# Fault Analysis of Shearer-Cutting Units Driven by Integrated Importance Measure
## 4 Effectiveness Discussion of IIM Ranking  
To verify the effectiveness of IIM ranking, the Birnbaum importance measure (BIM), criticality importance measure (CIM), and Fussell-Vesely importance measure (FVIM) are introduced to discuss their relative value distribution and the accuracy of their rankings. The calculation results of the three importance measures are as follows. (1) Birnbaum importance measure  
BIM is used to measure the influence of component reliability change on system reliability. The calculation model is shown in Equation (2) [31]. \[I(BIM)^{S}_{C_{i}}=P(S=0|C_{i}=0)-P(S=0|C_{i}=1) \tag{2}\]  
where \(S=0\) means that the top event does not occur, which represents that the system is in a normal state. \(C_{i}=0\) means the fault event does not occur, and \(C_{i}=1\) indicates that the fault event occurs. Equation (2) represents the difference between the system reliability when the bottom event \(C_{i}\) does not occur and the system reliability when the bottom event \(C_{i}\) occurs. (2) Criticality importance measure  
Based on BIM, CIM considers the influence of fault events on the occurrence of system fault time. The calculation model of CIM is shown in Equation (3), which is equal to the product of BIM and the degree of influence of fault events on system fault events [32]. \[I(CIM)^{S}_{C_{i}}=P(S=0|C_{i}=0)-P(S=0|C_{i}=1)\cdot\frac{P(C_{i}=1)}{P(S=1)} \tag{3}\]  
where \(P(C_{i}=1)\) represents the probability that the bottom event \(i\) occurs, and \(P(S=1)\) represents the top event occurs, which is the probability of system failure. (3) Fussell-Vesely importance measure  
FVIM indicates the contribution of the minimum cut set failure containing the bottom event \(i\) to the system failure, and its calculation model is shown in Equation (4) [33; 34]. \[I(FV)^{S}_{C_{i}}=\frac{P(C_{i}=1)P\big{(}S^{i}_{cut}=1\big{|}C_{i}=1\big{)}}{ P(S=1)} \tag{4}\]  
Figure 5: Integrated importance measure results of the bottom events. where \(P\big{(}S_{cut}^{i}=1\big{|}C_{i}=1\big{)}\) represents the probability that the minimum cut set containing the bottom event \(i\) fails when the bottom event \(i\) occurs. \(P(C_{i}=1)\) presents the probability that the bottom event \(i\) occurs, and \(P(S=1)\) represents the top event occurs, which is the probability of system failure. BIM, CIM, and FVIM are calculated by Equations (2)-(4), respectively. The calculation results of the four importance measures are shown in Table 6. We can find that BIM and FVIM are concentrated in the order of 0.1, CIM results are concentrated in the order of 0.01, and IIM is concentrated in the order of 10 \(\times\) 10\({}^{-07}\). As an indicator of identifying the weakest links in the system, IM focuses on its relative importance level. Moreover, the BIM ranking in descending order is C19 \(>\) C18 \(>\) C8 \(>\) C17 \(>\) C14 \(>\) C10 \(>\) C15 \(>\) C12 \(>\) C16 \(>\) C9 \(>\) C1 \(>\) C5 \(>\) C20 \(>\) C2 \(>\) C3 \(>\) C13 \(>\) C11 \(>\) C4 \(>\) C7 \(>\) C6. Under the condition of limited maintenance resources, the bottom events C19 (bearing wear), C17 (nonconformity of oil varieties or improper oil quantity), and C18 (cooling spray function failure) should be repaired as priorities. The CIM ranking in descending order is C19 \(>\) C18 \(>\) C8 \(>\) C17 \(>\) C14 \(>\) C10 \(>\) C15 \(>\) C12 \(>\) C16 \(>\) C9 \(>\) C1 \(>\) C5 \(>\) C20 \(>\) C2 \(>\) C3 \(>\) C13 \(>\) C11 \(>\) C4 \(>\) C7 \(>\) C6. Given the limited maintenance resources, the bottom events C19 (bearing wear), C18 (cooling spray function failure), and C8 (short circuit between coils, water in connection cavity, dampened circuit) should be repaired as priorities. The FVIM ranking in descending order is C15 \(>\) C19 \(>\) C18 \(>\) C8 \(>\)C17 \(>\)C14 \(>\) C10 \(>\) C12 \(>\) C16 \(>\) C9 \(>\) C1 \(>\) C5 \(>\) C20 \(>\) C2 \(>\) C3 \(>\) C13 \(>\)C11 \(>\) C4 \(>\) C7 \(>\) C6.