# Fuzzy-FMEA Theory Approach for Prioritizing Supply Chain Nervousness Factors
## 3 Research Methodology
### SCN Model
The following formula [68] is used for the defuzzification process to obtain the clear number or the crisp value \(H(k^{u}_{e})\) of semantic element \(u\):  
\[H(k^{u})=\frac{a^{u}}{2(1+N)}+\frac{(N+2 NM+M)b^{u}}{2(1+N)(1+M)}+\frac{c^{u}}{ 2(1+M)}\, \tag{1}\]  
where \(N\) and \(M\) are numbers identified based on the triangular fuzzy values \(a^{u}\), \(b^{u}\), and \(c^{u}\), such that \(N\) and \(M\) indicate how large \(b^{u}\) is relative to \(a^{u}\) and \(c^{u}\), respectively. Step 3: Forming the FMEA and level of nervousness. The weights of the elements of levels I, II, and III (interruption modes), denoted by \(\varphi^{l}\), \(\varphi^{l_{i}}\), \(\varphi^{l_{i}l_{j}}\), respectively, are calculated as the average values, such that we have the following:  
\[\varphi^{l}=\frac{1}{n}\sum_{e=1}^{n}\varphi^{l}_{e},\,\,\,\varphi^{l_{i}}= \frac{1}{n}\sum_{e=1}^{n}\varphi^{l_{i}}_{e},\,\,\,\varphi^{l_{i}l_{j}}=\frac {1}{n}\sum_{e=1}^{n}\varphi^{l_{i}l_{j}}_{e},\]  
where \(\varphi^{l}_{e}\), \(\varphi^{l_{i}}_{e}\), and \(\varphi^{l_{i}l_{j}}_{e}\) are the weights given by expert \(e,e=1,\cdots,\,n\), on elements of sets \(l\), \(l_{i}\), and \(l_{i}l_{j}\), respectively. \begin{table}
\begin{tabular}{p{56.9 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline \hline
**Semantic Element** & **Severity (Se)** & **Occurrence (Oc)** & **Detection (De)** \\ \hline EU & SC is not affected & Interruptions barely occur & Almost all interruptions \\  & & & are detected \\ \hline VU & SC is slightly affected & Interruptions & The probability that an \\  & & infrequently/rarely occur & error is not recognized is low \\ \hline UL & Some SC processes are & & \\  & slightly affected, but the & Interruptions occur less & The probability that a fault \\  & flows work appropriately & & will not be detected is quite low \\ \hline NE & The flows fairly work, & & \\  & but some important processes & Interruptions occur & The error is not \\  & are affected as long as there & occasionally/intermittently & recognized infrequently \\  & is no unsatisfied customer & & \\ \hline LK & The flows fairly work, & & \\  & but some important processes & Interruptions occur often/in & Not being recognized is \\  & are seriously affected, and some & moderate amounts & a common occurrence \\  & customers are not satisfied & & \\ \hline VL & The process and flows have & & \\  & problems and have lost & & \\  & their basic functions & recurrently & cannot be recognized \\ \hline EL & SC operations completely lose & & \\  & their basic functions and endanger & Interruptions are & \\  & personal safety or violate laws & almost unpredictable & Errors are almost undetectable \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The corresponding denotation of semantic elements. The above weights are used to calculate the RPN's coefficient of each interruption failure mode, \(\lambda_{l_{i}l_{j}}\), as follows:  
\[\lambda_{l_{i}l_{j}}=\varphi^{l_{i}}\varphi^{l_{i}l_{j}}.\]  
Then the RPN number of each failure mode, \(\text{RPN}_{l_{i}l_{j}}\), can be represented as a weighted RPN, as follows:  
\[\text{RPN}_{l_{i}l_{j}}=\lambda_{l_{i}l_{j}}\times\text{Se}_{l_{i}l_{j}}\times \text{Oc}_{l_{i}l_{j}}\times\text{De}_{l_{i}l_{j}},\]  
where \(\text{Se}_{l_{i}l_{j}}\), \(\text{Oc}_{l_{i}l_{j}}\), and \(\text{De}_{l_{i}l_{j}}\) are the average crisp values of Se, Oc, and De, respectively, given for failure mode \(l_{i}l_{j}\). Se, Oc, and De for each failure mode is assessed by each expert using one of the semantic elements. Then, the average of all assessments, i.e., the average crisp values, are used to calculate the values of \(\text{Se}_{l_{i}l_{j}}\), \(\text{Oc}_{l_{i}l_{j}}\), and \(\text{De}_{l_{i}l_{j}}\). Step 4: Calculating the gray correlations and ranking. Define \(F\) and \(F_{0}\) as the comparison and reference matrices, respectively. In \(F\) matrix, the evaluations of Se, Oc, and De are listed with the corresponding interruption failure mode. These evaluations are benchmarked to the corresponding ratings in the reference matrix \(F_{0}\).