## 3 Methodology
### Reliability Importance Measure Of Components  
#### 3.2.1 Analytical Method for the Imprecision of Components MTBF  
In engineering practice, the probabilistic behavior of complex equipment can not be fully characterized due to the uncertainties of working conditions. This problem can be tackled by implementing probabilistic methods, alike imprecise probability method [23]. [PERSON] et al. developed a nonparametric predictive inference (NPI) based imprecise reliability analysis method using survival signature [10]. In fact, global optimization procedures are adopted to obtain lower and upper bounds for a complex system with multiple fuzzy variables [24; 25].  
Assume lower distribution functions \((\underline{E})\) and upper distribution functions \((\overline{F})\) are nondecreasing functions mapping the real line \(\mathfrak{R}\) into [0,1] and \(\underline{E}(x)\leq\overline{F}(x)\) for all real values \(x\)[26]. According to the framework of a p-box [27], the lower and upper bounds of CDFs of the failure times of type \(k\) component can be represented as \(\underline{F}_{k}(t)\) and \(\overline{F}_{k}(t)\), respectively. Let \(\left[(\hat{s}_{i}^{k})^{L},(\hat{s}_{i}^{k})^{U}\right](i=1,2,\cdots,n)\) be intervals of type \(k\) component. The \(k\)th value of the reliability function \(R_{k}\) is also an interval.  
[PERSON] et al. presented nonparametric predictive inference (NPI) for system reliability using survival signature in [10]. This approach can be used to calculate the lower and upper bounds of the survival function for a system with \(K\) types of components. For \(k\in\{1,2,\cdots,K\}\), let \(n_{k}\) denote the number of components of type \(k\) and let \(C_{k}(t)\) represent the number of components of type \(k\) that function at time \(t\). This study assumes that the failure ends the functioning of a component and it can  
Figure 5: Reliability curves of the system considering swapping.  
not be repaired or replaced. Therefore, the lower and upper bounds of the survival function can be expressed as follows  
\[\begin{cases}\underline{R}_{Ts}(t)=\underline{p}(T_{s}>t)=\sum_{l_{1}=0}^{m_{1}} \cdots\sum_{l_{k}=0}^{m_{K}}\Phi(l_{1},\cdots,l_{K})\cdot\prod_{k=1}^{K} \overline{D}(C_{k}(t)=l_{k})\\ \overline{R}_{Ts}(t)=\overline{P}(T_{s}>t)=\sum_{l_{1}=0}^{m_{1}}\cdots\sum_{ l_{k}=0}^{m_{K}}\Phi(l_{1},\cdots,l_{K})\cdot\prod_{k=1}^{K}\underline{D}(C_{k} (t)=l_{k})\end{cases} \tag{5}\]  
where  
\[\begin{cases}\overline{D}(C_{k}(t)=l_{k})=\overline{P}(C_{k}(t)\leq l_{k})- \overline{P}(C_{k}(t)\leq l_{k}-1)\\ \underline{D}(C_{k}(t)=l_{k})=\underline{P}(C_{k}(t)\leq l_{k})-\underline{P}( C_{k}(t)\leq l_{k}-1)\end{cases} \tag{6}\]  
Taking the system presented in Figure 4 as example, let scale parameters of components be uncertain (\(\lambda_{1}\in[0.3,0.36]\) for type 1 and \(\lambda_{2}\in[0.2,0.3]\) for type 2). Hence, it is known that \(\underline{F}_{1}(t|\underline{\lambda}_{1})=1-e^{-(0.3t)^{4}}\), \(\overline{F}_{1}(t|\overline{\lambda}_{1})=1-e^{-(0.36t)^{4}}\), \(\underline{F}_{2}(t|\underline{\lambda}_{2})=1-e^{-(t/5)^{4}}\), and \(\overline{F}_{2}(t|\overline{\lambda}_{2})=1-e^{-(0.3t)^{4}}\). The lower and upper bounds of system reliability function become  
\[\begin{cases}\underline{R}_{Ts}(t)=\sum_{l_{1}=0}^{m_{1}}\sum_{l_{2}=0}^{m_{2} }\Phi(l_{1},l_{2})\prod_{k=1}^{2}(\genfrac{}{}{0.0 pt}{}{m_{k}}{l_{k}}F_{k}(t| \underline{\lambda}_{k})^{m_{k}-l_{k}}\left[1-F_{k}(t|\underline{\lambda}_{k} )\right]^{l_{k}}\\ \overline{R}_{Ts}(t)=\sum_{l_{1}=0}^{m_{1}}\sum_{l_{2}=0}^{m_{2}}\Phi(l_{1},l _{2})\prod_{k=1}^{2}(\genfrac{}{}{0.0 pt}{}{m_{k}}{l_{k}}F_{k}(t|\overline{ \lambda}_{k})^{m_{k}-l_{k}}\left[1-F_{k}(t|\overline{\lambda}_{k})\right]^{l_{ k}}\end{cases} \tag{7}\]  
Hence, the lower and upper bounds of system reliability of the example can be obtained using Equations (7), which are shown in Figure 6.