## 5 Experimental Results
### Performance
#### 5.2.1 Hypervolume  
Hypervolume is the metric that indicates the volume of the hyperspace dominated by a set of solutions. This metric is widely adopted in practice both for evaluation of genetic optimization algorithms and for guiding the evolutionary process in indicator-based methods [38]. The value of hypervolume depends on the chosen reference point. To simplify the computational process and due to the fact that we solve a maximization problem, we adopt the origin of coordinates as a reference point for hypervolume calculation. In our case, this point also corresponds to the worst possible solution. Larger hypervolume values indicate better performance.  
The values of the hypervolume indicator obtained for different numbers of knapsacks for _NSGA-II_ are presented in the second column of Table 3. The first part of the table corresponds to random selection and the second to tournament selection. We can see that the hypervolume values obtained with random selection are of the same scale as those obtained with tournament selection. In addition, when \(n_{k}<10\), random selection results in higher values of hypervolume. However, when the number of knapsacks reaches 10, tournament selection becomes more effective in terms of hypervolume maximization.  
Columns 3-6 show the relative increase (positive number, bold font) or decrease (negative number) in the hypervolume indicator for other algorithms. For a more intuitive representation, the values of these columns are also graphically presented in Figure 9. The first observation that we can make is that _NSGA-III_, while developed for many-objective optimization, almost always results in lower hypervolume values, even for larger numbers of objectives. This supports similar observation from [39]. _NSGA-III_ reaches its minimum in relative increase in hypervolume of approximately \(-10\%\) for \(n_{k}=7\) for random selection and for \(n_{k}=8\) for tournament selection. After that, it starts increasing if random selection is used, and it does not change much for tournament selection.  
Figure 9: Relative increase in hypervolume compared to _NSGA-II_ as a function of the number of knapsacks \(n_{k}\). The legend is shared among plots.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(n_{k}\) &
\begin{tabular}{c} _NSGA-II,_ \\ **h-Vol.** \\ \end{tabular} & _NSGA-III_ & _PO-Count_ & _PO-Prob_ & _PO-Prob\({}^{*}\)_ \\ \hline \multicolumn{5}{c}{Random selection} \\ \hline  
\hline \hline \end{tabular}
\end{table}
Table 3: Increase in hypervolume relative to NSGA-II, %.  
The values of relative hypervolume increase for _PO-count_ are very close to 0. This means that _PO-count_ results in a population covering the same hypervolume as _NSGA-II_. The largest difference between _NSGA-II_ and _PO-count_ is observed for \(n_{k}=25\) with tournament selection. In this case, the hypervolume of _PO-count_ is 1.86% less than that of _NSGA-II_. Contrarily, both _PO-prob_ and _PO-prob_* improve hypervolume significantly as compared to _NSGA-II_. This difference is visible for small \(n_{k}\) and is especially prominent for large numbers of knapsacks. When \(n_{k}<5\), _PO-prob_ and _PO-prob_* increase the hypervolume by up to 4%. After that, we observe an inverse pattern. However, the relative decrease in this case does not exceed \(-2.13\%\). Finally, starting from \(n_{k}=8\) for random selection and \(n_{k}=10\) for tournament selection, _PO-prob_ again results in larger hypervolume values. For _PO-prob_*, this happens even faster: \(n_{k}=7\) for random selection and \(n_{k}=8\) for tournament selection. From this point on, we see a rapid growth with a maximum relative increase observed for \(n_{k}=25\): +61.23% and +51.88% for random selection, and +30.35% and +28.59% for tournament selection. The fact that _PO-prob_* is closer to _NSGA-II_ than _PO-prob_ is due to _PO-prob_* turning into _NSGA-II_ during the last 150 generations. We can also notice that the effect of the _PO-prob_ algorithm is less visible in the case of tournament selection.