## Preliminaries
### Probabilistic Interval-Valued Hesitant Fuzzy Sets  
**Example 1**.: _Supplier selection is a very important problem in supply chain management. A network dealer invited ten experts to evaluate the comprehensive conditions of a supplier. Among them, four experts' satisfaction interval value of \([0.8,0.9]\), the satisfaction interval value given by two experts is \([0.6,0.8]\), the satisfaction interval value given by three experts is \([0.4,0.6]\), and the remaining experts gave a satisfaction interval value of \([0.3,0.5]\). Using interval-valued hesitant fuzzy element, the above decision information can be expressed as \(\{[0.8,0.9],[0.6,0.8],[0.4,0.9],[0.3,0.5]\}\). Obviously, interval-valued hesitant fuzzy elements can only represent all possible decision information, but cannot reflect the importance of each decision information. However, using PIVHFE to represent the above decision information is \(\{([0.8,0.9],0.4)([0.6,0.8],0.3),([0.4,0.9],0.2),([0.3,0.5],0.1)\}\), which comprehensively represents all the evaluation information given by the decision maker. In fact, the given number of real numbers in PIVHFE is not the same, and the element arrangement is unordered in the actual decision problem, so the following rules are made:_  
**Note 1.** The intervals in the probabilistic interval-valued hesitant fuzzy element, let PIVHFE \(h=\{\left(\left[\gamma^{(k)-},\gamma^{(k)+}\right],p^{(k)}\right)|\left[ \gamma^{(k)-},\gamma^{(k)+}\right]\in h\}\) be arranged in descending order, which \(\left(\left[\gamma^{(k)-},\gamma^{(k)+}\right],p^{(k)}\right)\) represents the interval corresponding to the membership degree of the largest K.  
**Note 2.** Let \(l_{1}\) and \(l_{2}\) be the number of elements in probabilistic interval-valued hesitant fuzzy elements (PIVHFE) \(h_{1}\) and \(h_{2}\) respectively. If \(l_{1}\
eq l_{2}\), the number of elements can be increased in the PIVHFE with fewer original numbers, so that the number of elements in the two PIVHFE is the same. For the convenience of computation, the number of elements in PIVHFE is assumed to be \(l\).  
**Definition 4**.: _Let \(X=\{x_{1},x_{2},\cdots,x_{m}\}\) be a given set. A probabilistic interval-valued hesitant fuzzy sets (PIVHFS) \(H\) over \(X\) is an object having the form: \(H=\{\langle x_{i},h(x_{i})\rangle|x_{i}\in X\}\), where \(h(x_{i})=\{([\gamma_{i}^{-},\gamma_{i}^{+}],p_{\gamma_{i}})|(\gamma_{i}^{-}, \gamma_{i}^{+})\in h(x_{i}),p_{\gamma_{i}}\in[0,1],\sum_{[\gamma_{i}^{-}, \gamma_{i}^{+}]\in h(x_{i})}p_{\gamma_{i}}=1\}\). here \(\gamma_{i}=[\gamma_{i}^{-},\gamma_{i}^{+}]\) is an interval number. Let \(\gamma_{i}^{-}\) and \(\gamma_{i}^{+}\) represent the lower and upper limits of \(\gamma_{i}\), respectively. \(p_{\gamma_{i}}\) is the probability associated with \(\gamma_{i}\)._  
For convenience, a probabilistic interval-valued hesitant fuzzy element (PIVHFE) will be denoted by \(h_{i}=h(x_{i})\), and \(\overline{P}\) is the set of all PIVHFE on \(X\).  
The score function for IVHFEs will be defined below.  
**Definition 5**.: _For a PIVHFE \(h\), the score function of \(h\) is formulated by_  
\[S(h)=\frac{1}{\#h}\sum_{k=1}^{l}{(\gamma^{(k)})p^{(k)}}, \tag{4}\]  
_with \(\#h\) being the number of the interval values in \(h\), and \(S(h)\) is an interval value belonging to \([0,1]\). For two IVHFEs \(h_{1}\) and \(h_{2}\), if \(S(h_{1})\geq S(h_{2})\), then \(h_{1}\geq h_{2}\). The PIVHFE's deviation function is defined as:_  
\[\phi(h)=\sum_{k=1}^{l}{[0.5(\gamma^{(k)-}+\gamma^{(k)+})-S(h)]}^{2}p^{(k)}. \tag{5}\]  
**Definition 6**.: _Let \(h_{1}=\{([\gamma_{1}^{(k)-},\gamma_{1}^{(k)+}],p_{1}^{(k)})|k=1,2,\cdots,l\}\) and \(h_{2}=\{([\gamma_{2}^{(k)-},\gamma_{2}^{(k)+}],p_{2}^{1.  If \(\phi(h_{1})>\phi(h_{2})\), then \(h_{1}<h_{2}\). That is, \(h_{1}\) is less than \(h_{2}\). 2.  If \(\phi(h_{1})<\phi(h_{2})\), then \(h_{1}>h_{2}\). That is, \(h_{1}\) is greater than \(h_{2}\). 3.  If \(\phi(h_{1})=\phi(h_{2})\), then \(h_{1}=h_{2}\). That is, \(h_{1}\) equals \(h_{2}\).