# Time-Dependent Probabilistic Model for Hierarchical Structure in Failure Mode and Effect Analysis
## 4 Example Cases  
The functional level FTA depicted in Figure 1 was adopted as a case study to demonstrate the utility of the proposed time-dependent probabilistic model for a hierarchical structure. Based on Figures 1 and 2, the HTD-FMEA sheet was constructed as Table 1. Numerical values were randomly assigned to parameters as follows: (i) The constant cost, \(\beta=700\), of repairing the SF; (ii) the opportunity cost, \(\theta=1,000\), of the failure effect due to \"print operation failure\"; (iii) the occurrence rate \(\lambda\) of each SSF; (iv) the detection rate \(\mu\) of the FC; (v) the constant cost \(\alpha\) of repairing the SSF; and (vi) the detection probability increment coefficient c at the time of the SF. It can be utilized as a maintenance policy coefficient. If c is set to more than 1, the current maintenance policy is focused on post maintenance rather than preventive maintenance. Here, c is set to 1 and 5.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**System Failure**} & \(\beta\) & **Failure Effect** & \(\theta\) & \\ \hline \multicolumn{4}{c}{**Powder Layer Deposit Failure**} & **700** & **Print Operation Failure** & **1000** & **E[L]** \\ \hline
**CASE** & **Level** & **Subsystem Failure** & \(\lambda\) & \(\mu\) & \(\alpha\) & **c = 1** & **c = 5** \\ \hline \multirow{4}{*}{A} & 1 st & No complete layer & \(5.850\times 10^{-1}\) & & & & \\  & 2 nd & Recoater does not move & \(3.611\times 10^{-3}\) & \(2.686\times 10^{-2}\) & & 300 & 2030 & 6730 \\  & 3 rd & Recoater encoder failure & \(7.215\times 10^{-4}\) & & & & \\ \hline \multirow{4}{*}{B} & 1 st & No complete layer & \(7.400\times 10^{-2}\) & & & & \\  & 2 nd & Piston does not move & \(3.611\times 10^{-3}\) & \(7.648\times 10^{-1}\) & & 500 & 523 & 602 \\ \cline{1-1}  & 3 rd & Drive train failure & \(1.926\times 10^{-1}\) & & & & & \\ \cline{1-1}  & 4 th & Pulley failure & \(5.850\times 10^{-1}\) & & & & & \\ \hline \hline \end{tabular}
\end{table}
Table 1: HTD-FMEA sheet from the example in Figure 2.  
In Table 1, Case A shows that the \(E[L]\) in a situation where the \"Recoater encoder failure\" is detected after the occurrence of the \"Powder layer deposit failure\" is calculated by applying Equation (A24). When \(c=1\), the \(E[L]\) is 2030, which is a predictable loss that reflects the values of \(\alpha\), \(\beta\), and \(\theta\) in proportion to the detection probability. On the other hand, when \(c=5\), the \(E[L]\) is 6730, which shows that the expected value of the loss is more than three times that of \(c=1\). The reason for this difference is that the detecting system for the \"Recoater encoder failure\" is not only very weak but, additionally, the maintenance policy is focused on post-maintenance, resulting in an increase in the opportunity cost \(\theta\).  
Case B shows that the \(E[L]\) in a situation where the \"Pulley failure\" was detected after the occurrence of the \"Powder layer deposit failure\" is calculated by applying Equation (A25). The \(E[L]\) values were 523 and 602 for \(c=1\) and 5, respectively. In this case, the detection system for the \"Pulley failure\" is well-equipped and, even if the maintenance policy is focused on post-maintenance, \(E[L]\) is not expected to increase significantly.  
In this case, the loss due to Case A is expected to be much larger, so it is necessary to improve the direction of reducing the occurrence of the \"Recoater encoder failure\" or strengthening the detection system. At this time, it is desirable to take remedial measures to further reduce the loss.