## 3 Case Study. The Applied Evaluation Processes
### Phase 4: Improvement
\begin{table}
\begin{tabular}{c c c c c c} \hline
**i** & **Y** & **R(t)** & **toi** & **Applied Effort** & **Minimal Resistance** & \\ \hline
1 & \(-\)3.4034833 & 0.9673 & 0.06751 & 0.619 & 135.764 & \\
2 & \(-\)2.491662 & 0.9206 & 0.13899 & 1.274 & 65.942 & Max= & 21 \\
3 & \(-\)2.0034632 & 0.8738 & 0.20460 & 1.875 & 44.796 & Min= & 4 \\
4 & \(-\)1.6616459 & 0.8271 & 0.26821 & 2.458 & 34.172 & Beta= & 1.26264954 \\
5 & \(-\)1.3943983 & 0.7804 & 0.33143 & 3.038 & 27.653 & Eta= & 9.16515139 \\
6 & \(-\)1.1720537 & 0.7336 & 0.39525 & 3.622 & 23.189 & From the maximum range of values: \\  & & & 0.704 & 0.43644 & 4.000 & 21.000 & 3.146 & Standard deviation \\
7 & \(-\)0.9793812 & 0.6869 & 0.46040 & 4.220 & 19.907 & 1.4353 & Variation \\
8 & \(-\)0.8074473 & 0.6402 & 0.52756 & 4.835 & 17.373 & & \\
9 & \(-\)0.6504921 & 0.5935 & 0.59739 & 5.475 & 15.342 & 22.435 & Upper limit for resistance \\  & & & & & & & & \\
10 & \(-\)0.5045088 & 0.5467 & 0.67061 & 6.146 & 13.667 & 19.565 & Lower limit for resistance \\  & & & & & & & & & \\
11 & \(-\)0.3665129 & 0.5 & 0.74806 & 6.856 & 12.252 & & \\
12 & \(-\)0.2341223 & 0.4533 & 0.83075 & 7.614 & 11.032 & & \\
13 & \(-\)0.1052851 & 0.4065 & 0.92000 & 8.432 & 9.962 & & \\  & & 0 & 0.3679 & 1.00000 & 9.165 & 9.165 & \\
14 & 0.0219284 & 0.3598 & 1.01752 & 9.326 & 9.007 & \\
15 & 0.1495258 & 0.3131 & 1.12572 & 10.317 & 8.142 & & \\
16 & 0.279845 & 0.2664 & 1.24811 & 11.439 & 7.343 & & \\
17 & 0.4159621 & 0.2196 & 1.39018 & 12.741 & 6.593 & & \\
18 & 0.562502 & 0.1729 & 1.56126 & 14.309 & 5.870 & & \\
19 & 0.7276158 & 0.1262 & 1.77937 & 16.308 & 5.151 & & \\
20 & 0.9293107 & 0.0794 & 2.08757 & 19.133 & 4.390 & & \\
21 & 1.2296598 & 0.0327 & 2.64818 & 24.271 & 3.461 & & \\ \hline \end{tabular}
\end{table} 
Table 17: Calculation of the \(\beta\) and \(\eta\) parameters using the minimum and maximum values of resistance. where \(\sigma_{1}\) and \(\sigma_{2}\) are the maximum and minimum efforts at which the machine is performing, and \(\mu y\) is the mean of the median rank approach estimated from the elements of the Y vector determined by using the sample size from [PERSON] et al., [32] given as  
\[\mathbf{n}=\frac{-1}{ln(R(t))} \tag{7}\]  
The \(n\) elements of the \(Y\) vector used to estimate \(\mu y\) were determined based on the median rank approach as  
\[Y_{i}=\ln\biggl{(}-\ln\biggl{(}1-\frac{(i-0.3)}{(n+0.4)}\biggr{)}\biggr{)} \tag{8}\]  
From these \(Y\) elements, the corresponding standardized base lifetime \(t_{0i}\) values that allowed us to determine both the applied effort and its corresponding minimal resistance (see Table 17) are given as  
\[t_{0i}=\exp\biggl{\{}\frac{Y_{i}}{\beta}\biggr{\}} \tag{9}\]  
Therefore, from Equations (5), (6), and (9) the applied effort and the minimal resistance are given as  
\[Applied\ effort=\eta*t_{0i} \tag{10}\]  
\[Minimal\ resistance=\eta/t_{0i} \tag{11}\]  
Finally, the mean and standard deviation for the maximum values of resistance and the corresponding 95% confidence interval are given as  
\[\mu_{max}=\sum_{i=1}^{n}\frac{\max\ values\ of\ resistance}{n} \tag{12a}\]  
\[\sigma_{max}=\sqrt{\sum_{i=1}^{n}\biggl{(}\frac{\max\ value\ of\ resistance-u_{max}}{n-1}\biggr{)}^{2}} \tag{12b}\]  
\[CI=\mu_{max}\pm\biggl{(}\frac{1.96}{\sqrt{n}}\biggr{)} \tag{13}\]  
As shown Table 17, the corresponding reliability indicator was established for each machine resistance. For a machine that has a minimum resistance of 21% over its nominal value, it has a reliability of 0.704 (the minimum resistance that the machine must have to have this reliability). When the effort is 9.1651 (\(\eta\)) and the process has a resistance of 21, the reliability is 0.7040. When the resistance is higher, the reliability is higher too. This method consists of monitoring the critical value of 21% through the confidence interval, hoping that the maximum value of the machine effort does not exceed the upper limit of 22.4353. It is also possible to evaluate whether the maximum efforts produced by the machine exceed the standard efforts considered in the confidence interval.