# Linear Programming-Based Non-Probabilistic Reliability Bounds Method for Series Systems
## 3 Proposed LPNRBM
### Identification of Zero Design Variables
#### 3.2.2 Identification of Zero Design Variables for the Tricomponent System  
In this section, for simplicity, we focus on the tricomponent systems for which any two components have the relationship shown in Figure 3a. The proposed identification method consists of two parts: determining the tricomponent systems and identifying the zero design variables. The procedure for determining the above tricomponent systems is illustrated below. As shown in Figure 4, each node represents a component, and each branch connecting any two nodes denotes the fact that the two components represented by the two nodes have the relationship shown in Figure 3a. The figure is separable, and each mode may have a different number of branches. Thus, the tricomponent systems that we concentrate on are represented by all the triangles consisting of three different branches. Figure 3: Three different cases of a bicomponent system: (**a**) non-empty MECE event case; (**b,c**) one empty MECE event cases. As also shown in Figure 2, a tricomponent system generally involves eight MECE events, namely \(F_{1}\overline{F_{2}}\overline{F_{3}}\), \(\overline{F_{1}}F_{2}\overline{F_{3}}\), \(\overline{F_{1}}\overline{F_{2}}F_{3}\), \(F_{1}F_{2}\overline{F_{3}}\), \(F_{1}\overline{F_{2}}F_{3}\), \(\overline{F_{1}}F_{2}F_{3}\), \(F_{1}F_{2}F_{3}\), and \(\overline{F_{1}}\overline{F_{2}}\overline{F_{3}}\). Similarly, we only investigate the following three cases, namely \(\overline{F_{1}}\overline{F_{2}}F_{3}=\varnothing\), \(F_{1}F_{2}\overline{F_{3}}=\varnothing\), and \(F_{1}F_{2}F_{3}=\varnothing\). This is because (1) \(\overline{F_{1}}\overline{F_{2}}\overline{F_{3}}=\varnothing\) implies that the system is in an entirely failure state, (2) \(F_{1}\overline{F_{2}}\overline{F_{3}}\), \(\overline{F_{1}}\overline{F_{2}}\overline{F_{3}}\), and \(\overline{F_{1}}\overline{F_{2}}F_{3}\), as well as \(\overline{F_{1}}F_{2}F_{3}\), \(F_{1}\overline{F_{2}}F_{3}\), and \(F_{1}F_{2}\overline{F_{3}}\), can also be treated as the same type of events, and (3) these three MECE events are sufficient to show all the empty events in the different position relationships of the tricomponent subsystems. As shown in Figure 5, the tricomponent systems can be fully demonstrated in a three-dimensional structural system. Figure 4: Schematic illustration for determining the tricomponent system (\(m=6\)). Figure 5: _Cont._  
For the three performance functions of the tricomponent system, two performance functions with the maximum correlation angle are also given by Equation (9), while the third performance function is given by the following:  
\[G_{3}(\mathbf{\delta})=\sum\limits_{i=1}^{n}c_{i}\delta_{i}+c_{0} \tag{13}\]  
where \(c_{0}\) and \(c_{i}\) are given constants. The corresponding correlation angles \(\gamma_{12}\), \(\gamma_{13}\), and \(\gamma_{23}\) are given by the following:  
\[\left\{ \begin{aligned} \gamma_{13}&=\arccos \left(\frac{\sum\limits_{i=1}^{n}a_{i}c_{i}}{\sqrt{\sum\limits_{i=1}^{n}a_{i} ^{2}}\sqrt{\sum\limits_{i=1}^{n}c_{i}^{2}}}\right)\\ \gamma_{23}&=\arccos\left(\frac{\sum\limits_{i=1}^{ n}b_{i}c_{i}}{\sqrt{\sum\limits_{i=1}^{n}b_{i}^{2}}\sqrt{\sum\limits_{i=1}^{n}c_{i }^{2}}}\right)\end{aligned} \right. \tag{14}\]  
The five cases in Figure 5 can be identified by the four special points \(\mathbf{P}_{12}\), \(\mathbf{P}_{13}\), \(\mathbf{P}_{23}\), and \(\mathbf{P}_{123}\). \(\mathbf{P}_{123}\) is the point at the intersection of three failure surfaces, \(G_{1}(\mathbf{\delta})=0\), \(G_{2}(\mathbf{\delta})=0\), and \(G_{3}(\mathbf{\delta})=0\), and the closest point to the origin. Similarly, \(\mathbf{P}_{12}\) is the point that is located at the intersection of two failure surfaces, \(G_{1}(\mathbf{\delta})=0\) and \(G_{2}(\mathbf{\delta})=0\), and has the minimum distance to the origin. According to Ref. [28], the coordinates of \(\mathbf{P}_{12}\), \(\mathbf{P}_{13}\), \(\mathbf{P}_{23}\), and \(\mathbf{P}_{123}\) are as follows:  
\[\left\{