## 3 Probabilistic Hesitant Fuzzy Evidence Theory
### The Concept of the Probabilistic Hesitant Fuzzy Evidence Theory  
First, referring to the definitions of PHFS, the mathematical expression and definition of the PHFBPA for focal elements are as follows:  
**Definition 11**. : _The PHFBPA of \(\Theta\) on the power set space \(2^{\Theta}\) is defined as follows:_  
\[m_{P}=\{\langle\theta,h_{p}(\theta)\rangle\mid\theta\in 2^{\Theta}\}, \tag{15}\]  
_where \(h_{p}(\theta)\) is a PHFE, \(h_{p}(\theta)=\{\gamma_{i}(p_{i})\mid i=1,2,\cdots,n\}\) and \(\gamma_{i}\in[0,1]\) represent the set of probability assignments of focal elements \(\theta\in 2^{\Theta}\), \(p_{i}\in[0,1]\) is the associated occurring probability, and \(\sum_{i=1}^{n}p_{i}=1\), \(n\) is the cardinality of \(h_{p}(\theta)\)._  
The PHFBPA embodies the hesitation between multiple quantitative single values and the importance of each expert opinion. It can be seen that if \(h_{p}(\theta)\) contains only one unique value, the PHFBPA of the focal element degenerates into the classical single-value probability assignment, so that the PHFBPA of the focal element can be regarded as a generalized BPA. The summation operation of PHFBPA is defined as the accumulation of the probability assignments and the probabilities, respectively:  
\[m_{P1}(\theta_{1})+m_{P2}(\theta_{2})=\{[\gamma_{1}^{\sigma(i)}+\gamma_{2}^{ \sigma(i)}](p_{1}^{\sigma(i)}+p_{2}^{\sigma(i)})\}_{i=1}^{\theta}, \tag{16}\]  
where \(\gamma_{1}^{\sigma(i)}\) and \(\gamma_{2}^{\sigma(i)}\) are the \(i\)-th largest values in \(m_{P1}(\theta_{1})\) and \(m_{P1}(\theta_{2})\), and \(p_{1}^{\sigma(i)}\), \(p_{2}^{\sigma(i)}\) are the associated probabilities. \(q=max\{n_{1},n_{2}\}\), in which \(n_{1}\), \(n_{2}\) are the cardinality of \(m_{P1}(\theta_{1})\), \(m_{P1}(\theta_{2})\). If \(n_{1}\
eq n_{2}\), several terms \(\gamma(p)\) must be provided based on the conservative criterion or the optimistic criterion [35] with the probability 0. Based on the multiplication operator Equation (12) of PHFS, PHFBPA operators are also given to deal with focal elements:  
\[m_{P1}(\theta_{1})\otimes m_{P2}(\theta_{2})=\bigcup_{\gamma_{1}\in m_{P1}( \theta_{1}).\gamma_{2}\in m_{P2}(\theta_{2})}\{[\gamma_{1}\gamma_{2}](p_{1}p _{2})\}. \tag{17}\]  
**Definition 12**. : _Let \(m_{P1}(\cdot)\), \(m_{P2}(\cdot)\) be two sources of evidence on \(\Theta\), and \(A,B,C\) are subsets of \(2^{\Theta}\). The combination rule of PHFET is defined as:_  
\[\forall A\in 2^{\Theta},\,m_{PM}(A)=\sum_{B,C\in 2^{\Theta},B\cap C=A}m_{P1}(B) \otimes m_{P2}(C). \tag{18}\]  
To make a final decision according to PHFBPAs, we use the score function Equation (11) to evaluate the probability of the focal element. The higher the score value, the higher the degree of trust in the proposition. Thus, the maximum support rule is adopted as the strategy of decision-making. Therefore, the maximum modulus of PHFBPA will be selected as the decision, denoted as:  
\[D=arg\max_{\theta\in D^{\Theta}}score(m_{P}(\theta)). \tag{19}\]  
**Example 1**. : _Suppose that there are two PHFBPs on the FOD \(\Theta=\{\theta_{1},\theta_{2}\}\); they are described as follows:_  
\[ \begin{split} m_{P1}&=\{m_{P1}(\theta_{1}),m_{P1}( \theta_{2}),m_{P1}(\theta_{1}\cup\theta_{2})\}\\ &=\{\{0.6(0.8),0.5(0.2)\},\{0.2(0.5),0.1(0.5)\},\{0.3(0.7),0.2(0.3) \}\},\\ \end{split}  \tag{20}\]  
\[ \begin{split} m_{P2}&=\{m_{P2}(\theta_{1}),m_{P2}( \theta_{2}),m_{P2}(\theta_{1}\cup\theta_{2})\}\\ &=\{\{0.6(0.3),0.4(0.7)\},\{0.4(0.6,0.3(0.4))\},\{0.2(0.8),0.1(0. 2)\}\}.\end{split}  \tag{21}\]  
_Then, following the combination rule of PHFET, the fusion result is calculated as follows :_  
\[