## 3 Results and Discussion
### Models Based on Decision Trees XGBoOST, LightGbmRegression, and FastTreeRegression
It transpired that the polynomial regression model was the best model, considering the cross-validation score and the normalized RMSE evaluated against the test set. Nevertheless, it was not sufficiently sensitive to changes in the input parameters. It also underestimated the longest observations, which is a common drawback with the created models. The AdaBoost regression model predicted these cracks with the lowest normalized RMSE. A study discussing the dimensional control of welded stamped steel arms [32] came to similar conclusions. The smallest mean absolute percentage error (MAPE) was achieved for the ANN, and it was noticed that the accuracy of the model increased as the size of the training dataset increased; meanwhile, increasing the number of data has the opposite effect for linear regression. In [33], regression algorithms and neural networks were selected to determine the value of information flow in the production chain model; the comparative criteria were MAE, MSE, RMSE, and R\({}^{2}\). However, it was noted that the size and nature of the company's production had a significant impact on the selection of the evaluation factors for the comparative analysis. Another study [34] focused on hybrid forecasting models for manufacturing systems--specifically, in terms of the three manufacturing system areas of production planning, maintenance, and quality control; it confirmed that manufacturing companies are willing to select the right forecasting method for their particular needs. The assessment of the accuracy of the analyzed forecasting methods was based on the average R\({}^{2}\), which was calculated for each case study separately. Another study, dealing with the reliability prediction of diesel engines [35], compared random forest and MLP; the RMSE is a metric commonly used for the cross-validation procedure. The results from the models discussed in this paper were compared and contrasted based on the criteria in terms of the performance of the final system. Regarding the different mechanisms of action of the methods used in this study, it was difficult to bring the results to a \"common denominator\" in order to reliably compare the effectiveness of their operations (the possibility of solving the described problem) and the outcomes obtained. Based on the literature analysis and the authors' experience in building models based on ANNs, two comparative measures were selected: R\({}^{2}\) and RMSE. The authors emphasized that the selected measures were used to evaluate the generalization abilities of the model. It was decided that the selection of the best possible ML for solving the non-linear regression problem in this case should not be rigidly based only on one metric. Based on the above-mentioned criteria, it was found that the best solution was the FastTreeRegression model (Table 7), which was characterized by the lowest RMSE (error assessment) and the highest R\({}^{2}\) value (prediction quality assessment). Additionally, this algorithm exhibited significant opportunities for improvement both in terms of functioning and in improving the achieved metric values [36; 37]. In real-world applications, as in our study, R\({}^{2}\) values exceeding 0.7 indicate a strong influence on the dependent variable. We tested various model solutions, and none of them resulted in a significant improvement in this respect; this means that intervention is \begin{table}
\begin{tabular}{c c c} \hline \hline
**Model** & **RMSE** & **R\({}^{2}\)** \\ \hline MLP 5-8-1 & 0.177 & 0.3024 \\ XGBoost & 0.116 & 0.5182 \\ LightGbmRegression & 0.086 & 0.7070 \\ FastTreeRegression & 0.085 & 0.7163 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Comparison of the solutions in the research based on the selected criteria. required in the data rather than in the model itself. Admittedly, R\({}^{2}\) can be increased by, for example, adding appropriate variables to the model, as additional variables introduce more flexibility into the model; this allows it to capture noise and random fluctuations in the data.