## 2 Kaggle's Airline Passenger Satisfaction Data
We introduce direct and computationally efficient RB's representation in the following form:  
\[RB=\{sw_{1}^{(r)},sw_{2}^{(r)},\ldots,sw_{n}^{(r)},j^{(r)}\}_{r=1}^{R}. \tag{3}\]  
In turn, DB contains: (i) a tunable part, i.e., parameters of fuzzy sets representing numerical attributes and (ii) a non-tunable part, i.e., domains of qualitative attributes \(X_{i}=\{x_{i1},x_{i2},\ldots,x_{ii_{i}}\}\), \(i\in\{1,2,\ldots,n\}\) and the set of class labels \(Y=\{y_{1},y_{2},\ldots,y_{c}\}\). We developed original crossover and mutation operators for the transformation of the RB population as well as we adopted some specialized crossover and mutation operators for the DBs processing--see [11; 12; 13] for details. Evolutionary optimization objectives--FRBC's accuracy: the accuracy measure (subject to maximization) is defined as follows:  
\[Q_{ACC}^{(lrn)}=1-Q_{RMSE}^{(lrn)}, \tag{4}\]  
where  
\[Q_{RMSE}^{(lrn)}=\sqrt{\frac{1}{Kc}\sum_{k=1}^{K}\sum_{j=1}^{c}\left[\mu_{B_ {k}^{(singl,(lrn)}}(y_{j})-\mu_{B_{k}^{\prime}}(y_{j})\right]^{2}}. \tag{5}\]  
\(Q_{RMSE}^{(lrn)}\in[0,1]\); \(B_{k}^{\prime}\) (characterized by its membership function \(\mu_{B_{k}^{\prime}}(y)\)) is the fuzzy-set response of system (2) for the learning data sample \(\mathbf{x}_{k}^{(lrn)}\), and \(B_{k}^{(singl,)(lrn)}\) (represented by its membership function \(\mu_{B_{k}^{(singl,)(lrn)}}(y)=1\) for \(y=y_{k}^{(lrn)}\) and 0 elsewhere) is the desired fuzzy-singleton response for that sample. Evolutionary optimization objectives--FRBC's interpretability: we use the notion interpretability in a broader sense including not only semantic aspects of FRBC but also its complexity. We proposed the following measure (subject to maximization) for the evaluation of the FRBC's complexity-related interpretability:  
\[Q_{INT}=1-Q_{CPLX}, \tag{6}\]  
where  
\[Q_{CPLX}=\frac{Q_{RINP}+Q_{INP}+Q_{FS}}{3}, \tag{7}\]  
and  
\[Q_{RINP}=\frac{1}{R}\sum_{r=1}^{R}\frac{n_{INP}^{(r)}-1}{n-1},\ n>1, \tag{8}\]  
\[Q_{INP}=\frac{n_{INP}-1}{n-1},\ Q_{FS}=\frac{n_{FS}-1}{\sum_{i=1}^{n}a_{i}-1}, \ n>1. \tag{9}\]  
The FRBC's complexity measure \(Q_{CPLX}\) (7) (\(Q_{CPLX}\in[0,1]\); 0 and 1 represent minimal and maximal complexities, respectively) is an average of three sub-indices that measure an average complexity of particular rules \(Q_{RINP}\) (8) and the complexity of the whole system in terms of its active inputs \(Q_{INP}\) (9) and active fuzzy sets \(Q_{FS}\) (9). \(n_{INP}^{(r)}\) in (8) is the number of active input attributes in the \(r\)-th rule. \(n_{INP}\) and \(n_{FS}\) in (9) are the numbers of active inputs and fuzzy sets (linguistic terms), respectively, in the whole system. In turn, the FRBC's semantic-related interpretability is addressed by us by implementing strong fuzzy partitions (SFPs) [25] of domains of all numerical attributes. SFPs are fuzzy partitions in which the sum of the values of all membership functions for any domain value is equal to 1. SFPs satisfy the desired semantics-related interpretability demands [25]. Simple and computationally efficient implementation of SFP requirements for trapezoidal membership functions can be formulated as follows (see Figure 2 for three-set SFP of \(x_{i}\)):  
\[\sigma_{ik_{i}}=\rho_{i,k_{i}-1}=d_{ik_{i}}-e_{i,k_{i}-1},\quad k_{i}=2,3, \ldots,a_{i} \tag{10}\]  
and, obviously,  
\[ \begin{array}{l}e_{i1}\leq d_{i2}\leq e_{i2}\leq\cdots\leq d_{i,a_{i}-1}\leq e _{i,a_{i}-1}\leq d_{ia_{i}},\\ i=1,2,\ldots,n.\end{array}  \tag{11}\]  
MOEOAs used: as already mentioned in the Introduction of the paper, for comparison purposes, two MOEOAs are used in our experiments. They include the well-known SPEA2 method and our generalization of SPEA2 referred to as SPEA3. Three indices are usually applied to evaluate and compare different MOEOAs [25]. These indices include: (i) the accuracy of non-dominated solutions obtained, (ii) the spread of solutions in the solution set, and (iii) the distribution of solutions in the solution set.