## 3 Materials and Methods
### Updating FMEA Using QMI  
In FMEA frameworks, only risk variables are considered to rank failure modes by their RPN number to prioritize failure modes. The focus is only on the failure mode itself. However, the ability to mitigate or eliminate such failure modes is not considered, even though it has a strong influence on the impact of the failure mode. In fact, the RPN number feeds into the detectability risk variable, which measures the probability of detection. This assumes that the available mitigation capabilities are sufficient to reduce or eliminate the effects of the failure mode. However, in some cases, this capability is insufficient, which affects the accuracy of the detectability ratings. Thus, if a failure mode is detected but nothing is done to correct it, it can be concluded that the detectability rating of the RPN number is inadequate. If the RPN number does not account for this problem in the FMEA risk assessment, there may be cases where the RPN underestimates the risk of failure modes.  
To overcome this problem, it is proposed to include _QMI_ in the risk assessment procedures to update the impact of the failure mode according to the available mitigation options. In this context, a qualitative risk assessment model (effective risk model) is proposed that balances risk and performance variables to assess the effective risk of a given failure mode within a given scenario. Figure 2 shows the conceptual framework of the proposed qualitative risk model. On the left are the risk variables used in the RPI model and on the right are the four performance variables used in the _QMI_ model described in the previous section.  
Figure 2: Effective risk model framework.  
The effective risk for each failure mode is calculated using Equation (15), where the RPI model expanded to four variables includes three risk variables and mitigation performance:  
\[Erisk_{i}=RPI_{A>B>C>D}(A,B,C,D) \tag{15}\]  
where subscript \(i\) indicates the failure mode number, and \(A,B,C,D\) represent the risk variables S, O, D, and _QMI_. The order of their importance, \(A>B>C>D\), is determined according to the risk scenario paradigm.  
_QMI_ is determined using Equation (16) and must be set according to the rating scale defined by range \([1:\alpha]\), which is also used to evaluate severity, occurrence, and detectability.  
Since the mitigation performance has a complementary meaning to the risk, it is necessary to determine the complementary values of the _QMI_, which are defined by \(\alpha^{4}-QPI_{\text{model}}\). Then, the reduction to scale \([1:\alpha]\) is done with Equation (16).  
\[QMI_{i}=\frac{\alpha^{4}-QMI_{i-\text{model}}}{\alpha^{3}} \tag{16}\]  
In this way, the effective risk for a given failure mode is calculated by Equation (17).  
\[Erisk_{i}=RPI_{S>O>QMI>D}(S,O,QMI,D) \tag{17}\]  
Typically, risk scenarios have more than one failure mode. Therefore, it is useful to evaluate the risk associated with a particular risk scenario, which can be calculated using Equation (18).  
\[Erisk_{scenario}=\frac{\sum\limits_{i=1}^{n}Erisk_{i}}{\alpha^{4}} \tag{18}\]  
In the numerator is the sum of the failure mode risks, where \(n\) denotes the number of failure modes. In the denominator is the upper limit of the rating scale used to evaluate the risk variables.