## Appendix C Average Distance to Diagonal  
A diagonal in the \(n\)-dimensional hyperspace is defined with the following equation: \(x_{1}=x_{2}=\cdots=x_{n}\). In the case of \(n=2\) and \(x\) and \(y\) being two axes, the conventional definition of the diagonal is the following: \(y=x\). The optimization problem considered in this paper is symmetric. This means that all objectives have the same scale, or,for all objectives, the values are distributed similarly. In such a case, the Euclidean distance between a solution and the diagonal shows how close (for large distances) or how far (for small distances) the solution is to the extremes. The values of the average distance to the diagonal for four different numbers of knapsacks, \(n_{k}\in\{2,4,6,8\}\), are shown in Figure A3.  
As we can see, on all figures, the values for _PO-prob_ and _PO-prob\({}^{*}\)_ are very close to each other up to generation 350. After that, we can observe different behaviors of plots for these two algorithms. This observation, similarly to previous ones, is explained by the definition of _PO-prob\({}^{*}\)_ as a combination of _PO-prob_ and _NSGA-II_.  
Let us analyze the results for **random selection**. In Figure 10(a), we can see that for two knapsacks, \(n_{k}=2\), the curves for _NSGA-II_, _NSGA-III_, and _PO-count_ are very close to each other, and the associated solutions are relatively close to the diagonal. On the other hand, for \(gen\geq 200\), the solutions produced by _PO-prob_ and _PO-prob\({}^{*}\)_ are more than two times farther away from the diagonal. It means that these algorithms produce solutions that are much closer to extreme values and cover less the central part of the Pareto-frontier. This observation also agrees with the results presented in Section 5.1.1, see Figure 3(b). As expected, after the number of generations reaches 350, the solutions of _PO-prob\({}^{*}\)_ tend to be closer to the diagonal, as the _NSGA-II_ selection procedure is used.  
We can notice, however, that this general trend changes when the number of knapsacks increases. For \(n_{k}=4\), all algorithms have similar values for the average distance to the diagonal, see Figure 10(c). For six knapsacks, the relation is already inverted: the solutions produced by _NSGA-II_ and _PO-count_ tend to extremes more than those for other algorithms, see Figure 10(e). For \(n_{k}=8\), starting from \(gen=200\), the solutions produced by _PO-prob_ are at least 25% closer to the diagonal than those produced by _NSGA-II_ and _PO-count_, see Figure 3(g). Also note that the solutions produced by _NSGA-III_ always have a relatively low distance to the diagonal, regardless of the number of knapsacks. This can be explained by the fact that all chosen solutions in _NSGA-III_ are attached to reference points that do not change during the whole evolutionary process. All this supports the relative findings from Section 5.1.2.  
In the case of **tournament selection**, we observe a similar dependency between the average distance to the diagonal and the number of knapsacks. The average distance to the di agonal for _PO-prob_ and _PO-prob_* decreases when \(n\_k\) increases. This effect becomes visible faster than for random selection; compare Figure A3d,c. Moreover, we can notice that for all algorithms, tournament selection tends to produce solutions that are farther away from the diagonal for larger numbers of knapsacks. In the case of \(n_{k}=2\), however, _PO-prob_ with random selection produces more extreme solutions than with tournament selection, see Figure A3a,b. This observation also corresponds to the results presented in Figures 4b and 5f of Section 5.1.1.