## 3 Case Study. The Applied Evaluation Processes
### Phase 4: Improvement
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Origin** & **Qty. of Failures** & **Relative** & **Cumulative** \\ \hline Mechanical & 31 & 53.45\% & 53.45\% \\ Human Error & 11 & 18.97\% & 72.42\% \\ Heat treatment & 9 & 15.51\% & 87.93\% \\ Material & 5 & 8.62\% & 96.55\% \\ Unexpected event & 2 & 3.45\% & 100.00\% \\ Total & 58 & 100.00\% & \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Frequency of kinds of failures by origin. Figure 5: Pareto diagram of the origins of failures and their contribution. Evaluation of the reliability of component-subsystem relationships. To evaluate the reliability from the period of 25 January to 30 January 2021, the statistical model represented by Equation (1) (Weibull distribution) was used because only failures of mechanical origin were present. To evaluate the reliability from the period of 17 May to 28 May 2021, both the Weibull distribution (Equation (1a)) and the exponential distribution (Equation (2)) were used because there were failures of mechanical and human-error origin. Table 9 shows the reliability indices for the component-subsystem relationships present in the machine tool during the period of 25 January to 30 January 2021 for five time periods considered within the 9.5-h shift. Table 10 shows the reliability indices for the component-subsystem relationships present in the machine tool during the period of 17 to 28 May 2021 for five time periods considered within the 9.5-h shift. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{2}{c}{**Weibull**} & \multicolumn{2}{c}{**Exponential**} \\ \cline{2-4}
**Combination** & \(\beta\) & \(\eta\) & \(\Lambda\) \\ \hline
1, 1 & 1 & 3.1667 & \\
1, 6 & 1 & 9.5 & \\
2, 1 & 1 & 9.5 & \\
3, 1 & 1.0688 & 3.1268 & \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Component-subsystem relationships presented in the failure analysis and their parameters from 25 to 30 January 2021. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**Weibull**} & \multicolumn{2}{c}{**Exponential**} \\ \cline{2-4}
**Combination** & \(\beta\) & \(\eta\) & \(\Lambda\) \\ \hline
1, 1 & 3.4615 & 7.9737 & \\
1, 6 & 2.18399 & 7.19726 & \\
2, 1 & 1.52633 & 5.32846 & \\
3, 1 & 1.90918 & 2.34728 & \\
3, 3 & & & 1/9.5 & \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Component-subsystem relationships presented in the failure analysis and their parameters from 17 May to 28 May 2021. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Periods** & \(\mathbf{R_{1,1}}\) & \(\mathbf{R_{1,6}}\) & \(\mathbf{R_{2,1}}\) & \(\mathbf{R_{3,1}}\) & \(\mathbf{R_{3,3}}\) \\ \hline
1 h & 0.7292 & 0.9 & 0.9 & 0.645 & 0.7292 \\
3 h & 0.3877 & 0.7292 & 0.7292 & 0.2683 & 0.3877 \\
5 h & 0.2061 & 0.5907 & 0.5907 & 0.1111 & 0.2061 \\
7 h & 0.1096 & 0.4786 & 0.4786 & 0.0464 & 0.1096 \\
9 h & 0.0583 & 0.3877 & 0.3877 & 0.0193 & 0.0583 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Reliability for the component-subsystem relationships during five time periods from 25 to 30 January 2021. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**Weibull**} & \multicolumn{2}{c}{**Exponential**} \\ \cline{2-4}
**Combination** & \(\beta\) & \(\eta\) & \(\Lambda\) \\ \hline
1, 1 & 3.4615 & 7.9737 & \\
1, 6 & 2.18399 & 7.19726 & \\
2, 1 & 1.52633 & 5.32846 & \\
3, 1 & 1.90918 & 2.34728 & \\
3, 3 & & & 1/9.5 & \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Component-subsystem relationships presented in the failure analysis and their parameters from 17 May to 28 May 2021. Evaluation of the reliability of the subsystems and the man-machine system. Using Equation (3), which considers a series configuration, the reliabilities of the subsystems that contributed to the failures were evaluated. Considering Equation (4), which considers a parallel configuration, the reliability of the man-machine system was evaluated.