## 5 Results and Discussion
### Numerical Analysis
#### 5.2.1 Coverage  
Coverage is the percentage of the total accommodated patient demand. In [20], the authors reported that the greedy algorithm had the best time performance, albeit with very weak coverage compared to the other methods. Gurobi was the best of the three methods in terms of coverage but was associated with the worst time performance, which is attributable to the fact that it requires an unacceptably long running time to achieve a feasible solution. However, 3 SH achieved good coverage (96.6% of Gurobi's solutions on average), whereas when compared with Gurobi's time performance, 3 SH took only approximately 27 s on average. Figure 4 shows a comparison of the maximum coverage of the MCGRH and the other three methods of [20], grouped by the number of opened facilities. The x-axis represents the number of opened facilities and available drones, while the y-axis represents the percentage coverage. The average maximum coverage obtained using each method was as follows: Gurobi (81.65%), greedy (63.57%), 3 SH (78.89%), and MCGRH (80.5%). Evidently, MCGRH is the closest to Gurobi in terms of the average maximum coverage (approximately 98.6% of Gurobi's solutions on average), and it exceeded Gurobi in 11 instances (see Table 3 and Figure 5). Because the objective is to maximize coverage, a comparison between the Gurobi and 3 SH methods is considered. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{\(p^{1}\)} & \multirow{2}{*}{\(1K1^{2}\)} & \multicolumn{3}{c}{**Gurobi**} & \multicolumn{3}{c}{**Greedy**} & \multicolumn{3}{c}{**3 SH**} & \multicolumn{3}{c}{**MCGRH**} \\ \cline{3-14}  & & & **Time** & **Coverage** & **Time** & **Coverage** & **Time (s)** & & **Coverage (\%)** & & **Time (s)** & & & & & & & & \\  & & **(s)** & **(\%)** & **(s)** & **(\%)** & **Avg** & **Min** & **Max** & **Avg** & **Min** & **Max** & **Avg** & **Min** & **Max** & **Avg** & **Min** & **Max** \\ \hline  
& 1.22 & 1.14 & 1.52 & 92.87 & 89.4 & **95.1** \\ \hline \hline \multicolumn{14}{c}{Average} & 4946.2 & **81.65** & **0.28** & 63.57 & \multicolumnFigure 5 shows the percentage deviation of coverage between Gurobi, 3 SH, and MCGRH for all instances, where a positive deviation indicates that the MCGRH's result is better for that instance. In half of the instances (11 of 22), MCGRH exceeded Gurobi's results, whereas in 17 of 22 instances, MCGRH outperformed 3 SH. We used the Wilcoxon signed-rank test to ensure that the analysis relied on a solid statistical basis. We applied the test using SPSS to compare the coverage obtained by MCGRH and Gurobi, on the one hand, and MCGRH and 3 SH, on the other. The null hypotheses and results for both tests are presented below. 1. Comparison of MCGRH and Gurobi. * Null hypothesis: The population distributions of the Gurobi algorithm and MCGRH were identical with respect to coverage. * Results: The results were comparable for both algorithms (11 negative and 11 positive ranks). In addition, the \(p\)-value = 0.262, where 0.262 > 0.05; thus, the null hypothesis was accepted. 2. Comparison of MCGRH and 3 SH
* Null hypothesis: The population distributions of the 3 SH and MCGRH are identical with respect to coverage. * Results: There were 5 negative and 17 positive ranks. Thus, most of the pairs were positive ranks, which means that MCGRH had a larger coverage than 3 SH. In addition, the \(p\)-value = 0.008, where 0.008 < 0.05; therefore, the null hypothesis was rejected. In other words, the results of the MCGRH were statistically significantly better than those of the 3 SH. Figure 4: Comparison between the maximum coverage of the four methods. Figure 5: Percent deviation of MCGRH from Gurobi and 3 SH. the same resources are used for each method, and because the objective is to maximize the patients covered, it is more important to maximize coverage and help more patients in a very fast time than to reduce the consumption of the drones' batteries.