## 5 Computational Experiments, Results, and Discussion
### Results for Small-Sized Instances  
In this section, computational experiments on small-sized instances are conducted to compare the performance of the developed MOSS with the best metaheuristic found in  
Figure 8: Progress of average performance over computational time in the second MOSS-tuning experiments. (**a**) Mean \(\mathcal{H}(\mathcal{D})\) at different \(Ref\)_SetSize_ values. (**b**) Mean \(\mathcal{H}(\mathcal{D})\) at different \(n_{maxitr}^{TS}\) values. (**c**) Mean \(TGD(\mathcal{D})\) at different \(Ref\)_SetSize_ values. (**d**) Mean \(TGD(\mathcal{D})\) at different \(n_{maxitr}^{TS}\) values. the literature for the DMOSP, which is based on NSGA-II [4], along with the comparison with the optimal Pareto front solutions obtained using the \(\varepsilon\)-constraint method [3]. These experiments utilized a set of 30 small-sized instances with \(|\mathcal{P}|\) that ranged from 1 to 23. Based on the results of the tuning experiments in the previous section, the levels of the MOSS parameters were chosen as \(\Theta^{X}=0.5\), \(TabuSize=10\), \(n^{no\_imp}_{maxitr}=20\), \(n^{best\_moves}=5\), \(n^{rand\_moves}=5\), \(RefSetSize=15\), and \(n^{TS}_{maxitr}=40\). Both NSGA-II and MOSS were run with a computational time limit of 60 s.  
For each DMOSP small instance, the average performance metrics, namely the mode of \(|\mathcal{D}|\) (\(|\overline{\mathcal{D}}|\)), its average (\(|\overline{\mathcal{D}}|\)), the average total gravitational distance (\(\overline{TGD}\)), and the average percentage hypervolume deviation (\(\overline{\mathcal{H}}\)), were calculated based on the thirty independent runs conducted using each metaheuristic. Table 5 lists these values. It is important to note that there are minor differences in the NSGA-II results reported here compared to Abdelmaguid [4] since the computational time limit here was 60 s instead of 30 s. Boldface is used in Table 5 to highlight the best value for each metric when both metaheuristics' results are compared. When comparing the two measures of the resultant number of non-dominated solutions (\(|\mathcal{D}|\)), namely the mode \(|\overline{\mathcal{D}}|\) and the average \(|\overline{\mathcal{D}}|\), as reported in Table 5, it is evident that the developed MOSS is capable of producing a larger number of non-dominated solutions in 18 instances, versus only 3 for the NSGA-II based on the mode. Meanwhile, for the average, MOSS resulted in better values in 24 instances versus 5 for NSGA-II. Therefore, it can be concluded that the developed MOSS has better search capability that enables it to find more non-dominated solutions compared to NSGA-II. As illustrated in Table 5, the developed MOSS resulted in better \(\overline{TGD}\) in 25 instances. Similarly, for \(\overline{\mathcal{H}}\), MOSS resulted in better values in 25 instances. For both metrics combined, MOSS resulted in better values in 23 instances versus only 3 for NSGA-II. As an overall average performance for the 30 instances, the average TGD of MOSS was 12.84, compared to 18.90 for NSGA-II, and the average percentage hypervolume deviation of MOSS was 1.83, as opposed to 2.74 for NSGA-II. To statistically compare these results, a paired \(t\)-test was conducted on \(\overline{TGD}\) and \(\overline{\mathcal{H}}\). The paired \(t\)-test was suitable for this comparison since the experiments were conducted on the same group of MOSS instances, and the calculated averages were approximately normally distributed due to the central limit theorem. Figure 9 shows the box-and-whisker plots for the mean differences of both metrics, along with the confidence interval at a 95% confidence level of the null hypothesis (\(H_{o}\)). Here, \(H_{o}\) is stated as the difference of the means equaling zero. Figure 9: Box-and-whisker plots and confidence intervals for the average difference in \(\overline{TGD}\) and \(\overline{\mathcal{H}}\), based on the results of a paired \(t\)-test for the computational experiments on small-sized instances at a 95% confidence level.