## 3 Materials and Methods
### Problem Formulation  
The reliability optimization problem aims to find optimum parameters for a structure, such as cross-sectional area or node coordinates, by considering the minimization of an objective such as cost or mass while satisfying constraints such as displacement, stress, or minimum probability of structure failure [61]. This process can be mathematically presented as follows:  
Find design variables such as cross-sectional area or nodal coordinates:  
\[\left(A^{T},x^{T}\right)=\left(A_{1},\ A_{2},\ \ldots,\ A_{n},\ x_{1},\ x_{2},\ \ldots,\ x_{n_{2}}\right)^{T} \tag{1}\]  
Figure 5: Overview of optimization algorithm and data handling for structural design.  
Figure 6: The results section of the software interface showing a detailed list of optimized variables and illustrating the outcome of the optimization algorithm.  
where \(A_{1},\ A_{2},\ldots,\ A_{n}\) represent the sizes of the structural elements and \(x_{1},\ x_{2},\ \ldots,\ x_{n_{2}}\) represent the coordinates of the nodes in the structure--that minimizes or maximizes objective function:  
\[\min f(A,x)=\sum_{m=1}^{n}A_{i}\cdot\rho_{i}\cdot l_{i} \tag{2}\]  
where \(\min f(A,x)\) represents the structure's mass or some other objective measure to be minimized or maximized, \(A_{i}\) is the cross-sectional size of the \(i\)-th element, \(\rho_{i}\) is the material density of the \(i\)-th element, \(l_{i}\) is the length of the \(i\)-th element, and \(n\) is the total number of elements--while satisfying:  
\[\text{prob}(g(A,x)\leq 0)\leq\Phi(-\beta)=P_{f_{\text{max}}} \tag{3}\]  
where \(g(A,x)\leq 0\) denotes LSF calculated based on a set of uncertain variables, demand, and capacity of the system. This function defines the success of a structure, and it usually checks if the demand \(D(A,x)\) from the structure or structural element exceeds the capacity \(C(A,x)\) of that element, depending on different variables \((A,x)\)[52]. \(\beta\) stands for reliability index, and, using this value, the probability of failure (\(P_{f_{\text{max}}}\)) can be calculated using the cumulative distribution function of the standard normal distribution (\(\Phi\)). Here, limit states depend on the designer's decision, and it can be chosen as SLS, which measures maximum displacement; or ULS, which measures maximum stresses due to external loads; or both [52], which can be formulated as follows:  
\[\begin{array}{l}g_{1}(A,x)=u(A,x)-u_{\text{max}}\leq 0\\ g_{2}(A,x)=\sigma(A,x)-\sigma_{\text{max}}\leq 0\end{array} \tag{4}\]  
where \(\sigma(A,x)\) is the stress values calculated, and \(\sigma_{\text{max}}\) is the allowable stress. Similarly, \(u(A,x)\) is the displacements on the nodes, and \(u_{\text{max}}\) is the allowable displacement. Moreover, the design variables \(A\) and \(x\) must have specific bounds to ensure practical and feasible designs:  
\[\begin{array}{l}A_{min}\leq A\leq A_{max}\\ x_{min}\leq x\leq x_{max}\end{array} \tag{5}\]  
\(A_{min}\) and \(A_{max}\) are the minimum and maximum allowable values for the cross-sectional areas of the structural elements. Similarly, \(x_{min}\) and \(x_{max}\) are the minimum and maximum allowable values for the coordinates of the nodes.