## 3 Preliminaries
### Z-Probabilistic Linguistic Term Set  
The concept of PLTS is very powerful, but one downside is that it ignores the credibility of decision-makers. This can lead to inaccurate results in the decision-making process. To overcome this, the Z-Probabilistic Linguistic Term Set (ZPLTS) was proposed [18]. Definition 5: Let \(X\) be a non-empty set. A ZPLTS, \(\hat{Z}^{\#}\), is defined as \(\hat{Z}^{\#}=\{\langle x,\hat{Z}\rangle\mid x\in X\}\), with \(\hat{Z}\) being a Z Probabilistic Value [18]. Definition 6: Let \(L_{A}(p)\) and \(L_{B}(p)\) be two PLTSs. A Z Probabilistic Linguistic Value (ZPLV) is defined as [18]:  
\[\hat{Z}=(\hat{A},\hat{B})=(L_{\hat{A}}(p),L_{\hat{B}}(p)) \tag{2}\]  
where \(L_{\hat{A}}(p)=\{L_{\hat{A}}^{m}(p_{A}^{m})\mid L_{\hat{A}}^{m}\in S,p_{A}^{m} \leq 0,m=1,2,\ldots,\#L_{\hat{A}}(p_{\hat{A}}),\sum_{m=1}^{\#L_{\hat{A}}(p_{ \hat{A}})}p_{\hat{A}}^{m}\leq 1\}\),  
\(L_{\hat{B}}(p)=\{L_{B}^{n}(p_{B}^{n})\mid L_{B}^{n}\in S^{\prime},p_{B}^{n}\leq 0,n=1,2,\ldots,\#L_{\hat{B}}(p_{\hat{B}}),\sum_{m=1}^{\#L_{\hat{B}}(p_{\hat{B}}) }p_{\hat{B}}^{n}\leq 1\}\), and  
\(S=\{-\zeta,\ldots,-1,0,1,\ldots,\zeta\}\) and \(S^{\prime}=\{-\zeta,\ldots,-1,0,1,\ldots,\zeta\}\) are LTSs. In order to normalize a ZPLV, both the value and the credibility sets, that is, \(L_{\hat{A}}(p)\) and \(L_{\hat{B}}(p)\), must be normalized. A ZPLV is normalized in two stages. Firstly, the probability distribution of the linguistic terms is normalized, and thereafter, the number of linguistic terms is normalized. The first stage is required whenever the sums of the probabilities are less than 1; then, the probabilities must be normalized to equate to 1. Definition 7: Let \(\hat{Z}\) be a ZPLV with \(\hat{Z}=(L_{\hat{A}}(p),L_{B}(p))\), and \(\sum_{m=1}^{\#L_{\hat{A}}(p_{\hat{A}})}p_{\hat{A}}^{m}\leq 1\) and \(\sum_{n=1}^{\#L_{\hat{B}}(p_{\hat{B}})}p_{\hat{B}}^{n}\leq 1\). The associated ZPLV after normalizing the probability distributions can be defined as [18]:  
\[\hat{Z}=(L_{\hat{A}}(p),L_{\hat{B}}(p))=(\{L_{\hat{A}}^{m}(\hat{p}_{\hat{A}}^{ m})\},\{L_{\hat{B}}^{n}(\hat{p}_{\hat{B}}^{n})\}) \tag{3}\]_where \(\dot{p}_{\hat{A}}^{m}=\frac{p_{\hat{A}}^{n}}{\sum_{n=1}^{n^{\hat{A}}_{n}\setminus \hat{p}_{\hat{A}}^{m}}\frac{p_{\hat{A}}^{n}}{p_{\hat{A}}^{n}}}\) and \(\dot{p}_{\hat{B}}^{n}=\frac{p_{\hat{B}}^{n}}{\sum_{n=1}^{n^{\hat{B}}_{n}\setminus \hat{p}_{\hat{B}}^{n}}p_{\hat{B}}^{n}}\), \(\dot{p}_{\hat{A}}^{n}=1,2,\ldots,\#L_{\hat{A}}(p_{\hat{A}})\) and \(n=1,2,\ldots,\#L_{\hat{B}}(p_{\hat{B}})\)._  
After the first normalization stage, that is, normalizing the probabilities to equate to \(1\), the second stage is to normalize the number of linguistic terms. This is required in order to calculate the distance between two ZPLVs. Definition 8: Let \(\hat{Z}_{1}\) and \(\hat{Z}_{2}\) be two ZPLVs with \(\hat{Z}_{1}=(L_{\hat{A}1}(p),L_{\hat{B}1}(p))\) and \(\hat{Z}_{2}=(L_{\hat{A}2}(p),L_{\hat{B}2}(p))\). The number of linguistic terms in a PLTS is given by \(\#L(p)\). In the case that \(\#L_{\hat{A}1}(p_{\hat{A}1})>\#L_{\hat{A}2}(p_{\hat{A}2})\), then the number of terms need to be normalized as follows [18]:  
1. \(L_{\hat{A}2}\) is increased by adding \(\phi\) terms to \(L_{\hat{A}2}\), where \(\phi=\#L_{\hat{A}1}(p_{\hat{A}1})-\#L_{\hat{A}2}(p_{\hat{A}2})\). The \(\phi\) linguistic terms to be added may be any linguistic terms in \(L_{\hat{A}2}(p)\). 2. The probabilities of the \(\phi\) linguistic terms that were added must be set to \(0\). The same process applies when \(\#L_{\hat{B}1}(p_{\hat{B}1})>\#L_{\hat{B}2}(p_{\hat{B}2})\). Example 2: Let \(\hat{Z}_{1}\) and \(\hat{Z}_{2}\) be two ZPLVs with \(\hat{Z}_{1}=(\{s_{0}(0.2),s_{1}(0.8)\},\{s_{1}^{\prime}(0.4),s_{2}^{\prime}(0. 2),s_{3}^{\prime}(0.2)\})\) 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 normalize the ZPLVs, firstly, all probabilities must add up to \(1\). This results in \(\hat{Z}_{1}=(\{s_{0}(0.2),s_{1}(0.8)\},\{s_{1}^{\prime}(0.5),s_{2}^{\prime}(0.