## 2 Materials and Methods
### Dominance-Based Rough Set Approach
#### 2.1.2 Inference of Decision Rules  
In the next step, knowledge is extracted from the already structured data. For this purpose, DRSA is coupled with decision rules, capturing the conditions capable of distinguishing the classes, hence relating performances on one or more criteria to a desired category. In the most common scenario that we implement in this work, the analysis involves inference of certain rules based on the lower approximations. This is due to their high reliability. In the case of non-monotonic criteria and binary classification, such rules take the following form: \(g_{j_{1}}(a_{i})\geq r_{j_{1}}\wedge\ldots\wedge g_{j_{k}}(a_{i})\leq r_{j_{k}} \wedge\ldots\Rightarrow a_{i}\in Cl_{r}\), where \(g_{j_{1}},\ldots,g_{j_{k}},\ldots\in G\) and \(r_{j_{1}},\ldots,r_{j_{k}},\ldots\) correspond to the performances of alternatives contained in \(\underline{P}(Cl_{r})\). To induce decision rules, we use DOMLEM (see Algorithm 1) [37]. It is applied separately to each decision class. The procedure is based on a greedy approach, iteratively constructing a conjunction of conditions to cover only alternatives from the given approximation. To select the best condition, firstly, its credibility, expressed as the fraction of correctly classified alternatives, is used. In case of a tie, generality, expressed as the total number of covered alternatives, is employed. Once the rule covering only alternatives from the lower approximation is created, a search for redundant conditions is performed. If, after removing some condition, the rule still covers only the desired alternatives, then such a condition is removed to ensure its minimality. The procedure of creating new rules is continued until all alternatives from the lower approximation are covered. Finally, if after removing some rule, the remaining ones still cover all alternatives from the lower approximation, then the rule is removed to ensure the minimality of the inferred set. To demonstrate the operating steps of DOMLEM, we refer to a decision matrix given in Table 2. There are four alternatives in class \(Cl_{0}\) (\(a_{1}-a_{4}\)) and three alternatives in class \(Cl_{1}\) (\(a_{5}-a_{7}\)). For both classes, the lower and upper approximations are equal. Let us first infer certain rules for \(Cl_{1}\). In the initial iteration, there are six potential conditions: \(g_{1}\geq 2\), \(g_{1}>1\), \(g_{1}\leq 2\), \(g_{1}\leq 2\), \(g_{1}\leq 1\), \(g_{2}\geq 2\), and \(g_{2}\leq 2\). Among them, \(g_{2}\geq 2\) has the highest credibility of \(\frac{3}{4}\) since it covers four alternatives, of which three belong to \(Cl_{1}\). Therefore, it is added to the working condition part of the rule. Since it does not exclusively cover alternatives from the lower approximation of \(Cl_{1}\), further conditions have to be included. There are five possible conditions left. Each is evaluated only on the set of alternatives covered by the constructed rule (in this case, \(a_{4}-a_{7}\)). For example, condition \(g_{1}\geq 2\) covers two out of four alternatives (\(a_{5}\) and \(a_{7}\)). In this iteration, conditions \(g_{1}\geq 1\) and \(g_{1}\geq 2\) share the highest credibility of one, but the former covers three alternatives while the latter only two. The conjunction of \(g_{2}\geq 2\) and \(g_{1}\geq 1\) covers only alternatives from \(\underline{P}(Cl_{1})\), so there is no need to add further conditions. Moreover, eliminating either condition will result in a rule covering alternatives from \(Cl_{0}\). Thus, rule \"if \(g_{2}\geq 2\) and \(g_{1}\geq 1\), then \(Cl_{1}\)\" is minimal. It also covers all alternatives from \(\underline{P}(Cl_{1})\), so no more rules need to be inferred. \begin{table}
\begin{tabular}{c c c c} \hline
**Alt.** & \(\boldsymbol{g_{1}}\) & \(\boldsymbol{g_{2}}\) & \(\boldsymbol{D}\) \\ \hline \(a_{1}\) & 3 & 2 & \(Cl_{0}\) \\ \(a_{2}\) & 5 & 1 & \(Cl_{0}\) \\ \(a_{3}\) & 5 & 1 & \(Cl_{0}\) \\ \(a_{4}\) & 3 & 2 & \(Cl_{1}\) \\ \(a_{5}\) & 4 & 3 & \(Cl_{1}\) \\ \hline \end{tabular}
\end{table}