## 3 Preliminaries
### Z-Probabilistic Linguistic Term Set
25),s_{3}^{\prime}(0.25)\})\) and \(\hat{Z}_{2}=(\{s_{-2}(0.25),s_{-1}(0.5),s_{0}(0.25)\},\{s_{-2}^{\prime}(0.25),s _{0}^{\prime}(0.25),s_{2}^{\prime}(0.5)\})\). The next step is to ensure that the cardinality of the evaluation and credibility values for each ZPLV is equal, resulting in \(\hat{Z}_{1}=(\{s_{0}(0.2),s_{1}(0.8),s_{1}(0)\},\{s_{1}^{\prime}(0.4),s_{2}^{ \prime}(0.2),s_{3}^{\prime}(0.2)\})\) and \(\hat{Z}_{2}=(\{s_{-2}(0.2),s_{-1}(0.4),s_{0}(0.2)\},\{s_{-2}^{\prime}(0.2),s_{ 0}^{\prime}(0.2),s_{2}^{\prime}(0.4)\})\). In order to make use of ZPLVs, they need to be comparable with each other. [PERSON] et al. [18] proposed the score and deviation methods to compare two ZPLVs. Definition 9: Let \(\hat{Z}=(\hat{A},\hat{B})=(L_{\hat{A}}(p),L_{\hat{B}}(p))\) be a ZPLV, where \((\{L_{\hat{A}}^{m}(p_{\hat{A}}^{m})\mid m=1,2,\ldots,\#L_{\hat{A}}(p_{\hat{A }})\},\{L_{\hat{B}}^{n}(p_{\hat{B}}^{n})\mid n=1,2,\ldots,\#L_{\hat{B}}(p_{\hat {B}})\})\). The subscripts of \(L_{\hat{A}}^{m}\) and \(L_{\hat{B}}^{n}\) are denoted as \(v_{\hat{A}}^{m}\) and \(v_{\hat{B}}^{n}\), respectively. The score of \(\hat{Z}\) is defined as [18]:  
\[S(\hat{Z})=(\bar{\alpha}+\varsigma)(\bar{\alpha}^{\prime}+\zeta) \tag{4}\]  
where \(\bar{\alpha}=\frac{\sum_{m=1}^{n^{\hat{A}}_{n}\setminus\hat{p}_{\hat{A}}^{m}}v _{\hat{A}}^{m}}{\sum_{n=1}^{n^{\hat{A}}_{n}\setminus\hat{p}_{\hat{A}}^{m}}p_{ \hat{A}}^{m}}\) and \(\bar{\alpha}^{\prime}=\frac{\sum_{n \begin{tabular}{l l}
1. & _If_ \(S(\hat{Z}_{1})>S(\hat{Z}_{2})\) _then_ \(\hat{Z}_{1}>\hat{Z}_{2}\)_._ \\
2. & _If_ \(S(\hat{Z}_{1})<S(\hat{Z}_{2})\) _then_ \(\hat{Z}_{1}<\hat{Z}_{2}\)_._ \\
3. & _If_ \(S(\hat{Z}_{1})=S(\hat{Z}_{2})\) _then the deviation degree is required to compare_ \(\hat{Z}_{1}\) _and_ \(\hat{Z}_{2}\) _as follows:_ \\  & _(a)_ _If_ \(D(\hat{Z}_{1})>D(\hat{Z}_{2})\) _then_ \(\hat{Z}_{1}<\hat{Z}_{2}\)_._ \\  & _(b)_ _If_ \(D(\hat{Z}_{1})<D(\hat{Z}_{2})\) _then_ \(\hat{Z}_{1}>\hat{Z}_{2}\)_._ \\  & _(c)_ _If_ \(D(\hat{Z}_{1})=D(\hat{Z}_{2})\) _then_ \(\hat{Z}_{1}\approx\hat{Z}_{2}\)_._ \\  & _(d)_ _If_ \(L_{\hat{A}1}(p)=L_{\hat{A}2}(p)\) _and_ \(L_{\hat{B}1}(p)=L_{\hat{B}2}(p)\) _then_ \(\hat{Z}_{1}=\hat{Z}_{2}\)_._ \\ \end{tabular}   
**Example 3**. : _Let \(\hat{Z}_{1}\) and \(\hat{Z}_{2}\) be two ZPLVs, with \(\hat{Z}_{1}=\{\{s_{0}(0.2),s_{1}(0.8)\},\{s^{\prime}_{1}(0.55),s^{\prime}_{2}(0.45)\}\}\) and \(\hat{Z}_{2}=\{\{s_{-1}(0.75),s_{0}(0.25)\},\{s^{\prime}_{1}(0.1),s^{\prime}_{2 }(0.9)\}\}\). There are two terms in \(L_{\hat{A}1}(p_{\hat{A}1})\), two terms in \(L_{\hat{B}1}(p_{\hat{B}1})\), two terms in \(L_{\hat{A}2}(p_{\hat{A}2})\), and two terms in \(L_{\hat{B}2}(p_{\hat{B}2})\). The \(\zeta\) and \(\zeta\) values are both 3. The scores of \(\hat{Z}_{1}\) and \(\hat{Z}_{2}\), denoted as \(S(\hat{Z}_{1})\) and \(S(\hat{Z}_{2})\), are calculated as follows._  
\[S(\hat{Z}_{1})=(\bar{\alpha}+3)(\bar{\alpha}^{\prime}+3)\text{, and }\bar{ \alpha}=\frac{0\times 0.2+1\times 0.8}{0.2+0.8}=0.8\text{, and }\bar{\alpha}^{\prime}=\frac{1\times 0.55+2 \times 0.45}{0.55+0.45}=1.45\]  
\[\therefore S(\hat{Z}_{1})=(0.8+3)(1.45+3)=16.91\]  
_Similarly, the score of \(\hat{Z}_{2}\) can be calculated as follows._  
\[S(\hat{Z}_{2})=(\bar{\alpha}+3)(\bar{\alpha}^{\prime}+3)\text{, and }\bar{ \alpha}=\frac{-1\times 0.75+0\times 0.25}{0.75+0.25}=-0.75\text{, and }\bar{\alpha}^{\prime}=\frac{1\times 0.1 +2\times 0.9}{0.1+0.9}=1.9\]  
\[\therefore S(\hat{Z}_{2})=(-0.75+3)(1.9+3)=11.025\]  
\[\therefore S(\hat{Z}_{1})>S(\hat{Z}_{2})\Rightarrow\hat{Z}_{1}>\hat{Z}_{2}.\]  
**Example 4**. : _Let \(\hat{Z}_{1}\) and \(\hat{Z}_{2}\) be two ZPLVs, with \(\hat{Z}_{1}=\{\{s_{0}(0.8),s_{1}(0.2)\},\{s^{\prime}_{0}(0.3),s^{\prime}_{1}( 0.7)\}\}\) and \(\hat{Z}_{2}=\{\{s_{0}(0.9),s_{2}(0.1)\},\{s^{\prime}_{0}(0.65),s^{\prime}_{2}( 0.35)\}\}\). There are two terms in \(L_{\hat{A}1}(p_{\hat{A}1})\), two terms in \(L_{\hat{B}1}(p_{\hat{B}1})\), two terms in \(L_{\hat{A}2}(p_{\hat{A}2})\), and two terms in \(L_{\hat{B}2}(p_{\hat{B}2})\). The \(\zeta\) and \(\zeta\) values are both 3.