# Case Study of Expected Loss Failure Mode and Effect Analysis Model Based on Maintenance Data
## 2 Research Methodology
### Risk Evaluation Modeling  
In this study, the probability of cause failure and system failure was assumed to be exponentially distributed with each failure rate. We calculated the expected value of loss for the cases in a system with detection sensors or does not have them. First, we considered a system of no detection sensor that the failure causes are checked only by a constant maintenance period. Changing or repairing faulted components is carried out to detect the abnormality of cause during regular maintenance. However, there are cases where a cause failure occurs and leads to system failure. In this case, changing or repairing faulted components is carried out instantly other than the maintenance period. Second, we considered a system of detection sensors. Then, regular maintenance are unnecessary. We assume that there is no system failure because alarms occur when the cause failure occurs, so preemptive response is possible.  
We calculated the expected value of loss when there is no detection sensor. When the regular maintenance period increases, the cost of cause failure decreases, but system failure increases. We calculated the expected loss by regular and corrective maintenance. The criteria for determining whether to set up the detection sensor are presented by comparing the cost of the detection sensor with the expected loss.  
In advance, we assume that the cause failures and system failures occur in random with exponential distribution. The causes are checked every maintenance cycle, and abnormal parts are exchanged or repaired to renew all causes after each maintenance. The probability that a cause failure occurred and was detected at the next maintenance period equals the cumulative probability of an exponential distribution during the regular maintenance cycle. This is empirically equivalent to the discovery of cause failures relative to the total number of tests. The summary of cause failures is as follows:  
* Cause failure \[\begin{array}{c}P\{\text{cause failure occurs within time }\tau\}=1-e^{-\lambda_{C}\tau}=\frac{N_{C}}{N}\\ \lambda_{C}=-\frac{\ln(1-N_{C}/N)}{N_{C}}\\ MTBF_{C}=\frac{1}{\lambda_{C}}\cong\frac{N\tau}{N_{C}}=\frac{\tau}{N_{C}}\\ \end{array}\text{ for }N_{C}\ll N\] (1)  
Similarly, the probability that a system failure occurred and was detected in a system uptime is equal to the cumulative probability of an exponential distribution. This is empirically equivalent to the number of discovery of system failures relative to the total number of tests. The summary of system failures is as follows:  
* System failure \[\begin{array}{c}P\{\text{system failure occurs within time }\tau\}=1-e^{-\lambda_{S}\tau}=\frac{N_{S}}{N}\\ \lambda_{S}=-\frac{\ln(1-N_{S}/N)}{N_{S}}\\ MTBF_{S}=\frac{1}{\lambda_{S}}\cong\frac{N\tau}{N_{S}}=\frac{\tau}{N_{S}}\\ \end{array}\text{ for }N_{S}\ll N\] (2)  
We calculated the case where the detection sensor is installed on the system and not on the system. First, the loss function of the case without the detection sensor is as follows.  
* Expected loss without detection sensor of 1-cause system: \[\begin{array}{c}L(\tau)=N_{C}\alpha+N_{S}\beta+N_{S}\theta\tau_{D}\\ =\frac{T}{MTBF_{C}}\alpha+\frac{T}{MTBF_{S}}\beta+\frac{T}{MTBF_{S}}\theta \tau_{D}\\ =T[\lambda_{C}\alpha+\lambda_{S}(\beta+\theta\tau_{D})]\\ \end{array}\] (3)
* Expected loss without detection sensor of \(k\)-causes system:  
in this case study, based on the EL-FMEA model, Items 1 to 4 estimate the expected loss (\(L_{\text{sens}}\)) when there is a detection sensor, and Items 5 to 8 estimate the expected loss (\(L\)) when there is no detection sensor. Finally, in Items 5 to 8, the expected loss (\(L_{\text{sens}}\)) is first estimated before the sensor is installed to determine whether to install the detection sensor. The coefficient of determination for the best alternative (\(m\)) is calculated.