# 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  
#### 2.1.1 Objective Occurrence Calculation Based on the Dempster-Shafer Evidence Theory  
Like conventional FTA, failure data are first preprocessed to calculated the frequencies of BEs. For data with Type I problem, experts' opinions are needed to assign probability weights to all BEs that may cause the failure. The results might vary depending on measurement precision or the uncertainty in statements of experts. The Dempster-Shafer evidence theory (D-S theory) [31] is a mathematical theory of evidence developed to combine information supplied by different experts or from other sources. It performs well in dealing with uncertain information. Therefore, an approach of processing a Type I problem based on the D-S theory is given as the following steps:  
Let \(\mathbf{X}=\{\mathbf{X}_{1},\,\mathbf{X}_{2},\,\ldots,\,\mathbf{X}_{q}\}\) denote the BE set obtained through fault tree construction, where \(\mathbf{X}_{i}\) (\(i=1,\,\ldots,\,q\)) is the \(i\)th BE. Step 1 is a frequency assessment based on failure data: analyze the failure data with clear causes. Trace the source and find all possible BEs causing the failure, then calculate the frequencies. Step 2: From this step, failure data with a Type I problem is processed: for a single failure with an uncertain cause, a group of experts (\(E_{1},\,\ldots,\,E_{m}\)) determine all possible BEs that can cause the failure, which are expressed as (\(D_{1},\,\ldots,\,D_{s}\)), to establish the frame of discernment, symbolized by \(\mathbf{D}=(D_{1},\,\ldots,\,D_{s})\), with \(D_{i}\in\mathbf{X}\) and \(\mathbf{D}\subseteq\mathbf{X}\). Figure 1: The framework of FTA for the hydraulic system of HCMT. \(P(D)\) is denoted as the power set composed of \(2^{D}\) elements of \(\mathbf{D}\); each element of \(2^{D}\) represents a proposition. A mass assignment to each subset of \(P(D)\) is known as the basic probability assignment (BPA) \(M\). If \(A_{1}\),..., \(A_{n}\) are the sets of interest with \(A_{j}\in P(D)\), then a BPA is defined as:  
\[M:P(D)\rightarrow[0,1],\ \sum\limits_{j=1}^{n}M\big{(}A_{j}\big{)}=1,\ M( \varnothing)=0 \tag{1}\]  
Step 3: Experts determine all possible combinations of \(D_{1}\),..., \(D_{s}\), which are expressed as \(A_{1}\),..., \(A_{n}\), as focal elements of the frame of discernment. It shows that \(A_{j}\subseteq\mathbf{D}\) (\(j=1\),..., \(n\)). Next, the assessment of each focal element is provided by each expert, as shown in Table 1, where \(M_{k}(A_{j})\) (\(j=1\),..., \(n\); \(k=1\),..., \(m\)) is the BPA provided by expert \(E_{k}\) on the assessment of \(A_{j}\). Step 4: Evidence combination: if the evidence shows agreement, combine it with [PERSON]'s combination rule as follows:  
\[\left\{ \begin{array}{ll}M(A)=\left\{\begin{array}{ll}\frac{\sum\limits_{ \alpha_{j}=A}\sum\limits_{k=1}^{m}M_{k}\big{(}A_{j}\big{)}}{1-K},&A\
eq \varnothing,A\subseteq D\\ 0,&A=\varnothing,A\subseteq D\end{array}\right.\end{array} \right. \tag{2}\]  
When evidence highly conflicts with each other, the above combination rule is not efficient; thus, a new combination rule [32] for conflict evidence is introduced as Equation (3):  
\[\left\{ \begin{array}{ll}M(A)=\left\{\begin{array}{ll}\frac{\sum\limits_{A_{ j}=A}\sum\limits_{k=1}^{m}M_{k}\big{(}A_{j}\big{)}\Big{(}\sum\limits_{l=1}^{m}M_{l} \big{(}A_{j}\big{)}\Big{)}}{m^{2}-K},&A\
eq\varnothing,A\subseteq D\\ 0,&A=\varnothing,A\subseteq D\end{array}\right.\end{array} \right. \tag{3}\]  
where \(K\) is the conflict coefficient and m is the amount of evidence in the frame of discernment D. Then, the aggregated assessment of focal elements with respect to each BPA can be calculated by the combination rule, a group assessment matrix \(M\) is constructed as \(\mathbf{M}=[M(A_{1}),M(A_{2}),\)..., \(M(A_{t})]\). Step 5: Calculate the belief and plausibility function of \(D_{i}\), which are defined as follows:  
\[\left\{