# Linear Programming-Based Non-Probabilistic Reliability Bounds Method for Series Systems
## 3 Proposed LPNRBM
### Establishment of LP Model  
Similar to [PERSON]'s method, the sample space of the component states can also be grouped into \(2^{m}\) mutually exclusive and collectively exhaustive (MECE) events, each of which is denoted by \(e_{i},i=1,2,\cdots,2^{m}\) and represented as a distinct intersection of the failure events \(F_{i}\) and their complements \(\overline{F_{i}}\). Figure 2 schematically shows the \(2^{3}=8\) MECE events of a tricomponent series system in a three-dimensional space. Each of the MECE events corresponds to a non-probabilistic measure, which is defined as the ratio of its corresponding multidimensional volume to the multidimensional volume of the entire convex model.  
The LP model can be formulated as follows. The non-probabilistic measures of the MECE events \(d_{i},i=1,2,\cdots,2^{m}\) are regarded as the design variables, the non-probabilistic failure degree of the series system serves as the objective function, and the unicomponent non-probabilistic failure degree and the bicomponent non-probabilistic joint failure degree that are generally computed are considered the linear equality constraints. Moreover, some additional inequality or equality constraints, i.e., all design variables are non-negative and their sum equals 1, are also considered. The LP model can be given as follows:  
\[\begin{array}{ll}\min\ (\max)&\mathbf{c}^{T}\mathbf{d}\\ \text{s.t.}&\mathbf{a}_{1}\mathbf{d}=\mathbf{b}_{1}\\ &\mathbf{a}_{2}\mathbf{d}\geq\mathbf{b}_{2}\end{array} \tag{6}\]  
where \(\mathbf{c}=(c_{1},c_{2},\cdots c_{2^{m}})^{T}\) is a vector whose elements are either 0 or \(1,\mathbf{d}=(d_{1},d_{2},\cdots,d_{2^{m}})^{T}\) is a vector collecting all the design variables, \(\mathbf{a}_{1}\) and \(\mathbf{a}_{2}\) denote the coefficient matrices, and \(\mathbf{b}_{1}\) and \(\mathbf{b}_{2}\) denote the coefficient vectors. For example, as shown in Figure 2, for this tricomponent system it can be found that the following equations hold.  
\[\left\{\begin{array}{ll}d_{1}+d_{2}+d_{3}+d_{4}+d_{5}+d_{6}+d_{7}=f_{\text{ series}}\\ d_{1}+d_{4}+d_{5}+d_{7}=f_{1}\\ d_{2}+d_{4}+d_{6}+d_{7}=f_{2}\\ d_{3}+d_{5}+d_{6}+d_{7}=f_{3}\\ d_{4}+d_{7}=f_{12}\\ d_{5}+d_{7}=f_{13}\\ d_{6}+d_{7}=f_{23}\\ d_{7}=f_{123}\\ d_{1}+d_{2}+d_{3}+d_{4}+d_{5}+d_{6}+d_{7}+d_{8}=1\end{array}\right. \tag{7}\]  
Figure 2: MECE events of a tricomponent series system in a three-dimensional space.  
Thus, the LP model can be formulated as follows:  
\[\min\ (\max) \sum\limits_{i=1}^{7}d_{i}\] s.t. \[\left[\begin{array}{cccccccc}1&0&0&1&1&0&1&0\\ 0&1&0&1&0&1&1&0\\ 0&0&1&0&1&1&1&0\\ 0&0&0&1&0&0&1&0\\ 0&0&0&0&1&0&1&0\\ 0&0&0&0&0&1&1&0\\ 1&1&1&1&1&1&1\end{array}\right]\left[\begin{array}{c}d_{1}\\ d_{2}\\ d_{3}\\ d_{4}\\ d_{5}\\ d_{6}\\ d_{7}\\ d_{8}\end{array}\right]=\left[\begin{array}{c}f_{1}\\ f_{2}\\ f_{3}\\ f_{13}\\ f_{23}\\ 1\end{array}\right] \tag{8}\] \[d_{i}\geq 0\ \ (i=1,2,\cdots,8)\]  
According to the LP model in Equation (6), the minimum and maximum of the objective function obtained are treated as the lower and upper bounds of the system non-probabilistic failure degree, respectively.  
It should be noted that the LP model has a main limitation; that is, the size of the LP problem increases exponentially with the number of components. For instance, the LP problem involves 1024 and 1,048,576 design variables in the cases of the number of components being 10 and 20, respectively. In the following, we identify some zero design variables and remove them to overcome this limitation.