# Extended Fuzzy-Based Models of Production Data Analysis within AI-Based Industry 4.0 Paradigm
## 2 Materials and Methods
### Specificity of the Rules  
The specificity of the presented method is that the benchmark data give us indication of the certain elements, but it is not complete knowledge. We accept that is not complete data. Therefore, we do not define linguistic numbers with typical covering with the fuzzy numbers. What we have is the universe of the variable, and the indication of the intervals' (granules) characteristic for the given output data. Fuzzy sets defined with such intervals do not fulfill typical expectations from the linguistic variables as membership sum to 1 for all elements of the input space. Figure 1 shows one of the generated input variables regarding source data and spread.  
As we attempt to recognize known data classes without a complete knowledge about whole input parameters, we may propose only rules that point us to the information we are looking for. Therefore, having 47 reference datasets, we transform them into 47 rules.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Input No.** & **Input Description** & \multicolumn{6}{c}{**Numeric Values and Corresponding Linguistic Values**} \\ \hline \multirow{3}{*}{1} & Surface roughness & 6.3 & 10 & 20 & & & & \\ \cline{2-10}  & Fuzzy sets—cores & low & low & high & & & & \\ \hline \multirow{3}{*}{2} & Shape of milling cutter & 10 & 20 & 25 & 80 & 100 & 250 & & \\ \cline{2-10}  & Fuzzy set & low & low & medium & medium & high & high & & \\ \hline \multirow{3}{*}{3} & Milling tool length & 32 & 50 & 55 & 92 & 100 & 125 & 165 & 175 \\ \cline{2-10}  & Fuzzy sets—cores & low & low & medium & medium & high & very high & very high \\ \hline \multirow{3}{*}{4} & Average cutting speed & 60 & 120 & 127.5 & 210 & 222 & 270 & 318 & 375 \\ \cline{2-10}  & Fuzzy sets—cores & low & low & medium & medium & high & very high & very high \\ \hline \multirow{3}{*}{5} & Cutting depth & 2.5 & 8 & 9 & 12.5 & 15 & 20 & 25 & 75 \\ \cline{2-10}  & Fuzzy sets—cores & low & low & medium & medium & high & very high & very high \\ \hline \multirow{3}{*}{6} & Milling width & 0.15 & 0.25 & 0.3 & 0.5 & 1 & & \\ \cline{2-10}  & Fuzzy sets—cores & low & low & medium & medium & high & & \\ \hline \multirow{3}{*}{7} & Cutting feed & 178 & 249 & 270 & 338 & 361 & 722 & 795 & 1032 \\ \cline{2-10}  & Fuzzy sets—cores & low & medium & medium & high & high & very high & very high \\ \hline \multirow{3}{*}{8} & Operating cost & 75 & 260 & 315 & 455 & 486 & 765 & 1700 & \\ \cline{2-10}  & Fuzzy sets—cores & low & medium & medium & high & high & very high & \\ \hline \hline \end{tabular}
\end{table}
Table 2: Detailed data of input variables for variant 2.  
The rules came from the reference data and have form:  
\[\text{IF}v^{1}isFS^{1}_{j1}\text{ and }v^{2}\text{ is }FS^{2}_{j2}\text{ and }\dots\text{ and }v^{k}\text{ is }FS^{k}_{j\kappa}\text{THENY}^{l}=Y^{l}_{yes},\]  
where: \(v\)--inputs, \(k\)--the number of input spaces, \(FS^{k}_{jk}\)--\(j\)-th fuzzy set of \(k\)-th input space, \(\gamma^{l}\)--\(l\)-th output space, \(Y^{l}_{yes}\)--fuzzy set describing confirmation of recognition a class.  
The structure of the system is presented in Figure 2.