## 5 Case Study
### Implementation and Results  
First, the knowledge parameters of the PUL-DUCG are determined according to the proposed knowledge acquisition method. Then, the reasoning algorithm of the proposed PUL-DUCG model is implemented to find out the root cause event with the maximum probability. We use the knowledge parameter \(a\) as an example to explain the knowledge acquisition process. **Step 1:** By applying Equation (12), the collective knowledge assessment matrix \(\widetilde{B}\) is derived as shown in Table 3. \begin{table}
\begin{tabular}{c c} \hline \hline
**Technical Parameter** & **Role Analysis** \\ \hline Aluminum height (\(AH\)) & \(AH\) can stabilize battery voltage. \\ Electrolyte level (\(EL\)) & \(EL\) maintains cell energy stability. \\ Molecular ratio (\(MR\)) & \(MR\) will influence the dissolution of alumina in electrolyte. \\ Heat insulation capacity (\(IP\)) & Heat insulation performance of aluminum reduction cell. \\ Electrolyte temperature (\(ET\)) & The appropriate temperature to ensure the normal operation of aluminum batteries. \\ Voltage fluctuation (\(VF\)) & \(VF\) is influenced by \(AH\). \\ Crystallization temperature (\(CT\)) & The primary temperature of crystallization. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Selected technical parameters. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Experts** & \multicolumn{3}{c}{**States**} \\ \cline{3-5}  & & **High** & **Normal** & **Low** \\ \hline \multirow{5}{*}{\(AH\)} & \(E_{1}\) & \(<\)[s\({}_{3}\), s\({}_{4}\)], (0.18,0.82)\(>\) & \(<\)[s\({}_{2}\), s\({}_{3}\)], (0.72,0.28)\(>\) & \(<\)[s\({}_{1}\), s\({}_{2}\)] (0.00,1.00)\(>\) \\  & \(E_{2}\) & \(<\)[s\({}_{5}\), s\({}_{6}\)], (0.18,0.82)\(>\) & \(<\)[s\({}_{3}\), s\({}_{4}\)], (0.65,0.57)\(>\) & \(<\)[s\({}_{1}\), s\({}_{2}\)], (0.02,0.99)\(>\) \\  & \(E_{3}\) & \(<\)[s\({}_{4}\), s\({}_{5}\)], (0.18,0.82)\(>\) & \(<\)[s\({}_{3}\), s\({}_{4}\)], (0.71,0.37)\(>\) & \(<\)[s\({}_{2}\), s\({}_{3}\)], (0.01,0.98)\(>\) \\  & \(E_{4}\) & \(<\)[s\({}_{4}\), s\({}_{5}\)], (0.18,0.82)\(>\) & \(<\)[s\({}_{3}\), s\({}_{4}\)], (0.61,0.39)\(>\) & \(<\)[s\({}_{2}\), s\({}_{3}\)], (0.00,1.00)\(>\) \\  & \(E_{5}\) & \(<\)[s\({}_{4}\), s\({}_{5}\)], (0.18,0.82)\(>\) & \(<\)[s\({}_{3}\), s\({}_{4}\)], (0.71,0.37)\(>\) & \(<\)[s\({}_{2}\), s\({}_{3}\)], (0.00,1.00)\(>\) \\ \hline \multirow{5}{*}{\(EL\)} & \(E_{1}\) & \(<\)[s\({}_{3}\), s\({}_{4}\)], (0.10,0.70)\(>\) & \(<\)[s\({}_{2}\), s\({}_{3}\)], (0.75,0.20)\(>\) & \(<\)[s\({}_{1}\), s\({}_{2}\)], (0.00,1.00)\(>\) \\  & \(E_{2}\) & \(<\)[s\({}_{5}\), s\({}_  
**Step 2:** The expression of target child events \(X_{nk}\) under different states could be simplified as  
\[X_{4k} =F_{4k;1}B_{1}=(r_{4;1}/r_{4})a_{4k;1}B_{1}\] \[X_{5k} =F_{5k;1}B_{1}+F_{5k;2}B_{2}=(r_{5;1}/r_{5})a_{5k;1}B_{1}+(r_{52}/r_ {5})a_{5k;2}B_{2}\] \[X_{6k} =F_{6k;1}B_{1}+F_{6k;2}B_{2}=(r_{6;1}/r_{6})a_{6k;1}B_{1}+(r_{62}/r_ {6})a_{6k;2}B_{2}\] \[X_{7k} =F_{7k;3}B_{3}=(r_{7;3}/r_{7})a_{7k;3}B_{3}\]  
**Step 3:** Via Equation (17), the occurrence probabilities of target child events \(X_{nk}\) under different states are calculated as listed in Table 4.  
For example, when events \(X_{42}\), \(X_{52}\), \(X_{61}\) and \(X_{73}\) occur, we need to acquire the different states of \(B_{1}\), \(B_{2}\) and \(B_{3}\), namely the fuzzy representation of the posterior probability of events \(B_{1k}\mid X_{43}X_{51}X_{62}X_{71}\), \(B_{2k}\mid X_{43}X_{51}X_{62}X_{71}\) and \(B_{3k}\mid X_{43}X_{51}X_{62}X_{71}\), where \(k=1\), \(2\), \(3\).