## 2 Kaggle's Airline Passenger Satisfaction Data
Learning data set \(L\): it is a basis for the FRBC's design from data and contains \(K\) input-output samples:  
\[L=\{x_{k}^{(lrn)},y_{k}^{(lrn)}\}_{k=1}^{K}, \tag{1}\]  
where \(x_{k}^{(lrn)}=(x_{1k}^{(lrn)},x_{2k}^{(lrn)},\ldots,x_{nk}^{(lrn)})\in\mathbf{X}=X _{1}\times X_{2}\times\cdots\times X_{n}\) (\(\times\) stands for Cartesian product of ordinary sets) is the set of input attributes and \(y_{k}^{(lrn)}\) is the corresponding class label (\(y_{k}^{(lrn)}\in\Upsilon\)) for the \(k\)-th data sample, \(k=1,2,\ldots,K\). Attribute representation: each numerical attribute \(x_{i},i\in\{1,2,\ldots,n\}\) is represented by \(a_{i}\) fuzzy sets \(A_{i_{k}}\in F(X_{i})\) (\(F(X_{i})\) denotes a family of all fuzzy sets defined in the universe \(X_{i}\)), \(k_{i}=1,2,\ldots,a_{i}\). \(A_{i_{1}}\) is an \(S\)-type fuzzy set (representing linguistic term \"_Small_\"), \(A_{i_{2}},A_{i_{3}},\ldots,A_{i,a_{i}-1}\) are \(M\)-type sets (representing linguistic terms \"_Medium_ 1_\", \"_Medium_ 2_\",..., \"_Medium_ \(a_{i}-2\)\"). For simplicity, \(A_{ik_{i}}\)s denote also the corresponding linguistic terms. Trapezoidal membership functions of \(S\)-, \(M\)-, and \(L\)-type fuzzy sets are shown in Figure 1. Each qualitative attribute \(x_{i}\), \(i\in\{1,2,\ldots,n\}\) (\(x_{i}\in X_{i}=\{x_{i1},x_{i2},\ldots,x_{ia_{i}}\}\)) is characterized by \(a_{i}\) fuzzy singletons \(A_{ik_{i}}=A_{ik_{i}}^{(singl. )}\), \(k_{i}=1,2,\ldots,a_{i}\) defined for particular \"values\" \(x_{ik_{i}}\) of \(x_{i}\) as follows: \(\mu_{A_{ik_{i}}^{(singl. )}}(x_{i})=1\) for \(x_{i}=x_{ik_{i}}\) and \(0\) elsewhere. Similarly, class labels \(y_{j^{(r)}}\) in particular rules (2) (\(j^{(r)}\in\{1,2,\ldots,c\}\)) are represented by appropriate fuzzy singletons \(B_{j^{(r)}}^{(singl. )}\) with the following membership functions: \(\mu_{B_{j^{(r)}}^{(singl. )}}(y)=1\) for \(y=y_{j^{(r)}}\) and \(0\) elsewhere. FRBC's knowledge base: it contains \(R\) genetically optimized fuzzy rules discovered in the learning data set (1). We introduced the following form of the \(r\)-th rule, \(r=1,2,\ldots,R\) (\(R\) changes during the learning process):  
\[ \begin{split}&\textbf{IF}\ [x_{1}\ \text{is}\ [\text{not}]_{(sw_{1}^{(r)}<0)}\ A_{1,|sw_{1}^{(r)}|} ]_{(sw_{1}^{(r)}\
eq 0)}\ \textbf{AND}...\textbf{AND}\\ &\ [x_{n}\ \text{is}\ [\text{not}]_{(sw_{n}^{(r)}<0)}\ A_{n,|sw_{n}^{( r)}|}]_{(sw_{n}^{(r)}\
eq 0)}\\ &\ \textbf{THEN}\ y\ \text{is}\ B_{j^{(r)}}^{(singl.)},\end{split}  \tag{2}\]  
where: (i) \([\textit{expression}]_{(condition)}\) in (2) denotes conditional inclusion of \([\textit{expression}]\) into a given rule if and only if (\(\textit{condition}\)) is fulfilled, (ii) \(|\cdot|\) returns the absolute value,  
Figure 1: Trapezoidal membership functions of \(S\)-type, \(M\)-type, and \(L\)-type fuzzy sets and their parameters. and (iii) \(sw_{i}^{(r)}\) is a switch which controls the presence/absence of the \(i\)-th input attribute in the \(r\)-th rule, \(i=1,2,\ldots,n\). \(sw_{i}^{(r)}\in\{0,\pm 1,\pm 2,\ldots,\pm a_{i}\}\), where \(a_{i}\) is the number of fuzzy sets (linguistic terms) defined for the \(i\)-th attribute. For \(sw_{i}^{(r)}=0\), the \(i\)-th attribute is excluded from (not active in) that rule, whereas for \(sw_{i}^{(r)}>0\) the component \([x_{i}\) is \(A_{ik_{i}}]\) (\(k_{i}=|sw_{i}^{(r)}|\)) is included (active) and for \(sw_{i}^{(r)}<0\) the component \([x_{i}\) is not \(A_{ik_{i}}]\) is used in that rule (not \(A_{ik_{i}}=\bar{A}_{ik_{i}}\) and \(\mu_{A_{ik_{i}}}(x_{i})=1-\mu_{A_{ik_{i}}}(x_{i})\); \(\mu_{A_{ik_{i}}}(x_{i})\) and \(\mu_{A_{ik_{i}}}(x_{i})\) are membership functions of fuzzy sets \(A_{ik_{i}}\) and \(\bar{A}_{ik_{i}}\)). In our approach two entities, i.e., rule base structure RB and data base DB represent FRBC's knowledge base.