## 2 Preliminaries
### Some Associated Definitions on Preference Relations  
**Definition 1**. : [27] _An IVFPR \(R\) on the alternative set \(X=\left\{x_{1},x_{2},\ldots,x_{n}\right\}\) is presented by an interval-valued fuzzy judgment matrix \(R=\left(r_{ij}\right)_{n\times n}\subset X\times X\) with \(r_{ij}=\left[z_{ij},\overline{r}_{ij}\right],\) where the preference degree to which alternative \(x_{i}\) over \(x_{j}\) is between \(\underline{r}_{ij}\) and \(\overline{r}_{ij}\). Moreover, \(\underline{r}_{ij}\) and \(\overline{r}_{ij}\) fulfill the following conditions:_  
\[0\leq\underline{r}_{ij}\leq\overline{r}_{ij}\leq 1,\ \underline{r}_{ij}+ \overline{r}_{ji}=1,\ \underline{r}_{ii}=\overline{r}_{ii}=0.5\ \text{for all}\ i,j=1,2,\ldots,n.\]  
**Definition 2**. : [28] _Let \(X=\left\{x_{1},x_{2},\ldots,x_{n}\right\}\) be a non-empty alternative set. An IVIF set \(\widetilde{A}\) in \(X\) is denoted by \(\widetilde{A}=\left\{X,\mu_{\widetilde{A}}(x),\
u_{\widetilde{A}}(x)\big{|}x \in X\right\}\), where \(\mu_{\widetilde{A}}(x)=\left[\underline{\mu}_{\widetilde{A}}(x),\overline{ \mu}_{\widetilde{A}}(x)\right]\subseteq\left[0,1\right]\), and \(v_{\widetilde{A}}(x)=\left[\underline{v}_{\widetilde{A}}(x),\overline{v}_{ \widetilde{A}}(x)\right]\subseteq\left[0,1\right]\). \(\mu_{\widetilde{A}}(x)\) and \(v_{\widetilde{A}}(x)\) are interval values, indicating that the membership degree and non-membership degree of element \(X\) to IVFIV \(\widetilde{A}\), respectively, meeting \(\overline{\mu}_{\widetilde{A}}(x)+\overline{v}_{\widetilde{A}}(x)\leq 1\) for any \(x\in X\). Therefore, the IVFV \(\widetilde{A}\) can be expressed as \(\widetilde{A}=\left\{\left(X,\left[\underline{\mu}_{\widetilde{A}}(x), \overline{\mu}_{\widetilde{A}}(x)\right],\left[\underline{v}_{\widetilde{A}}( x),\overline{\
u}_{\widetilde{A}}(x)\right]\right)\right|x\in X\right\}\)._  
_Similarly, \(\widehat{\pi}_{\widetilde{A}}(x)=\left[\underline{\mu}_{\widetilde{A}}(x), \overline{\pi}_{\widetilde{A}}(x)\right]\) is named the interval hesitancy preference degree for any \(x\in X\), where \(\underline{\pi}_{\widetilde{A}}(x)=1-\overline{\mu}_{\widetilde{A}}(x)- \overline{v}_{\widetilde{A}}(x)\) and \(\overline{\pi}_{\widetilde{A}}(x)=1-\underline{\mu}_{\widetilde{A}}(x)- \underline{v}_{\widetilde{A}}(x)\)._  
_The pair \(\theta=\left(\left[\underline{\mu},\overline{\mu}],\left[\underline{v}, \overline{v}\right]\right)\) is call an interval-valued intuitionistic fuzzy value (IVFV) [29], where \(\left[\underline{\mu},\overline{\mu}\right]\subseteq\left[0,1\right]\) and \(\left[\underline{v},\overline{v}\right]\subseteq\left[0,1\right]\). [PERSON] and [PERSON] [15] proposed the concepts of the score function and accuracy function, which is one of the feasible methods to rank IVIFV._  
**Definition 3**. : [15] _For an IVIFV \(\theta=\left(\left[\underline{\mu},\overline{\mu}\right],\left[\underline{v}, \overline{v}\right]\right)\), the score function \(S(\theta)\) is_  
\[S(\theta)=\frac{1}{2}\Big{(}\underline{\mu}-\overline{v}+\overline{\mu}- \overline{v}\Big{)} \tag{1}\]  
_and accuracy function \(H(\theta)\) is_  
\[H(\theta)=\frac{1}{2}\Big{(}\underline{\mu}+\overline{\mu}+\underline{v}+ \overline{v}\Big{)} \tag{2}\]  
_For any two IVIFVs \(\theta_{1}=\left(\left[\underline{\mu}_{1},\overline{\mu}_{1}\right],\left[ \underline{v}_{1},\overline{v}_{1}\right]\right)\) and \(\theta_{2}=\left(\left[\underline{\mu}_  
**Remark 1**. : _Owing to the increasingly sophisticated decision-making environment and the vagueness of the human mind, it is hard to provide certain preferences on the pairwise comparison of alternatives regarding \(([0,1],[0,1])\) or \(([1,1],[1,1])\). Meanwhile, decision makers are usually experts in the decision-making issues in which they participate and the complete stubbornness rarely occurs. Therefore, the preference value \(\left([0,0],[0,0]\right)\) seldom appears.