## 5 Results and Discussion
### Numerical Analysis  
In this study, we developed a new heuristic for solving an MCFLPD, referred to as the MCGRH. We compared the results of MCGRH with those of the Gurobi mixed-integer programming solver (MIP), greedy heuristic, and three-stage heuristic (3 SH) methods used in [20] to achieve the best maximum coverage within the minimum time. In terms of coverage paired with time performance, the method that achieved the best results in [20] was 3 SH. As described in Section 2, 3 SH comprises three stages. In the first stage, a weight \(\frac{w_{i}}{b_{ij}}\)\(a_{ij}\), which is the weight of assigning each potential demand point \(i\in I\) to each facility \(j\in J\), where \(b_{ij}<B\). At this stage, the algorithm attempts to maximize the total weights assigned to each facility. The second stage deals with the problem of assigning drones to facilities and demand points as a maximum profit knapsack problem. In the third stage, the solution is improved by performing a local exchange heuristic, which removes the facility with the lowest demand and replaces it with the available ones at random. In contrast, the MCGRH overpasses 3 SH, where it attempts to work on elements that maximize coverage with fewer processes. Thus, in MCGRH, the \(p\) facilities to be opened are selected from the facilities with the highest demand weights for their potential patients \(\sum_{i\in~{}I}\ w_{i}\ c_{ij}\ \forall\ j\in J\), which is achieved using a randomized-greedy approach. In addition, the assignment of potential patients to the opened facilities is performed in a greedy manner by selecting the nearest patient with available capacity \(U\). This is intended to reduce the amount of battery consumed by the drone to serve patient \(i\), enabling as many patients as possible to be served using the drone. In other words, because the two factors that affect the consumed battery \(b_{ij}\) are the demand weight \(w_{i}\) and the distance \(d_{ij}\) between patient \(i\) and facility \(j\), and because \(w_{i}\) is fixed and \(d_{ij}\) is variable, selecting the facility with the shortest distance from \(i\) gives more importance to the demand weight than the distance when calculating the amount of consumed battery. This helps achieve the objective of maximizing the total demand served. Hence, the MCGRH addresses the problem of assigning drones to facilities as a maximum-profit knapsack problem. In addition, the proposed method addresses the problem of assigning patients to drones by prioritizing patients with the lowest consumed battery \(b_{ij}\), followed by the highest demand \(w_{i}\). These features account for the fact that MCGRH outperformed 3 SH, as shown in detail in due course. Table 3 shows the results of all the considered methods on 22 different instances reported in [20], as well as the MCGRH. The instances were grouped by the number of opened facilities \(p\) followed by the number of available drones \(K\). The table shows the performance of the four methods as measured by time (in seconds) and coverage (i.e., the percentage of the total patient demand that is satisfied). The Gurobi MIP is an exact and deterministic solution method that runs until a solution is found or a limit of 7200 s is reached. 3 SH and MCGRH are not deterministic, so the table shows their statistical results (average, minimum, and maximum) over 30 different runs of each instance. In addition, in the last row of the table, the average time and coverage of the 22 instances are indicated (since we are concerned with the minimum time and maximum coverage, we considered them when computing the overall average results of the 3 SH and MCGRH in the last \begin{table}
\begin{tabular}{c c} \hline \hline
**Power Transfer Efficiency (\(\eta\))** & 0.66 \\
**Lift-to-Drag Ratio (\(\theta_{s}\))** & 3.5 \\
**Tare Weight** & 10.1 kg \\
**Maximum Payload** & 5 kg \\
**Battery Capacity** & 777 W h \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The parameters of the drone. row of Table 3).