## 6 Results and Discussion
### Solutions Quality  
Table 4 presents the experimental results obtained by both GA-VRPTW and GGA with nine Solomon benchmark problems. In this table, NV indicates the number of delivery vehicles used in the optimum solution, Optimum is the optimal solution of the problem, Time is the time in seconds used to reach the reported result, and RE is the relative error previously described. The numbers in parentheses refer to the ranking reached by each method for each problem. The best results were obtained using 60 processes and a single cluster of the MiniGrid. It can be observed that for all instances, GGA is closer to optimal than GA-VRPTW, and the results are reached with relatively short times. Figure 11: Speedup of the GGA in the Autonomous University of Morelos State (UAEM) cluster. The Wilcoxon signed ranks test is used in this experiment, and the \(p\)-value computed is 0.003843, indicating that GGA has one significant improvement over GA-VRPTW. Table 5 shows the results obtained with the GGA using 10 Gehring and Homberger benchmark problems. In this table, UB is the upper bound best known to the problem (both for the number of vehicles and for the distance traveled), Best and Worse are the best and the worst values found by GGA for each problem, RE is the relative error, and Mean, Median, and Mode are the statistical measures of the results obtained in 30 runs. GGA best results were obtained using 120 processes. The two clusters of the MiniGrid were used, and the MiniGrid had a balanced process distribution. It can be observed that for half the instances, GGA obtained the UB, and in one case (the C1_10_8 problem), the UB was improved (41,954.51 versus 42,660.70), because a better solution is found, the RE is represented in Table 5 with *. The execution times for GGA were an average of 15,000 to 16,000 s. Table 5 shows that the failure rate to achieve one known result was no higher than 0.501. This value indicates that GGA is very competitive. On the other hand, none of the significant size problems had a mode value equal to the best UB. The most significant difference between the mode value and the UB is 51, and the smallest difference is 0.4. If the median value is equal to the optimum, this indicates that at least half of the 30 tests reached the optimal solution. It is observed that none of the median values of the problems is equal to the UB. The problems that reach results close to optimal, in their mode and median values, were the C1_10_2 problem and the C1_10_7 problem. C1_10_1, C1_10_3, and C1_10_6 problems are the most challenging regarding reaching good results concerning these statistical values. Table 6 show a comparison of both NV values and the travel costs obtained by the GGA with those of other similar approaches. Table 6 depicted a comparison of the experimental results with the Solomon benchmark problems. The optimum values of the number of vehicles and travel costs listed \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Problem**} & \multicolumn{2}{c}{**Optimum**} & \multicolumn{2}{c}{**GA-VRPTW**} & \multicolumn{2}{c}{**GGA**} \\ \cline{2-7}  & **NV** & **Optimum** & **Time** & **RE** & **Time** & **RE** \\ \hline C101 & 10 & 827.3 & 2223 & 0.0119 (2) & 130 & 0.0111 (1) \\ C102 & 10 & 827.3 & 2734 & 0.1141 (2) & 127 & 0.0100 (1) \\ C103 & 10 & 826.3 & 2196 & 0.1434 (2) & 220 & 0.1141 (1) \\ C104 & 10 & 822.9 & 2515 & 0.1374 (2) & 190 & 0.1310 (1) \\ C105 & 10 & 827.3 & 2313 & 0.0020 (2) & 127 & 0.0001 (1) \\ C106 & 10 & 827.3 & 2131 & 0.0224 (2) & 300 & 0.0203 (1) \\ C107 & 10 & 827.3 & 2048 & 0.0707 (2) & 354 & 0.0704 (1) \\ C108 & 10 & 827.3 & 2051 & 0.0041 (2) & 210 & 0.0002 (1) \\ C109 & 10 & 827.3 & 1985 & 0.0453 (2) & 120 & 0.0451 (1) \\ \hline \multicolumn{7}{c}{Average ranking} & 2 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Results comparison between GA-VRPTW and GGA with Solomon benchmark problems.