## 5 Computational Experiments
### Numerical Results  
Average results for the 37 test instances are shown for the different versions in Table 9, respectively for _NSGAII_, _MS-NSGAII_, and _NSILS_. In this table, the minimum (min), maximum (max), and the average values (av.) for the number of solutions in the first front (\(|F_{0}|\)), the hyper-volume metric (\(HV\)), the spacing measure (\(SP\)), the percentage of the deviation from the optimal solution, and the \(CPU\) time reported in seconds are given. As can been seen in Table 9, all versions with the local search procedure were better than versions without it, at a cost of a computing time about three-times higher on average. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Name** & **Value** \\ \hline Population size & \(NP\) & 100 \\ \hline Maximum generations in NSGAII & \(NG\) & 500 \\ \hline Maximum generations in MS-NSGAII & \(NG_{1}\) & 100 \\ No. of restarts in MS-NSGAII & \(nbstart\) & 5 \\ \hline Maximum generations in NSILS & \(NG\) & 50 \\ No. of ILS iterations in \(NSILS\) & \(max_{p}\) & 20 \\ Perturbation level of the ILS & \(C_{max2}\) & 5 \\ \hline RCLsize in PRH & \(NCand\) & 4 \\ Perturbation level of repair & \(C_{max}\) & 2 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Fixed parameters. PRH, Parallel Randomized constructive Heuristic. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c c c c c} \hline \hline  & & \multicolumn{3}{c}{\(|F_{0}|\)} & \multicolumn{3}{c}{\(HV\)} & \multicolumn{3}{c}{\(SP\)} & \multicolumn{3}{c}{\(DO_{1}\)} & \multicolumn{3}{c}{\(DO_{2}\)} & \multicolumn{3}{c}{\(CPU\)} \\ \hline  & **Version** & **min** & **max** & **av.** & **min** & **max** & **av.** & **min** & **max** & **av.** & **min** & **max** & **av.** & **min** & **max** & **av.** & **min** & **max** & **av.** & **min** & **max** & **av.** \\ \hline \multirow{3}{*}{\(NSGAII\)} & 1 & 20.54 & 34.10 & 27.23 & 0.90 & 0.94 & 0.93 & 0.03 & 0.09 & 0.06 & 1.90 & 5.99 & 3.93 & 4.88 & 10.87 & 7.90 & 55.75 & 57.47 & 56.66 \\  & 2 & 18.96 & 28.97 & 24.06 & 0.93 & 0.95 & 0.94 & 0.05 & 0.10 & 0.07 & -0.08 & 2.20 & 0.86 & 1.88 & 4.81 & 3.25 & 139.31 & 160.45 & 149.50 \\  & 3 & 21.43 & 31.89 & 26.91 & 0.93 & 0.95 & 0.94 & 0.04 & 0.09 & 0.06 & 0.01 & 3.14 & 1.40 & 1.65 & 5.42 & 3.35 & 139.67 & 160.86 & 149.86 \\ \hline \multirow{3}{*}{\(MS-NSGAII\)} & 1 & 12.72 & 23.54 & 18.16 & 0.88 & 0.95 & 0.92 & 0.05 & 0.12 & 0.08 & 2.30 & 8.03 & 5.39 & 7.11 & 15.03 & 10.91 & 69.01 & 70.12 & 69.53 \\  & 2 & 11.48 & 18.08 & 14.44 & 0.93 & 0.95 & 0.94 & 0.06 & 0.15 & 0.11 & 0.18 & 4.17 & 2.08 & 1.47 & 6.09 & 3.76 & 166.23 & 176.34 & 172.14 \\  & 3 & 11.71 & 20.17 & 15.30 & 0.93 & 0.95 & 0.94 & 0.07 & 0.16 & 0.11 & \(-\)0.27 & 3.32 & 1.50 & 2.78 & 7.38 & 4.89 & 166.47 & 176.58 & 172.39 \\ \hline \multicolumn{3}{c}{\(NSILS\)} & 10 & 8.67 & 13.98 & 11.15 & 0.93 & 0.96 & 0.94 & 0.07 & 0.16 & 0.11 & \(-\)1.05 & 3.72 & 0.90 & 5.94 & 10.19 & 8.02 & 110.89 & 121.60 & 117.53 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Average results for the 7 versions on 37 instances. av., average. In terms of the \(|F_{0}|\) indicator, globally, _NSGAII-1_ found more solutions in the non-dominated Pareto front, offering the decision maker a set of solutions with a good compromise between the total travel cost and the customer preferences. Regarding the \(SP\) metric, all versions of the _NSGAII_ and _NSILS-2_ methods were almost equivalent with a good coverage in the objective space by members from the non-dominated fronts. According to the \(HV\) metric, from _NSILS_, applying \(LS1\) was better. However, _MS-NSGAII_ was practically equivalent to _NSGAII_ for both local search-based methods. Furthermore, _NSGAII-1_ was the worst one according to this performance indicator. It is important to mention that the position of the solutions in the search space did not have a direct impact on the \(HV\) metric. Even when the \(SP\) metrics and \(|F_{0}|\) were better, this did not guarantee better values for the \(HV\) metric.