## 2 Preliminaries  
### Dempster-Shafer Evidence Theory  
D-S evidence theory [31; 32; 37] has many advantages in handling uncertain information and can be applied to many fields such as decision-making [38; 39], risk assessment [40], reliability analysis [41], and so on. Firstly, D-S evidence theory allows probability masses to be assigned to not only singletons, but also multiple hypotheses, rather than only singleton subsets in comparison to the probability theory. Secondly, information from different sources can be combined without a prior distribution. Thirdly, instead of being forced to be assigned to some singleton subsets, a certain degree of ignorance can be allowed in some situations. A few basic concepts are introduced as follows:  
Let \(\Theta\) be a set of mutually exclusive and collectively exhaustive events, indicated by:  
\[\Theta=\left\{\theta_{1},\theta_{2},\ldots,\theta_{i},\ldots,\theta_{N}\right\} \tag{1}\]  
The set \(\Theta\) is called a frame of discernment. The power set of \(\Theta\) is indicated by \(2^{\Theta}\), namely:  
\[2^{\Theta}=\left\{\emptyset,\left\{\theta_{1}\right\},\left\{\theta_{2}\right\},\ldots,\left\{\theta_{N}\right\},\left\{\theta_{1},\theta_{2}\right\},\ldots, \left\{\theta_{1},\ldots,\theta_{i}\right\},\ldots,\left\{\theta_{1},\ldots, \theta_{N}\right\}\right\} \tag{2}\]  
If \(A\in 2^{\Theta}\), \(A\) is called a proposition. In the power set \(2^{\Theta},\emptyset\) is called the empty set, the singletons are \(\left\{\theta_{1}\right\},\left\{\theta_{2}\right\},\ldots,\left\{\theta_{N}\right\}\), and the multiple hypotheses are \(\left\{\theta_{1},\theta_{2}\right\},\ldots,\left\{\theta_{1},\ldots,\theta_{ 1}\right\},\ldots,\left\{\theta_{N}\right\}\).  
For a frame of discernment \(\Theta\), a mass function is a mapping \(m\) from \(2^{\Theta}\) to \([0,1]\), formally defined by:  
\[m:2^{\Theta}\rightarrow[0,1] \tag{3}\]  
which satisfies the following condition:  
\[m(\varnothing)=0\quad\text{and}\quad\sum\limits_{A\in 2^{\Theta}}m(A)=1 \tag{4}\]  
In D-S evidence theory, a mass function is also called a basic probability assignment (BPA). BPA reflects the degree of support for the proposition A in the recognition framework. If \(m(A)>0\), A is called a focal element, and the union is called the core of the mass function.  
Associated with each BPA, the belief function \(Bel\) and plausibility function \(Pl\) are defined as:  
\[Bel(A)=\sum\limits_{B\subseteq A}m(B) \tag{5}\]  
\[Pl(A)=1-Bel(\overline{A})=\sum\limits_{B\cap A\
eq\varnothing}m(B) \tag{6}\]  
where \(\overline{A}=\Theta-A\). Obviously, \(Pl(A)\geq Bel(A)\), for each \(A\subseteq 2^{\Theta}\).  
Assume there are two BPAs indicated by \(m_{1}\) and \(m_{2}\); [PERSON]'s rule of combination is used to combine them as follows:  
\[m(A)=\left\{\begin{array}{ll}\frac{1}{1-K}\sum\limits_{B\cap C=A}m_{1}(B)m_{ 2}(C),&A\
eq\varnothing\\ 0,&A=\varnothing\end{array}\right. \tag{7}\]  
where:  
\[K=\sum\limits_{B\cap C=\varnothing}m_{1}(B)m_{2}(C) \tag{8}\]  
In D-S evidence theory, \(K\) is a coefficient to measure the conflict between pieces of evidence. Note that [PERSON]'s rule of combination is only applicable to two such BPAs that satisfy the condition \(K<1\), and there are many other combination rules [42; 43]. It should also be noted that the conflict in D-S evidence theory is an open issue. Many methods have been proposed to address this issue [33; 44].