## 2 Robustness Analysis Using Evidence Theory  
Consider that uncertainty problem is given as \(f(\mathbf{Z})\), where \(\mathbf{Z}\) represents the \(n_{Z}\)-dimensional uncertain vector, \(f\) is the performance function which is uncertain due to \(\mathbf{Z}\). Conventional methods [16; 17; 18] of robust optimization employ probability theory to deal with the uncertainties. The typical strategy is to consider the uncertain parameters of a problem as random variables, and thereby the performance value is also a random variable. The mean and variance are used to formulate the robustness model. In practical engineering, it is sometimes hard to construct accurate probability models due to the limited information. Thus, evidence theory [20; 21] is adopted to model the robustness. In evidence theory, the frame of discernment (FD) needs to be established first, which contains several independent basic propositions. It is similar to the sample space of a random parameter in probability theory. Here, \(2^{\Theta}\) denotes the power set of the FD (namely \(\Theta\)), and \(2^{\Theta}\) consists of all possible propositions contained in \(\Theta\). For example, for a FD with the two basic propositions of \(\Theta_{1}\) and \(\Theta_{2}\), the corresponding power set is \(2^{\Theta}=\{\varnothing,\{\Theta_{1}\},\{\Theta_{2}\},\{\Theta_{1},\Theta_{2}\}\}\). Evidence theory adopts a basic probability assignment (BPA) to measure the confidence level of each proposition. For a certain proposition \(A\), the BPA is a mapping function that satisfies the following axioms:  
\[ \begin{array}{l}0\leq m(A)\leq 1,\,\forall A\in 2^{\Theta}\\ m(\Phi)=0\\ \sum\limits_{A\in 2^{\Theta}}m(A)=1\end{array}  \tag{1}\]  
where if \(m(A)\geq 0\), \(A\) is called a focal element of \(m\). The BPA of \(m(A)\) denotes the extent to which the evidence supports Proposition A. When the information comes from multiple sources, \(m(A)\) can be obtained by evidence combination rules [28]. Evidence theory uses an interval consisting of the belief measure (\(Bel\)) and the plausibility measure (\(Pl\)) to describe the true extent of the proposition. The two measures are defined as:  
\[ \begin{array}{l}Bel(A)=\sum\limits_{C\subseteq A}m(C)\\ Pl(A)=\sum\limits_{C\cap A\
eq\Phi}m(C)\end{array}  \tag{2}\]  
As can be seen from Equation (2), \(Bel(A)\) is the summary of all the BPA that totally support Proposition \(A\), while \(Pl(A)\) is the summary of the BPA that support Proposition \(A\) totally or partially. A two-dimensional design problem is taken as the example to illustrate the process of robustness analysis using evidence theory. The performance function contains two uncertain parameters \((a,b)\), which are both considered as evidence variables. The FDs of \(a\), \(b\) are the two closed intervals, i.e., \(A=\left[A^{L},A^{R}\right]\) and \(B=\left[B^{L},B^{R}\right]\). \(A\) contains \(n_{A}\) number of focal elements, and the subinterval of \(A_{i}=\left[B_{i}^{L},B_{i}^{R}\right]\) represents the \(i\)-th focal element of \(A\). The definitions of \(n_{B}\) and \(B_{j}\) are similar. Thus, a Cartesian product can be constructed:  
\[D=A\times B=\left\{D_{k}=\left[A_{i},B_{j}\right],A_{i}\in A,B_{j}\in B\right\} \tag{3}\]  
where \(D_{k}\) is the \(k\)-th focal element of \(D\), and the total number of focal elements is \(n_{A}\times n_{B}\). For ease of presentation, assuming that \(a\), \(b\) are independent, a two-dimensional joint BPA is obtained:  
\[m(D_{k})=m(A_{i})\cdot m\left(B_{j}\right) \tag{4}\]  
More general problems with parametric correlation can be handled using the mathematical tool of copula functions [29]. As analyzed above, the performance function of \(f\) is uncertain. The performance threshold of \(v\) is given to evaluate its robustness. Given that the design objective is to minimize the value of \(f\), the higher the trueness of Proposition \(f\leq v\), the higher the robustness of \(f\) relative to \(v\).