## 5 A Case Study
### Calculating the RIF for Each Subsystem  
Five criteria are used to determine the RIF: structural complexity (S), technical maturity (T), manufacturing stability (M), assembly stability (A) and environmental conditions (E). Five experts were invited to provide their judgments: two of the experts had a Master's degree in engineering and had been working in the design of servo turrets for five years, two had been in charge of after-sales for six years, and the last expert obtained his doctorate in reliability analysis and also had been working for three years on reliability evaluation. Table 1 shows the \"best\" and \"worst\" criteria based on the experts' judgments. In this paper, we use the scale (1, 2,..., 9) to describe the relevant importance of the \"best\" or \"worst\" criteria. The matrices of \"best-to-others\" and \"others-to-worst\" are shown in Table 1. The five experts had different judgments for \"best\"--T, S, A, M, E--while the judgments for \"worst\" were almost the same--E, E, E, E, A. Furthermore, the \"best-to-others\" matrices \(A_{B}^{i}\) and \"worst-to-others\" matrices \(A_{W}^{i}\) were determined by expert \(i\), where a scale from 1 to 9 was used to evaluate the relative importance, reflecting the preference of the best over others or the others over the worst (the higher the number, the more important the factor). Then, the Bayesian BWM was used to obtain weights as shown in Equation (5) by MCMC: the weights were 0.263, 0.290, 0.184, 0.159 and 0.105. In this work, we used the linguistic variable as shown in Table 2; then, each expert gave the RIF corresponding to each criterion according to the update to the servo turret and their knowledge. Usually, it is assumed that the lifetime of a product follows some common distribution, and the Weibull distribution is most prominently used. In this work, we used the Weibull distribution to describe the reliability model of the servo turret. Because the working principle of the servo turret did not change, we assumed that the shape parameter did not change in the process of the update but that only the scale parameter possibly did, as shown in Table 3.  
Figure 3: Two failures in the reliability test: (**a**) a failure of the lock and loosen component, (**b**) a failure of the signal component. Figure 2: Reliability test system of the servo turret. One criterion is usually judged to be more important than the other, and we only compare the weights. [PERSON] and [PERSON] (2019) introduced the notion of creadal ranking that can describe the degree to which one criterion is superior to another, and a directed graph is used for visualization where the nodes are each criterion and each edge \(M\Rightarrow M^{\prime}\) indicates that criterion \(M\) is more important than \(M^{\prime}\) at a confidence level of \(v\). Figure 4 shows the credal ranking of criteria by a weight-directed graph. For example, structural complexity (S) is certainly more important than environmental conditions (E), but it is more desirable than manufacturing stability (M) with a confidence of 0.71. The weights obtained by the arithmetic mean are found to be equal for S and T, while the weights obtained by Bayesian BWM are obtained from the probability perspective, as shown in Figure 4. It can be found that the importance of criterion T is higher than S with a confidence interval of 0.63, which means that it is more reasonable for criterion T to be higher than S. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{6}{c}{**(a) Best-to Others Vectors**} \\ \hline Expert No. & best & S & T & M & A & E \\ \hline
1 & T & 3 & 1 & 4 & 4 & 5 \\
2 & S & 1 & 2 & 3 & 4 & 6 \\
3 & A & 2 & 1 & 4 & 3 & 6 \\
5 & E & 1 & 3 & 4 & 5 & 4 \\ \hline \multicolumn{6}{c}{**(b) Others-to-Worst Vectors**} \\ \hline Expert No. & 1 & 2 & 3 & 4 & 5 \\ \hline Worst & E & E & E & E & A \\ S & 4 & 4 & 3 & 2 & 3 \\ T & 5 & 4 & 5 & 3 & 2 \\ M & 2 & 3 & 3 & 3 & 2 \\ A & 3 & 3 & 2 & 2 & 1 \\ E & 1 & 1 & 1 & 1 & 2 \\ \hline \hline \end{tabular}
\end{table}