## 3 Case Study. The Applied Evaluation Processes
### Phase 3: Reliability Evaluation Models
\[h(t)=\frac{\beta}{\eta}\left(\frac{t}{\eta}\right)^{\beta-1} \tag{1b}\]  
From Equation (1b), notice that because, for the Weibull distribution, the instantaneous risk is dependent on time, then any failure mode whose probability depends on time should be modeled by using the Weibull distribution. In particular, observe that the time-dependent behavior implies that the event is generated by a non-homogeneous Poisson process (see [Rine] for details), and, consequently, the exponential distribution should not be used to represent time-dependent events. The exponential distribution does not depend on time, and it is generated by a homogeneous Poisson process. In Equation (2), the exponential reliability is given, where \(\lambda\) is the non-time-dependent risk function. \[R(t)=e^{-\lambda t} \tag{2}\] \begin{table}
\begin{tabular}{l l l l l l l l l l} \hline \hline \multirow{2}{*}{**Date**} & \multirow{2}{*}{**Machine**} & \multirow{2}{*}{**Part**} & \multirow{2}{*}{**Failure** \#} & **QTY. Fail-** & **Total Fail-** & **Time to** & \multirow{2}{*}{**Failure**} & **Cause** & **Subsystem** & **Component** \\  & & & **Number** & & **ures/Day** & & **ures/Day** & & **Failure** & **Cause** & **Subsystem** & **Component** \\ \hline jan/06 & MT & 138 & 37 & 5 & & & Operation & 3 & 1 \\ jan/06 & MT & 138 & 19 & 2 & 7 & 0.73684 & Adjustment & 3 & 1 \\ jan/07 & MT & 138 & 33 & 1 & & & Adjustment & 3 & 1 \\ jan/07 & RS3 & 591 & 16 & 1 & & & Program setting & 3 & 1 \\ jan/07 & TMO & 18 & 33 & 1 & & & Adjustment & 3 & 1 \\ jan/07 & RAB & 584 & 24 & 1 & 4 & 0.42105 & Adjustment & 3 & 1 \\ jan/08 & RS3 & 591 & 16 & 2 & 2 & 0.21053 & Program setting & 3 & 1 \\ jan/09 & LOP & 82 & 6 & 1 & 1 & 0.10526 & Tool selection & 3 & 1 \\ jan/12 & MT & 954 & 19 & 1 & & Adjustment & 3 & 1 \\ jan/12 & RS3 & 563 & 19 & 1 & 2 & 0.21053 & Adjustment & 3 & 1 \\ jan/13 & LM & 131 & 4 & 1 & 1 & 0.10526 & Tool selection & 2 & 1 \\ jan/14 & R2 & 121 & 8 & 1 & & Adjustment & 3 & 1 \\ jan/14 & R2 & 246 & 8 & 1 & & Adjustment & 3 & 1 \\ jan/14 & TUI & 563 & 21 & 1 & 3 & 0.31579 & Mat handling & 3 & 3 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Database used to collect and analyze failures that occurred in machine tools. Evaluation of the reliability of a subsystem. The evaluation of the reliability of a subsystem affected by one or more components that participated in the failure of the product assumes that components have a series configuration within the subsystem, as they are independent and not mutually exclusive. The statistical model of Equation (3) is used to evaluate the reliability of a subsystem with a series configuration. In Equation (3), \(R\)s is the reliability of the subsystem, and \(R_{i}\) is the reliability of the component-subsystem relationship. \[R_{s}=\prod\
olimits_{i=1}^{n}R_{i} \tag{3}\]  
Evaluation of the reliability of the human-machine system. To evaluate the reliability of the human-machine system, the assumption is made that the subsystems have a parallel configuration since they are independent and mutually exclusive. Using the statistical model of Equation (4), the reliability index of the human-machine system is evaluated. It is worth mentioning that the subsystems that do not have any component affecting them will have a reliability of 1 and will not affect the calculation of the system's reliability, given by  
\[R_{S}=1\ -\prod\
olimits_{i=1}^{n}(1-R_{i}) \tag{4}\]  
In Equation (4), \(R_{S}\) is the reliability of the man-machine system, and \(R_{i}\) is the reliability of the subsystem. The equation for its calculation assumes a parallel system. The reliability index obtained by the statistical models represented by Equations (1)-(4) represents the probability of not finding defective parts to be scrapped after a period t. Depending on the reliability obtained, it may be necessary to identify the lowest reliability presented in the system and perform improvement actions, such as those presented in the next section.