# Time-Dependent Probabilistic Approach of Failure Mode and Effect Analysis
## 4 Conclusions  
So far, we have proposed a time-dependent FMEA model (TD-FMEA model) and obtained an expected loss value \(E[L]\) according to the occurrence of failure-cause through accurate calculation (Equation (9)). Then, we derived risk prioritization metric (_RPM_), which is a defined index measuring the risk of failure-cause in this study, multiplying by the failure-cause occurrence rate \(\varepsilon\) and the expected loss value \(E[L]\) (Equation (11)). The mathematical model proposed in this study performs an associated calculation on system failure, failure-cause, and its detection. The probability density function of failure and failure-cause are exponential by homogeneous Poisson process (HPP) assumption (Equations (1) and (2)), and that of failure-cause detection is a case function combined in a uniform and exponential manner (Equation (4)). The loss function is also a case function conditioned by detection time of failure-cause and occurrence time of failure (Equation (8)). Time-dependent probabilistic approach pursues logical improvement compared to previous models. Conventional _RPN_ approach assumes that \(O\) (occurrence of failure), \(S\) (severity of failure), and \(D\) (detection of failure-cause) are independent. However, _RPM_ in our model does not assume the independence because it combines failure occurrence rate, detection rate of failure-cause, and loss function to derive the risk of failure-cause. Table 4 compared the traditional FMEA and our TD-FMEA model. The _RPM_ derived in this way provides room to supplement the disadvantages of traditional risk prioritization number (_RPN_). The coefficient of determination \(R^{2}\) between the priority of _RPM_ and _RPN_ has a range of 0.686-0.889 in our example (Figure 4). That is the rank correlation has a range of 0.828-0.943, so that _RPM_ and _RPN_ have very strong correlations. From this, we can say that the validity of our model has been empirically proven. Furthermore, our model has the advantage that the _RPM_ value represents the relative size of the actual risk. In contrast, the product of \(O\), \(S\), and \(D\) derived from _RPN_ of conventional approaches is only an auxiliary value to derive the priority without having any substantial meaning (Table 3). Utility of both indicators \(E[L]\) and _RPM_ with regard to evaluation of the risk associated with each failure has been described in great detail in Section 3 of this paper. For a reasonable calculation, we provide a standardization of parameters for failure-cause occurrence rate \(\varepsilon\), failure occurrence rate \(\lambda\), and failure-cause detection rate \(\mu\), which is a variation of our model from the guideline of Ford Motor Company (Table 2). In addition, the results of the sensitivity analysis of _RPM_ can be found in Appendix B. In future endeavors, our future prospect is to examine actual cases based on actual parameters. The purpose of the study is to derive correlation coefficients of _RPN_ and _RPM_ rankings to infer the validity of _RPM_, and to verify the effectiveness by deriving risks from _RPM_ values. Moreover, we intend to focus on development of an FMEA risk-assessment methodology for a single failure with multiple causes along with determination of an optimal inspection period to enhance the detection rate based on \(E[L]\) values. In addition, if the failure-causes are analyzed hierarchically, a time-dependent probabilistic model will be developed based on undetected time. 1. In traditional FMEAs, risks can be assessed on a ranking basis, making it an easy-to-use approach when data is difficult to obtain. In the big data world, however, the application of a numerical-based TD-FMEA approach will enable more accurate assessment of risk. 2. In traditional FMEAs, the results of a risk assessment may depend on the subjective opinion of the expert. On the other hand, in the case of the TD-FMEA approach, the objectivity of the result is secured based on probability. Conceptualization, H.-a.J. and S.M. ; Methodology, H.-a.J. and S.M. ; Modeling, S.M.