## 4 Performance Metrics  
Many metrics have been proposed in the literature to assess the performance of multi-objective optimization algorithms [49]. The simplest of them is the number of non-dominated solutions that a metaheuristic generates (\(|\mathcal{D}|\)). This metric is an indication of the capability of a metaheuristic to generate alternative non-dominated solutions. However, it does not provide any clue about their quality. Whenever the set of optimal Pareto front solutions (\(\mathcal{P}\)) is known for an instance, the gravitational distance (\(GD\)) and the inverted gravitational distance (\(IGD\)) can be used to assess the quality of the generated non-dominated solutions via a heuristic (\(\mathcal{D}\)) with respect to the solutions in set \(\mathcal{P}\). These metrics are evaluated using Equations (1) and (2) below. \[GD(\mathcal{D}) =\frac{1}{|\mathcal{D}|}\sqrt{\sum_{s\in\mathcal{D}}\min_{q\in \mathcal{P}}\{\Delta OF(s,q)\}} \tag{1}\] \[IGD(\mathcal{D}) =\frac{1}{|\mathcal{P}|}\sqrt{\sum_{q\in\mathcal{P}}\min_{s\in \mathcal{D}}\{\Delta OF(s,q)\}} \tag{2}\]  
where,  
\[\Delta OF(s,q)=(C_{\text{max}}(s)-C_{\text{max}}(q))^{2}+(MWFT(s)-MWFT(q))^{2}\]  
Both metrics are suitable for the studied DMOSP since it is a discrete optimization problem characterized by a small number of non-dominated solutions for the considered objectives [3; 4]. Small values for both metrics are favorable, and zero values for both indicate that \(\mathcal{D}=\mathcal{P}\). The main difference between \(GD\) and \(IGD\) is that the former indicates how successful the heuristic is in reaching solutions close to a subset of the solutions in set \(\mathcal{P}\). However, it fails to indicate whether or not a heuristic can find solutions with proximity to all solutions in set \(\mathcal{P}\) (i.e., covering all optimal Pareto front solutions). Meanwhile, the latter metric measures the closeness of solutions in set \(\mathcal{P}\) to their nearest ones in set \(\mathcal{D}\). This inherently checks for the ability of a heuristic to cover all solutions in set \(\mathcal{P}\). However, unlike \(GD\), it fails to account for the negative effect of inferior non-dominated solutions in set \(\mathcal{D}\) that can be quite far from the optimal Pareto front. This paper proposes a combined metric, the total gravitational distance (\(TGD\)), calculated as \(TGD(\mathcal{D})=GD(\mathcal{D})+IGD(\mathcal{D})\). The \(TGD\) attempts to combine the benefits of \(GD\) and \(IGD\) metrics and subdue their shortcomings. As a result, small values for \(TGD\) indicate that the heuristic results in solutions close to the Pareto front with good coverage. For large-sized instances with unknown Pareto front solutions, the performance of a multi-objective optimization algorithm can be compared to benchmark algorithms using hypervolume (HV), which was introduced by [50]. The reference point used here to calculate the hypervolume is arbitrarily selected as \(\overline{C_{\text{max}}}=1.5\times LB_{Cmax}\) and \(\overline{MWFT}=1.8\times LB_{MWFT}\), where \(LB_{Cmax}\) and \(LB_{MWFT}\) are the lower bounds for the makespan and the MWFT of a DMOSP instance, respectively. The evaluation formulas for the lower bounds are provided in [4]. In this paper, the percentage hypervolume measure (\(HV\%\)) introduced by [4] is used to compare the performance of the developed MOSS with the previously developed NSGA-II approach. The \(HV\%\) is a relative measure in which the value of \(HV(\mathcal{D})\) is divided by the area of the rectangle defined according to the two vertexes (\(\overline{C_{\text{max}}},\overline{MWFT}\)) and (\(LB_{C_{\text{max}}},LB_{MWFT}\)), as expressed in Equation (3). \[HV\%(\mathcal{D})=\frac{HV(\mathcal{D})}{0.5 LB_{Cmax}\times 0.8 LB_{MWFT}} \times 100 \tag{3}\]  
When comparing two multi-objective optimization algorithms, a higher \(HV\%(\mathcal{D})\) value signifies better performance.