## 5 Computational Experiments and Discussion
All algorithms are performed for 10 independent runs and the mean, standard deviation, coefficient of variation (C.V.), and the best result of each algorithm are reported in column 3-6 respectively. The Wilcoxon signed-rank test with \(\alpha~{}=~{}0.05\) is performed to test the significance of the results. As the Wilcoxon signed-rank test does not assume normality in the data, it can be used when this assumption has been violated and the use of the dependent \(t\)-test is inappropriate. The \(p\)-value of the test is reported in the last column of the tables. The best result of each problem instances is highlighted in bold. In addition, the best result of the all problem instances are summarized as a bar chart in Figure 6. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Instance** & **Algorithm** & **Mean** & **Std. Dev.** & **C.V.** & **Best** & \(p\)**-Value** \\ \hline \multirow{3}{*}{Perl 1} & SHS & 263.27 & \(5.99\times 10^{-14}\) & \(2.28\times 10^{-16}\) & **263.3** & 1.000 \\  & PHS & 263.27 & \(5.99\times 10^{-14}\) & \(2.28\times 10^{-16}\) & **263.3** & 1.000 \\  & HS-SA & 263.27 & \(5.99\times 10^{-14}\) & \(2.28\times 10^{-16}\) & **263.3** & – \\ \hline \multirow{3}{*}{Perl 2} & SHS & 1693.18 & 3.2646 & 0.0019 & 1689.5 & 0.005 \\  & PHS & 1646.25 & 2.1141 & 0.0013 & 1643.9 & 0.005 \\  & HS-SA & 1485.27 & 1.9309 & 0.0013 & **1482.4** & – \\ \hline \multirow{3}{*}{Perl 3} & SHS & 2271.26 & 8.9040 & 0.0039 & 2257.7 & 0.005 \\  & PHS & 2099.91 & 5.5006 & 0.0026 & 2096.2 & 0.005 \\ \cline{1-1}  & HS-SA & 2082.90 & 4.5094 & 0.0022 & **2078.0** & – \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Comparative results of Perl dataset in LIRP for SHS, PHS, and HS-SA. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Instance** & **Algorithm** & **Mean** & **Std. Dev.** & **C.V.** & **Best** & \(p\)**-Value** \\ \hline \multirow{3}{*}{Gaskell 1} & SHS & 1592.13 & 7.1444 & 0.0045 & 1586.2 & 0.005 \\  & PHS & 1585.23 & 4.9927 & 0.0031 & 1578.5 & 0.028 \\  & HS-SA & 1579.26 & 3.7438 & 0.0024 & **1575.7** & – \\ \hline \multirow{3}{*}{Gaskell 2} & SHS & 1042.76 & 3.7271 & 0.0036 & 1039.0 & 0.005 \\  & PHS & 1009.80 & 1.9408 & 0.0019 & 1008.9 & 0.022 \\  & HS-SA & 1007.32 & 1.8396 & 0.0018 & **1006.1** & – \\ \hline \multirow{3}{*}{Gaskell 3} & SHS & 829.58 & 5.9608 & 0.0072 & 820.2 & 0.005 \\  & PHS & 797.70 & 5.6214 & 0.0070 & 790.3 & 0.013 \\  & HS-SA & 789.51 & 5.2295 & 0.0066 & **780.9** & – \\ \hline \multirow{3}{*}{[PERSON] 4} & SHS & 1032.74 & 3.3067 & 0.0032 & 1030.6 & 0.005 \\  & PHS & 1021.37 & 2.6579 & 0.0026 & 1018.1 & 0.028 \\ \cline{1-1}  & HS-SA & 1018.49 & 1.5920 & 0.0016 & **1017.0** & – \\ \hline \multirow{3}{*}{[PERSON] 5} & SHS & 610.80 & 6.4625 & 0.0106 & 604.7 & 0.005 \\ \cline{1-1}  & PHS & 602.50 & 3.9302 & 0.0065 & 599.2 & 0.005 \\ \cline{1-1}  & HS-SA & 571.79 & 3.6303 & 0.0063 & **566.9** & – \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Comparative results of Gaskell dataset in LIRP for SHS, PHS, and HS-SA. The Wilcoxon signed-rank test elicits a statistical significant difference in the performance of finding the minimum cost of the LIRP between the hybrid HS-SA and the SHS and PHS across all the datasets (\(p\)-value \(<\) 0.005) based on the difference in the median cost. These tests are used to show that the results are significant, supporting that the proposed algorithm will produce results that are different from the results of the SHS and PHS. Apart from the Perl 1 instance where the proposed algorithm produced the same results as SHS and PHS, all other datasets showed \(p\)-values of less than 0.05. This means the results generated by the hybrid HS-SA is significant and different from the values from the SHA and PHS. So, the results from the proposed algorithm is significant from the other works, but is it better? From Tables 6-8, the hybrid HS-SA outperforms the solutions in SHS and PHS for all cases of datasets of Perl, Gaskell, and Christofides. For this, we use the C.V. values. The lower the value of the C.V., the more precise the estimate.