## 6 Numerical Experiments
### Solving the TOP with Our Agile Optimization Algorithm
Increasing the number of threads increases the total computing time (as more computing resources are running in parallel) but not the wall-clock time, which is still 1 s. To assess the effectiveness of the proposed approach, the obtained solutions have been compared against the BKS from the literature. In particular, the performance of the algorithm is measured using the current BKS reported by [51]. Tables 3 and 4 show instances and their found solutions. The BKS is tabulated as the obtained reward and the computational time--in seconds--to reach it. Our best solution (OBS) for a specific number of threads is tabulated. The computational time using 128 threads is only recorded in Tables 3 and 4. We have used this time because 128 threads found the best OBS among the different number of threads. The average percentage gap between the OBS for 128 threads and the BKS is calculated. Notice that increasing the number of used threads might find better solutions. For example, the solution found using 128 threads is better than the one found using 16 threads for instance p1.3.m (Table 3). These results are discussed in the next section. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Instance**} & \multicolumn{3}{c}{**BKS ([51])**} & \multicolumn{8}{c}{**Our Best Solution (OBS)**} & \multirow{2}{*}{**Gap (\%)**} \\ \cline{2-2} \cline{4-13}  & **Reward** & **Time** & & & & & & & & & & **[a] – [b]** \\ \hline p1.2.b & 15 & 3.00 & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 0.09 & 0.00 \\ p1.2.c & 20 & 2.00 & 20 & 20 & 20 & 20 & 20 & 20 & 20 & 20 & 0.19 & 0.00 \\ p1.2.d & 30 & 4.30 & 30 & 30 & 30 & 30 & 30 & 30 & 30 & 30 & 0.74 & 0.00 \\ p1.2.e & 45 & 3.40 & 45 & 45 & 45 & 45 & 45 & 45 & 45 & 45 & 0.49 & 0.00 \\ p1.2.f & 80 & 4.90 & 80 & 80 & 80 & 80 & 80 & 80 & 80 & 80 & 0.04 & 0.00 \\ p1.2.g & 90 & 3.00 & 85 & 85 & 85 & 85 & 85 & 85 & 85 & 85 & 90 & 0.62 & 0.00 \\ p1.2.h & 110 & 5.10 & 105 & 105 & 105 & 105 & 105 & 105 & 105 & 110 & 0.25 & 0.00 \\ p1.2.i & 135 & 12.10 & 130 & 130 & 130 & 130 & 130 & 130 & 130 & 135 & 0.44 & 0.00 \\ p1.2.j & 155 & 8.30 & 150 & 150 & 150 & 150 & 150 & 150 & 150 & 155 & 0.98 & 0.00 \\ p1.2.k & 175 & 31.00 & 170 & 170 & 175 & 175 & 175 & 175 & 175 & 175 & 0.24 & 0.00 \\ p1.2.l & 195 & 2.70 & 185 & 185 & 185 & 185 & 190 & 190 & 195 & 0.29 & 0.00 \\ p1.2.m & 215 & 45.30 & 200 & 205 & 205 & 205 & 205 & 205 & 215 & 0.56 & 0.00 \\ p1.2.n & 235 & 6.40 & 230 & 230 & 230 & 230 & 230 & 230 & 230 & 235 & 0.58 & 0.00 \\ p1.2.o & 240 & 10.30 & 235 & 235 & 235 & 235 & 235 & 235 & 235 & 235 & 0.27 & 2.13 \\ p1.2.p & 250 & 10.10 & 240 & 240 & 240 & 240 & 240 & 240 & 240 & 240 & 245 & 0.61 & 2.04 \\ p1.2.q & 265 & 7.80 & 250 & 250 & 250 & 250 & 255 & 255 & 255 & 255 & 0.33 & 3.92 \\ p1.2.r & 280 & 9.20 & 275 & 275 & 275 & 275 & 275 & 275 & 275 & 275 & 0.43 & 1.82 \\ p1.3.c & 15 & 3.00 & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 0.73 & 0.00 \\ p1.3.d & 15 & 3.00 & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 0.60 & 0.00 \\ p1.3.e & 30 & 2.00 & 30 & 30 & 30 & 30 & 30 & 30 & 30 & 30 & 0.88 & 0.00 \\ p1.3.f & 40 & 2.00 & 40 & 40 & 40 & 40 & 40 & 40 & 40 & 40 & 0.36 & 0.00 \\ p1.3.g & 50 & 4.30 & 50 & 50 & 50 & 50 & 50 & 50 & 50 & 50 & 0.51 & 0.00 \\ p1.3.h & 70 & 4.60 & 70 & 70 & 70 & 70 & 70 & 70 & 70 & 70 & 0.16 & 0.00 \\ p1.3.i & 105 & 2.00 & 95 & 95 & 95 & 95 & 100 & 100 & 100 & 105 & 0.50 & 0.00 \\ p1.3.j & 115 & 16.90 & 110 & 110 & 110 & 115 & 115 & 115 & 115 & 0.09 & 0.00 \\ p1.3.k & 135 & 5.80 & 125 & 125 & 130 & 130 & 130 & 130 & 130 & 130 & 0.53 & 3.85 \\ p1.3.l & 155 & 6.50 & 155 & 155 & 155 & 155 & 155 & 155 & 155 & 0.49 & 0.00 \\ p1.3.m & 175 & 19.40 & 160 & 160 & 160 & 160 & 165 & 165 & 165 & 170 & 0.37 & 2.94 \\ p1.3.n & 190 & 6.70 & 185 & 185 & 190 & 190 & 190 & 190 & 190 & 0.21 & 0.00 \\ p1.3.o & 205 & 8.30 & 205 & 205 & 205 & 205 & 205 & 205 & 205 & 205 & 0.76 & 0.00 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Results for set \(p1\).