# Failure Mode and Effects Analysis Considering Consensus and Preferences Interdependence
## 3 IVPFGBM and IVPFWGBM Operators
### Basic Concepts of IVPFS  
**Definition 1**.: _[_48_]_ _Let \(R=\{r_{1},r_{2},\cdots,r_{n}\}\) be an ordinary finite nonempty set and an IVPFS in \(R\) is defined as_  
\[P=\{<r,\mu_{P}(r),\
u_{P}(r)>r\in R\}, \tag{1}\]  
_where \(\mu_{P}(r)=[\underline{\mu}_{P}(r),\overline{\mu}_{P}(r)]\), \(\
u_{P}(r)=[\underline{\
u}_{P}(r),\overline{\
u}_{P}(r)]\), \(\mu_{P}(r)\subseteq[0,1]\), and \(\
u_{P}(r)\subseteq[0,1]\) are interval values and \(\underline{\mu}_{P}(r)\in[0,1]\), \(\overline{\mu}_{P}(r)\in[0,1]\), \(\underline{\
u}_{P}(r)\in[0,1]\), and \(\overline{\
u}_{P}(r)\in[0,1]\) satisfy \(\overline{\mu}_{P}^{2}(r)+\overline{\
u}_{P}^{2}(r)\leq 1\)._  
For every \(r\in R\), we designate \(\pi_{P}(r)=[\underline{\pi}_{P}(r),\overline{\pi}_{P}(r)]\) as the degree of indeterminacy of the IVPFS, where \(\underline{\pi}_{P}(r)=\sqrt{1-\overline{\mu}_{P}^{2}(r)-\overline{\
u}_{P}^{2 }(r)}\) and \(\overline{\pi}_{P}(r)=\sqrt{1-\underline{\mu}_{P}^{2}(r)-\underline{\
u}_{P}^ {2}(r)}\). For convenience, \(P=([\underline{\mu}_{P},\overline{\mu}_{P}],[\underline{\
u}_{P},\overline{ \
u}_{P}])\) is called an interval-valued Pythagorean fuzzy number (IVPFN), where \(\overline{\mu}_{P}^{2}+\overline{\
u}_{P}^{2}\leq 1\). Notability, the IVPFS is reduced into PFS when \(\underline{\mu}_{P}=\overline{\mu}_{P}\) and \(\underline{\
u}_{P}=\overline{\
u}_{P}\).  
**Definition 2**.: _[_48_]_ _Let \(P=([\underline{\mu}_{P},\overline{\mu}_{P}],[\underline{\
u}_{P},\overline{ \
u}_{P}])\), \(P_{1}=([\underline{\mu}_{P_{1}},\overline{\mu}_{P_{1}}],[\underline{\
u}_{P_{ 1}},\overline{\
u}_{P_{1}}])\), and \(P_{2}=([\underline{\mu}_{P_{2}},\overline{\mu}_{P_{2}}],[\underline{\
u}_{P_{ 2}},\overline{\
u}_{P_{2}}])\) be three IVPFNs, and \(\lambda>0\). Then, the basic operational laws are defined as follows:_  
**Definition 3**.: _[_48_]_ _Suppose \(P_{1}=([\underline{\mu}_{P_{1}},\overline{\mu}_{P_{1}}],[\underline{\
u}_{P_{ 1}},\overline{\
u}_{P_{1}}])\) and \(P_{2}=([\underline{\mu}_{P_{2}},\overline{\mu}_{P_{2}}],[\underline{\
u}_{P_{ 2}},\overline{\
u}_{P_{2}}])\) are two IVPFNs, then the distance between \(P_{1}\) and \(P_{2}\) is defined as:_  
\[d(P_{1},P_{2})=\frac{1}{4}\Big{(}\Big{|}\underline{\mu}_{P_{1}}^{2}-\underline {\mu}_{P_{2}}^{2}\Big{|}+\Big{|}\overline{\mu}_{P_{1}}^{2}-\overline{\mu}_{P_{ 2}}^{2}\Big{|}+\Big{|}\overline{\
u}_{P_{1}}^{2}-\overline{\
u}_{P_{2}}^{2} \Big{|}+\Big{|}\overline{\mu}_{P_{1}}^{2}-\overline{\mu}_{P_{2}}^{2}\Big{|}+ \Big{|}\overline{\mu}_{P_{1}}^{2}-\overline{\mu}_{P_{2}}^{2}\Big{|}\Big{)}. \tag{2}\]  
**Definition 4**.: _[_49_]_ _Let \(P=([\underline{\mu}_{P},\overline{\mu}_{P}],[\underline{\
u}_{P},\overline{ \
u}_{P}])\) be an IVPFN. Then, the score and accuracy functions of \(P\) are defined, respectively, as follows_  
\[s(P) =\frac{1}{2}\left(\underline{\mu}_{P}^{2}+\overline{\mu}_{P}^{2}- \underline{\
u}_{P}^{2}-\overline{\
u}_{P}^{2}\right), \tag{3}\] \[a(P) =\frac{1}{2}\left(\underline{\mu}_{P}^{2}+\overline{\mu}_{P}^{2}+ \underline{\
u}_{P}^{2}+\overline{\
u}_{P}^{2}\right). \tag{4}\]  
According to Equations (3) and (4), the order relation for two IVPFNs [49] is defined as follows:  
**(1)**: If \(s(P_{1})>s(P_{2})\), then \(P_{1}\) is superior to \(P_{2}\), \(P_{1}>P_{2}\);
**(2)**: If \(s(P_{1})=s(P_{2})\), then  
**a.**: If \(a(P_{1})>a(P_{2})\), then \(P_{1}\) is superior to \(P_{2}\), \(P_{1}>P_{2}\);
**b.**: If \(a(P_{1})=a(P_{2})\), then \(P_{1}\) is equivalent to \(P_{2}\), \(P_{1}=P_{2}\).