## 5 Computational Experiments, Results, and Discussion
### Tuning MOSS Parameters  
Before conducting performance assessment experiments for the developed MOSS, tuning experiments were performed to guide the selection of the best values of its parameters. This was done here using selected small-sized test instances for which the optimal Pareto front solutions were known. These instances were selected from the testbed provided in Abdelmaguid [3], for which the \(\varepsilon\)-constraint method was used in finding the optimal Pareto front solutions. Six small instances from the thirty instances provided in Abdelmaguid [3] were selected for the tuning experiments. They were selected such that there was diversity in their \(|\mathcal{P}|\) values, which equal 2, 4, 8, 11, 14, and 17. This diversity was the result of variation in the structures of these six instances that was necessary to avoid biased results. The MOSS parameters included in the tuning experiments and their selected levels are listed in Table 4.  
Another MOSS parameter is \(\bar{\Delta}\), which defines the distance limit that must be exceeded by a solution to be added to the reference set. This parameter is problem-specific and determined using the formula provided in Abdelmaguid [29], as provided in Equation (5). The value of the constant \(\mathbf{A}\) was set to 3 for small-sized instances and 5 for large-size ones. A small instance was arbitrarily identified with a total number of operations less than 50. \[\bar{\Delta}=\left\lceil\sum_{w\in\mathcal{W}}\frac{|J_{w}|}{\mathbf{A}}+\sum_{ j\in\mathcal{J}}\frac{|W_{j}|}{\mathbf{A}}\right\rceil \tag{5}\]  
Three independent runs were conducted for each treatment combination. Therefore, a total of \(3\times 6\times 3^{7}\) = 39,366 independent runs were conducted in a random order. For each run, \(itr^{max}=100\), and there was no time limit. For each run, the values of \(TGD(\mathcal{D})\), \(\mathcal{H}(\mathcal{D})\) and the computational time (\(CT\)) were recorded. Since the number of iterations was fixed for all runs, the reported average total computational time for a treatment combination \begin{table}
\begin{tabular}{l c} \hline \hline
**Parameter** & **Selected Levels** \\ \hline \(RefSetSize\) & 10, 20, 30 \\ \(\Theta^{X}\) & 0.2, 0.5. 0.8 \\ _TabuSize_ & 10, 20, 40 \\ \(n^{TS}_{max}\) & 20, 50, 100 \\ \(n^{no,imp}_{max}\) & 5, 10, 20 \\ \(n^{best,moss}\) & 5, 10, 20 \\ \(n^{rand,moss}\) & 5, 10, 20 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Factors considered in the parameter-tuning experiments and their levels. allowed for estimating the average computational time requirement at the selected levels of the MOSS parameters. Figure 7a shows the main effect plots for \(\mathcal{H}(\mathcal{D})\). It was noticed that the main effects plots for \(TGD(\mathcal{D})\) demonstrated the same behavior; therefore, they were not included here. Meanwhile, Figure 7b shows the main effects plots for the computational time. Based on these plots and the analysis of variance (ANOVA) results at 95% confidence level, the levels of the variables that will result in the lowest average \(\mathcal{H}(\mathcal{D})\) and \(TGD(\mathcal{D})\), such that they do not affect the average computational time, were selected as follows: \(\Theta^{X}=0.5\), \(TabuSize=10\), \(n_{maxitr}^{no\_imp}=20\), \(n^{best\_moves}=5\), and \(n^{rand\_moves}=5\). There is an apparent tradeoff between the solution quality and the computational time requirement for the remaining two parameters, \(RefSetSize\) and \(n_{maxitr}^{TS}\). This means that, if there is a computational time limit, the selected levels of both parameters, which resulted in low-quality solutions yet low computational times in these tuning experiments, can result in good-quality solutions because they allow the algorithm to conduct a larger number of iterations. Therefore, additional tuning experiments were conducted to help select appropriate levels for these two parameters.