## 3 Generalized Probabilistic Interval-Valued Hesitant Fuzzy Information Aggregation Operators
### GPIVHFOWA Operator  
**Definition 8**. : _Let \(h_{j}(j=1,2,\cdots,n)\) be a set of PIVHFes, a generalized probabilistic interval-valued hesitant fuzzy ordered weighted averaging (GPIVHFOWA) operator is a mapping GPIVHFOWA: \(H^{n}\to H\), such that_  
\[GPIVHFOWA(h_{1},h_{2},\cdots,h_{n})=\mathop{\oplus}\limits_{j=1}^{n}w_{j}h_{ \sigma(j)}, \tag{6}\]  
_where let \(w=(w_{1},w_{2},\cdots,w_{n})^{T}\) be the related weight vectors, and satisfies the condition \(w_{j}\geq 0\) and \(\sum_{j=1}^{n}w_{j}=1\)._  
**Theorem 2**. : _Let \(h_{j}(j=1,2,\cdots,n)\) be a set of PIVHFes, then the aggregated value by the GPIVHFOWA operator is also PIVHFE, and_  
\[\footnotesize \begin{array}{c}\mbox{\sc GPIVHFOWA}(h_{1},h_{2},\cdots,h_{n} )=\mathop{\oplus}\limits_{j=1}^{n}w_{j}h_{\sigma(j)}=\bigg{\{}\left(\left[f^{- 1}\left(\mathop{\sum}\limits_{j=1}^{n}w_{j}/(\gamma_{\sigma(j)}^{(k)-}\right) \right),f^{-1}\left(\mathop{\sum}\limits_{j=1}^{n}w_{j}f(\gamma_{\sigma(j)}^{( k)+})\right)\right],\mathop{\overline{\sum}\limits_{j=1}^{n}p_{j}^{(k)}} \bigg{)}\bigg{|}k=1,2,\cdots,l\bigg{\}}.\end{array}  \tag{7}\]  
_where \(\left[\gamma_{\sigma(j)}^{(k)-}\gamma_{\sigma(j)}^{(k)+}\right]\) is the \(j\) interval in descending order of \([\gamma_{1}^{(k)-},\gamma_{1}^{(k)+}],[\gamma_{2}^{(k)-},\gamma_{2}^{(k)+}], \cdots,[\gamma_{m}^{(k)-},\gamma_{n}^{(k)+}]\), and \(\mathop{\overline{\sum}\limits_{j=1}^{n}p_{j}^{(k)}}=\mathop{\sum}\limits_{j =1}^{n}p_{j}^{(k)}/\mathop{\sum}\limits_{k=1}^{l}\mathop{\sum}\limits_{j=1}^{ n}p_{j}^{(k)}\)._  
Proof. :  
(1) It is proved that the aggregation result obtained by an operator is still PIVHFE, then  
\[\left[f^{-1}\left(\sum\limits_{j=1}^{n}w_{j}f(\gamma_{\sigma(j)}^{(k)-}) \right),f^{-1}\left(\sum\limits_{j=1}^{n}w_{j}f(\gamma_{\sigma(j)}^{(k)+}) \right)\right]\in[0,1],\mathop{\sum}\limits_{j=1}^{\overline{n}}p_{j}^{(k)} \in[0,1],\mathop{\sum}\limits_{k=1}^{l}\mathop{\sum}\limits_{j=1}^{ \overline{n}}p_{j}^{(k)}=1.\]  
According to the computational formula of \(\mathop{\overline{\sum}\limits_{j=1}^{\overline{n}}p_{j}^{(k)}}\), we know that \(\mathop{\overline{\sum}\limits_{j=1}^{\overline{n}}p_{j}^{(k)}}\in[0,1]\), \(\mathop{\sum}\limits_{k=1}^{l}\mathop{\sum}\limits_{j=1}^{\overline{n}}p_{j}^ {(k)}=1\) are clearly established. Since \(0\leq\gamma_{\sigma(j)}^{(k)-}\leq\gamma_{\sigma(j)}^{(k)+}\leq 1\) for all \(k=1,2,\cdots,l,j=1,2,\cdots,n\). \(f(t)\) and \(f^{-1}(t)\) are strictly monotone increasing functions, then we have \(f(0)\leq f(\gamma_{\sigma(j)}^{(k)-})\leq f(\gamma_{\sigma(j)}^{(k)+})\leq f(1)\), and then \(w_{j}f(0)\leq w_{j}f(\gamma_{\sigma(j)}^{(k)-})\leq w_{j}f(\gamma_{\sigma(j)}^{ (k)+})\leq w_{j}f(1)\). It pointed out that  
\[f(0)=\sum\limits_{j=1}^{n}w_{j}f(0)\leq\sum\limits_{j=1}^{n}w_{j}f\Big{(}\gamma _{\sigma(j)}^{(k)-}\Big{)}\leq\sum\limits_{j=1}^{n}w_{j}f\Big{(}\gamma_{\sigma (j)}^{(k)+}\Big{)}\leq\sum\limits_{j=1}^{n}w_{j}f(1)=f(1). \tag{8}\]  
The simultaneous acting function \(f^{-1}(t)\) on both sides of the above formula, then  
\[ \begin{array}{c}0=f^{-1}(f(0))\leq f^{-1}\left(\mathop{\sum}\limits_{j=1}^{ n}w_{j}f\Big{(}\gamma_{\sigma(j)}^{(k)-}\Big{)}\right)\leq f^{-1}\left(\mathop{ \sum}\limits_{j=1}^{n}w_{j}f\Big{(}\gamma_{\sigma(j)}^{(k)+}\Big{)}\right) \leq f^{-1}(f(1))=1,\\ \left[f^{-1}\left(\mathop{\sum}\limits_{j=1}^{n}w_{j}f\Big{(}\gamma_{\sigma(j) }^{(k)-}\Big{)}\right),f^{-1}\left(\mathop{\sum}\limits_{j=1}^{n}w_{j}f \Big{(}\gamma_{\sigma(j)}^{(k)+}\Big{)}\right)\right]\in[0,1].\end{array}  \tag{9}\]  
Therefore, our proof of Theorem 2 is over. Next, the mathematical induction will be used to prove the Equation (7). When \(n=2\), we have:  
\[