## 2 Theoretical Background
### Uncertain Programming  
Suppose that \(\xi\) is an uncertain vector, \(x\) is a decision vector, and \(f(x,\ \xi)\) is an uncertain objective function. Since uncertain constraints \(g_{i}(x,\xi)\leq 0,\ i=1,2,\cdots,l\) do not always hold completely, it is naturally hoped that the uncertain constraints hold under confidence levels \(\alpha_{1},\alpha_{2},\cdots,\alpha_{l}\) as follows:  
\[\mathcal{M}\{g_{i}(x,\ \xi)\leq 0\}\geq\alpha_{i},\ i=1,2,\cdots,l \tag{5}\]  
The uncertain objective function \(f(x,\ \xi)\) cannot be directly minimized. Therefore, the decision-maker may minimize the expected value of \(f(x,\ \xi)\), i.e., \(\min_{x}E[f(x,\xi)]\).  
To obtain an optimal solution with minimum uncertain expected objective value subjected to a set of chance constraints, [PERSON] [37] presented the following chance-constrained uncertain programming model:  
\[\left\{\begin{array}{l}\min_{x}E[f(x,\xi)]\\ \mathrm{s.t.}\ \mathcal{M}\{g_{i}(x,\xi)\leq 0\}\geq\alpha_{i},\ i=1,2,\cdots,l \end{array}\right. \tag{6}\]  
The characteristic of this chance-constrained uncertain programming model is that the objective function and constraints are consisted of uncertain variables instead of random variables. [PERSON] [4] also proposed the following crisp equivalent mathematical model for uncertain programming under special conditions.  
**Theorem 2.**_Consider independent uncertain variables \(\xi_{1}\), \(\xi_{2}\), \(\cdots\), \(\xi_{n}\)with regular uncertainty distributions \(\Phi_{1}\), \(\Phi_{2}\), \(\cdots\), \(\Phi_{n}\), respectively. Assume \(f(x;\xi_{1},\cdots,\xi_{n})\) is continuous, strictly increasing for \(\xi_{1}\), \(\xi_{2}\), \(\cdots\), \(\xi_{m}\),and strictly decreasing for \(\xi_{m+1}\), \(\xi_{m+2}\), \(\cdots\), \(\xi_{n}\). Moreover, \(g_{i}(x;\xi_{1},\xi_{2},\cdots,\xi_{n})is characterized by a LSF \(g(\mathbf{\eta})\). Moreover, all input factors of the structure are considered as random variables \(\mathbf{\eta}=(\eta_{1},\eta_{2},\cdots,\eta_{n})^{\mathrm{T}}\). Nevertheless, epistemic uncertainties cannot be accurately explained by probability theory or statistical methods under insufficient sample data. In this section, input factors containing epistemic uncertainties are treated as uncertain variables. Then, a new metric based on uncertainty theory is defined to quantify structural reliability.  
**Definition 6**.: _Let \((\Gamma,\ \mathcal{L},\ \mathcal{M})\) be an uncertainty space. Input factors of LSF \(g(\mathbf{\xi})\) are uniformly characterized by an uncertain vector \(\mathbf{\xi}=(\xi_{1},\xi_{2},\cdots,\xi_{n})^{\mathrm{T}}\), in which \(\ \xi_{i}\) are uncertain variables. The performance state space of structure is divided by the LSF into two domains: the safety domain \(\{g(\mathbf{\xi})>0\}\) and failure the domain \(\{g(\mathbf{\xi})\leq 0\}\). Then, the uncertainty reliability indicator (URI) can be defined as the uncertain measure of the safety domain:_  
\[\mathrm{URI}=\mathcal{M}\{g(\xi_{1},\xi_{2},\cdots,\xi_{n})>0\} \tag{8}\]  
Due to the self-duality of the uncertain measure, the corresponding uncertainty failure indicator (UFI) can be calculated as follows:  
\[\begin{split}\mathrm{UFI}&=\mathcal{M}\{g(\xi_{1}, \xi_{2},\cdots,\xi_{n})\leq 0\}\\ &=1-\mathcal{M}\{g(\xi_{1},\xi_{2},\cdots,\xi_{n})>0\}\\ &=1-\mathrm{URI}\end{split} \tag{9}\]  
The URI is a meaningful tool for quantifying structural reliability under epistemic uncertainties. A higher \(\mathrm{URI}\in[0,1]\) indicates that structural engineers believe the safety event is more possible to happen.  
In many circumstances, it is often relatively difficult for structural engineers to directly obtain the uncertainty distribution \(\mathbb{F}_{g(\mathbf{\xi})}(x)\) of LSF \(g(\mathbf{\xi})\). To address this issue, two URI solving approaches are proposed for engineering structural reliability problems in the following two subsections.