Risk Assessment Analysis of Multiple Failure Modes Using the Fuzzy Rough FMECA Method: A Case of FACDG

Abstract

With the increasing operating mileage and ownership of high-speed electric multiple units (EMU), a reasonable operation and maintenance strategy is the key to ensure their safe and reliable operation. As a key component of recombined EMU, creating a reasonable and effective risk assessment method for the fully automatic coupler draft gear (FACDG) is the first task. Therefore, based on fuzzy rough number theory, combined with the analytic hierarchy process (AHP), entropy weight method (EWM), technique for order performance by similarity to ideal solution (TOPSIS) and grey relational analysis (GRA), a risk priority indicator of comprehensive nearness degree is developed. Furthermore, a novel multi-criteria decision making (MCDM) failure modes, effects and criticality analysis (FMECA) assessment method is proposed. The effectiveness and rationality of the risk assessment method proposed are verified by the analysis of data and failure modes of a certain FACDG at fourth-level engineering maintenance.

1. Introduction

With the rapid development of high-speed railway technology, high-speed electric multiple units (EMU) have become an indispensable means of transportation in people’s daily life due to their high speed, low energy consumption, light pollution and stability [1,2,3,4,5,6]. In China, there are two operation modes: short EMU trains with 8 cars (standard units) and long EMU trains with 16 cars (two standard units) [7]. By the end of 2021, the total operating mileage of China’s high-speed railway (CHSR) reached more than 40,000 km, and it is served by more than 4000 EMU trains (standard units). However, the short EMU trains are often programmed to operate as long EMU trains according to operating assignment and maintenance scheduling. The fully automatic coupler draft gear (FACDG) is the key device to achieve the recombined EMU, which realizes the mechanical, electrical and pneumatic linkage and transfers the traction force and braking force of a standard unit. Its failure leads to major safety risks such as decoupling, control failure and emergency braking failure [8,9,10]. The FACDG is visually inspected and lubricated in first-level to third-level maintenance, and disintegrate maintenance is carried out in fourth-level and fifth-level maintenance. Therefore, using accurate failure modes analysis and the determination based on risk assessment method is the most effective way to develop reasonable maintenance scheduling and ensure the safe and effective operation of EMU trains.

Failure modes, effects and criticality analysis (FMECA) is an effective method for multi-failure modes reliability risk assessment. All potential failure modes occurring in each component of a system are identified, and the cause of each failure mode and its impact on system operation are determined [10,11,12,13]. Therefore, the FMECA method is widely used to carry out reliability risk assessment in many fields, such as railway rolling stock [10], electrical system [11], aeronautical [13], engineering machinery [14], nuclear engineering [15], maritime [16,17], wind turbines [13,18,19], etc. Appoh and Yunusa-Kaltungo [8] proposed a dynamic mixture model for the comprehensive risk assessment of multiple failure sources. The robustness and accuracy of the model were verified via the application analysis of a coupler in the rolling stock. The criticality of their failure modes and weaknesses were obtained using the criticality matrix method. Then, a shorter-time and lower-cost preventive maintenance schedule was determined based on the analysis results [20,21,22,23]. Deng et al. [24] proposed a framework based on the FMECA method and network theory. The results of a case analysis showed that the most dangerous failure in bogie systems came from the wheel size. Fu et al. [25] proposed a risk prioritization model with the cumulative prospect theory and type 2 intuitionistic fuzzy. In the risk assessment, the risk sensitivity and decision-making psychology were analyzed using the cumulative prospect theory, and the uncertainty was described using the type 2 intuitionistic fuzzy number. In order to solve the high-dimensional problem, Mohammadzadeh et al. [26] proposed a simple and effective deep learning type 2 fuzzy neural network based on Fourier. Bognár and Hegedűs [27] studied the PRISM method and aggregation functions, and presented the risk assessment and prioritization suitable for different situations methodology. Based on the shortages of traditional failure mode and effects analysis, Ouyang et al. [28] proposed a novel failure mode and effects analysis classification method for risk assessment by combining risk factors in pairs. Huang et al. [29] established an analysis system for multiple operating failures of EMU, and obtained the interaction between the failure modes and the failure propagation chain. Based on the analysis of China’s high-speed railway control system, a fuzzy FMECA method with a cloud model was proposed, which can effectively reduce the subjective influence of expert factors on the assessment results [30,31]. In order to improve the rationality of the FMECA analysis results, many scholars have carried out research on method improvement. Wang et al. [32] developed a risk assessment method based on fuzzy weighted geometric mean. The assessment was carried out using fuzzy terms and fuzzy ratings, which solved the problem that the original method cannot adequately prioritize failure modes. The technique for order performance by similarity to ideal solution (TOPSIS) method cannot directly decide the sorting of alternatives in the decision-making process; for this problem, Zhang et al. [33,34] established an improved TOPSIS method that can deal with the sorting and classification of alternatives simultaneously by introducing the decision theory of fuzzy rough sets. Song et al. [35] developed a novel risk priority method based on rough number and TOPSIS theory, which can not only control fuzziness and subjectivity in an uncertain environment, but can also consider the importance of risk factors in the determination of failure modes risk priority. Li et al. [36] proposed a normalization algorithm on account of the analytic hierarchy process (AHP) by introducing the weights of risk assessment factors. The failure modes and effects analysis of floating offshore wind turbines is completed by using the proposed algorithm, and the results showed that the catastrophic failures can be minimized. Zhu et al. [37] integrated the rough number theory, AHP and compromise sorting method to improve the objectivity of FMECA assessment. In this method, the subjective uncertainty of expert evaluation was solved using rough number theory, the subjective weight of risk factors was obtained by the AHP and the compromise sorting method of failure modes was obtained using the compromise sorting method. Gupta et al. [38] proposed a risk assessment model for FMECA based on fuzzy logic and Dempster–Shafer theory. Wen et al. [39] presented a risk assessment method considering the weights of the risk factors. The fuzzy language term sets were used to deal with the hesitation information, the known information was used to supplement the missing information and the statistical variance was used to ensure the weights of the risk factors. Fata et al. [40] proposed an improved FMECA technique based on multi-criteria decision making (MCDM) of the Elimination et Choice Translating Reality (ELECTRE) TRI method. Failure modes were classified into different risk levels, and the appropriate threshold control was used to achieve a gradual transition from indifference to a preference for decision makers. Wang et al. [41] proposed a hybrid MCDM model based on TOPSIS, the simple additive weight (SAW) method and grey relational analysis, which can solve the problem of conflicts among alternatives of MCDM. Poongavanam et al. [42] carried out studies on three different kinds of MCDM and identified the best solution to meet the environmental agreement from 14 alternatives. Many scholars have made a lot of contributions in the research of risk assessment methods, but more attention is paid to solving the shortcomings in some aspects of assessment.

Based on the failure modes and data statistics of fourth-level overhaul, combined with the analysis results of the RPN method used, it was found that there is an inconsistency between the results of the RPN method and the failure modes of the actual operation and maintenance [43]. The lack of the RPN method used was found, and the corresponding solution was obtained. In this paper, the fuzzy sets theory is used to deal with the subjective uncertainty of decision makers. The uncertainty among decision makers is resolved using the rough number theory. The weights of risk assessment factors are calculated using the AHP-EWM. The risk priority ranking is carried out by means of TOPSIS-GRA. An FMECA assessment method is proposed. The validity and rationality of the risk assessment method proposed are verified via the analysis of operation and maintenance of a certain FACDG. The proposed method lays a good foundation for the reasonable formulation of maintenance strategy in China’s operating EMUs.

Herein, in Section 2, the fuzzy sets theory and the rough number theory are firstly introduced, the comprehensive weights method is presented based on analyzing of risk assessment factors and the comprehensive nearness degree of the risk priority ranking is proposed based on TOPSIS-GRA. In Section 3, a practical case study on the application of the proposed method is carried out. The conclusions of this paper and future research works are presented in Section 4.

2. Methodology

2.1. Fuzzy Sets Theory

The fuzzy sets theory is applied to solve the problem of uncertainty and accuracy of decision makers in an assessment. Based on the method of information expression and quantization, a convex fuzzy set characterized by a given interval of real numbers is constructed, and the membership degree of each real number is determined to be between 0 and 1 [44,45]. The membership functions as a triangle and trapezoid are shown in Figure 1. Therefore, they can, respectively, be expressed as [46]

$$\tilde{A}(x)=f\left(a,b,c\right)=\{\begin{array}{cc}(x-a)/(b-a)& x\in [a,b)\\ (c-x)/(c-b)& x\in [b,c]\\ 0& \mathrm{otherwise}\end{array}$$

(1)

$$\tilde{A}(x)=f\left(a,b,c,d\right)=\{\begin{array}{cc}(x-a)/(b-a)& x\in [a,b)\\ 1& x\in [b,c)\\ (d-x)/(d-c)& x\in [c,d]\\ 0& \mathrm{otherwise}\end{array}$$

(2)

Figure 1. The membership function: (a) triangle and (b) trapezoid.

For finding an exact value which can best represent the fuzzy sets from the membership function, the Alpha-cut defuzzification (ACD) method proposed by Amir is used for defuzzification [47]. Therefore, the defuzzification equations of the membership function as a triangle and trapezoid can, respectively, be described as

$$\mathit{A}C{D}_{\mathrm{tri}}=\frac{a+4b+c}{6}$$

(3)

$$\mathit{A}C{D}_{\mathrm{tra}}=\frac{a+2b+2c+d}{6}$$

(4)

Based on the fuzzy assessment language, the decision results are obtained by the decision maker’s assessment of each risk factor of the failure modes. The defuzzification decision matrix X is obtained via the ACD method. This is,

$$\mathit{X}=\left[\begin{array}{cccc}{x}_{11}& x_{12}& \cdots & x_{1n}\\ x_{21}& x_{22}& \cdots & x_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ x_{m1}& x_{m2}& \cdots & x_{mn}\end{array}\right]$$

(5)

where x_ij is the defuzzification value of each failure mode, i = 0, 1, 2, …, m; j = 0, 1, 2, …, n, m is the number of failure modes; and n is the number of risk assessment factors.

2.2. Rough Number Theory

By using interval values instead of exact values to indicate the uncertainty among decision makers, the decision makers’ individual cognitive bias and randomness are effectively avoided. The uncertainty and subjectivity among decision makers can be reduced effectively, and the objectivity of assessment results can be improved [48]. Herein, the rough number theory is applied to solve the weight of the decision makers’ assessment in the FMECA for FACDG.

The rough number of any class C_h in the domain of discourse U is defined as

$$RN\left(C_h\right)=\left[\underset{\_}{Lim}\left(C_h\right),\overline{Lim}\left(C_h\right)\right]$$

(6)

The aggregating rough number RN(C) is expressed as

$$RN(C)=\left[\frac{{\displaystyle {\displaystyle \sum _{h=1}^q}\underset{\_}{Lim}\left(C_h\right)}}{q},\frac{{\displaystyle {\displaystyle \sum _{h=1}^q}\overline{Lim}\left(C_h\right)}}{q}\right]$$

(7)

where $\overline{Lim}\left(C_h\right)$ and $\underset{\_}{Lim}\left(C_h\right)$ are the upper approximation and lower approximation of the C_h, $\overline{Lim}\left(C_h\right)=\frac{1}{N_U}\sum R(Y)\mid Y\in \overline{\mathit{A}pr}\left(C_h\right)$, $\underset{\_}{Lim}\left(C_h\right)=\frac{1}{N_L}\sum R(Y)\mid Y\in \underset{\_}{\mathit{A}pr}\left(C_h\right)$; N_U and N_L are the number of objects included in the upper approximation and lower approximation of the C_h; and h = 1, 2, …, q, q is the number of decision makers [35].

The uncertainty is expressed using the lower approximation and upper approximation in rough number theory. The size of the interval of the boundary region characterizes the size of the uncertainty among decision makers. Therefore, even though the data size is small and the distribution is unknown, it can handle the uncertainty among domain decision makers and maintain the objectivity of the original information.

2.3. Comprehensive Weight Analysis

The weights of the risk assessment factors include the subjective weights and objective weights in order to assign full play to the advantages of the subjective and objective weights analysis methods. Herein, the AHP is used to determine the subjective weights, and the EWM is applied to calculate the objective weights.

2.3.1. Subjective Weight Analysis

The AHP is a systematic, hierarchical and multi-criteria decision analysis method, which is applied to solve the comparison problem of multiple indicators and schemes to select the optimal scheme [36]. Its main steps are as follows:

Step 1: Divide the hierarchical structure. Considering the objectives, objects and criteria of decision making, the hierarchical structure can be divided into target level and indicator level.

Step 2: Construct pairwise comparison matrix A. Saaty’s method shown in Table 1 is used to determine the importance of different attributes relative to the target [36]. a_j and a_g represent the risk assessment factors, a_jg indicates the relative importance of a_j to a_g and a_jg is the integer from 1 to 9 or its reciprocal.

Table 1. Pairwise comparisons of Saaty’s method.

Scale	Description	Comment
1	aj and ag are equally important	agj and ajg are reciprocal of each other $a_{jg}=\frac{1}{a_{gj}}$
3	aj is weakly more important than ag
5	aj is strongly more important than ag
7	aj is very strongly more important than ag
9	aj is absolutely more important than ag
2, 4, 6, 8	represent the median of adjacent judgments

Then, the pairwise comparison matrix A can be described as

$$\mathit{A}={\left(a_{jg}\right)}_{n\times n}$$

(8)

where j, g = 1, 2, …, n. Herein, the risk assessment factors contain the Severity (S), Occurrence (O), Detection (D) and Maintenance (M).

Step 3: Solve for the relative weights. The maximum eigenvalue λ_max and the corresponding eigenvector ω are received according to the pairwise comparison matrix A. The equation is expressed as

$$\mathit{A}\mathsf{\omega }={\lambda }_{\max }\mathsf{\omega }$$

(9)

Step 4: Consistency test. The consistency is verified by calculating the consistency ratio (CR) of the comparison matrix A. The CR is described as

$$CR=CI/RI$$

(10)

where CI is the consistency indicator of pairwise comparison matrix, $CI=\left({\lambda }_{\max }-n\right)/\left(n-1\right)$, and RI is the average random consistency indicator of the same order, which is determined by the order of the pairwise comparison matrix [37].

Generally, the pairwise comparison matrix A can be accepted when CR < 0.1. Otherwise, the comparison value should be adjusted until CR < 0.1.

Step 5: Determine the subjective weights of the risk assessment factors. The aggregated comparison matrix ${\mathit{A}}^{*}$ is determined as

$${\mathit{A}}^{*}=\left[\begin{array}{cccc}\left[1,1\right]& \left[a_{12}^L,a_{12}^U\right]& \cdots & \left[a_{1n}^L,a_{1n}^U\right]\\ \left[a_{21}^L,a_{21}^U\right]& \left[1,1\right]& \cdots & \left[a_{2n}^L,a_{2n}^U\right]\\ ⋮& ⋮& \ddots & ⋮\\ \left[a_{n1}^L,a_{n1}^U\right]& \left[a_{n2}^L,a_{n2}^U\right]& \cdots & \left[1,1\right]\end{array}\right]$$

(11)

Then, the subjective weight w_sj of each risk assessment factor can be calculated as

$$w_{sj}=\left[\sqrt[\begin{array}{c}n\\ \end{array}]{{\displaystyle \prod _{g=1}^n}{a}_{jg}^L},\sqrt[\begin{array}{c}n\\ \end{array}]{{\displaystyle \prod _{g=1}^n}{a}_{jg}^U}\right]$$

(12)

The normalization form of the subjective weights is described as

$$w_s^{\prime }=\frac{w_s}{\max \left(w_s^U\right)}=\left\{[w_{s\mathrm{S}}^{L\prime },w_{s\mathrm{S}}^{U\prime }],[w_{s\mathrm{O}}^{L\prime },w_{s\mathrm{O}}^{U\prime }],[w_{s\mathrm{D}}^{L\prime },w_{s\mathrm{D}}^{U\prime }],[w_{s\mathrm{M}}^{L\prime },w_{s\mathrm{M}}^{U\prime }]\right\}$$

(13)

2.3.2. Objective Weight Analysis

The EWM is applied to obtain the objective weights of the risk assessment factors. The values of dispersion can be measured using the implied interactions among risk factors in decision making; then, the objective weight of each risk factor is obtained [49]. The greater the value dispersion of the risk assessment factor, the smaller the entropy value, and the greater the weight should be. The input data of EWM are the decision matrix. The input data should be normalized for calculating entropy, and the indicator of positive and negative can be determined in the normalization process.

The normalized decision matrix ${\mathit{X}}^{*}$ can be expressed as

$${\mathit{X}}^{*}=\left[\begin{array}{cccc}{x}_{11}^{*}& x_{12}^{*}& \cdots & x_{1n}^{*}\\ x_{21}^{*}& x_{22}^{*}& \cdots & x_{2n}^{*}\\ ⋮& ⋮& \ddots & ⋮\\ x_{m1}^{*}& x_{m2}^{*}& \cdots & x_{mn}^{*}\end{array}\right]$$

(14)

The positive indicator and negative indicator are described as [50]

$$x_{ij}^{*}=\frac{x_{ij}-x_{j\min }}{x_{j\max }-x_{j\min }},x_{ij}^{*}=\frac{x_{j\max }-x_{ij}}{x_{j\max }-x_{j\min }}$$

(15)

where x_jmax and x_jmin are the maximum and minimum values of the same risk assessment factor, respectively.

Then, the entropy S_j of each risk assessment factor is defined as

$$S_j=-\frac{1}{\ln m}{\displaystyle \sum _{i=1}^m}\left(\frac{x_{ij}^{*}}{{\displaystyle {\displaystyle \sum _{i=1}^m}{x}_{ij}^{*}}}\cdot \ln \frac{x_{ij}^{*}}{{\displaystyle {\displaystyle \sum _{i=1}^m}{x}_{ij}^{*}}}\right)\begin{array}{cc}& i=1,2,\cdots ,m; j=\end{array}1,2,\cdots ,n$$

(16)

The information entropy redundancy H_j is described as

$$H_j=1-S_j$$

(17)

Based on the rough number theory, the information entropy redundancy of each decision maker’s evaluation data is determined as

$$H=\{[H_{\mathrm{S}}^L,H_{\mathrm{S}}^U],[H_{\mathrm{O}}^L,H_{\mathrm{O}}^U],[H_{\mathrm{D}}^L,H_{\mathrm{D}}^U],[H_{\mathrm{M}}^L,H_{\mathrm{M}}^U]\}$$

(18)

The information entropy redundancy represents the objective weight of each risk assessment factor, i.e., H = w_o. Therefore, the normalization form of the objective weight of the risk assessment factor is expressed as

$$w_o^{\prime }=\frac{w_o}{\max \left(w_o^U\right)}=\{[w_{o\mathrm{S}}^{L\prime },w_{o\mathrm{S}}^{U\prime }],[w_{o\mathrm{O}}^{L\prime },w_{o\mathrm{O}}^{U\prime }],[w_{o\mathrm{D}}^{L\prime },w_{o\mathrm{D}}^{U\prime }],[w_{o\mathrm{M}}^{L\prime },w_{o\mathrm{M}}^{U\prime }]\}$$

(19)

Consequently, the comprehensive weight w is established as

$$w=w_s^{\prime }\cdot w_o^{\prime }$$

(20)

2.4. Risk Priority Ranking

The MCDM method can solve complex problems with high uncertainty and multiple viewpoints by ranking the failure modes according to existing decision information [41]. The TOPSIS method determines the fault risk by determining the distance between the evaluation sequence and the positive and negative ideal solution vectors. However, if the total distance is the same, the risk of failure modes cannot be judged. The GRA judges the fault risk using the similarity of reference sequence, and the limitation of the TOPSIS can be solved. Therefore, the combination of the TOPSIS and GRA can satisfy the risk ranking of multi-failure modes of the FACDG.

2.4.1. TOPSIS Analysis

The TOPSIS method was proposed by Hwang and Yoon in 1981 for solving the MCDM problem. Since then, scholars have proposed improved methods according to their research. Saxena et al. [51] proposed an optimal software reliability growth selected model based on criteria importance through inter-criteria correlation and TOPSIS. Atenidegbe and Mogaji [52] developed a TOPSIS-Entropy assessment model for contamination risk. Yeo et al. [53] proposed an improved fuzzy TOPSIS risk assessment technique for engine systems, which effectively improves the assessment efficiency through a high-hazard risk control scheme. According to the rough TOPSIS method, the positive ideal solution (PIS) and the negative ideal solution (NIS) should be determined [35]. In the benefit criterion, the larger the indicator value, the closer to the PIS, while it is the opposite in the cost criterion. The PIS and the NIS can, respectively, be expressed as

$$x^{+}(j)=\{\begin{array}{l}{ \max }_{i=1}^m\left(x_{ij}^U\right),\begin{array}{cc}& j\in B\end{array}\\ { \min }_{i=1}^m\left(x_{ij}^U\right),\begin{array}{cc}& j\in C\end{array}\end{array}$$

(21)

$$x^{-}(j)=\{\begin{array}{l}{ \min }_{i=1}^m\left(x_{ij}^L\right),\begin{array}{cc}& j\in B\end{array}\\ { \max }_{i=1}^m\left(x_{ij}^L\right),\begin{array}{cc}& j\in \end{array}C\end{array}$$

(22)

where $x^{+}(j)$ and $x^{-}(j)$ are, respectively, the values of the criteria PIS and NIS, B is the benefit criterion and C is the cost criterion.

In general, the Euclidean distance is used to calculate the distance between each data sequence and the PIS and the NIS. The distance between the data sequence and PIS and NIS are, respectively, expressed as

$$d_i^{+}={\left\{\sum _{j\in B}{\left(x_{ij}^L-x^{+}(j)\right)}^2+\sum _{j\in C}{\left(x_{ij}^U-x^{+}(j)\right)}^2\right\}}^{1/2}$$

(23)

$$d_i^{-}={\left\{\sum _{j\in B}{\left(x_{ij}^U-x^{-}(j)\right)}^2+\sum _{j\in C}{\left(x_{ij}^L-x^{-}(j)\right)}^2\right\}}^{1/2}$$

(24)

where $d_i^{+}$ is the distance between the data sequence and PIS, and $d_i^{-}$ is the distance between the data sequence and NIS.

The dimensionless PIS $D_i^{+}$ and NIS $D_i^{-}$ are, respectively, defined as

$$D_i^{+}=d_i^{+}/{\displaystyle \sum _{i=1}^m}{d}_i^{+}$$

(25)

$$D_i^{-}=d_i^{-}/{\displaystyle \sum _{i=1}^m}{d}_i^{-}$$

(26)

2.4.2. Grey Relational Analysis

The grey relational analysis (GRA) is an effective analysis method for solving the decision problem in an uncertain environment and multi-attribute situation. The process is to plot each factor as a sequence curve, and the relational grade of each risk assessment factor by the similarity of its geometric shape can be obtained [54]. The rough GRA method is applied as follows.

The positive reference sequence and negative reference sequence are set as

$${\mathit{X}}_0^{+}=\left(x_0^{+}(1),x_0^{+}(2),\dots ,x_0^{+}(n)\right)$$

(27)

$${\mathit{X}}_0^{-}=\left(x_0^{-}(1),x_0^{-}(2),\dots ,x_0^{-}(n)\right)$$

(28)

Then, the compared sequence is described as

$${\mathit{X}}_i=\left([x_i^L(1),x_i^U(1)],[x_i^L(2),x_i^U(2)],\dots ,[x_i^L(n),x_i^U(n)]\right)$$

(29)

The benefit criterion is described as

$$\{\begin{array}{l}{\gamma }^{+}\left(x_0^{+}(j),x_i^L(j)\right)=\frac{\underset{i}{\min }\underset{j}{\min }{D}_B^{+}+\xi L_B^{+}}{D_B^{+}+\xi L_B^{+}}, \mathrm{herein} D_B^{+}=\left|x_0^{+}(j)-x_i{}^L(j)\right|,L_B^{+}=\underset{i}{\max }\underset{j}{\max }\left|x_0^{+}(j)-x_i{}^L(j)\right|\\ {\gamma }^{-}\left(x_0^{-}(j),x_i^U(j)\right)=\frac{\underset{i}{\min }\underset{j}{\min }{D}_B^{-}+\xi L_B^{-}}{D_B^{-}+\xi L_B^{-}},\mathrm{herein} D_B^{-}=\left|x_0^{-}(j)-x_i{}^U(j)\right|,L_B^{-}=\underset{i}{\max }\underset{j}{\max }\left|x_0^{-}(j)-x_i{}^U(j)\right|\end{array}$$

(30)

The cost criterion is described as

$$\{\begin{array}{l}{\gamma }^{+}\left(x_0^{+}(j),x_i^U(j)\right)=\frac{\underset{i}{\min }\underset{j}{\min }{D}_C^{+}+\xi L_C^{+}}{D_C^{+}+\xi L_C^{+}},\mathrm{herein} D_C^{+}=\left|x_0^{+}(j)-x_i{}^U(j)\right|,L_C^{+}=\underset{i}{\max }\underset{j}{\max }\left|x_0^{+}(j)-x_i{}^U(j)\right|\\ {\gamma }^{-}\left(x_0^{-}(j),x_i^L(j)\right)=\frac{\underset{i}{\min }\underset{j}{\min }{D}_C^{-}+\xi L_C^{-}}{D_C^{-}+\xi L_C^{-}},\mathrm{herein} D_C^{-}=\left|x_0^{-}(j)-x_i{}^L(j)\right|,L_C^{-}=\underset{i}{\max }\underset{j}{\max }\left|x_0^{-}(j)-x_i{}^L(j)\right|\end{array}$$

(31)

where ξ is the resolution coefficient, which is used to decrease the effect of the excessive maximum value on the distortion of the correlation coefficient, and is generally taken as 0.5 [55].

Then,

$${\gamma }^{+}\left(X_0,X_i\right)=\{\begin{array}{l}\frac{1}{n}{\displaystyle \sum _{j=1}^n}{\gamma }^{+}\left(x_0^{+}(j),x_i^L(j)\right) j\in B\\ \frac{1}{n}{\displaystyle \sum _{j=1}^n}{\gamma }^{+}\left(x_0^{+}(j),x_0^U(j)\right) j\in C\end{array}$$

(32)

$${\gamma }^{-}\left(X_0,X_i\right)=\{\begin{array}{l}\frac{1}{n}{\displaystyle \sum _{j=1}^n}{\gamma }^{+}\left(x_0^{-}(j),x_i^U(j)\right)j\in B\\ \frac{1}{n}{\displaystyle \sum _{j=1}^n}{\gamma }^{+}\left(x_0^{-}(j),x_0^L(j)\right)j\in C\end{array}$$

(33)

where ${\gamma }^{+}\left(X_0,X_i\right)$ and ${\gamma }^{-}\left(X_0,X_i\right)$ are, respectively, the positive and negative relational grade of $X_0$ and $X_i$; the closer the value is to 1, the better the correlation.

The dimensionless positive correlation degree ${\Psi }_i^{+}$ and negative correlation degree ${\Psi }_i^{-}$ are defined as

$${\Psi }_i^{+}={\gamma }_i^{+}/{\displaystyle \sum _{i=1}^m}{\gamma }_i^{+},\begin{array}{cc}& {\Psi }_i^{-}={\gamma }_i^{-}/{\displaystyle \sum _{i=1}^m}{\gamma }_i^{-}\end{array}$$

(34)

The analysis indicates that the larger the values of ${\Psi }_i^{+}$ and $D_i^{-}$, the closer they are to the PIS, namely, the lower the risk of failure mode. Similarly, the larger the values of ${\Psi }_i^{-}$ and $D_i^{+}$, the closer they are to the NIS, namely, the higher the risk of failure mode. A comprehensive indicator ${\Omega }^{+}$ or ${\Omega }^{-}$ can be established as

$${\Omega }_i^{+}={\alpha }_1{D}_i^{-}+{\beta }_1{\Psi }_i^{+},\begin{array}{cc}& {\Omega }_i^{-}={\alpha }_2{D}_i^{+}+{\beta }_2{\Psi }_i^{-}\end{array}$$

(35)

where α₁ and β₁ are the weights of the indicators $D^{-}$ and ${\Psi }^{+}$, and α₂ and β₂ are the weights of the indicators $D^{+}$ and ${\Psi }^{-}$.

The comprehensive nearness degree δ_i is established as

$${\delta }_i=\frac{{\Omega }_i^{-}}{{\Omega }_i^{+}+{\Omega }_i^{-}}$$

(36)

Therefore, based on the above analysis, the risk assessment process of the fuzzy rough number FMECA is shown in Figure 2.

Figure 2. Flowchart of assessment process of the fuzzy rough number FMECA.

3. Application Example: The Case of a Certain FACDG

In this section, we use a certain FACDG, widely used in China’s operating EMUs, as an application example. The methodology is proposed by analyzing the shortcomings of the RPN method and the characteristics of other risk assessment methods [49]. The methodology established in Section 2 is applied to carry out the risk assessment analysis.

3.1. Failure Modes Analysis

Figure 3 shows a certain FACDG that is widely used in China’s operating EMUs. It includes a connected system, electric coupler and pusher, air circuit system, telescopic and cushioning devices, electrical components and mounting alignment device. The FACDG is used to realize the connection between short EMU trains to form long EMU trains. The convex cone at the front of the FACDG is introduced into the concave cone of the opposite FACDG, and the coupling rods are interlocked with the notch of the coupler tongue. Meanwhile, the pipeline valves are squeezed to complete the air circuit connection. After that, the electric coupler cover is opened under the pneumatic action, and the circuit connection is completed. When uncoupling, the electrical coupler is first uncoupled, and then the separation of mechanical and pneumatic circuits is realized.

Figure 3. Model of the FACDG. 1—Coupling rod, 2—Coupler tongue, 3—Connecting snap ring, 4—Single electric solenoid valve, 5—Double electric solenoid valve, 6—Telescopic cylinder, 7—Collapsed pipe, 8—Rubber ring, 9—Mounting base, 10—Master pin, 11—Rubber bearing, 12—Centering device, 13—Bracket, 14—Rubber support, 15—Push cylinder, 16—Electric coupler, 17—Electric coupler cover, 18—Uncoupling pipeline valve, 19—Main pipeline valve, 20—Brake pipeline valve.

Table 2 shows the failure modes and analysis of a type of FACDG with fourth-level overhaul from November 2015 to June 2020, and the images of high-risk failure components are shown in Figure 4.

Table 2. Failure modes analysis of the FACDG.

Subsystem	Component	Code	Failure Mode	Effect	Failure Rate/%
Coupling system	Trigger composition	0101A	Deflection of guide rod	Affecting task completion	3.50
		0101B	Spring failure and force drop		1.60
	Coupler tongue	0102A	Cracks on locating pin rod	Affecting automatic coupling and decoupling	19.00
	Coupler body	0103A	Surface linear defect	Fault symptoms	5.50
	Uncoupling cylinder	0104A	Air leakage of cylinder	Mechanical coupler uncoupling failure	0.07
	Coupler tongue extension spring	0105A	Spring failure and force drop	Mechanical coupler coupling failure	0.70
	Connecting snap ring	0106A	Cracks on upper connecting snap ring	Fracture, leading to decoupling of EMU	1.10
Electric coupler and pusher system	Contact	0201A	Corrosion	Electrical coupling failure	0.58
	Sealing ring of contact elements	0202A	Cracking and aging	Electrical coupling failure	5.17
		0202B	Cracking and aging	Short circuit and burning loss of circuit	5.17
	Rubber sealing ring of coupler head	0203A	Cracking and aging	Short circuit and burning loss of circuit	4.67
	Shaft of tension spring	0204A	Deformation	Electrical coupling failure	1.50
	Electric coupler cover	0205A	Damage of moving components of the opening and closing structure	Electrical coupling failure	2.50
	Bearing of protective cover	0206A	Deformation and damage	Electrical coupling failure	5.00
	Spring of protective cover	0207A	Deformation and damage	Electrical coupling failure	1.00
	Shaft sleeve of protective cover	0208A	Oil loss	Electrical coupling failure	8.10
	Push cylinder	0209A	Air leakage at rod end	Electrical coupling failure	40.00
	Fastener groups	0210A	Shedding and fracture	Electrical coupling failure	5.40
Air circuit system	Mechanical valves	0301A	Valve element rusted and unable to move	Electrical coupling failure	3.50
	Pipeline	0302A	Air leakage	Functional failure	0.14
	Brake pipeline valve	0303A	Ventilation leakage	Functional failure	7.50
	Main pipeline valve	0304A	Ventilation leakage	Functional failure	7.50
	Uncoupling pipeline valve	0305A	Ventilation leakage	Functional failure	7.50
Electrical components system	Magnetic proximity switch	0401A	Damage	Affecting task completion	0.35
	Single electric solenoid valve	0402A	Not exceeding the standard leakage	Unlocking failure of locking device	9.21
	Double electric solenoid valve	0403A	Exceeding the standard leakage	Failure of extension and contraction	53.35
	Limit switch	0404A	Damage of gold-plated contacts	Affecting task completion	0.14
Crushing devices of expansion and buffering	Rubber ring	0501A	Cracks and damages on the surface	Fault symptoms	17.00
	Crushing pipeline	0502A	Severe crushing	Decrease in overload protection capacity	1.00
	Telescopic cylinder	0503A	Deformation of cylinder barrel	Fault symptoms	0.42
	Unlock cylinder	0504A	Sliding thread of cylinder barrel thread	Loss of action	1.49
		0504B	Corrosion of piston assembly	Loss of action	0.20
		0504C	Collision and deformation of inner wall of cylinder barrel	Fault symptoms	0.42
		0504D	Scratches on inner wall of cylinder barrel	Fault symptoms	0.86
Alignment devices of installation of hanging	Cam disc	0601A	Wear	Position offset	23.50
	Bracket	0602A	Surface crack	Fault symptoms	18.50
	Rubber support	0603A	Too-small electrical impedance	Fault symptoms	100.00
	Installation frame	0604A	Linear surface defect	Fault symptoms	7.00
		0604B	Poor size and appearance	Fault symptoms	0.50
	Rubber bearing	0605A	Too-small stiffness	Noise during operation	42.86
		0605B	Too-small electrical impedance	Fault symptoms	100.00
	Master pin	0606A	Poor Teflon coating on the surface	Fault symptoms	100.00

Figure 4. Images of failure components: (a) Cracks on upper connecting snap ring; (b) Cracks on locating pin rod; (c) Cracks and damages on the surface; (d) Valve element rusted and unable to move; (e) Air leakage at rod end.

3.2. Analysis of Fuzzy Assessment

The membership function of FACDG fuzzy assessment is established by using the trapezoidal and triangular fuzzy numbers, as shown in Figure 5. The fuzzy assessment language of risk factors is classified into five levels, VL, L, M, H and VH, based on the statistical analysis of 42 failure modes. The assessment criteria of determined risk assessment factors is illustrated in Table 3. According to the membership function established, assessment criteria and the 42 failure modes of FACDG, the fuzzy numbers are obtained by combining the fuzzy linguistic assessment of three decision makers for S, D and M. The O is determined by the failure rate of the fourth-level maintenance. Then, the defuzzification numbers are obtained using the ACD method. The determined fuzzy numbers and defuzzification numbers are shown in Table 4.

Figure 5. The membership function of fuzzy linguistic variables.

Table 3. The assessment criteria of risk assessment factors.

Linguistic Variable (Level)	O	S	D	M
Very low (VL)	Almost no occurrence p ≤ 0.1%	Almost no damage	Very easy to find and identify	Very easy to maintain
Low (L)	Rare occurrence 0.1%< p ≤ 1.0%	Mild damage	Easy to find and identify	Easy to maintain
Medium (M)	Occasional occurrence 1.0% < p ≤ 10%	Moderate damage	Generally easy to find and identify	Generally easy to maintain
High (H)	Sometimes occurrence 10% < p ≤ 20%	Serious damage	Difficult to find and identify	Difficult to maintain
Very high (VH)	Constant occurrence p > 20%	Significant damage	Very difficult to find and identify	Very difficult to maintain

Table 4. The fuzzy numbers and defuzzification numbers.

Level	Fuzzy Number	Defuzzification Number
VL	(0, 0, 1, 3)	0.8333
L	(1, 3, 5)	3
M	(3, 5, 7)	5
H	(5, 7, 9)	7
VL	(7, 9, 10, 10)	9.167

Therefore, the defuzzification decision matrices are obtained.

$$X_1=\begin{array}{c}SDM\hfill \\ {\left[\begin{array}{ccc}3.000& 5.000& 3.000\\ 3.000& 5.000& 3.000\\ 9.167& 7.000& 7.000\\ 5.000& 5.000& 9.167\\ \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot \\ \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot \\ 0.833& 3.000& 3.000\\ 3.000& 5.000& 7.000\\ 0.833& 5.000& 7.000\\ 0.833& 3.000& 3.000\end{array}\right]}_{42\times 3}\end{array}, X_2=\begin{array}{c}SDM\hfill \\ {\left[\begin{array}{ccc}3.000& 5.000& 5.000\\ 5.000& 5.000& 5.000\\ 7.000& 9.167& 7.000\\ 3.000& 5.000& 9.167\\ \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot \\ \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot \\ 3.000& 3.000& 5.000\\ 3.000& 5.000& 5.000\\ 0.833& 7.000& 5.000\\ 0.833& 5.000& 5.000\end{array}\right]}_{42\times 3}\end{array}, X_3=\begin{array}{c}SDM\hfill \\ {\left[\begin{array}{ccc}0.833& 3.000& 5.000\\ 3.000& 3.000& 5.000\\ 7.000& 7.000& 7.000\\ 5.000& 7.000& 9.167\\ \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot \\ \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot \\ 0.833& 3.000& 7.000\\ 3.000& 7.000& 7.000\\ 0.833& 7.000& 7.000\\ 0.833& 5.000& 5.000\end{array}\right]}_{42\times 3}\end{array}, X_O={\left[\begin{array}{c}5.000\\ 5.000\\ 7.000\\ 5.000\\ \cdot \cdot \cdot \\ \cdot \cdot \cdot \\ 3.000\\ 9.167\\ 9.167\\ 9.167\end{array}\right]}_{42\times 1}$$

The aggregated decision matrix X_c is obtained according to rough number theory.

$$\begin{array}{c}SODM\hfill \\ X_c={\left[\begin{array}{cccc}[1.7961,2.7592]& [5.0000,5.0000]& [3.8889,4.7778]& [3.8889,4.7778]\\ [3.2222,4.1111]& [5.0000,5.0000]& [3.8889,4.7778]& [3.8889,4.7778]\\ [7.2408,8.2039]& [7.0000,7.0000]& [7.2408,8.2039]& [7.0000,7.0000]\\ [3.8889,4.7778]& [5.0000,5.0000]& [5.2222,6.1111]& [9.1670,9.1670]\\ \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot \\ \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot \\ [1.0738,2.0369]& [3.0000,3.0000]& [3.0000,3.0000]& [4.1111,5.8889]\\ [3.0000,3.0000]& [9.1670,9.1670]& [5.2222,6.1111]& [5.8889,6.7778]\\ [0.8330,0.8330]& [9.1670,9.1670]& [5.8889,6.7778]& [5.8889,6.7778]\\ [0.8330,0.8330]& [9.1670,9.1670]& [3.8889,4.7778]& [3.8889,4.7778]\end{array}\right]}_{42\times 4}\end{array}$$

Compared with the weighting approach specified [12], the interval-expressed aggregated decision matrix can effectively solve the uncertainty problem in the evaluation process. For improving the comparability of the assessment indicator, each assessment indicator is evaluated using the interval normalization method. Herein, the maximum value of the upper limit of the rough number of each risk assessment factor is selected as the phase reference point. Then, the normalized equations are expressed as

$$x_{ij}^{L\prime }=\frac{x_{ij}^L}{{\displaystyle \underset{}{\overset{}{{\max }_{i=1}^m}}}\left\{\max \left[x_{ij}^L,x_{ij}^U\right]\right\}}$$

(37)

$$x_{ij}^{U\prime }=\frac{x_{ij}^U}{{\displaystyle \underset{}{\overset{}{{\max }_{i=1}^m}}}\left\{\max \left[x_{ij}^L,x_{ij}^U\right]\right\}}$$

(38)

The normalized decision matrix X_R of the aggregated decision matrix is calculated.

$$\begin{array}{c}SODM\hfill \\ {\mathit{X}}_R={\left[\begin{array}{cccc}[0.1959,0.3010]& [0.5454,0.5454]& [0.4357,0.5353]& [0.4242,0.5212]\\ [0.3515,0.4485]& [0.5454,0.5454]& [0.4357,0.5353]& [0.4242,0.5212]\\ [0.7899,0.8949]& [0.7636,0.7636]& [0.8112,0.9191]& [0.7636,0.7636]\\ [0.4242,0.5212]& [0.5454,0.5454]& [0.5850,0.6846]& [1.0000,1.0000]\\ \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot \\ \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot \\ [0.1171,0.2222]& [0.3273,0.3273]& [0.3361,0.3361]& [0.4485,0.6424]\\ [0.3273,0.3273]& [1.0000,1.0000]& [0.5850,0.6846]& [0.6424,0.7394]\\ [0.0909,0.0909]& [1.0000,1.0000]& [0.6597,0.7593]& [0.6424,0.7394]\\ [0.0909,0.0909]& [1.0000,1.0000]& [0.4357,0.5353]& [0.4242,0.5212]\end{array}\right]}_{42\times 4}\end{array}$$

3.3. Weights of Risk Assessment Factors

3.3.1. Calculation of Subjective Weights

Based on the nine-scale method and the pairwise comparison on four risk assessment factors, the pairwise comparison matrix of three decision makers are obtained according to Equation (8).

$${\mathit{A}}_1=\left[\begin{array}{cccc}1& 2& 5& 4\\ 1/2& 1& 5& 3\\ 1/5& 1/5& 1& 1/2\\ 1/4& 1/3& 2& 1\end{array}\right], {\mathit{A}}_2=\left[\begin{array}{cccc}1& 3& 6& 4\\ 1/3& 1& 5& 4\\ 1/6& 1/5& 1& 1/3\\ 1/4& 1/4& 3& 1\end{array}\right], {\mathit{A}}_3=\left[\begin{array}{cccc}1& 2& 5& 3\\ 1/2& 1& 4& 3\\ 1/5& 1/4& 1& 1/3\\ 1/3& 1/3& 3& 1\end{array}\right]$$

The maximum eigenvalues of the pairwise comparison matrix are, respectively, obtained according to Equation (9).

$${\lambda }_{1\max }=4.0566, {\lambda }_{2\max }=4.2097, {\lambda }_{3\max }=4.1076$$

The eigenvectors corresponding to the maximum eigenvalues are obtained, respectively.

$$\begin{gathered} {\mathsf{\omega }}_1=(-0.8131,-0.5316,-0.1231,-0.2039) \\ {\mathsf{\omega }}_2=(-0.8551,-0.4728,-0.0937,-0.1910) \\ {\mathsf{\omega }}_3=(-0.7934,-0.5364,-0.1222,-0.2606) \end{gathered}$$

The consistency indicators CI of the comparison matrix are, respectively, calculated.

$$C{I}_{A1}=0.0189, C{I}_{A2}=0.0699, C{I}_{A3}=0.0359$$

Herein, RI is taken as 0.89 [37]; therefore, the consistency ratios, CR, of the comparison matrices are, respectively, obtained.

$$C{R}_{A1}=0.0212, C{R}_{A2}=0.0785, C{R}_{A3}=0.0403$$

Therefore, because the values of CR_A1, CR_A2 and CR_A3 are all less than 0.1, all pairwise comparison matrices pass the consistency test.

Then, the rough number comparison matrix is obtained.

$$\begin{array}{l}SODM\hfill \\ {\mathit{A}}^{*}=\left[\begin{array}{cccc}\left[1.0000,1.0000\right]& \left[2.1111,2.5556\right]& \left[5.1111,5.5556\right]& \left[3.4444,3.8889\right]\\ \left[0.4074,0.4815\right]& \left[1.0000,1.0000\right]& \left[4.4444,4.8889\right]& \left[3.1111,3.5556\right]\\ \left[0.1815,0.1963\right]& \left[0.2056,0.2278\right]& \left[1.0000,1.0000\right]& \left[0.3519,0.4259\right]\\ \left[0.2593,0.2963\right]& \left[0.2870,0.3241\right]& \left[2.4444,2.8889\right]& \left[1.0000,1.0000\right]\end{array}\right]\end{array}$$

The subjective weight of each risk assessment factor is obtained according to Equation (12).

$$w_s=\{[2.4691,2.7259],[1.5406,1.7009],[0.3385,0.3715],[0.6531,0.7257]\}$$

Then, the normalization form of the subjective weight is obtained according to Equation (13).

$$w_s^{\prime }=\{[0.9058,1],[0.5652,0.6240],[0.1242,0.1363],[0.2396,0.2662]\}$$

3.3.2. Calculation of Objective Weights

The evaluation information from decision makers’ defuzzification of failure modes is used as the numerical information for calculating the objective weights. The target layer is the criticality of the failure modes. Therefore, the target layer is isotropic with the risk assessment factors; that is, the greater the value of the risk assessment factor, the greater the criticality of the failure mode. Then, the normalized matrices for fuzzification ${\mathit{X}}_1^{*}$, ${\mathit{X}}_2^{*}$, ${\mathit{X}}_3^{*}$ and ${\mathit{X}}_4^{*}$ are, respectively, obtained.

$${\mathit{X}}_1^{*}=\begin{array}{l}SDM\hfill \\ {\left[\begin{array}{ccc}0.2600& 0.6757& 0.2600\\ 0.2600& 0.6757& 0.2600\\ 1& 1& 0.7400\\ 0.5000& 0.6757& 1\\ \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot \\ \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot \\ 0& 0.3514& 0.2600\\ 0.2600& 0.6757& 0.7400\\ 0& 0.6757& 0.7400\\ 0& 0.3514& 0.2600\end{array}\right]}_{42\times 3}\end{array}, {\mathit{X}}_2^{*}=\begin{array}{l}SDM\hfill \\ {\left[\begin{array}{ccc}0.2600& 0.5000& 0.5000\\ 0.5000& 0.5000& 0.5000\\ 0.7400& 1& 0.7400\\ 0.2600& 0.5000& 1\\ \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot \\ \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot \\ 0.2600& 0.2600& 0.5000\\ 0.2600& 0.5000& 0.5000\\ 0& 0.7400& 0.5000\\ 0& 0.5000& 0.5000\end{array}\right]}_{42\times 3}\end{array}, {\mathit{X}}_3^{*}=\begin{array}{l}SDM\hfill \\ {\left[\begin{array}{ccc}0& 0.3514& 0.5000\\ 0.2600& 0.3514& 0.5000\\ 0.7400& 1& 0.7400\\ 0.5000& 1& 1\\ \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot \\ \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot \\ 0& 0.3514& 0.7400\\ 0.2600& 1& 0.7400\\ 0& 1& 0.7400\\ 0& 0.6757& 0.5000\end{array}\right]}_{42\times 3}\end{array}, {\mathit{X}}_O^{*}={\left[\begin{array}{c}0.5000\\ 0.5000\\ 0.7400\\ 0.5000\\ \cdot \cdot \cdot \\ \cdot \cdot \cdot \\ 0.2600\\ 1\\ 1\\ 1\end{array}\right]}_{42\times 1}$$

The information entropy redundancy H₁, H₂, H₃ and H_O can be obtained using the Equations (16) and (17).

$$H_1=\left\{0.0783,0.0500,0.0449\right\}, H_2=\left\{0.0616,0.0300,0.0431\right\}, H_3=\left\{0.0800,0.0380,0.0539\right\}, H_O=0.0360$$

The information entropy redundancy of aggregation H is obtained using Equation (18).

$$H=\{[0.0692,0.0774],[0.0360,0.0360],[0.0349,0.0438],[0.0449,0.0497]\}$$

The normalized objective weights can be obtained using Equation (19).

$$w^{\prime }{}_o=\{[0.8941,1],[0.4651,0.4651],[0.4509,0.5659],[0.5801,0.6421]\}$$

Based on the results of subjective and objective weighting analyses of the AHP method and the EWM method, it can be seen that the use of a single AHP method cannot satisfy the research of this paper [36,37]. Therefore, integrating the normalized decision matrix ${\mathit{X}}_R$ and the comprehensive weight w, the weighted decision matrix ${\mathit{X}}_{R^{\prime }}$ can be obtained.

$$\begin{array}{l}SODM\hfill \\ {\mathit{X}}_{R^{\prime }}={\left[\begin{array}{cccc}[0.1587,0.3010]& [0.1434,0.1583]& [0.0244,0.0413]& [0.0590,0.0891]\\ [0.2847,0.4485]& [0.1434,0.1583]& [0.0244,0.0413]& [0.0590,0.0891]\\ [0.6397,0.8949]& [0.2007,0.2216]& [0.0454,0.0709]& [0.1061,0.1305]\\ [0.3436,0.5212]& [0.1434,0.1583]& [0.0328,0.0528]& [0.1390,0.1709]\\ \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot \\ \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot \\ [0.0949,0.2222]& [0.0860,0.0950]& [0.0188,0.0259]& [0.0623,0.1098]\\ [0.2650,0.3273]& [0.2629,0.2902]& [0.0328,0.0528]& [0.0893,0.1264]\\ [0.0736,0.0909]& [0.2629,0.2902]& [0.0369,0.0586]& [0.0893,0.1264]\\ [0.0736,0.0909]& [0.2629,0.2902]& [0.0244,0.0413]& [0.0590,0.0891]\end{array}\right]}_{42\times 4}\end{array}$$

3.4. Risk Ranking of Failure Modes

It is necessary to analyze whether the risk assessment factor is the benefit criterion or the cost criterion. For risk priority ranking, the PIS indicates a low risk of failure mode, and the NIS indicates a high risk of failure mode. Herein, the higher the value of the four risk assessment factors, the higher the risk. Therefore, the cost criterion is applied. The PIS and NIS of the four risk assessment factors are calculated according to Equations (21) and (22). Table 5 shows the calculation results.

Table 5. PIS and NIS of the risk assessment factors.

Ideal Solution	S	O	D	M
PIS	0.0736	0.0239	0.0052	0.0163
NIS	1	0.2902	0.0771	0.1709

The Euclidean distances ${\mathit{d}}^{+}$ and ${\mathit{d}}^{-}$ are obtained using Equations (23) and (24). Then, the dimensionless Euclidean distances ${\mathit{D}}^{+}$ and ${\mathit{D}}^{-}$ are determined using Equations (25) and (26).

$${\mathit{d}}^{+}={\left[\begin{array}{c}0.2764\\ 0.4065\\ 0.8550\\ 0.4945\\ \cdot \cdot \cdot \\ \cdot \cdot \cdot \\ 0.1905\\ 0.3869\\ 0.2936\\ 0.2790\end{array}\right]}_{42\times 1}, {\mathit{d}}^{-}={\left[\begin{array}{c}0.8629\\ 0.7406\\ 0.3782\\ 0.6748\\ \cdot \cdot \cdot \\ \cdot \cdot \cdot \\ 0.9360\\ 0.7413\\ 0.9313\\ 0.9350\end{array}\right]}_{42\times 1}, {\mathit{D}}^{+}={\left[\begin{array}{c}0.0154\\ 0.0226\\ 0.0476\\ 0.0275\\ \cdot \cdot \cdot \\ \cdot \cdot \cdot \\ 0.0106\\ 0.0215\\ 0.0164\\ 0.0155\end{array}\right]}_{42\times 1}, {\mathit{D}}^{-}={\left[\begin{array}{c}0.0287\\ 0.0246\\ 0.0126\\ 0.0224\\ \cdot \cdot \cdot \\ \cdot \cdot \cdot \\ 0.0311\\ 0.0246\\ 0.0309\\ 0.0311\end{array}\right]}_{42\times 1}$$

The PIS and NIS are used as the reference sequences in the gray correlation method. The gray correlation matrices of PIS and NIS are calculated. Thereby, the gray correlations ${\gamma }^{+}$ and ${\gamma }^{-}$ are obtained using Equations (32) and (33). Then, the dimensionless gray correlations ${\mathsf{\Psi }}^{+}$ and ${\mathsf{\Psi }}^{-}$ are determined using Equation (34).

$${\mathsf{\gamma }}^{+}={\left[\begin{array}{c}0.8129\\ 0.7833\\ 0.6878\\ 0.7382\\ \cdot \cdot \cdot \\ \cdot \cdot \cdot \\ 0.8570\\ 0.7522\\ 0.8295\\ 0.8514\end{array}\right]}_{42\times 1}, {\mathsf{\gamma }}^{-}={\left[\begin{array}{c}0.7456\\ 0.7557\\ 0.8504\\ 0.7995\\ \cdot \cdot \cdot \\ \cdot \cdot \cdot \\ 0.7227\\ 0.8187\\ 0.8066\\ 0.7888\end{array}\right]}_{42\times 1}, {\mathsf{\Psi }}^{+}={\left[\begin{array}{c}0.0244\\ 0.0235\\ 0.0207\\ 0.0222\\ \cdot \cdot \cdot \\ \cdot \cdot \cdot \\ 0.0258\\ 0.0226\\ 0.0249\\ 0.0256\end{array}\right]}_{42\times 1}, {\mathsf{\Psi }}^{-}={\left[\begin{array}{c}0.0231\\ 0.0234\\ 0.0263\\ 0.0248\\ \cdot \cdot \cdot \\ \cdot \cdot \cdot \\ 0.0224\\ 0.0254\\ 0.0250\\ 0.0244\end{array}\right]}_{42\times 1}$$

FACDG has many failure modes; if a single TOPSIS method is used here [35], there is a situation where the Euclidean distances are the same but the failure modes are different. According to the analysis, it can be seen that the larger the values of ${\mathsf{\gamma }}^{+}$ and ${\mathit{d}}^{-}$, the larger the target layer ${\mathsf{\Omega }}_i^{+}$ and the larger the values of ${\mathsf{\gamma }}^{-}$ and ${\mathit{d}}^{+}$, the larger the target layer ${\mathsf{\Omega }}_i^{-}$. So, the dimensionless Euclidean distance and dimensionless gray correlations are all positive indicators. The dimensionless weight coefficients ${\alpha }_1$, ${\beta }_1$, ${\alpha }_2$ and ${\beta }_2$ are determined. The relationship of the target layer ${\mathsf{\Omega }}_i^{+}$ to ${\mathsf{\gamma }}^{+}$ and ${\mathit{d}}^{-}$, and the target layer ${\mathsf{\Omega }}_i^{-}$ to ${\mathsf{\gamma }}^{-}$ and ${\mathit{d}}^{+}$, are shown in Figure 6.

$${\alpha }_1=0.4163, {\beta }_1=0.5837, {\alpha }_2=0.4712, {\beta }_2=0.5288$$

Figure 6. Variation in target layer vs. dimensionless Euclidean distance and dimensionless gray correlation: (a) ${\mathsf{\Omega }}_i^{+}$; (b) ${\mathsf{\Omega }}_i^{-}$.

Therefore, the nearness degree of each failure mode is calculated. Table 6 shows the results of the fuzzy rough number FMECA based on the comprehensive nearness degree and the RPN method [43]. Based on the results of the RPN method, it can be found that the extreme difference in RPN values of failure modes reaches 3594, which can lead to the deviation of risk ranking of failure modes due to the instability of assessment sensitivity. Likewise, there are seven groups of failure modes with the same RPN values, namely, 0303A and 0304A; 0101B, 0105A and 0603A; 0101A and 0606A; 0104A and 0402A; 0202A, 0203A and 0302A; 0401A and 0504B; and 0204A and 0208A. This is due to assigning the same weight to the decision makers. However, it can be found that the risk priority of failure modes can be better distinguished via the comprehensive nearness degree, and the same criticality of failure mode does not appear. The comprehensive nearness degree of 42 failure modes is maintained within the range of an order of magnitude. Additionally, the risk level of failure modes has a good agreement with the engineering maintenance analysis [43]. The shortcomings of the traditional RPN method are better solved, such as the uncertainty of subjective assessment by decision makers, uncertainty among decision makers, the same criticality of failure modes due to different combinations of risk assessment factors and the unstable sensitivity. It can be found that the most critical failure mode is the fatigue damage of the upper connecting snap ring, as shown in Table 6. The proposed FMECA assessment methodology for FACDG was verified to be reasonable and effective through the comparative analysis of the actual engineering maintenance statistics, the fuzzy rough number method and the traditional RPN method. The proposed methodology lays a good foundation for the development of the reliability maintenance strategy of FACDG.

Table 6. PIS and NIS of the risk assessment factors.

No.	Code	δ	Ranking of δ	Values of RPN [49]
1	0101A	0.42654	33	160
2	0101B	0.49024	23	240
3	0102A	0.67814	2	3136
4	0103A	0.53940	9	640
5	0104A	0.50083	22	144
6	0105A	0.52959	15	240
7	0106A	0.70048	1	3600
8	0201A	0.46459	28	72
9	0202A	0.48204	24	120
10	0202B	0.53480	11	150
11	0203A	0.51223	20	120
12	0204A	0.44018	31	48
13	0205A	0.38827	38	16
14	0206A	0.44224	30	60
15	0207A	0.42651	34	54
16	0208A	0.41521	37	48
17	0209A	0.56608	7	1890
18	0210A	0.53118	14	50
19	0301A	0.57931	5	750
20	0302A	0.47635	25	120
21	0303A	0.57855	6	720
22	0304A	0.54776	8	720
23	0305A	0.52488	18	576
24	0401A	0.52623	17	90
25	0402A	0.44927	29	144
26	0403A	0.52249	19	360
27	0404A	0.53321	12	250
28	0501A	0.62189	4	3456
29	0502A	0.64926	3	504
30	0503A	0.35910	41	4
31	0504A	0.52664	16	288
32	0504B	0.47513	26	90
33	0504C	0.36888	40	32
34	0504D	0.33977	42	75
35	0601A	0.53185	13	1134
36	0602A	0.47498	27	384
37	0603A	0.42313	35	240
38	0604A	0.53660	10	864
39	0604B	0.37598	39	20
40	0605A	0.50144	21	756
41	0605B	0.43281	32	490
42	0606A	0.42096	36	160

4. Conclusions

Integrating the fuzzy rough theory, AHP-EWM and TOPSIS-GRA, a multi-criteria decision-making FMECA assessment methodology is proposed. In this method, the fuzzy sets theory is adopted to resolve the subjective uncertainty of decision makers. The uncertainty among decision makers is resolved using the rough number theory. For the different contributions of risk assessment factors, the AHP and EWM are, respectively, applied to determine the subjective weights and objective weights. The comprehensive weight model is developed. The TOPSIS and GRA are applied to resolve the instability of risk assessment sensitivity, and a comprehensive nearness degree is established to deal with the risk priority ranking. Based on the analysis of the structure and working principle of a certain FACDG, combined with the statistical analysis of the fourth-level engineering maintenance, a statistical classification of 42 failure modes is established. The comparative analysis of the proposed method with engineering maintenance and the RPN method are carried out. The results show that the major failure modes obtained from the risk priority ranking of the proposed risk assessment method are in good agreement with the major failure modes of the engineering maintenance, which verifies that the proposed risk assessment method is valid and reasonable. Simultaneously, it lays a basis for the development of a reliable maintenance strategy for the FACDG in the EMU.

Based on the study in this paper, future work can be carried out as follows: (1) The research of other risk assessment methods in FACDG should be carried out. (2) The applicability of the proposed method for other systems of EMUs should be examined. (3) Research on risk assessment methods in fifth-level FACDG should be carried out.