## 5 Calculation Example  
Assume you are evaluating the health of a fleet of aircraft's landing gear retraction hydraulic system, the evaluated item follows the above four evaluation principles, and the evaluation-related index parameters are listed in Supplementary Table S3. The proposed relative deterioration index \(d\) can be obtained from Table S3, and Equation (30) can be obtained by substituting \(d\) into the membership function.  
Fuzzy statistics are used as an auxiliary to the affiliation function in this study. To calculate the affiliation degree of each fuzzy level, the data is rounded to four decimal places, and the fuzzy evaluation matrix is calculated as \(R_{P}\), \(R_{S}\), \(R_{D}\), \(R_{M}\). The histogram reflecting the fuzzy level is shown in Figure S1.  
\[R_{P}=\left[\begin{array}{cccc}0&0&0&1&0\\ 0&0&0&0.2500&0.7500\\ 0&0&1&0&0\\ 0&0&0.0271&0.9729&0\\ 0&0&0&1&0\\ 0&0.7009&0.2991&0&0\\ 0&0&1&0&0\\ 0&1&0&0&0\\ 0&1&0&0&0\\ \end{array}\right]R_{S}=\left[\begin{array}{cccc}0&0&0.2992&0.7008&0\\ 1&0&0&0&0\\ 0&0&1&0\\ 0&0&1&0&0\\ 0&0.0603&0.9397&0&0\\ 0&0&0&0.0302&0.9698\\ 0&0&0&0.8830&0.1170\\ 0&0.9620&0.0380&0&0\\ 0&0.1816&0.7005&0.1179\\ 0&0.1093&0.7317&0.1590\\ \end{array}\right]\]  
\[R_{D}=\left[\begin{array}{cccc}0&0&0.0096&0.9904&0\\ 0&0&0.9157&0.0843&0\\ 0&0&0.7735&0.2265&0\\ 0&0&0&0&1\\ 0&0.08386&0.1614&0&0\\ 0&0&0.0603&0.9397&0\\ 0&0.01054&0.8946&0\\ 0&0&0.1170&0.8830\\ \end{array}\right]R_{M}=\left[\begin{array}{cccc}0.0843&0.9157&0&0&0\\ 0&0&0&0.9619&0.0381\\ 0&0&1&0&0\\ 0&0&0.9932&0.0068&0\\ 0&1&0&0&0\\ 0&0.0068&0.9932&0&0\\ 0&0.0271&0.9729&0\\ 0&0&0&0&1\\ 0&0&1&0\\ \end{array}\right] \tag{30}\]  
The weight vectors \(w_{1}\) and \(w_{2}\) are obtained by the entropy weight method:  
\[\begin{array}{ll}w_{1}=(0.075,0.075,0.098,0.152,0.121,0.075,0.169,0.075,0.08 6,0.075)\\ w_{2}=(0.281,0.180,0.212,0.327)\\ \end{array} \tag{31}\]  
The order of the data in \(w_{1}\) is from \(u_{1}\) to \(u_{10}\), and the order of the data in \(w_{2}\) is \(P\), \(S\), \(D\), \(M\). Make a first-level fuzzy judgment on \(w_{1}\):  
\[\begin{array}{ll}B_{P}=w_{1}\times R_{P}=(0.00000,0.12757,0.36171,0.45547,0. 05625)\\ B_{S}=w_{1}\times R_{S}=(0.07500,0.07945,0.31481,0.41717,0.11457)\\ B_{D}=w_{1}\times R_{D}=(0.00000,0.10262,0.35069,0.32947,0.21823)\\ B_{M}=w_{1}\times R_{M}=(0.00632,0.14483,0.53985,0.22114,0.08886)\\ \end{array} \tag{32}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**HD Grade** & **Evaluation Criteria** & **Measures** \\ \hline A & \(1.00000\leq HD<1.50000\) & Grounded overhaul \\ B & \(1.50000\leq HD<2.50000\) & Workshop and base maintenance \\ C & \(2.50000\leq HD<3.50000\) & Route troubleshooting and replacement \\ D & \(3.5000\leq HD<4.50000\) & Route maintenance and general monitoring \\ E & \(4.50000\leq HD\leq 5.00000\) & General service \\ \hline \hline \end{tabular}
\end{table}
Table 12: HD grade and measures.  
From the first-level fuzzy judgment, the second-level fuzzy judgment \(R\) can be obtained:  
\[R=\begin{bmatrix}B_{P}\\ B_{S}\\ B_{D}\\ B_{M}\end{bmatrix}=\begin{bmatrix}0&0.12757&0.36171&0.45547&0.05625\\ 0.07500&0.07945&0.31481&0.41717&0.11457\\ 0&0.10262&0.35069&0.32947&0.21823\\ 0.00632&0.14483&0.53985&0.22114&0.08886\end{bmatrix} \tag{33}\]  
\[B=w_{2}\times R=(0.01557,0.11926,0.40918,0.34524,0.11175) \tag{34}\]  
\[HD=B\times V^{T}=3.42134 \tag{35}\]  
Among them \(V=[1,2,3,4,5]\), that is grades I, II, III, IV, V.  
As a result, the fleet's total landing gear retraction system is 3.42134. The health status may be determined from Table 12 as C. As a result, the fleet's general state of health is considerable. Focus on route maintenance and troubleshooting, regularly replace faulty and aging parts, and monitor key parts and areas.