## 3 Generalized Probabilistic Interval-Valued Hesitant Fuzzy Information Aggregation Operators
### The Properties of the GPIVHFOWA Operator and the GPIVHFOWG Operator  
This section will explore some basic properties of the GPIVHFOWA operator and the GPIVHFOWG operator, including idempotency, monotonicity and boundedness and commutativity. **Property 1 (idempotency)**. : _Let \(h_{j}(j=1,2,\cdots,n)\) be a set of PIVHFEs. If \(\forall i=1,2,\cdots,n,h_{i}=h\), then:_  
\[ \begin{array}{l}\text{{GPIVHFOWA}}(h_{1},h_{2},\cdots,h_{n})=h,\end{array}  \tag{14}\]  
_where \((\widetilde{h}_{1},\widetilde{h}_{2},\cdots,\widetilde{h}_{n})\) is any permutation of \((h_{1},h_{2},\cdots,h_{n})\)._  
Relationship among the Probabilistic Interval-Valued Hesitant Fuzzy Information Aggregation Operators  
In this section, we first study some common information aggregation operators when additive operator \(g(t)\) take different functions. Then, the relationships among these common information aggregation operators are discussed. **Case 1:** When \(g(t)=-\ln(t)\), the GPIVHFOWA operator and the GPIVHFOWG operator are transformed into probabilistic interval-valued hesitant fuzzy ordered weighted averaging (PIVHFOWA) operators and probabilistic interval-valued hesitant fuzzy ordered weighted geometric (PIVHFOWG) operators respectively, as follows:  
\[PIVHFOWA(h_{1},h_{2},\cdots,h_{n})=\Bigg{\{}\Bigg{(}\Bigg{[}1-\prod\limits_{j= 1}^{n}(1-\gamma_{e(j)}^{(k)-})^{w_{j}},1-\prod\limits_{j=1}^{n}(1-\gamma_{e(j) }^{(k)+})^{w_{j}}\Bigg{]}\cdot\ \ \frac{\overline{n}}{j=1}p_{j}^{(k)}\Bigg{)}\Bigg{|}k=1,2,\cdots,l\Bigg{\}}, \tag{25}\]  
\[PIVHFOWG(h_{1},h_{2},\cdots,h_{n})=\Bigg{\{}\Bigg{(}\Bigg{[}\prod\limits_{j=1} ^{n}(\gamma_{e(j)}^{(k)-})^{w_{j}},\prod\limits_{j=1}^{n}(\gamma_{e(j)}^{(k)+ })^{w_{j}}\Bigg{]}\,\ \ \sum\limits_{j=1}^{n}p_{j}^{(k)}\Bigg{)}\Bigg{|}k=1,2,\cdots,l\Bigg{\}}, \tag{26}\]  
Similar operators are proposed in the interval-valued hesitant fuzzy environment of literature [28], namely interval-valued hesitant fuzzy ordered weighted averaging (IVHFOWA) operator and interval-valued hesitant fuzzy ordered weighted geometric (IVHFOWG) operator, as follows:  
\[ \begin{array}{l}IVHFOWA(\widetilde{h}_{1},\widetilde{h}_{2},\cdots,\widetilde {h}_{n})=\sum\limits_{j=1}^{n}(w_{j}\widetilde{h}_{e(j)})\\ =\Bigg{\{}\Bigg{[}1-\prod\limits_{j=1}^{n}(1-\widetilde{\gamma}_{e(j)}^{(k)} )^{w_{j}},1-\prod\limits_{j=1}^{n}(1-\widetilde{\gamma}_{e(j)}^{(k)})^{w_{j} }\Bigg{]}\Bigg{|}\widetilde{\gamma}_{e(1)}\in\widetilde{h}_{e(1)},\widetilde{ \gamma}_{e(2)}\in\widetilde{h}_{e(2)},\cdots,\widetilde{\gamma}_{e(n)}\in \widetilde{h}_{e(n)}\Bigg{\}},\end{array}  \tag{27}\]  
\[ \begin{array}{l}IVHFOWG(\widetilde{h}_{1},\widetilde{h}_{2},\cdots, \widetilde{h}_{n})=\sum\limits_{j=1}^{n}(w_{j}\widetilde{h}_{e(j)})\\ =\Bigg{\{}\Bigg{[}\prod\limits_{j=1}^{n}(\widetilde{\gamma}_{e(j)}^{L})^{w_{ j}},\prod\limits_{j=1}^{n}(\widetilde{\gamma}_{e(j)}^{L})^{w_{j}}\Bigg{]}\Bigg{|} \widetilde{\gamma}_{e(1)}\in\widetilde{h}_{e(1)},\widetilde{\gamma}_{e(2)} \in\widetilde{h}_{e(2)},\cdots,\widetilde{\gamma}_{e(n)}\in\widetilde{h}_{e( n)}\Big{\}},\end{array}  \tag{28}\]  
**Case 2:** When \(g(t)=-\ln[(2-t)/t]\), the GPIVHFOWA operator and GPIVHFOWG operator are transformed into probabilistic interval-valued hesitant fuzzy [PERSON] ordered weighted averaging (PIVHFEOWA) operator and the probabilistic interval-valued hesitant fuzzy [PERSON] ordered weighted geometric (PIVHFEOWG) operator, as follows:  
\[ \begin{array}{l}PIVHFEOWA(h_{1},h_{2},\cdots,h_{n})\\ =\Bigg{\{}\Bigg{(}\Bigg{[}\frac{\prod\limits_{j=1}^{n}(1+\gamma_{e(j)}^{(k)-} )^{w_{j}}-\prod\limits_{j=1}^{n}(1-\gamma_{e(j)}^{(k)-})^{w_{j}}}{\prod\limits _{j=1}^{n}(1+\gamma_{e(j)}^{(k)-})^{w_{j}}+\prod\limits_{j=1}^{n}(1-\gamma_{e(j )}^{(k)-})^{w_{j}}},\ \frac{\prod\limits_{j=1}^{n}(1+\gamma_{e(j)}^{(k)+})^{w_{j}}-\prod \limits_{j=1}^{n}(1-\gamma_{e(j)}^{(k)})^{w_{j}}}{\prod\limits_{j=1}^{n}(1+ \gamma_{e(j)}^{(k)+})^{w_{j}}+\prod\limits_{j=1}^{n}(1-\gamma_{e(j)}^{(k)+})^{ w_{j}}}\Bigg{]},\ \frac{\overline{n}}{j}p_{j}^{(k)}\Bigg{)}\Bigg{|}k=1,2,\cdots,l\Bigg{\}},\end{array}  \tag{29}\]  
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