## 5 Numerical Experiments
### Results of Search Strategy before Obtaining a Feasible Solution
From the results, the \(p\)-values are smaller than 0.05 except for instance No.6. Thus, we could find a significant difference between the search methods 1A-S and 1C-S for almost all instances from the viewpoint of a searching speed. To summarize the results, there is no difference in the performance between the search methods 1A-S and 1C-S (Tables 4 and 5). On the other hand, as for the search speed, the search method 1C-S (54.53 [s]) can find feasible solutions with shorter time (approxi \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{4}{c}{**(a)**} \\ \hline
**No.** & **Avg. [min]** & **Best [min]** & **Worst [min]** & **\# of Feasible Solution** \\ \hline
1 & 243.76 & 230.85 & 259.07 & 50 \\
2 & 238.16 & 230.87 & 252.95 & 50 \\
3 & 228.47 & 218.98 & 237.25 & 50 \\
4 & 242.56 & 232.38 & 252.36 & 50 \\
5 & 249.06 & 240.95 & 260.69 & 50 \\
6 & 270.81 & 260.50 & 284.04 & 31 \\
7 & 238.52 & 225.12 & 252.72 & 34 \\
8 & 213.15 & 207.43 & 226.01 & 50 \\
9 & 241.64 & 231.13 & 255.18 & 49 \\
10 & 243.28 & 234.40 & 252.54 & 45 \\ \hline Avg. & 240.94 & 231.26 & 253.28 & 45.9 \\ \hline \multicolumn{4}{c}{**(b)**} \\ \hline
**No.** & **Avg. [min]** & **Best [min]** & **Worst [min]** & **\# of Feasible Solution** \\ \hline
1 & 245.46 & 234.80 & 258.63 & 40 \\
2 & 242.59 & 228.66 & 260.92 & 50 \\
3 & 234.57 & 222.44 & 249.94 & 50 \\
4 & 246.81 & 236.23 & 268.79 & 49 \\
5 & 254.91 & 241.88 & 269.26 & 42 \\
6 & 268.45 & 260.94 & 276.57 & 16 \\
7 & 238.02 & 222.71 & 247.85 & 10 \\
8 & 213.98 & 207.79 & 237.93 & 49 \\
9 & 245.18 & 233.40 & 260.38 & 44 \\
10 & 244.20 & 236.92 & 257.74 & 29 \\ \hline Avg. & 243.42 & 232.58 & 258.80 & 37.9 \\ \hline \multicolumn{4}{c}{**(c)**} \\ \hline
**No.** & **Avg. [min]** & **Best [min]** & **Worst [min]** & **\# of Feasible Solution** \\ \hline
1 & 241.63 & 233.74 & 251.90 & 50 \\
2 & 236.61 & 227.33 & 246.43 & 50 \\
3 & 228.31 & 217.29 & 242.36 & 50 \\
4 & 242.15 & 233.15 & 254.30 & 50 \\
5 & 249.59 & 241.04 & 259.27 & 50 \\
6 & 269.55 & 256.99 & 284.94 & 32 \\
7 & 236.17 & 225.00 & 252.39 & 30 \\
8 & 213.14 & 205.85 & 224.93 & 50 \\
9 & 239.44 & 228.72 & 252.01 & 50 \\
10 & 239.80 & 229.91 & 253.65 & 44 \\ \hline Avg. & 239.64 & 229.90 & 252.22 & 45.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Results of search methods before obtaining a feasible solution. (**a**) Search method 1A-S (search for all neighborhood solutions of the Or-opt and the CROSS-exchange). (**b**) Search method 1B-S (search only normal order insertion of the Or-opt). (**c**) Search method 1C-S (search for normal order insertion of the Or-opt and normal order exchange of the CROSS-exchange). mately 30 [s]) than the search method 1A-S (85.29 [s]), which means that by using the search method 1C-S, we can reduce the computation time by \(\left(85.29-54.53\right)/85.29\times 100=36.1\%\). Therefore, we concluded that the search method 1C-S (the search for the normal order insertion of the Or-opt and the normal order exchange of the CROSS-exchange) is the best search method before finding a feasible solution. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{2}{c}{**1A-S**} & \multicolumn{2}{c}{**1C-S**} & \\
**No.** & **Avg. [s]** & **SD** & **Avg. [s]** & **SD** & **_p_** \\ \hline \(1\) & 78.76 & 24.84 & 44.67 & 16.83 & 0.000 \\ \(2\) & 72.83 & 16.30 & 32.84 & 5.60 & 0.000 \\ \(3\) & 66.37 & 10.68 & 30.22 & 5.79 & 0.000 \\ \(4\) & 75.37 & 13.50 & 36.85 & 9.72 & 0.000 \\ \(5\) & 72.86 & 10.15 & 42.23 & 15.70 & 0.000 \\ \(6\) & 123.70 & 41.62 & 109.59 & 45.82 & 0.205 \\ \(7\) & 115.36 & 37.28 & 89.80 & 45.26 & 0.018 \\ \(8\) & 68.81 & 19.88 & 32.74 & 8.61 & 0.000 \\ \(9\) & 75.99 & 16.51 & 42.16 & 16.41 & 0.000 \\ \(10\) & 102.82 & 39.51 & 84.23 & 42.42 & 0.035 \\ \hline Avg. & 85.29 & 23.03 & 54.53 & 21.22 & 0026 \\ \hline \hline \end{tabular}
\end{table}