# Time-Dependent Probabilistic Approach of Failure Mode and Effect Analysis
## 3 Application
### Comparative Analysis of RPM and RPN
In the case of \(RPM\), the FC\({}_{22}\) (score \(=401\)) demonstrates the highest priority and can be considered the most critical risk followed by FC\({}_{23}\) (score \(=182\)) and FC\({}_{11}\) (score \(=22\)) ranked second and third, respectively. The difference in score corresponding to the FC\({}_{22}\) and FC\({}_{23}\) is 219 (\(401-182\)), whereas that between the FC\({}_{23}\) and FC\({}_{11}\) is 160 (\(182-22\)). This means that 219 is about 1.37 times larger 160 from the viewpoint of risk size. Moreover, as depicted in Figure 4 for case (a), the FC\({}_{11}\) and FC\({}_{23}\) are prioritized as three and two, respectively, when employing the \(RPM\)-based approach, whereas the same failure-causes are prioritized as two and three, respectively, in accordance with \(RPN\) values. This is because the occurrence rate \(\varepsilon\) for the FC\({}_{23}\) is considerably higher compared to that for FC\({}_{11}\), although the parameter \(\mu\) is not significantly different for both risks (refer Table 1). This result is also reflected in \(RPM\) scores corresponding to the two cases. The score corresponding to the FC\({}_{23}\) (score \(=182\)) equals eight times that corresponding to FC\({}_{11}\) (score \(=22\)); that is, the loss incurred per hour in the event of a failure-caused by FC\({}_{23}\) would be eight times that incurred in the event of a failure that was caused by FC\({}_{11}\). On the other hand, this result is different in terms of \(E[L]\). As described in Table 3, the FC\({}_{23}\) with \(E[L]\) = 945 assumes the least priority. This implies that there is a high probability of detecting the FC\({}_{23}\) before the failure occurs; thus, its corresponding \(E[L]\) score is low. Figure 4: Rank correlation between \(RPM\) and \(RPN\) obtained by design FMEA: \(RPM\) and \(RPN\) are strongly correlated except for slight variations in priority rank. In this case, square root of \(R^{2}\) is Spearman correlation coefficient, i.e., Pearson correlation coefficient of rank. Regarding case (b), in Table 3, FC\({}_{22}\), FC\({}_{13}\), and FC\({}_{21}\) are consistently assigned priorities of 1, 5, and 6, respectively, using both the _RPN_- and _RPM_-based approaches. As depicted in Figure 4 for case (b), the square root of \(R^{2}\) between _RPN_ and _RPM_ was 0.829, indicating a strong correlation. Additionally, the FC\({}_{11}\), FC\({}_{23}\), and FC\({}_{12}\) are assigned of 4 (score = 23), 2 (score = 428), and 3 (score = 29), respectively, when employing the _RPM_-based approach, whereas the same failure-causes are prioritized as two (score = 210), 3 (score = 192), and 4 (score = 168), respectively, when employing the _RPN_-based approach. This is because, as described in Table 1, variable parameters \(\alpha,\ \beta,\) and \(\theta\) for FC\({}_{23}\) and FC\({}_{12}\) exceed those for fixed parameters. On the other hand, in terms of \(E[L]\), the FC\({}_{11}\) with the smallest value of parameter \(\mu\) assumes the highest priority, whereas risk FC\({}_{12}\) with the largest values of \(\mu\) assumes the lowest priority (refer Table 1). In case (c), compared to _RPM_-based results obtained for case (a) (refer Figure 4), it can be realized that the _RPM_-based priority assigned to the FC\({}_{22}\) changes from one to two while that of the FC\({}_{23}\) changes from two to one. This is because the value of parameter \(\lambda\) for the FC\({}_{23}\) exceeds that for the FC\({}_{22}\) by a fairly large magnitude (refer Table 1). In contrast, in terms of \(E[L]\), the FC\({}_{11}\) with medium value of \(\lambda\) and the smallest value of \(\mu\) assumes the highest priority, whereas the FC\({}_{22}\) with small value of \(\lambda\) and large value of \(\mu\) is assigned the lowest priority. Lastly, in case (d), compared to _RPM_-based results, for case (d) are identical to those obtained for case (c) in terms of priority, and they are quite similar on the score side. In Figure 4, the square root of \(R^{2}\) between _RPN_ and _RPM_ was also 0.829, indicating a strong correlation.