\) is increased by \(\delta_{i}\) and decreased by one if cluster \(i\) was drawn in time \(t-1\). Therefore, if cluster \(i\) is not chosen often, its \(q_{i}(t)\) will be doubled by its UD density value, and vice versa. This prioritizes it for being drawn next time step.