## 4 Experimental Setup  
Our experimental setup is summarized in Table 2. To perform the experimental evaluations, we chose the 0/1 multi-objective knapsack problem with independent objectives as defined in [35]. The mathematical formulation of this problem is presented in Appendix A. As stated in [35], this optimization problem is easy to formulate, but at the same time, it is rather general. It is also representative of a certain class of real-world problems. This problem was used in numerous research papers to compare evolutionary algorithms and to study different aspects of their performance, see [4; 36], for example. Finally, it scales easily to higher dimensions, preserving its properties.  
We use 10 different test problems with the number of objectives \(n\_k\in\{2-8,10,15,25\}\) and 250 items. We adopt two selection schemes: random selection and binary tournament selection with replacement. For all algorithms and all settings, we execute genetic optimization for 500 generations. We also use uniform crossover with mutation probability 0.01 and population size 250.  
We implement our approach within the framework of the _NSGA-II_ algorithm by redefining the sorting procedure using _PO-count_ and _PO-prob_ as an alternative to _PD_, see step 4.1 in Figure 2. We refer to these algorithms by the name of the chosen sorting method. Additionally, we experiment with a combination of _PO-prob_ and the classical _NSGA-II_. In this approach, we run _PO-prob_ during the first 350 generations and then switch to _NSGA-II_ during the last 150 generations. We refer to the latter approach as _PO-prob\({}^{*}\)_.  
\begin{table}
\begin{tabular}{c c} \hline \hline Problem & \(0/1\) knapsack, \(n_{k}\in\{2-8,10,15,25\}\), 250 \\ \hline Baseline & _NSGA-II_, _NSGA-III_ \\ Algorithms & _PO-count_, _PO-prob_, _PO-prob\({}^{*}\) \\ Implementation & python package _DEAP_ \\ Parent selection & random, binary tournament \\ Crossover & uniform \\ Parameters & \(n\_gen=500\), \(pop\_size=250\), \(mut\_prob=0.01\) \\ Runs & 30 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Experimental setup.  
As it was shown in Section 3, sorting based on Pareto optimality results in less incomparable solutions (ties). In addition, theoretical results from [14] guarantee maximization of diversity when sorting by \(PO\). Thereby, we hypothesize that \(PO\)-based ranking can produce better results for many-objective problems. To demonstrate this, we compare our approaches with the basic _NSGA-II_ algorithm and its modification designed for many-objective optimization, _NSGA-III_. The _NSGA-III_ algorithm additionally takes reference points as an input parameter. We use deap.tools.uniform_reference_points function to generate uniformly distributed reference points. As suggested by the authors of _NSGA-III_, the number of reference points is chosen to be close the value of population size, see [16]. We limit the number of baseline approaches to these two classical algorithms because, as stated in [36; 37], many new algorithms are overspecialized and can perform poorly in general settings.  
We implement the proposed algorithms as an extension to the \(DEAP\) python library. Unless stated otherwise, the values reported in this paper are averages of 30 independent runs.