# Failure Mode and Effects Analysis Considering Consensus and Preferences Interdependence
## 3 IVPFGBM and IVPFWGBM Operators
### Consensus-ReachingProcess  
To increase the agreement degree of FMEA team experts, a consensus-reaching process is introduced into the risk assessment process of FMEA. According to the evaluation information provided by experts, we will define the consensus degree on four levels, which include the consensus degree on elements of failure modes, the consensus degree on failure modes, the consensus degree on the matrix, and the consensus degree on team experts. **Definition 8**. : _Let \(A^{k}\) be the assessment matrix provided by the k expert and C be the comprehensive assessment matrix. The consensus degree between assessment matrix \(A^{k}\) and comprehensive assessment matrix \(C\) on the element of failure modes FM\({}_{i}\) under risk factors RF\({}_{j}\) is given as follows:_  
\[CDE^{k}_{ij}(A^{k},C)=\frac{1}{2}(1+grc(a^{k}_{ij},c_{ij})), \tag{10}\]  
_where \(grc(a^{k}_{ij},c_{ij})\) is the gray relational coefficient between \(a^{k}_{ij}\) and \(c_{ij}\). The \(CDE^{k}_{ij}(A^{k},C)\) is called the consensus degree on elements of failure modes. Given \(A^{k}\) and C, the elaborated expression of \(CDE^{k}_{ij}(A^{k},C)\) is_  
\[CDE^{k}_{ij}(A^{k},C)=\frac{1}{2}\Bigg{(}1+\frac{\min\min\min|s(a^{k}_{ij})-s( c_{ij})|+\rho\max\max\max|s(a^{k}_{ij})-s(c_{ij})|}{|s(a^{k}_{ij})-s(c_{ij})|+ \rho\max\max\max|s(a^{k}_{ij})-s(c_{ij})|}\Bigg{)}, \tag{11}\]  
_where \(\rho\) is the identification coefficient, \(\rho\in[0,1]\). The smaller the value of the identification coefficient, the larger the range of the gray relational coefficient, but it will not affect the final priority of failure modes [50]. Generally, \(\rho=0.5\)[51]._  
**Definition 9**. : _Let \(A^{k}\) be the assessment matrix provided by the k expert and C be the comprehensive assessment matrix. The consensus degree between assessment matrix \(A^{k}\) and comprehensive assessment matrix \(C\) on the failure modes FM\({}_{i}\) is defined as:_  
\[CDF^{k}_{i}(A^{k},C)=\frac{1}{n}{\sum_{j=1}^{n}CDE_{ij}(A^{k},C)}, \tag{12}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Linguistic Variables** & **Abbreviation** & **IVPFNs** \\ \hline Very high & VH & ([0.8000,0.9000],[0.1000,0.2000]) \\ High & H & ([0.70000,0.8000],[0.2000,0.3000]) \\ Medium & M & ([0.5000,0.6000],[0.4000,0.5000]) \\ Low & L & ([0.3000,0.4000],[0.6000,0.7000]) \\ Very low & VL & ([0.1000,0.2000],[0.8000,0.9000]) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Linguistic terms for rating the weights of risk factors [49]. \begin{table}
\begin{tabular}{c c c} \hline \hline
**LinguisticVariables** & **Abbreviation** & **IVPFNs** \\ \hline Extremely high & EH & ([0.9000,1.0000],[0.0000,0.1000]) \\ Very high & VH & ([0.8000,0.9000],[0.1000,0.2000]) \\ High & H & ([0.7000,0.8000],[0.2000,0.3000]) \\ Medium high & MH & ([0.6000,0.7000],[0.3000,0.4000]) \\ Medium & M & ([0.5000,0.6000],[0.4000,0.5000]) \\ Medium low & ML & ([0.4000,0.5000],[0.5000,6.000]) \\ Low & L & ([0.3000,0.4000],[0.6000,0.7000]) \\ Very low & VL & ([0.2000,0.3000],[0.7000,0.8000]) \\ Extremely low & EL & ([0.1000,0.2000],[0.8000,0.9000]) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Linguistic terms for rating the failure modes [49]. then \(CDF_{i}^{k}(A^{k},C)\) is called the consensus degree on failure modes._  
**Definition 10**. : _Let \(A^{k}\) be the assessment matrix provided by the \(k\) expert and \(C\) be the comprehensive assessment matrix. The consensus degree between assessment matrix \(A^{k}\) and comprehensive assessment matrix \(C\) on the matrix \(A^{k}\) is calculated as_  
\[CDM^{k}(A^{k},C)=\frac{1}{mn}{\sum_{i=1}^{m}{\sum_{j=1}^{n}{CDE_{ij}(A^{k},C)}}}, \tag{13}\]  
_then \(CDM^{k}(A^{k},C)\) is called the consensus degree on decision-makers._  
**Definition 11**. : _Let \(A^{k}\) be the assessment matrix provided by the \(k\) expert and \(C\) be the comprehensive assessment matrix. The consensus degree on the team experts is determined as_  
\[CDT(A^{k},C)={\sum_{k=1}^{L}{\lambda_{k}CDM^{k}(A^{k},C)}}. \tag{14}\]  
Then, we will construct a consensus model.