# Linear Programming-Based Non-Probabilistic Reliability Bounds Method for Series Systems
## 2 Review of the NRBM  
A series system with \(m\) components, each of which can be defined by a performance function, is considered:  
\[g_{j}(\mathbf{X}),j=1,2,\ldots,m \tag{1}\]  
where \(\mathbf{X}=\left(X_{1},X_{2},\ldots,X_{n}\right)^{\mathrm{T}}\) denotes an \(n\)-dimensional vector with uncertain variables. The failure surface \(g_{j}(\mathbf{X})=0\) is the boundary between the safe and failure domains. In other words, \(g_{j}(\mathbf{X})\geq 0\) and \(g_{j}(\mathbf{X})<0\) indicate that the structure is in a safe state and a failure state, respectively. In this paper, a multidimensional ellipsoid is used to describe the uncertainty domain of \(\mathbf{X}\):  
\[(\mathbf{X}-\mathbf{X}^{\mathrm{c}})^{\mathrm{T}}\mathbf{\Omega}_{x}(\mathbf{ X}-\mathbf{X}^{\mathrm{c}})\leq 1 \tag{2}\]  
where \(\mathbf{X}^{\mathrm{c}}\) represent the midpoint of the ellipsoid, and \(\mathbf{\Omega}_{x}\) denotes an \(n\times n\) characteristic matrix. Each variable \(X_{i},i=1,2,\ldots,n\) has a range of values, namely the marginal interval \(\left[X_{i}^{L},X_{i}^{U}\right]\). The vector \(\mathbf{X}\) will be first mapped into the normalized vector \(\mathbf{\delta}\) using a linear transformation, which may geometrically involve translation, scaling, and rotation, through which the ellipsoid becomes the sphere \(\mathbf{\delta}^{\mathrm{T}}\mathbf{\delta}\leq 1\), and the performance function \(g_{j}(\mathbf{X})\) can be rewritten as its normalized form \(G_{j}(\mathbf{\delta})\). This type of treatment makes it easy to conduct subsequent system reliability analysis. This is because of the following two reasons: first, a dimensionless variable is produced, which can facilitate the measurement of reliability; second, it is more convenient to compute the volumes related to system reliability in \(\mathbf{\delta}\) space than in \(\mathbf{X}\) space. In a series system, the entire system fails if any of its components fail. Similar to the probabilistic reliability of a series system, the non-probabilistic reliability of a series system \(R_{\mathrm{series}}\) can be defined as the ratio of the volume inside the safe domain to the volume of the entire convex model:  
\[R_{\mathrm{series}}=\frac{A_{w}-A_{s}}{A_{w}} \tag{3}\]where \(A_{w}-A_{s}\) and \(A_{w}\) denote the multidimensional volume of the part of the convex model inside the safe domain and the volume of the entire convex model, respectively. Correspondingly, the non-probabilistic failure degree \(f_{\text{series}}\) can be defined as follows:  
\[f_{\text{series}}=\frac{A_{\text{s}}}{A_{w}} \tag{4}\]  
It can be seen that \(R_{\text{series}}+f_{\text{series}}=1\). Figure 1a presents the two-dimensional case of \(f_{\text{series}}\). Due to the dependence between the components, a precise system reliability prediction is generally impossible for all the component reliabilities. In our recent work, we proposed an NRBM for series systems that employs a unicomponent non-probabilistic failure degree and a bicomponent non-probabilistic joint failure degree, as follows:  
\[f_{1}+\sum_{j=2}^{m}\max\left[\left(f_{j}-\sum_{k=1}^{j-1}f_{jk}\right),0 \right]\leq f_{\text{series}}\leq\sum_{j=1}^{m}f_{j}-\sum_{j=2}^{m}\max\left( \begin{subarray}{c}f_{jk}\\ k<j\end{subarray} \right) \tag{5}\]  
where \(f_{i}=\frac{A_{i}}{A_{w}}(i=1,2,\cdots,m)\) and \(f_{jk}=\frac{A_{jk}}{A_{w}}\) denote the unicomponent non-probabilistic failure degree and the bicomponent non-probabilistic joint failure degree, respectively. \(A_{i}\) and \(A_{jk}\) represent the multidimensional volume of the part of the convex model in the failure domain (\(G_{i}(\delta)<0\)) and that for the joint failure domain (\(G_{j}(\delta)<0\) and \(G_{k}(\delta)<0\)), respectively. Figure 1b,c respectively provide the diagrams of \(A_{i}\) and \(A_{jk}\). Compared with the traditional MCS, the proposed NRBM achieves a balance between efficiency and accuracy and is therefore more applicable to practical engineering problems. However, as mentioned previously, the NRBM has two main limitations.