# Failure Analysis for Hydraulic System of Heavy-Duty Machine Tool with Incomplete Failure Data
## 2 Fault Tree Analysis Method Combining Incomplete Failure Data and Experts' Opinions
### Occurrence Calculation
#### 2.1.2 Subjective Occurrence Estimation Based on Rough Set Theory  
For BEs with a Type II problem, human judgments become an essential requirement. Hence, experts' opinions are also introduced to estimate the occurrence probability. Unlike fuzzy set theory, which defines a set by a partial membership without a clear boundary, the rough set theory utilizes the boundary region of a set to express vagueness. Additionally, there is no need for it to require additional subjective information to analyze data [33], which remains objective. The steps of subjective occurrence estimation based on rough set theory are proposed as follows [34]:  
Step 1: Define the problem. Find all possible BEs with insufficient failure data to form a set **T** as **T** = {\(T_{1}\), \(T_{2}\),..., \(T_{p}\)}, where \(T_{i}\) is the \(i\)th BE with Type II problem, \(T_{i}\in\textbf{X}\), \(I=1,\ldots,p\). Assume there are three experts, including designers \(F_{1}\), maintainers \(F_{2}\), and users \(F_{3}\), which give an estimation of the occurrence of each BE, which is expressed by the interval rough number, as shown in Table 3, where \(\xi_{ij}=\left(\left[a_{ij},b_{ij}\right],\left[c_{ij},d_{ij}\right]\right)\), \(c_{ij}<a_{ij}<b_{ij}<d_{ij}\), indicating the occurrence of \(T_{i}\) given by expert \(F_{j}\), \(j=1,2,3\). Step 2: Determine the expert weight. When determining the weight of each expert, the influence of expert risk preferences is considered and reflected by the distance between the expert estimation and its positive and negative ideal point. (1) Determine the positive and negative ideal point: For expert \(F_{j}\), the positive ideal point is defined as the maximum estimation of the occurrence of all \(T_{i}\) by Equation (6), while the negative ideal point is defined as the minimum estimation of the occurrence of all \(T_{i}\) by Equation (7):  
\[\xi_{j}^{+}=\left(\left[\underset{i}{\max}a_{ij},\underset{i}{\max}b_{ij} \right],\left[\underset{i}{\max}c_{ij},\underset{i}{\max}d_{ij}\right]\right) \tag{6}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Combination** & \(\boldsymbol{D_{1}}\) & \(\boldsymbol{D_{2}}\) & \(\ldots\) & \(\boldsymbol{D_{s}}\) \\ \hline Frequency interval & \([Bel(D_{1})\), \(Pl(D_{1})]\) & \([Bel(D_{2})\), \(Pl(D_{2})]\) & \(\ldots\) & \([Bel(D_{s})\), \(Pl(D_{s})]\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Frequency interval of \(D_{i}\). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**BE** & **Designers \(\boldsymbol{F_{1}}\)** & **Maintainers \(\boldsymbol{F_{2}}\)** & **Users \(\boldsymbol{F_{3}}\)** \\ \hline \(T_{1}\) & \(\xi_{11}\) & \(\xi_{12}\) & \(\xi_{13}\) \\ \(T_{2}\) & \(\xi_{21}\) & \(\xi_{22}\) & \(\xi_{23}\) \\ \(\cdots\) & \(\cdots\) & \(\cdots\) & \(\cdots\) \\ \(T_{p}\) & \(\xi_{p1}\) & \(\xi_{p2}\) & \(\xi_{p3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Estimation of occurrence given by experts. \[\xi_{j}^{-}=\left(\left[\underset{i}{\min}a_{ij},\underset{i}{\min}b_{ij}\right], \left[\underset{i}{\min}c_{ij},\underset{i}{\min}d_{ij}\right]\right) \tag{7}\]  
(2) Calculate the distance between the occurrence estimation and its positive and negative ideal point by Equations (8) and (9), respectively:  
\[d_{ij}^{+}=d\left(\xi_{ij},\xi_{j}^{+}\right) \tag{8}\]  
\[d_{ij}^{-}=d\left(\xi_{ij},\xi_{j}^{-}\right) \tag{9}\]  
where for interval rough number \(\xi_{1}=\left([a_{1},b_{1}],[c_{1},d_{1}]\right)\) and \(\xi_{2}=\left([a_{2},b_{2}],[c_{2},d_{2}]\right)\), the operator \(d(\bullet)\) is defined as follows:  
\[d(\xi_{1},\xi_{2})=\sqrt{\frac{\left(a_{1}-a_{2}\right)^{2}+\left(b_{1}-b_{2} \right)^{2}+\left(c_{1}-c_{2}\right)^{2}+\left(d_{1}-d_{2}\right)^{2}}{4}} \tag{10}\]  
(3) The risk preference coefficient is fused into the above distances to produce a new distance, called the preference distance, which represents the difference in the subjective assessment of different types of experts.