## 6 Experimental Results
### Quality of Solutions  
Regarding the quality of the solutions achieved, in this section, a statistical analysis based on the results is given. Given that in this work, the computational power of a GPU is utilized, large population sizes are used. In Table 8, the minimum, the maximum and the average costs of the solutions achieved for all of the datasets and a population size of 256 are given. In Table 9, the same metrics are given for a population size of 16,384. Furthermore, Figures 8 and 9 show descriptive statistics for the measurements. There was a need for normalization, since the costs were not of the same magnitude. Thus, the average cost has taken the value of 100 for all of the datasets, and the remaining statistics were calculated. For the _sta-f-83_ dataset in Figures 8 and 9, there is no box, as the initial and the best solution are very close and with a high cost (157-160). The measurements were done with identical seeds (1-100) to the \(rand\) function of the C programming language. The mutation probability was 0.6%, the crossover probability 50% and the number of generations 200 for all of the experiments. It can be seen that all of the results gained for the bigger population size are better. This is something expected, since by having a larger population size, a bigger part of the solution space is explored. \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Dataset** & \(\mathbf{pop}=\mathbf{256}\) & \(\mathbf{pop}=\mathbf{2048}\) & \(\mathbf{pop}=\mathbf{16},\mathbf{384}\) \\ \hline car-f-92 & 81.40 & 61.34 & 46.17 \\ car-s-91 & 94.92 & 60.79 & 50.19 \\ ear-f-83 & 65.67 & 56.00 & 49.19 \\ hec-s-92 & 71.63 & 54.45 & 54.68 \\ kfu-s93 & 57.49 & 41.70 & 38.46 \\ lse-f-91 & 77.11 & 53.93 & 50.95 \\ pur-s93 & INF & INF & INF \\ rye-s93 & 123.55 & 81.60 & 83.19 \\ sta-f-83 & 19.43 & 19.43 & 18.84 \\ tre-s-92 & 48.74 & 39.40 & 35.03 \\ uta-s92 & 80.55 & 61.57 & 48.31 \\ ute-s92 & 31.05 & 25.25 & 17.71 \\ yor-f-83 & 38.58 & 32.22 & 27.03 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Average difference (%) from the best reported solution in the bibliography with the genetic algorithm and a tournament selection size of \(population\ size/32\). \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Dataset** & **Minimum Cost** & **Maximum Cost** & **Average Cost** \\ \hline car-f-92 1 & 4.86 & 5.32 & 5.04 \\ car-s-91 1 & 5.72 & 6.33 & 5.92 \\ ear-f-83 1 & 39.05 & 42.52 & 40.24 \\ hec-s-92 1 & 11.34 & 12.91 & 12.27 \\ kfu-s-93 1 & 15.45 & 16.98 & 16.14 \\ lse-f-91 1 & 12.06 & 13.37 & 12.68 \\ pur-s93 1 & 10.75 & 33.64 & 21.51 \\ rye-s93 1 & 9.20 & 10.48 & 9.63 \\ sta-f-83 1 & 157.38 & 157.70 & 157.47 \\ tre-s-92 1 & 9.15 & 9.75 & 9.51 \\ uta-s-92 1 & 4.05 & 4.33 & 4.20 \\ ute-s-92 1 & 25.43 & 27.18 & 26.19 \\ yor-f-83 1 & 39.52 & 42.41 & 41.32 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Minimum, maximum and average cost with a population size of 256. Figure 8: Descriptive statistics chart for population size 256. Figure 9: Descriptive statistics chart for a population size of 16,384. \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Dataset** & **Minimum Cost** & **Maximum Cost** & **Average Cost** \\ \hline car-f-92 I & 4.77 & 5.06 & 4.92 \\ car-s-91 I & 5.32 & 5.84 & 5.60 \\ ear-f-83 I & 37.27 & 39.82 & 38.65 \\ hec-s-92 I & 10.88 & 12.14 & 11.39 \\ kfu-s-93 I & 14.48 & 16.00 & 15.33 \\ lse-f-91 I & 11.22 & 12.62 & 11.93 \\ pur-s-93 I & 5.25 & 14.99 & 8.36 \\ rye-s-93 I & 8.64 & 9.32 & 8.98 \\ sta-f-83 I & 157.20 & 157.45 & 157.39 \\ tre-s-92 I & 8.72 & 9.61 & 9.17 \\ uta-s-92 I & 3.64 & 4.13 & 3.90 \\ ute-s-92 I & 25.11 & 26.71 & 25.58 \\ yor-f-83 I & 38.80 & 41.21 & 40.05 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Minimum, maximum and average cost with a population size of 16,384. The approach of this paper achieved competitive results for many of the Carter/Toronto datasets. In Table 10, the best costs from this work, from other evolutionary techniques and the best available in the literature are shown.