## 2 Preliminaries
### Interval-Valued Intuitionistic Fuzzy Set
#### 2.3.2 IVIFS and IVIFNs  
The interval-valued intuitionistic fuzzy set (IVIFS) was first proposed by [PERSON] and [PERSON] (2017). In the IVIFS, the concept of interval is introduced to extend the IFS, so that the membership and non-membership functions in the IVIFS no longer function but are interval values. The concept of interval values can better address uncertain information in practice. **Definition 3**. : _IVIFS and IVIFNs:_  
_Let \(X\) be a non-empty set. An IVIFS \(\widetilde{\alpha}\) in \(X\) is given in Equation (4)._  
\[\widetilde{\alpha}=\left\{\langle x,\widetilde{\mu}_{\widetilde{\alpha}}(x), \widetilde{\
u}_{\widetilde{\alpha}}(x)\rangle|x\in X\right\} \tag{4}\]  
_where \(\widetilde{\mu}_{\widetilde{\alpha}}(x)=\left[\mu_{\widetilde{\alpha}}^{-},\mu _{\widetilde{\alpha}}^{+}\right]\subset[0,\,1]\) and \(\widetilde{\
u}_{\widetilde{\alpha}}(x)=\left[\
u_{\widetilde{\alpha}}^{-}, \
u_{\widetilde{\alpha}}^{+}\right]\subset[0,\,1],\ \forall x\in X\). Here, \(\widetilde{\mu}_{\widetilde{\alpha}}(x)\) and \(\widetilde{\
u}_{\widetilde{\alpha}}(x)\) are the intervals with the following conditions:_  
\[\mu_{\widetilde{\alpha}}^{-}=\inf(\widetilde{\mu}_{\widetilde{\alpha}}(x)),\ \mu_{ \widetilde{\alpha}}^{+}=\sup(\widetilde{\mu}_{\widetilde{\alpha}}(x)) \tag{5}\]  
_and_  
\[v_{\widetilde{\alpha}}^{-}=\inf(\widetilde{\
u}_{\widetilde{\alpha}}(x)),\ \
u_{ \widetilde{\alpha}}^{+}=\sup(\widetilde{\
u}_{\widetilde{\alpha}}(x)) \tag{6}\]  
\[0\leq\sup(\widetilde{\mu}_{\widetilde{\alpha}}(x))+\sup(\widetilde{\
u}_{ \widetilde{\alpha}}(x))\leq 1 \tag{7}\]  
Figure 2: IFS explanation of real number \(R\). _Additionally,_  
\[\widetilde{\pi}_{\widetilde{\alpha}}(x)=\big{[}\pi_{\widetilde{\alpha}}^{-},\pi_{ \widetilde{\alpha}}^{+}\big{]},\ x\in X \tag{8}\]  
_where \(\pi_{\widetilde{\alpha}}^{-}=1-\mu_{\widetilde{\alpha}}^{+}-\
u_{\widetilde{ \alpha}}^{+},\ \pi_{\widetilde{\alpha}}^{+}=1-\mu_{\widetilde{\alpha}}^{-}-\
u_{ \widetilde{\alpha}}^{-}.\)_  
The intervals \(\widetilde{\mu}_{\widetilde{\alpha}}(x)\), \(\widetilde{\
u}_{\widetilde{\alpha}}(x)\), and \(\widetilde{\pi}_{\widetilde{\alpha}}(x)\) denote the degrees of membership, non-membership, and hesitation of the element \(x\) in the set \(X\), respectively. Here, the interval pair \((\widetilde{\mu}_{\widetilde{\alpha}}(x),\widetilde{\
u}_{\widetilde{\alpha} }(x))\) is called the interval-valued intuitionistic fuzzy number (IVIFN). For convenience, it can be denoted as \(\widetilde{\alpha}=\big{(}\big{[}\mu_{\widetilde{\alpha}}^{-},\mu_{ \widetilde{\alpha}}^{+}\big{]},\big{[}\
u_{\widetilde{\alpha}}^{-},\
u_{ \widetilde{\alpha}}^{+}\big{]}\big{)}\). If \(\mu_{\widetilde{\alpha}}^{-}=\mu_{\widetilde{\alpha}}^{+}\) and \(\
u_{\widetilde{\alpha}}^{-}=\
u_{\widetilde{\alpha}}^{+}\), then the IVIFN degenerates to an intuitionistic fuzzy number (IFN). **Definition 4**. : _Basic arithmetic operations of IVIFNs:_  
_Let \(\widetilde{\alpha}_{1}=\Big{(}\Big{[}\mu_{\widetilde{\alpha}_{1}}^{-},\mu_{ \widetilde{\alpha}_{1}}^{+}\Big{]},\Big{[}\
u_{\widetilde{\alpha}_{1}}^{-},\
u _{\widetilde{\alpha}_{1}}^{+}\Big{]}\Big{)}\) and \(\widetilde{\alpha}_{2}=\Big{(}\Big{[}\mu_{\widetilde{\alpha}_{2}}^{-},\mu_{ \widetilde{\alpha}_{2}}^{+}\Big{]},\Big{[}\
u_{\widetilde{\alpha}_{2}}^{-},\
u _{\widetilde{\alpha}_{2}}^{+}\Big{]}\Big{)}\) be two IVIFNs; \(\lambda\) is a positive real number.