## 4 Research Method
### Indicators Transformation and Failure Analysis Process Based on Fuzzy-FMEA
#### 4.2.2 Failure Analysis Process Based on Fuzzy-FMEA  
The assessment models and procedures are outlined below.  
1. Build a fuzzy term set.  
The evaluation indicators used to assess the subject's performance are generally ambiguous. Fuzzy linguistic evaluation utilizes evaluation values and linguistic variables to quantify the performance of indicators, such as \"should\", \"seem\", \"very prominent\", \"small\", and so on. These may be represented using fuzzy mathematics, which completely expresses the fuzziness of people's ideas and expressions, making assessment more objective. A consistent evaluation standard, namely the fuzzy term set, has been built for the new risk assessment factors, as shown in Table 3.  
Determine the fuzzy number of fuzzy terms.  
Triangular fuzzy numbers serve as a method for transforming ambiguous and uncertain linguistic variables into precise numerical values [45]. This method helps solve the issues in which an object's performance cannot be precisely quantified but can only be fuzzily evaluated using linguistic terms. Triangular fuzzy numbers are often used to quantify the fuzzy term set in Table 1. Formula (1) illustrates the use of the triangular fuzzy number \(\tilde{A}=(L,M,U)\), and the membership function for quantification. And M is the median fuzzy number, whereas L and U denote the lower and upper bounds of the fuzzy number, respectively.  
\[\mu_{\tilde{A}}(x)=\left\{\begin{array}{ll}\frac{x-L}{M-L},\ (L\leq x\leq M)\\ \frac{U-x}{L-M},(M\leq x\leq U)\\ 0,(x<L\ or\ x>U)\end{array}\right. \tag{1}\]  
In Formula (1), \(u_{\tilde{A}}(x)\) represents the score \(x\) subordinate to a certain fuzzy term set. This function's value ranges from 0 to 1, with increasing subordination bringing the number closer to 1.  
The membership function curves determined in accordance with Formula (1) are shown in Figure 2. The horizontal axis represents numerical values, whereas the vertical axis represents membership degree. The membership function is 0 at L and U, reaching a maximum value of 1 at M, thereby forming a triangular shape. This membership function enables triangular fuzzy numbers to represent uncertainty flexibly and underpins subsequent defuzzification processes.  
Assume there are \(m\) experts, the \(j\)th expert's experience value is \(a_{j}\) (\(j=1,2,\ldots,m\)), and the expert's judgment of a variable in a failure mode is \(A_{j}\). The triangular fuzzy number  
Figure 2: Membership function curves of fuzzy terms.  
\begin{table}
\begin{tabular}{c c c c c} \hline
**Fuzzy** & **Severity (S)** & **Occurrence** & **Detection** & **Fuzzy** \\
**Terminology** & & **(O)** & **(D)** & **Numbers** \\ \hline Very high (VH) & Significant impact & Very high & Very high & (L, M, U) \\ High (H) & Moderate impact & High & High & (L, M, U) \\ Medium (M) & Medium impact & Medium & General & (L, M, U) \\ Low (L) & Slight impact & Low & Low & (L, M, U) \\ Very low (R) & Minimal impact & Very low & scarcely & (L, M, U) \\ \hline \end{tabular}
\end{table}
Table 3: The fuzzy term set.  
is then represented as \(\tilde{A}_{j}=\left(L_{j},M_{j},U_{j}\right)\), and Formula (2) is used to determine the fuzzy numbers for different terms. In Formula (2), \(\sum_{j=1}^{m}a_{j}=1\), \(a_{j}\in(0,1)\).  
\[\left\{\begin{array}{l}L=\sum_{j=1}^{m}a_{j}L_{j}\\ M=\sum_{j=1}^{m}a_{j}L_{j}\\ U=\sum_{j=1}^{m}a_{j}L_{j}\end{array}\right. \tag{2}\]  
(3) Defuzzification.  
Defuzzification is a critical stage in the fuzzy-MCDM process for evaluating service performance. There are several methods for defuzzification, the most prevalent of which is the center-of-gravity method. This method uses the center-of-gravity of the area measured by the ROC curve and abscissa as the final fuzzy inference output value [46].  
\[x(\tilde{A})=\frac{\int_{L}^{U}x\mu_{\tilde{A}}(x)dx}{\int_{L}^{U}\mu_{\tilde{ A}}(x)dx} \tag{3}\]  
Formula (3) calculates the defuzzification value, or failure risk priority value of the secondary indication.