# Time-Dependent Probabilistic Approach of Failure Mode and Effect Analysis
## Appendix B  
A sensitivity analysis was performed in this study to investigate the influence of distribution-parameter values on those of _RPM_. Sensitivity of _RPM_ values to values of parameters--\(\mu\) and \(\lambda\)--can be readily analyzed. In this study, sensitivity of the value of _RPM_ was analyzed by assigning parameters \(\varepsilon\), \(\alpha\), \(\beta\), and \(\theta\) with constant of 1.81E\(-\)02, 200, 1000, and 1200, respectively. Variations inthe value of _RPM_ corresponding to changes in values of \(\mu\) and \(\lambda\) are listed in Table 11 and depicted in Figure 11, respectively. Results obtained demonstrate that smaller values of \(\mu\) result in enhanced sensitivity of _RPM_. Figure 11a demonstrates that the value of _RPM_ steeply increases with reduction in the value of \(\mu\). Secondly, as depicted in Figure 11b, it can be realized that the value of _RPM_ is nearly insensitive to small values of \(\lambda\), and that the corresponding sensitivity increases with increase in the value of \(\lambda\).  
Figure 11 demonstrates that the maximum value of _RPM_ is realized for the case wherein \(\mu=10\) and \(\lambda=10\). This implies that maximization of a given risk of failure with the corresponding failure-cause is hardly detectable whilst having a high occurrence probability. As depicted in Figure 11, there exists a big difference between _RPM_ values obtained for each \(\mu\). This can be verified in terms of the graph slope. In Figure 11a, the difference between _RPM_ values is large for cases with \(\mu=10\) and \(\lambda\) assuming values in the range of 2-7. On the other hand, the difference between _RPM_ values is small when \(\mu=1\) and \(\lambda\) assumes values in the range of 1-6. This implies that the _RPM_ value does not demonstrate much difference when the failure probability is either very high or very low. Inferences drawn from the above sensitivity analysis are that (i) each failure-cause with a high occurrence probability must be taken care of; and (ii) a cause with a small occurrence probability need not be assigned high priority.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline  & \(\mu\) & **1 st** & **2 nd** & **3 rd** & **4 th** & **5 th** & **6 th** & **7 th** & **8 th** & **9 th** & **10 th** \\ \(\lambda\) & & **1.00 E+00** & **7.65 E –01** & **4.39 E –01** & **2.72 E –01** & **1.35 E –01** & **6.01 E –02** & **2.69 E –02** & **9.81 E –03** & **3.10 E –03** & **9.81 E –04** \\ \hline
**1 st** & **9.62 E –07** & 3.6 & 3.6 & 3.6 & 3.6 & 3.6 & 3.7 & 3.9 & 5.2 & 17.6 & 122.3 \\
**2 nd** & **9.62 E –06** & 3.6 & 3.6 & 3.6 & 3.7 & 3.7 & 4.1 & 5.5 & 15.6 & 101.8 & 779.1 \\
**3 rd** & **9.62 E –05** & 3.7 & 3.7 & 3.7 & 3.8 & 4.4 & 6.7 & 16.6 & 81.7 & 582.8 & 3957.5 \\
**4 th** & **7.22 E–04** & 3.8 & 3.9 & 4.2 & 4.8 & 7.5 & 19.1 & 65.1 & 335.4 & 2070.7 & 1113.27 \\
**5 th** & **3.61 E –03** & 4.3 & 4.5 & 5.7 & 8.0 & 17.0 & 53.0 & 182.3 & 833.3 & 4159.8 & 17,546.2 \\
**6 th** & **1.81 E –02** & 5.9 & 6.8 & 10.5 & 17.3 & 42.4 & 131.6 & 405.0 & 1519.6 & 5981.7 & 20,942.4 \\
**7 th** & **7.40 E–02** & 9.6 & 11.9 & 20.3 & 34.9 & 83.5 & 232.4 & 621.3 & 1965.1 & 6720.4 & 21,883.9 \\
**8 th** & **1.93 E–01** & 14.2 & 17.9 & 30.9 & 52.1 & 116.8 & 296.2 & 725.7 & 2119.6 & 6912.5 & 22,092.6 \\
**9 th** & **5.85 E–01** & 21.9 & 27.3 & 45.4 & 72.7 & 149.7 & 345.8 & 791.1 & 2198.9 & 6999.4 & 22,182.3 \\
**10 th** & **1.00 E+00** & 26.0 & 32.2 & 52.0 & 81.1 & 160.9 & 360.0 & 807.3 & 2216.5 & 7017.6 & 22,204.6 \\ \hline \end{tabular}
\end{table}
Table 10: Risk-prioritization metric (_RPM_) table with respect to the failure occurrence rate \(\lambda\) and failure-cause detection rate \(\mu\) (\(c=1\), \(\epsilon=0.0181\), \(\alpha=200\), \(\beta=1000\), and \(\theta=1200\)).