## 3 Methodology Application
### Modeling Application
#### 3.4.2 Regression Methods  
Regression methods were chosen for estimating the manufacturing CT since this time is conditioned by the sum of the individual M/J CTs. In linear regression, the expected value \(\mathbf{y}\) to be found depends on several input attributes, \(\mathbf{X}\), and the problem is solved in a vector space:  
\[\mathbf{y}=\mathbf{\beta}\mathbf{X}+\varepsilon; \tag{4}\]  
where each attribute has its own \(\beta\)-coefficient and \(\varepsilon\) is the error of prediction; it is the difference between the training value and the predicted value. Least squares regression divides the data into subsets for which it builds linear regression models. It selects the model with the smallest RMSE as the output model. The M5 algorithm combines a decision tree and a linear regression function. J48 is used to select the best linear regression model [49]. A total of three experiments were run for items of type A, for which 39 method settings were run on 38 datasets with a final number of 24,884 models. A summary of the experiments is shown in Table 6. A total of 24 models with the best metrics in CCI, or RAE, were selected for tests on independent test data. For the modeling of M/J CT, unpruned decision trees with different numbers of classes for M/J/CT estimation of item machine base were selected, as shown in Table 7. Table 8 summarizes the results of the tests for models of manufacturing CT. The results of the modeling experiment and the LT model tests resulted in disappointing CCI values, as shown in Table 9. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Modeling**} & **Number of Models** & **Number of Evaluated Models** & **Selected PAs** \\  & **in Experiment** & **Models** & \\ \hline M/J CT & 384 & 10 & Unpruned J48 \\ Manufacturing & 12,600 & 7 & Linear regression, and least square \\ CT & 12,600 & 7 & regression \\ LT & 11,900 & 7 & Visualization \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Summary of experiments. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Machine/Job** & **Selected NC** & **CCI (\%)** \\ \hline Laser Programming & 4 & 50 \\ Laser & 3 & 70 \\ Drilling & 4 & 65 \\ Milling & 3 & 55 \\ Welding & 5 & 72 \\ Cleaning works & 3 & 90 \\ Additional works & 3 & 78 \\ Assembly & 3 & 56 \\ Bending & 4 & 65 \\ Cutting & 5 & 50 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Results of testing the M/J CT models of A type item “machine base”. For type B items, no experiments were performed, but PAs with the best performance from modeling time estimation of type A items were selected, specifically, decision trees for M/J CT estimation, as shown in Table 10, and linear regression for manufacturing CT, as shown in Table 11. In the LT modeling experiment, no suitable models were found for the type A items. Therefore, LT modeling was not performed, and data behavior visualization was used for type B items, as shown in Figure 5. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Item** & **Machine/Job** & **Customer** & **CCI (\%)** \\ \hline Steel plate & Cutting & D10183 & 72 \\ Steel plate & Cutting & D10605 & 72 \\ Steel plate & Shearing & D10206 & 50 \\ Steel plate & Shearing & D10605 & 60 \\ Lasered and bent part & Laser programming & D10216 & 50 \\ Lasered and bent part & Laser programming & D10607 & 75 \\ Lasered and bent part & Laser programming & D10605 & 75 \\ Lasered and bent part & Laser programming & D10600 & 75 \\ Lasered and bent part & Laser & D10216 & 50 \\ Lasered and bent part & Laser & D10607 & 87 \\ Lasered and bent part & Laser & D10605 & 80 \\ Lasered and bent part & Laser & D10600 & 87 \\ Lasered and bent part & Bending & D10216 & 75 \\ Lasered and bent part & Bending & D10607 & 75 \\ Lasered and bent part & Bending & D10605 & 75 \\ Lasered and bent part & Bending & D10600 & 67 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Results of testing the M/J CT models of type B items.