## 4 Statistical Analysis on Computational Results
### Statistical Analysis on aTSP Results  
In Table 13 we can see that for all cases \(p\)-value is less than 0.05, which means that there is a statistically significant difference between the three methods in all cases. For further examination, pairwise Wilcoxon tests were performed. \begin{table}
\begin{tabular}{l c c c} \hline  & \(\mathbf{X^{2}}\) & **df** & \(\boldsymbol{p}\)**-Value** \\ \hline
**FI\_1 min** & 6.8689 & 2 & 0.0322 \\
**FI\_2 mins** & 9.0314 & 2 & 0.0109 \\
**BI\_1 min** & 9.2739 & 2 & 0.0097 \\
**BI\_2 mins** & 9.6658 & 2 & 0.008 \\ \hline \end{tabular}
\end{table} 
Table 13: Kruskal–Wallis rank sum test. \begin{table}
\begin{tabular}{l c c c c c c c} \hline
**Instance** & **zOpt** & **GVNS\_1** & **GVNS\_2** & **GVNS\_3** & **GAP\_1 (\%)** & **GAP\_2 (\%)** & **GAP\_3 (\%)** \\ \hline ar9152.tsp & 837479 & 1372411 & 1648596 & 1648596 & 63.87 & 96.85 & 96.85 \\ gr9882.tsp & 300899 & 377898 & 388944 & 388944 & 25.59 & 29.26 & 29.26 \\ eg7146.tsp & 172387 & 209515 & 220315 & 220315 & 21.54 & 27.80 & 27.80 \\ fi10639.tsp & 520527 & 635913 & 649604 & 649604 & 22.17 & 24.80 & 24.80 \\ ho14473.tsp & 177105 & 449416 & 484812 & 474609 & 153.76 & 173.74 & 167.98 \\ ei8246.tsp & 206171 & 248441 & 258889 & 258889 & 20.50 & 25.57 & 25.57 \\ ja9847.tsp & 491924 & 602674 & 612304 & 612304 & 22.51 & 24.47 & 24.47 \\ kz9976.tsp & 1061882 & 1327647 & 1358247 & 1358247 & 25.03 & 27.91 & 27.91 \\ lu980.tsp & 11340 & 12151 & 12458 & 12559 & 7.15 & 9.86 & 10.75 \\ mo14185.tsp & 427377 & 527335 & 529729 & 529729 & 23.39 & 23.95 & 23.95 \\ nu3496.tsp & 96132 & 115162 & 108803 & 109804 & 19.80 & 13.18 & 14.22 \\ mu1979.tsp & 86891 & 94072 & 95874 & 94358 & 8.26 & 10.34 & 8.59 \\ a1941.tsp & 9352 & 9525 & 9757 & 9759 & 1.85 & 4.33 & 4.35 \\ rw1621.tsp & 26051 & 28784 & 29118 & 29131 & 10.49 & 11.77 & 11.82 \\ tz6117.tsp & 394718 & 475863 & 501184 & 501184 & 20.56 & 26.97 & 26.97 \\ uy734.tsp & 79114 & 84585 & 86358 & 84686 & 6.92 & 9.16 & 7.04 \\ wi29.tsp & 27603 & 27603 & 27603 & 27612 & 0.00 & 0.00 & 0.03 \\ sym7663.tsp & 238314 & 295414 & 308747 & 308747 & 23.96 & 29.55 & 29.55 \\ zi929.tsp & 95345 & 101161 & 104327 & 100881 & 6.10 & 9.42 & 5.81 \\ ca4663.tsp & 1290319 & 1573320 & 1646889 & 1470076 & 21.93 & 27.63 & 13.93 \\ \hline
**Average** & **327546.50** & **428444.50** & **454127.90** & **444501.70** & **25.27** & **30.33** & **29.08** \\ \hline \end{tabular}
\end{table} 
Table 12: Results using the Best Improvement search strategy and an execution time limit of 2 min. In accordance with the pairwise tests which are summarized in Table 14, it is clear that the GVNS_1 has significant differences with the other two schemes in all cases. Both GVNS_2 and GVNS_3 perform equivalently using the First Improvement search strategy independently of the execution time limit, while using the Best Improvement strategy they have significant reported differences with both time limits [14]. In respect to this Kruskal-Wallis statistical analysis, we have four box-plots illustrated in Figures 1a,b and 2a,b four box plots. Each one depicts either First Improvement or Best Improvement for one minute, as well as for two minutes runs. Moreover, based on the corresponding of the previous statistical summary, box plots, it is confirmed that the GVNS_1 produces much better results than the other two algorithms in all cases for the aTSP, while the GVNS_2 outperform the GVNS_3 in also all cases. Also, by checking the medians at the box plots it can be seen that the GVNS_2 performs significantly better on Best Improvement, as it produces results that are \"close\" to the results of GVNS_1. \begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{1}{c}{**FI\_1 min**} & \\ \hline  & GVNS1 & GVNS2 \\ GVNS2 & 0.00064 & \\ GVNS3 & 0.00064 & 0.4488 \\ \hline \multicolumn{1}{c}{**BI\_1 min**} & \\ \hline  & GVNS1 & GVNS2 \\ GVNS2 & 0.00109 & \\ GVNS3 & 0.00064 & 0.00064 \\ \hline \multicolumn{1}{c}{**BI\_2 mins**} & \\ \hline GVNS2 & 0.00109 & \\ GVNS3 & 0.00064 & 0.00064 \\ \hline \hline \end{tabular}
\end{table}