# Time-Dependent Probabilistic Approach of Failure Mode and Effect Analysis
## 3 Application
### Comparative Analysis of RPM and RPN  
To illustrate the effectiveness of the proposed risk-evaluation technique, the above-described case study was used to analyze and compare results obtained using the proposed TD-FMEA model and conventional design FMEA techniques. The results are described in Table 3, that lists values of the conventional _RPN_, proposed _RPM_, and \(E[L]\) for each probable failure-cause in terms of both score and priority. To facilitate the robust evaluation of risks during the case study, results obtained for four cases ((a)-(d)) have been discussed herein considering, in each case, the effect of variation in the value of any one parameter among \(\lambda\), \(\alpha\), \(\beta\), and \(\theta\). Figure 4 shows the correlation of priority of _RPN_ and _RPM_ in Table 3, respectively, in four cases. The square root of \(R^{2}\) is 0.8 or higher, it is determined that there is a strong correlation [31]. The square root of \(R^{2}\) always satisfies \(-1\leq r\leq 1\), the positive correlation is \(r>0\), the negative correlation is \(r<0\). Moreover, when it is free, it becomes \(r=0\). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Case**} & \multirow{2}{*}{**Failure-Causes**} & \multicolumn{2}{c}{_RPN_} & \multicolumn{2}{c}{_RPM_} & \multicolumn{2}{c}{_E[L]_} \\ \cline{3-8}  & & **Score** & **Priority** & **Score** & **Priority** & **Score** & **Priority** \\ \hline \multirow{4}{*}{(a) Fixed \(\lambda\), \(\alpha\), \(\beta\) and \(\theta\)} & FC11 & 210 & 2 & 22 & 3 & 6180 & 1 \\  & FC12 & 168 & 4 & 17 & 4 & 958 & 4 \\  & FC13 & 140 & 5 & 2 & 5 & 2103 & 2 \\  & FC21 & 72 & 6 & 9-E–02 & 6 & 945 & 5 \\  & FC22 & 240 & 1 & 401 & 1 & 2081 & 3 \\  & FC23 & 192 & 3 & 182 & 2 & 945 & 5 \\ \hline \multirow{4}{*}{(b) Fixed \(\lambda\) and \(\theta\)} & FC11 & 210 & 2 & 23 & 4 & 6263 & 1 \\  & FC12 & 168 & 4 & 29 & 3 & 1597 & 6 \\  & FC13 & 140 & 5 & 3 & 5 & 4147 & 2 \\  & FC21 & 72 & 6 & 2:E–01 & 6 & 1725 & 5 \\  & FC22 & 240 & 1 & 532 & 1 & 2760 & 3 \\  & FC23 & 192 & 3 & 428 & 2 & 2224 & 4 \\ \hline \multirow{4}{*}{(c) Fixed \(\alpha\), \(\beta\) and \(\theta\), and \(\theta\)} & FC11 & 210 & 2 & 22 & 3 & 6180 & 1 \\  & FC12 & 168 & 4 & 7 & 4 & 404 & 4 \\ \cline{1-1}  & FC13 & 140 & 5 & 4 & 5 & 5465 & 2 \\ \cline{1-1}  & FC21 & 72 & 6 & 4:E–02 & 6 & 402 & 5 \\ \cline{1-1}  & FC22 & 240 & 1 & 74 & 2 & 383 & 6 \\ \cline{1-1}  & FC23 & 192 & 3 & 330 & 1 & 1713 & 3 \\ \hline \multirow{4}{*}{(d) Variable \(\lambda\), \(\alpha\), \(\beta\) and \(\theta\)} & FC11 & 210 & 2 & 23 & 3 & 6263 & 2 \\  & FC12 & 168 & 4 & 10 & 4 & 551 & 4 \\ \cline{1-1}  & FC13 & 140 & 5 & 9 & 5 & 11,911 & 1 \\ \cline{1-1}  & FC21 & 72 & 6 & 3:E–02 & 6 & 289 & 6 \\ \cline{1-1}  & FC22 & 240 & 1 & 57 & 2 & 297 & 5 \\ \cline{1-1}  & FC23 & 192 & 3 & 897 & 1 & 4658 & 3 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison between _RPN_ of conventional FMEA model and _RPM_ of TD-FMEA model for design FMEA: The priority of _RPN_ and _RPM_ is similar, so we may confirm the validity of the TD-FMEA model to some extent. (% set \(\mu\) and \(\varepsilon\) to the modified values in Table 1). In Figure 4, regarding case (a), the square root of \(R^{2}\) between \(RPN\) and \(RPM\) was 0.943, indicating a strong correlation. In Table 3, however, in terms of score, they are different. Based on \(RPN\) values, the FC\({}_{22}\) (score \(=8\times 6\times 5=240\)) demonstrates the highest priority and can be considered the most critical risk closely followed by FC\({}_{11}\) (score \(=210\)) and FC\({}_{23}\) (score \(=192\)) ranked second and third, respectively. The difference in \(RPN\) scores corresponding to the FC\({}_{22}\) and FC\({}_{11}\) is 30 (\(=240-210\)), whereas that between the FC\({}_{11}\) and FC\({}_{23}\) is 18 (\(210-192\)). This, however, does not imply that 30 equals two times 18 from the viewpoint of risk size.