## 4 URI-Based Structural Design Optimization  
Generally speaking, reliability analysis and design of engineering structures often face the problem of epistemic uncertainty. To handle this epistemic uncertainty problem with insufficient sample data, a new URBDO model based on URI and chance-constrained uncertain programming model is presented in this section. Due to the lack of sample data, uncertain variables are employed in structural design optimization to uniformly describe input factors. To search for the lowest cost or lightest mass by changing design variables while maintaining the safety requirements of structures, a general URBDO model combines uncertainty reliability demands into design optimization constraints.  
The LSF \(g(\mathbf{d},\mathbf{x};\mathbf{\xi})\) of engineering structures depends on deterministic design variables, uncertain design variables, and uncertain variables, which are represented by \(\mathbf{d}=(d_{1},d_{2}\cdots,d_{n_{d}})^{\text{T}},\mathbf{x}=(x_{1},x_{2}\cdots,x_ {m})^{\text{T}}\), and \(\mathbf{\xi}=(\xi_{1},\xi_{2},\cdots,\xi_{n})^{\text{T}}\), respectively. Structural engineers tend to ensure that uncertainty reliability is higher than the target reliability level \(R_{\text{U}}\). Uncertainty reliability constraint combined with the URI definition can be expressed as follows:  
\[\text{URI}=\mathcal{M}\{g(\mathbf{d},\mathbf{x};\mathbf{\xi})>0\}\geq R_{\text{U}} \tag{12}\]  
In practical engineering structures, design problems may contain multiple objectives that require to be simultaneously considered. Uncertain multi-objective optimization belongs to uncertain programming in which the constraints and objectives contain uncertain variables. In such instances, typical formulation of the multi-objective URBDO model is defined as follows.  
**Definition 7**.: _Assume that \(g_{j}(\mathbf{d},\mathbf{x};\mathbf{\xi})\) is the \(j\)th LSF of a structure. If the design optimization problem has \(N_{\text{t}}\) objective functions \(f_{i}(\mathbf{d},\mathbf{x};\mathbf{\xi}),\ i=1,2,\cdots,N_{\text{t}}\) that conflict with each other, then the general URBDO model subjected to uncertainty reliability constraint can be established by weighting the objective functions:_  
\[\begin{split}&\text{find }\mathbf{d},\mathbf{x}\\ &\min\sum\limits_{i=1}^{N_{\text{t}}}\omega_{i}E[f_{i}(\mathbf{d},\bm {x};\mathbf{\xi})]\\ &\text{s. t.URI}f_{j}=\mathcal{M}\{g_{j}(\mathbf{d},\mathbf{x};\mathbf{\xi})>0 \}\geq R_{\text{U}_{j}}\\ &\mathbf{d}^{Lower}\leq\mathbf{d}\leq\mathbf{d}^{Upper},\mathbf{x}^{Lower}\leq \mathbf{x}\leq\mathbf{x}^{Upper},j=1,2,\cdots,N_{\text{g}}\end{split} \tag{13}\]  
_where \(\mathbf{d}\) is the vector of deterministic design variables, \(\mathbf{x}\) is the vector of uncertain design variables, \(\mathbf{\xi}\) is the vector of uncertain variables, \(\omega_{1},\omega_{2},\cdots,\omega_{N_{\text{t}}}\) are positive weights with\(\sum\
olimits_{i=1}^{N_{\text{f}}}\omega_{i}=1\), and \(R_{\text{U}_{j}}\) denotes the prescribed reliability level for the \(j\)th constraint. Since the objective functions are also uncertain variables, it is relatively difficult to directly minimize \(f_{i}(\mathbf{d},\mathbf{x};\mathbf{\xi})\). Therefore, minimizing its uncertain mean value \(E[f_{i}(\mathbf{d},\mathbf{x};\mathbf{\xi})]\) is reasonable._  
To solve the constructed URBDO model, the theorem described in Section 4.1 provides a crisp equivalent programming model for some special cases.