## 6 Results and Discussion
### Solutions Quality
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Problem**} & \multicolumn{2}{c}{**UB**} & \multicolumn{4}{c}{**GGA**} \\ \cline{2-10}  & **NV** & **Value** & **NV** & **Best** & **Worse** & **Mean** & **RE** & **Median** & **Mode** \\ \hline C1\_10\_1 & 100 & 42,478.95 & 100 & 42,478.95 & 42,550.95 & 42,514.31 & 0 & 42,515.52 & 42,530.01 \\ C1\_10\_2 & 90 & 42,222.96 & 90 & 42,278.45 & 42,300.37 & 42,286.83 & 0.0571 & 42,284.32 & 42,283.14 \\ C1\_10\_3 & 90 & 40,101.36 & 90 & 40,207.71 & 40,287.27 & 40,244.82 & 0 & 40,243.31 & 40,251.22 \\ C1\_10\_4 & 90 & 39,468.60 & 90 & 39,468.60 & 39,502.36 & 39,502.64 & 0 & 39,493.83 & 39,490.70 \\ C1\_10\_5 & 100 & 42,469.18 & 100 & 42,469.18 & 42,534.18 & 42,502.83 & 0 & 42,499.45 & 42,498.11 \\ C1\_10\_6 & 99 & 43,830.21 & 100 & 43,832.10 & 43,900.36 & 43,872.31 & 0.0043 & 43,870.90 & 43,870.53 \\ C1\_10\_7 & 97 & 43,372.03 & 97 & 43,453.92 & 43,468.22 & 43,457.34 & 0.1888 & 43,455.94 & 43,453.32 \\ C1\_10\_8 & 92 & 42,660.70 & 94 & 41,954.51 & 41,996.55 & 41,984.76 & * & 41,986.47 & 41,973.45 \\ C1\_10\_9 & 90 & 40,341.06 & 91 & 40,572.31 & 40,602.31 & 40,591.12 & 0.5010 & 40,590.52 & 40,588.44 \\ C1\_10\_10 & 90 & 39,852.44 & 90 & 39,933.06 & 40,059.01 & 39,970.21 & 0.1781 & 39,968.01 & 39,952.03 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Statistical results for 10 Gehring and Homberger benchmark problems. in this table are those reached by the KDMSS algorithm. The GGA algorithm is better than PHGA and LC03 algorithms since it has a lower relative error. Concerning the KDMSS algorithm, GGA is competitive since it has a minimal relative error. The smallest relative error (0.0002) is for the C108 problem, and the most significant relative error (0.1310) is with the C104 problem. Table 7 shows a comparison of the results with the Gehring and Homberger benchmark problems. Only some algorithms described in the existing literature are compared since they mostly report results for some problems, and it is difficult to make a comparison involving all benchmark instances. Due to this difficulty, in this work, a comparison of the relative error between the GGA solution and the best-known solution is described. GGA is better than the LC03 algorithm since it has a less relative error. Concerning the HM4 algorithm, GGA shows better results in most cases, and the HM4 algorithm is better than GGA in two problems only: C1_10_6 (UB = 42479.15) and C1_10_7 (UB = 42711.39), in which a better upper bound is obtained than the best results reported in the literature (43830.21 and 43372.03), because a better solution is found in HM4, the RE is represented in Table 7 with *. The number of vehicles does not improve, the existing literature reports 99 (Q) and 97 (SCR) vehicles, and the HM4 algorithm reports 100 and 99, respectively. When using GGA, the first problem uses 100 vehicles, and the second uses 97, which are equal to those reported in the literature. GGA find a new upper bound to the C1_10_8 problem (41954.51), but the number of vehicles does not improve since the existing literature reports 92 (SCR) and the GGA achieves 94 vehicles. The Friedman test is run with this relative error, and its resulting statistic value is 20.2 for five methods and nine problems, which has a \(p\)-value of 0.000456. When evaluating this \(p\)-value with a