# Linear Programming-Based Non-Probabilistic Reliability Bounds Method for Series Systems
## 5 Conclusions  
In this paper, an LPNRBM is proposed to estimate the non-probabilistic failure degree of series systems. The proposed method consists of three main parts: the construction of an LP model, the identification of zero design variables, and the removal of zero design variables from the LP model. The results for three numerical examples show that the proposed method can not only provide the narrowest possible interval of the non-probabilistic failure degree of series systems for the given information, but also overcome the ordering-dependency problem existing in the NRBM. As stated in Section 3.2, the LPNRBM can identify most but not all zero design variables so that it is an approximate but effective method instead of a precise one. Moreover, it is worth noting that according to the division of the sample space in this research, the number of design variables increases exponentially with the increase in the number of components. Although this research overcomes this problem to some extent by removing some zero design variables, the method is still limited for engineering problems with  
Figure 12: Reliability analysis results by NRBM, MCS, and LPNRBM for Example 3. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Method** & **Lower Bounds** & **Upper Bounds** & **Maximum Relative Error** \\ \hline NRBM [26] & \(5.0486\times 10^{-2}\) & \(5.3906\times 10^{-2}\) & 4.5991\% \\ LPNRBM & \(5.0486\times 10^{-2}\) & \(5.3553\times 10^{-2}\) & 3.9143\% \\ MCS [21] & \(5.1536\times 10^{-2}\) & — \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Reliability analysis results for Example 3.  
numerous components. Therefore, a potential idea for future work is to consider dividing the sample space in other ways. Conceptualization, X.Q. ; methodology, X.Q. and Z.L; software, Z.L. ; validation, X.Q., Z.L. and X.F. ; formal analysis, P.L. ; writing--original draft preparation, X.Q. and Z.L. ; writing--review and editing, X.Q. and X.F. ; visualization, X.F. and P.L. ; project administration, Z.L., X.F. and P.L. All authors have read and agreed to the published version of the manuscript. This research was funded by the National Natural Science Foundation of China (Grant No. 51775427). Data Availability Statement: Data is contained within the article. The authors would like to thank all of the reviewers who participated in the review during the preparation of this manuscript. The authors declare no conflicts of interest. The performance functions of the wooden roof truss structure can be obtained as follows. Figure 11: Overall force figure of the wooden roof truss structure. As shown in Figure A1, two force equilibrium equations can be given as follows:  
\[\left\{ \begin{matrix}F_{A_{y}}+F_{B_{y}}=2F_{1}+2F_{2}+F_{3}\\ F_{A_{y}}=F_{B_{y}}\end{matrix} \right.\] (A1)  
from which we have the following:  
\[\left\{ \begin{matrix}F_{A_{y}}=F_{1}+F_{2}+\frac{1}{2}F_{3}\\ F_{B_{y}}=F_{1}+F_{2}+\frac{1}{2}F_{3}\end{matrix} \right.\] (A2)  
where \(F_{A_{y}}\) denotes the orthogonal component of the hinge support in the \(y\)-direction, and \(F_{B_{y}}\) denotes the orthogonal component of the rolling bearing in the \(y\)-direction. Let \(F_{N_{i}}\) be the internal force of the \(i\)-th bar. As shown in Figure A2a, \(F_{N_{1}}\) and \(F_{N_{2}}\) can be obtained from the following two force equilibrium equations:  
\[\left\{ \begin{matrix}F_{N_{1}}=2F_{A_{y}}\\ F_{N_{2}}=\sqrt{5}F_{A_{y}}\end{matrix} \right.\] (A3)  
As show in Figure A6b, \(F_{N_{11}}\) can be obtained as follows:  
\[F_{N_{11}}+F_{3}=\frac{2\sqrt{5}}{5}F_{N_{8}} \tag{111}\]  
After substituting Equation (110) into Equation (111), we have the following:  
\[F_{N_{11}}=\frac{2}{3}F_{1}+\frac{4}{3}F_{2} \tag{112}\]  
Finally, 10 performance functions are obtained as follows:  
\[