## Intelligent System to Support the Decision-Making Process in Lean Maintenance Management
### The Theory of Rough Sets to Support the Lean Maintenance Assessment  
The rough set theory is one of the fastest growing branches of data exploration. It allows for a formal approach to all phenomena related to knowledge processing, therefore it is used as a methodology in the process of knowledge discovery from data. In particular, it can be used to test the imprecision and uncertainty in the data analysis process. It enables finding the relationship between explanatory variables (conditional attributes) and explained variables (decision attributes), which facilitates supporting decision-making based on data. It is also used to reduce dimensionality, consisting of removing from the dataset those explanatory variables that do not significantly affect the explained variables. Knowledge derived from data based on the rough set theory is recorded in the form of decision rules [55]. Details on the formal description of the rough set theory can be found, among others, at work [56]. Often, the purpose of the decision-making system based on rough sets is to search for hidden, and therefore, implicit rules that have not worked well during the selection made by an expert (or experts) [55; 56; 57]. Approximate sets are used to process the so-called unclear data with the use of intuitively understood inference rules. They can be used to search for hidden dependencies in input data, including decision support in the scope of cases that can be described with discrete attributes. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline \multicolumn{2}{c}{**Indicators**} & \multicolumn{2}{c}{**Acc**} & **TPR** & **TNR** & **PPV** & **NPV** & **MCC** & **F1** & **J** & **Err** & **FPR** & **FDR** & **FNR** \\ \hline \multirow{2}{*}{**Classifier an average**} & **Marked** & **30–50\%** & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 0.00 & 0.00 & 0.00 & 0.00 \\  & **class** & **70–85\%** & 0.85 & 0.83 & 0.86 & 0.71 & 0.92 & 0.66 & 0.77 & 0.69 & 0.15 & 0.67 & 0.29 & 0.17 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Indicators used to test the quality of classifiers. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Indicator** & **Designation** & **Formula** \\ \hline Acc & Accuracy & \(Acc=\frac{TP+TN}{TP+TN+FP+FN}\) \\ \hline Err & Overall error rate & \(Err=\frac{TP+TN}{TP+TN+FP+FN}\) \\ \hline TPR & True positives rate & \(TPR=\frac{TP}{TP+FN}\) \\ \hline TNR & True negatives rate & \(TNR=\frac{TN}{TN+FP}\) \\ \hline PPV & Positive predictive value & \(PPV=\frac{TP}{TN+FP}\) \\ \hline NPV & Negative predictive value & \(NPV=\frac{TN}{TN+FN}\) \\ \hline FPR & False positive rate & \(FPR=\frac{FP}{TP+TN}=1-TNR\) \\ \hline FDR & False discovery rate & \(FDR=\frac{FP}{TP+TP}\) \\ \hline FNR & False negatives rate & \(FNR=\frac{FN}{TP+TN}=1-TPR\) \\ \hline MCC & [PERSON]’s correlation coefficient & \(MCC=\frac{TP+TN-FP+FN}{\sqrt{(TP+FN)(TP+FP)(FN+TN)(FP+TN)}}\) \\ \hline F1 & F1-score & \(F1=\frac{2\times PPV+TPR}{PPV+TPR}\) \\ \hline J & Youden’s J statistic & \(J=TPR+TNR-1\) \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Indicators used to test the quality of classifiers [54]. In this paper, the rough set theory was used to assess the degree of lean maintenance use. The same set of input data was used for the assessment as in the decision trees. Due to the presence of the so-called incomplete data, the use of rough sets improved the accuracy of the solution. Various types of algorithms were used to interfere with the rules. The use of the rough set theory, and thus incomplete data, increased the number of analyzed enterprises from 24 included in decision trees to 34. An additional 10 analyzed enterprises were characterized by a set of variables, for which at least one variable did not have a specific value (no answer). By using decision tables in the rough set theory, it is possible to include more data when generating rules. This allows for the identification of new dependencies between the variables. To make the assessment, the rules were validated.