# Failure Mode and Effects Analysis Considering Consensus and Preferences Interdependence
## 3 IVPFGBM and IVPFWGBM Operators
### Risk Factors Weight  
Entropy was developed by [PERSON] and weaver [PERSON] and [PERSON] (1955) and is well-suited to reflect the relative importance of the criteria that represent the intrinsic information for assessing issues [PERSON] (1955). Therefore, the entropy method has been used by many researchers for calculating the objective weight of criteria in an MCDM problem [PERSON] (1955); [PERSON] (1955). In order to reflect the importance of risk factors, we will define an interval-valued Pythagorean fuzzy entropy (IVPFE), which is parallel to interval-valued intuitionistic fuzzy entropy [PERSON] (1955), to compute the weight of risk factors.  
**Definition 12**.: _Let \(P\) be an IVPFS defined in the universe of discourse \(U\). The IVPFE is given as follows:_  
\[E(P)=\frac{1}{n}\sum_{i=1}^{n}\frac{2-\left|\mu_{P_{i}}^{2}-\
u_{P_{i}}^{2} \right|-\left|\overline{\mu}_{P_{i}}^{2}-\overline{\
u}_{P_{i}}^{2}\right|+ \underline{\pi}_{P_{i}}^{2}+\overline{\pi}_{P_{i}}^{2}}{2+\left|\underline{ \mu}_{P_{i}}^{2}-\
u_{P_{i}}^{2}\right|+\left|\overline{\mu}_{P_{i}}^{2}- \overline{\
u}_{P_{i}}^{2}\right|+\underline{\pi}_{P_{i}}^{2}+\overline{\pi} _{P_{i}}^{2}}. \tag{20}\]  
**Step 3:** Aggregate the experts' preferences into a comprehensive evaluation matrix.  
In order to depict the interdependent relationships between experts' preferences, we can aggregate all individual evaluation information into a collective evaluation matrix \(D=(d_{ij})m\times n\) by the IVPFWGBM operator, where \(d_{ij}=IVPFWGBM_{\lambda}^{x,y}(a_{ij}^{1},a_{ij}^{2},\cdot\cdot\cdot,a_{ij}^{ L})\).  
**Step 4:** Calculate the combination weights of risk factors.  
**Step4.1:** Determine the subjective weights of risk factors.  
According to Table 1, we can convert all linguistic variables elements in matrix \(Z^{k}\) into the IVPFN to construct matrix \(H^{k}=\left(h_{j}^{k}\right)_{1\times n}\). Subsequently, the IVPFWGBM operator is applied to aggregate all individual valuations for risk factors to construct a collective evaluation matrix \(H=\left(h_{j}\right)_{1\times n^{\prime}}\) where \(h_{j}=IVPFWGBM_{\lambda}^{x,y}(h_{j}^{1},h_{j}^{2},\cdot\cdot\cdot,h_{j}^{L})\).  
Then, the normalized subjective weight of each risk factor is computed as follows:  
\[w_{j}^{s}=\frac{s(h_{j})}{\sum_{j=1}^{n}s(h_{j})},\ (j=1,2,\cdot\cdot\cdot,n). \tag{21}\]  
**Step 4.2:** Compute the objective weights of risk factors using the entropy method.  
\[E(d_{ij})=\frac{2-\left|\underline{\mu}_{P_{ij}}^{2}-\
u_{P_{ij}}^{2}\right|- \left|\overline{\mu}_{P_{ij}}^{2}-\overline{\
u}_{P_{ij}}^{2}\right|+ \underline{\pi}_{P_{ij}}^{2}+\overline{\pi}_{P_{ij}}^{2}}{2+\left|\underline{ \mu}_{P_{ij}}^{2}-\
u_{P_{ij}}^{2}\right|+\left|\overline{\mu}_{P_{ij}}^{2}- \overline{\
u}_{P_{ij}}^{2}\right|+\underline{\pi}_{P_{ij}}^{2}+\overline{\pi }_{P_{ij}}^{2}},(i=1,2,\cdot\cdot\cdot,m;j=1,2,\cdot\cdot\cdot,n). \tag{22}\]Then, the normalized objective weight of each risk factor can be calculated by applying the following formula.  
\[w_{j}^{o}=\frac{\sum_{i=1}^{m}\left(1-E(d_{ij})\right)}{\sum_{j=1}^{n}\sum_{i=1}^ {m}\left(1-E(d_{ij})\right)},(i=1,2,\cdot\cdot\cdot,m;j=1,2,\cdot\cdot\cdot,n). \tag{23}\]  
**Step 4.3:** Obtain the integration weights of risk factors:  
\[w_{j}=\sigma w_{j}^{s}+(1-\sigma)w_{j}^{o},(i=1,2,\cdot\cdot\cdot,m;j=1,2, \cdot\cdot,n), \tag{24}\]  
where parameter \(\sigma\) is the relative importance coefficient of the subjective weight and satisfies \(0\leq\sigma\leq 1\). Generally, the subjective weight and objective weight are assumed to be equally important, namely, \(\sigma=0.5\).