## 5 Experimental Results
### Characteristics of the Proposed Approach
#### 5.1.3 Fraction of Non-Dominated Solutions  
The fraction of the population that belongs to the first front shows how difficult it is for an algorithm to differentiate between the solutions and select the most fitted ones. If all solutions belong to the first front, then they are all incomparable for the given algorithm, in the sense that none of them is preferred over another. The inability of an algorithm to differentiate between solutions indicates that it cannot direct the evolution process. For the small example in Section 3, we show that sorting by _PO-prob_ with \(\epsilon>0\) results in a first front with only one element, and in the lowest number of ties (incomparable solutions) in general. This observation is also supported by our experimental results. Apart from some rare cases, _PO-prob_ with \(\epsilon=1/pop\_size\) assigns every solution to a separate front. However, when the population size increases, the probability of having more than one solution per front quickly tends to zero. This is also supported by the results presented in Table 1, _PO-prob_ with \(\epsilon=0.1\) also assigns one point per front, with the last front being an exception. _PO-prob_ with \(\epsilon=0\) has the same properties with one exception: the number of solutions in the first front is equal to the number of objectives \(n_{k}\). Indeed, in this case, all solutions that are the best according to one of the objectives will have the value of Pareto optimality equal to \(0\), see the discussion for _PO-prob_ in Section 3. This observation holds true for all setups and for every generation. It also demonstrates the ability of _PO-prob_ to differentiate between solutions in both multi- and many-objective optimization. As opposed to _PO-prob_, sorting the given population by _PO-count_ and by _PD_ always results in the same first front. Indeed, all nondominated solutions have Pareto optimality values equal to \(0\) and are assigned to the first front, see Section 3. The remaining fronts, however, are typically different. This results in different solutions generated over the evolution process. To illustrate this difference, we show in Figure 8 the fraction of solutions belonging to the first non-dominated front for all studied algorithms. Note, that the first non-dominated front here is identified with sorting by Pareto dominance and is referred to as _first ND-front_. However, every algorithm uses its own sorting procedure to evolve the population. We present results for the following number of knapsacks: \(n_{k}=2\), \(n_{k}=6\), and \(n_{k}=10\). The relative distributions for random and tournament selection have different trends only in the case of two knapsacks. That is why, for \(n_{k}=6\) and \(n_{k}=10\), we report the results only for random selection. Figure 8a,b show that for two knapsacks, the size of the first non-dominated front is relatively low for all algorithms. In the case of random selection, see Figure 8a, the maximum value of 30% is reached by _PO-prob\({}^{*}\)_ for _gen_num_ close to 500. The fraction of solutions in the fist ND-front is approximately 10% for _PO-prob_ and does not exceed 25% for _NSGA-II_, _NSGA-III_, and _PO-count_. The large size of the first non-dominated front produced by _PO-prob\({}^{*}\)_ is due to the combination of extreme solutions constructed by _PO-prob_ during the first 350 iterations and solutions spread out in the middle, which are constructed by _NSGA-II_ during the last 150 generations, see Figure 4b. For tournament selection, all algorithms produce the first non-dominated front of almost the same size, see Figure 8b. The maximum value in this case does not exceed 20%. It is visible, however, that the size of the first ND-front is slightly less for _PO-prob_ and is slightly larger for _PO-prob\({}^{*}\)_. This agrees with the results presented in Figure 5b. As before, _PO-prob\({}^{*}\)_ identifies more extreme solutions than _NSGA-II_, but this difference is not as large as in the case of random selection, see Figure 4b for comparison.