An Improved Assessment Method for FMEA for a Shipboard Integrated Electric Propulsion System Using Fuzzy Logic and DEMATEL Theory

Abstract

Shipboard integrated electric propulsion systems (IEPSs) are prone to suffer from system failures and security threats because of their complex functional structures and poor operational environments. An improved assessment method for failure mode and effects analysis (FMEA), integrating fuzzy logic and decision–making trial and evaluation laboratory (DEMATEL) theory, is proposed to enhance the system’s reliability and handle the correlation effects between failure modes and causes. In this method, information entropy and qualitative analysis are synthesized to determine the credibility weights of domain experts. Each risk factor and its relative importance are evaluated by linguistic terms and fuzzy ratings. The benchmark adjustment search algorithm is designed to obtain the alpha-level sets of fuzzy risk priority numbers (RPNs) for defuzzification. The defuzzified RPNs are regarded as the inputs for the DEMATEL technique to investigate the causal degrees of failure modes and causes. Accordingly, the risk levels of the failure modes are prioritized with respect to the causal degrees. The practical application to the typical failure modes of the propulsion subsystem is provided. The assessment results show that this system contributes to risk priority decision-making and disastrous accident prevention.

1. Introduction

Shipboard integrated electric propulsion systems (IEPSs) can supply power for various loads, such as propulsion system, regional loads, and high-energy weapons. By integrating propulsion power with the consumption of other electricity, this compact structure is characterized by its reduced volumes of power units and enhanced operation reliability, together with unified energy utilization and management [1,2]. The application requirements of high–power propulsion and pulse weapon are fulfilled by the next–generation medium-voltage DC IEPS for all–electric ships because of its regional power distribution features [3]. In addition, it has been recognized as one of the most promising power plants because of its remarkable advantages like high power density, small vibration and noise, strong ship vitality and combat effectiveness, and reduced life cycle maintenance costs. The shipboard IEPS is a large–scale system with high integration, a complicated structure, diversified functions, and intelligent control. Offshore working environments are harsh, usually featuring drastic temperature changes, strong mechanical impacts, and high humidity, accompanied by salt spray and oil mist, as well as other corrosive substances and explosive gases. These complex structures and hostile environments may lead to the increase in system failure rate and serious safety threats for shipboard medium voltage DC IEPSs [4,5,6], which poses a huge challenge to risk control and reliability assessment.

Failure mode and effects analysis (FMEA) is a widely used risk assessment and optimization design tool used to improve the safety and reliability of systems involved in many industries [7,8]. The so–called risk priority number (RPN) technique is employed to assess the risk levels of known or potential failure modes in traditional FMEA, defined as the product of the probability of severity (S), occurrence (O), and detectability (D) [9,10]. According to the priority ranking of failure modes, preventive and compensatory actions are introduced for the key equipment. When applied to risk assessments for safety-critical systems, like shipboard IEPSs, the traditional RPN has suffered from several weaknesses [11,12]. For example, it neglects the relative importance among risk factors S, O, and D. These three factors are assumed to have the same importance, so this may not conform to realistic situations in the FMEA process for safety-critical systems. Another prominent disadvantage is that various combinations of risk factors may produce an identical value for the RPN. However, the risk implication may be completely different. This difference will result in wasting resources and time or, in some cases, a high–risk event going unnoticed. In addition, the relative importance of different experts is not taken into consideration, thus making the risk assessment results less objective and accurate. Moreover, the traditional RPN is obtained by the product of three risk factors S, O, and D, where the assessment results are too sensitive to risk factor changes. Further, the correlation effects between the failure modes and failure causes are not taken into account, resulting in incomplete outcomes. Furthermore, each risk factor indicator is evaluated by the exact numerical values, ranging from 1 to 10, which heavily depend on the knowledge experience of domain experts and system fault databases. Nevertheless, this expertise cannot meet the requirements of newly developed systems, such as shipboard medium-voltage DC IEPSs.

Over the years, several advanced theories and techniques such as fuzzy logic theory [13,14,15,16], the decision–making trial and evaluation laboratory (DEMATEL) technique [17,18,19], the grey correlation method [20,21], the data envelopment analysis [22,23], technique for order preference by similarity to the ideal solution [24,25], quality function deployment [26,27], etc., have been developed and applied as enhancements to traditional FMEA in order to overcome these aforementioned shortcomings. However, there are still some problems in real applications, such as the credibility of domain experts, the uncertainty of fuzzy information, and the difficulty in considering the correlations between failure modes and causes. It should be noted that due to the complexity of shipboard IEPSs and the advancement of zonal medium-voltage DC distribution, there are few studies on risk and hazard assessments with respect to the systems involved [28,29]. Therefore, it is of great theoretical significance and practical value to undertake risk assessments for the typical failure modes in shipboard medium-voltage DC IEPSs.

The motivation of this study is to present an improved assessment method for FMEA by combining fuzzy logic and DEMATEL theory. Although the related approaches have already been described in some scientific studies in different fields, studies on shipboard medium-voltage DC IEPSs in the marine industry are very scarce. In the proposed method, fuzzy logic is employed to evaluate the risk factors and their relative importance, while DEMATEL theory is utilized to examine the correlation effects between the failure modes and causes. Furthermore, the credibility weights of domain experts are calculated according to information entropy and qualitative analysis. Additionally, the benchmark adjustment search algorithm is designed to obtain the alpha-level sets of fuzzy RPNs for the corresponding defuzzification. An example of propulsion subsystem failure is provided, and reasonable risk prioritization is produced. The assessment results can be used for risk priority decision-making, maintenance resource optimization, and disastrous accident prevention.

The remainder of this paper is organized as follows. The hierarchical structure of shipboard IEPSs is illustrated in Section 2. In Section 3, the proposed assessment method integrating fuzzy logic and DEMATEL theory is described in detail. The practical application of the propulsion subsystem is provided to test and validate the feasibility of the presented method in Section 4. The final section gives the related conclusions and contributions of this study.

2. System Hierarchy Description

The first step in the FMEA process is the description of the considered system and a definition of its hierarchy. To acquire an exhaustive system description, it is necessary to gather information about the functional structure of the system’s components with their own positions and connections. Owing to the complexity of shipboard IEPSs, the concept of system function and hardware hybrid analysis is introduced to establish the hierarchy model. In other words, failure modes and effects are preliminarily analyzed from the perspective of system function, and then the hardware failure analysis is focused on devices with significant functional failures. In the topology prototype given by the U.S. Navy [30,31], depicted in Figure 1 where PCM is the abbreviation of power conversion modules and VSD is short for variable speed drive, the shipboard medium-voltage DC IEPS can be segmented into a generation subsystem, a propulsion subsystem, a transmission and distribution subsystem, an energy storage subsystem, and other load subsystems.

Figure 1. Topology structure of a shipboard medium-voltage DC integrated electric propulsion system (IEPS).

The degree of details in implementing an FMEA is determined by the hierarchical structure described. In order to facilitate the failure mode analysis and system boundary definition according to CEI IEC 60812 standard and MIL–STD–1629A [32,33], it is suggested that the initial defined level and minimum defined level are divided [34,35], as shown in Figure 2. The initial defined level is the shipboard medium-voltage DC IEPS studied, and the minimum defined level is a key piece of equipment, such as a gas turbine, diesel engine, or synchronous motor. Smaller component analyses and lower-level structural decomposition are not considered here. In addition, this hierarchy will contribute to the tracking and positioning of failure modes, as well as investigating the interactions between failure modes and causes.

Figure 2. Hierarchical structure diagram of a shipboard IEPS. PWM represents pulse width modulation.

3. The Proposed Method Integrating Fuzzy Logic and DEMATEL Theory

In practical applications of the FMEA process, there are common cause failure phenomena and complicated failure mechanisms induced by multiple causes. However, traditional FMEA and RPN techniques cannot allow for mutual influences between the failure modes and causes. Consequently, an improved assessment method for the FMEA procedure using fuzzy logic and DEMATEL theory is proposed to solve the inherent defects stated previously. It is noteworthy that the correlation effects between the failure modes and causes are considered herein.

3.1. Fuzzy Linguistic Term Sets

In the real world, the risk ratings and the relative importance of three factors (S, O, and D) are difficult to be measured accurately. To this end, fuzzy logic is introduced to improve the assessment results of a traditional RPN. Fuzzy linguistic term sets and fuzzy number theory are exploited to evaluate risk factors and their relative weights. By referring to the fuzzy evaluation criteria of risk factors in the marine industry and combining the system characteristics of shipboard IEPSs, fuzzy term sets and linguistic descriptions can be determined. Specifically, the risk factor linguistic terms and relative weights are treated as trapezoidal fuzzy numbers and triangular fuzzy numbers, respectively. After investigating the literature and consulting experts from the research field of shipboard medium-voltage DC IEPSs, trapezoidal fuzzy numbers and triangular fuzzy numbers were defined. The fuzzy evaluation criteria and corresponding fuzzy numbers for each risk factor and its relative weight are depicted in Table 1, Table 2, Table 3 and Table 4.

Table 1. Fuzzy evaluation criteria of severity.

Linguistic Term	Description	Rating	Fuzzy Number
None (N)	No effect	1	(0.5, 0.5, 1, 1.5)
Very minor (VM)	System operable with negligible interference	2	(0.5, 1.5, 2,3)
Minor (M)	System operable with slight degradation of performance	3	(1.5, 2.5, 3,4)
Very low (VL)	System operable with significant degradation of performance	4	(2.5, 3.5, 4,5)
Low (L)	System inoperable but safe	5	(3.5, 4.5, 5,6)
Moderate (M)	System inoperable with minor damage	6	(4.5, 5.5, 6,7)
High (H)	System inoperable with obvious damage	7	(5.5, 6.5, 7,8)
Very high (VH)	System inoperable with severe damage	8	(6.5, 7.5, 8,9)
Hazardous with warning (HWW)	Disaster resulting in casualties and equipment destruction with warning	9	(7.5, 8.5, 9,10)
Hazardous without warning (HWOW)	Disaster resulting in casualties and equipment destruction without warning	10	(8.5, 9.5, 10, 10)

Table 2. Fuzzy evaluation criteria of occurrence.

Linguistic Term	Description	Rating	Fuzzy Number
Very remote (VR)	Unlikely failures	1	(0.5, 0.5, 1, 1.5)
Remote (R)	Rare failures	2	(0.5, 1.5, 2, 3)
Very low (VL)	Very few failures	3	(1.5, 2.5, 3, 4)
Low (L)	Relatively few failures	4	(2.5, 3.5, 4, 5)
Moderately low (ML)	Occasional failures	5	(3.5, 4.5, 5, 6)
Moderate (M)	Failures happen sometimes	6	(4.5, 5.5, 6, 7)
Moderately high (MH)	Repeated failures	7	(5.5, 6.5, 7,8)
High (H)	Frequent failures	8	(6.5, 7.5, 8, 9)
Very high (VH)	Failures happen almost always	9	(7.5, 8.5, 9, 10)
Extremely high (EH)	Inevitable failures	10	(8.5, 9.5, 10, 10)

Table 3. Fuzzy evaluation criteria of detectability.

Linguistic Term	Description	Rating	Fuzzy Number
Almost certain (AC)	Almost certainty	1	(0.5, 0.5, 1, 1.5)
Very high (VH)	Very high chance	2	(0.5, 1.5, 2, 3)
High (H)	High chance	3	(1.5, 2.5, 3, 4)
Moderately high (MH)	Moderately high chance	4	(2.5, 3.5, 4, 5)
Moderate (M)	Moderate chance	5	(3.5, 4.5, 5, 6)
Moderately low (ML)	Moderately low chance	6	(4.5, 5.5, 6, 7)
Very low (VL)	Very low chance	7	(5.5, 6.5, 7,8)
Remote (R)	Remote chance	8	(6.5, 7.5, 8, 9)
Very remote (VR)	Very remote chance	9	(7.5, 8.5, 9, 10)
Absolutely uncertain (AU)	No chance	10	(8.5, 9.5, 10, 10)

Table 4. Fuzzy evaluation criteria of risk factor weights.

Linguistic Term	Description	Fuzzy Number
Very low (VL)	Very low importance	(0, 0.1, 0.3)
Moderately low (ML)	Moderately low importance	(0.1, 0.3, 0.5)
Moderate (M)	Moderate importance	(0.3, 0.5, 0.7)
Moderately high (MH)	Moderately high importance	(0.5, 0.7, 0.9)
Very high (VH)	Very high importance	(0.7, 0.9, 1)

3.2. Fuzzy Evaluations of Risk Factors and Relative Weights

Assume that there are $M$ experts to be consulted in the FMEA assessment team, and let $h_j$ denote the credibility weight of specialist opinions ($j=1,2,\dots ,M$). Expert weights are usually qualitatively analyzed, thus making credibility results less objective and accurate. Thus, information entropy theory and qualitative analyses are synthesized to determine the credibility weights of the related experts. According to the scores of the expert members on work experience, technical field, and professional level, the entropy and its weight for each expert are calculated as

$$\{\begin{array}{l}{r}_j={\displaystyle \sum _{k=1}^3{r}_{jk}}\\ {\pi }_{jk}=r_{jk}/r_j\\ H(r_j)=-{\displaystyle \sum _{k=1}^3{\pi }_{jk}\ln {\pi }_{jk}}/\ln 3\end{array}$$

(1)

where $r_{jk}$ is the score of expert $j$ associated with the $k\mathrm{th}$ performance index, and $r_j$ and $H(r_j)$ are the total score and entropy value for expert $j$, respectively.

Then, the total entropy value and entropy weights of expert credibility can be determined respectively:

$$\{\begin{array}{l}{E}_e={\displaystyle \sum _{j=1}^{M}H(r_j)}\\ {\theta }_j=H(r_j)/E_e\end{array}.$$

(2)

Suppose ${\chi }_j$ is the credibility weight coefficient, given by the qualitative analysis considering the assessment environment, membership qualifications, and system familiarity. Accordingly, the comprehensive credibility weight of expert $j$, defined as the normalized form of the sum of the entropy weights and qualitative weights, is represented as

$$h_j=({\theta }_j+{\chi }_j)/({\displaystyle \sum _{j=1}^M{\theta }_j+{\chi }_j}).$$

(3)

The trapezoidal fuzzy number evaluations of the three risk factors by expert $j$, with respect to the $i\mathrm{th}$ failure mode, are defined as $S_{ij}=(S_{ij}^L,S_{ij}^{M_1},S_{ij}^{M_2},S_{ij}^U)$, $O_{ij}=(O_{ij}^L,O_{ij}^{M_1},$ $O_{ij}^{M_2},O_{ij}^U)$, and $D_{ij}=(D_{ij}^L,D_{ij}^{M_1},D_{ij}^{M_2},D_{ij}^U)$, respectively. Consequently, the overall fuzzy number evaluations of the risk factors related to the $i\mathrm{th}$ failure mode can be expressed as

$$\left\{\begin{array}{c}{S}_i=\sum _{j=1}^M{h}_j{S}_{ij}=(\sum _{j=1}^M{h}_j{S}_{ij}^L,\sum _{j=1}^M{h}_j{S}_{ij}^{M_1},\sum _{j=1}^M{h}_j{S}_{ij}^{M_2},\sum _{j=1}^M{h}_j{S}_{ij}^U)\hfill \\ O_i=\sum _{j=1}^M{h}_j{O}_{ij}=(\sum _{j=1}^M{h}_j{O}_{ij}^L,\sum _{j=1}^M{h}_j{O}_{ij}^{M_1},\sum _{j=1}^M{h}_j{O}_{ij}^{M_2},\sum _{j=1}^M{h}_j{O}_{ij}^U)\hfill \\ D_i=\sum _{j=1}^M{h}_j{D}_{ij}=(\sum _{j=1}^M{h}_j{D}_{ij}^L,\sum _{j=1}^M{h}_j{D}_{ij}^{M_1},\sum _{j=1}^M{h}_j{D}_{ij}^{M_2},\sum _{j=1}^M{h}_j{D}_{ij}^U)\hfill \end{array}\right..$$

(4)

Likewise, let $w_{Sj}=(w_{Sj}^L,w_{Sj}^M,w_{Sj}^U)$, $w_{Oj}=(w_{Oj}^L,w_{Oj}^M,w_{Oj}^U)$, and $w_{Dj}=(w_{Dj}^L,w_{Dj}^M,w_{Dj}^U)$ be the triangular fuzzy number evaluations of the relative importance weights for the three risk factors by expert $j$, respectively. Afterwards, the overall fuzzy number evaluations of the risk factor weights can be represented as

$$\{\begin{array}{l}{w}_S={\displaystyle \sum _{j=1}^M{h}_j{w}_{Sj}}=({\displaystyle \sum _{j=1}^M{h}_j{w}_{Sj}^L},{\displaystyle \sum _{j=1}^M{h}_j{w}_{Sj}^M},{\displaystyle \sum _{j=1}^M{h}_j{w}_{Sj}^U})\\ w_O={\displaystyle \sum _{j=1}^M{h}_j{w}_{Oj}}=({\displaystyle \sum _{j=1}^M{h}_j{w}_{Oj}^L},{\displaystyle \sum _{j=1}^M{h}_j{w}_{Oj}^M},{\displaystyle \sum _{j=1}^M{h}_j{w}_{Oj}^U})\\ w_D={\displaystyle \sum _{j=1}^M{h}_j{w}_{Dj}}=({\displaystyle \sum _{j=1}^M{h}_j{w}_{Dj}^L},{\displaystyle \sum _{j=1}^M{h}_j{w}_{Dj}^M},{\displaystyle \sum _{j=1}^M{h}_j{w}_{Dj}^U})\end{array}.$$

(5)

3.3. Alpha-Level Set Calculation with Benchmark Adjustment Search Algorithm

The fuzzy RPN is defined as the fuzzy weighted geometric mean of the three risk factors for the $i\mathrm{th}$ failure mode [36]:

$$\begin{array}{ll}\mathrm{FRPN}& ={\mathrm{S}}_i^{\frac{w_S}{w_S+w_O+w_D}}\cdot O_i^{\frac{w_O}{w_S+w_O+w_D}}\cdot D_i^{\frac{w_D}{w_S+w_O+w_D}}\\ & =R_1^{\frac{w_1}{w_1+w_2+w_3}}\cdot R_2^{\frac{w_2}{w_1+w_2+w_3}}\cdot R_3^{\frac{w_3}{w_1+w_2+w_3}}\end{array}$$

(6)

Redefine $R=\left[\begin{array}{ccc}{R}_1& R_2& R_3\end{array}\right]=\left[\begin{array}{ccc}{S}_i& O_i& D_i\end{array}\right]$ and $w=$ $\left[\begin{array}{ccc}{w}_1& w_2& w_3\end{array}\right]=\left[\begin{array}{ccc}{w}_S& w_O& w_D\end{array}\right]$, and it holds that

$$\begin{array}{ll}\mathrm{FRPN}& =\exp [\ln (R_1^{\frac{w_1}{w_1+w_2+w_3}}\cdot R_2^{\frac{w_2}{w_1+w_2+w_3}}\cdot R_3^{\frac{w_3}{w_1+w_2+w_3}})]\\ & =\exp [\frac{w_1}{w_1+w_2+w_3}\ln R_1+\frac{w_2}{w_1+w_2+w_3}\ln R_2+\frac{w_3}{w_1+w_2+w_3}\ln R_3]\\ & =\exp [{\displaystyle \sum _{\lambda =1}^3(w_{\lambda }\ln R_{\lambda })}/{\displaystyle \sum _{\lambda =1}^3{w}_{\lambda }}]\end{array}$$

(7)

Since $R_{\lambda }$ and $w_{\lambda }$ are fuzzy numbers ($\lambda =1,2,3$), they cannot be computed directly in Equation (7). Alpha-level set theory is introduced to calculate the fuzzy RPNs, which can be expressed as

$$\mathrm{FRPN}={\displaystyle \sum _{\alpha }\alpha }[{(\mathrm{FRPN})}_{\alpha }^L,{(\mathrm{FRPN})}_{\alpha }^U]$$

(8)

$$\{\begin{array}{l}{(\mathrm{FRPN})}_{\alpha }^L=\exp [{\displaystyle \sum _{\lambda =1}^3({(w_{\lambda })}_{\alpha }^L\ln {(R_{\lambda })}_{\alpha }^L)}/{\displaystyle \sum _{\lambda =1}^3{(w_{\lambda })}_{\alpha }^L}]\\ {(\mathrm{FRPN})}_{\alpha }^U=\exp [{\displaystyle \sum _{\lambda =1}^3({(w_{\lambda })}_{\alpha }^U\ln {(R_{\lambda })}_{\alpha }^U)}/{\displaystyle \sum _{\lambda =1}^3{(w_{\lambda })}_{\alpha }^U}]\end{array}$$

(9)

where $\alpha$ is the confidence level, and $[{(FRPN)}_{\alpha }^L,{(FRPN)}_{\alpha }^U]$ is the alpha-level set of the fuzzy RPNs.

The upper and lower limits of the alpha-level sets for the three fuzzy risk factors and their relative weights are written as

$$\{\begin{array}{l}{(R_{\lambda })}_{\alpha }^L=R_{\lambda }^L+\alpha (R_{\lambda }^{M_1}-R_{\lambda }^L)\\ {(R_{\lambda })}_{\alpha }^U=R_{\lambda }^U-\alpha (R_{\lambda }^U-R_{\lambda }^{M_2})\\ {(w_{\lambda })}_{\alpha }^L=w_{\lambda }^L+\alpha (w_{\lambda }^M-w_{\lambda }^L)\\ {(w_{\lambda })}_{\alpha }^U=w_{\lambda }^U-\alpha (w_{\lambda }^U-w_{\lambda }^M)\end{array}$$

(10)

where $\left[{(R_{\lambda })}_{\alpha }^L,{(R_{\lambda })}_{\alpha }^U\right]$ and $\left[{(w_{\lambda })}_{\alpha }^L,{(w_{\lambda })}_{\alpha }^U\right]$ are the alpha-level sets of the fuzzy risk factors and their relative weights, respectively.

A benchmark adjustment search algorithm is adopted to determine the alpha-level sets of the fuzzy RPNs. The fundamental design steps are briefly described as follows.

Step 1: The confidence level $\alpha$ is evenly and discretely distributed within the range of $[0,1]$, that is, the confidence level set is defined as $A=\left\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_N\right\}$, with ${\alpha }_0=0$, ${\alpha }_N=1$, ${\alpha }_0<{\alpha }_1<\cdot \cdot \cdot {\alpha }_N$ and $\Delta \alpha ={\alpha }_{l+1}-{\alpha }_l=1/N$ ($l=0,1,\dots ,$ $N-1$).

Step 2: Let $\alpha ={\alpha }_0$ and search for the alpha-level sets of the three fuzzy risk factors and their relative weights respectively, i.e., $\left[{(R_{\lambda })}_{\alpha }^L,{(R_{\lambda })}_{\alpha }^U\right]$ and $\left[{(w_{\lambda })}_{\alpha }^L,{(w_{\lambda })}_{\alpha }^U\right]$.

Step 3: The initial benchmark values ${\eta }_0$ and ${\rho }_0$ can be obtained by Equation (11). Define $I=\left\{1,2,3\right\}$, $I_0=\left\{\lambda \in I|\ln {(R_{\lambda })}_{\alpha }^L<{\eta }_0\right\}$, and $J_0=\left\{\lambda \in I|\ln {(R_{\lambda })}_{\alpha }^U>{\rho }_0\right\}$, which means that each element in set $I_0$ is the subscript value of $\ln {(R_{\lambda })}_{\alpha }^L$, satisfying $\ln {(R_{\lambda })}_{\alpha }^L<{\eta }_0$, and each element in set $J_0$ is the subscript value of $\ln {(R_{\lambda })}_{\alpha }^U$, satisfying $\ln {(R_{\lambda })}_{\alpha }^U>{\rho }_0$:

$$\{\begin{array}{l}{\eta }_0={\beta }_{L0}/{\gamma }_{L0}={\displaystyle \sum _{\lambda =1}^3{(w_{\lambda })}_{\alpha }^L\ln {(R_{\lambda })}_{\alpha }^L}/{\displaystyle \sum _{\lambda =1}^3{(w_{\lambda })}_{\alpha }^L}\\ {\rho }_0={\beta }_{U0}/{\gamma }_{U0}={\displaystyle \sum _{\lambda =1}^3{(w_{\lambda })}_{\alpha }^U\ln {(R_{\lambda })}_{\alpha }^U}/{\displaystyle \sum _{\lambda =1}^3{(w_{\lambda })}_{\alpha }^U}\end{array}.$$

(11)

Step 4: The benchmark value ${\eta }_p$ can be adjusted to compute the lower limits of the alpha-level sets for the three fuzzy risk factors and their relative weights according to Equation (12), where $I_p=\left\{\lambda \in I_{p-1}|\ln {(R_{\lambda })}_{\alpha }^L<{\eta }_p\right\}$, $\Delta I_p=I_{p-1}\backslash I_p$. If $\Delta I_p=\varnothing$, the optimal testing is satisfied, and ${(FRPN)}_{\alpha }^L=\exp ({\eta }_p)$. Otherwise, the benchmark value needs to be recalculated in this step:

$$\{\begin{array}{l}{\eta }_{p=1}={\beta }_{L1}/{\gamma }_{L1}=\frac{{\beta }_{L0}+{\displaystyle \sum _{\lambda \in I_0}[{(w_{\lambda })}_{\alpha }^U-{(w_{\lambda })}_{\alpha }^L]\ln {(R_{\lambda })}_{\alpha }^L}}{{\gamma }_{L0}+{\displaystyle \sum _{\lambda \in I_0}[{(w_{\lambda })}_{\alpha }^U-{(w_{\lambda })}_{\alpha }^L]}}\\ {\eta }_{p\ge 2}={\beta }_{Lp}/{\gamma }_{Lp}=\frac{{\beta }_{L(p-1)}-{\displaystyle \sum _{\lambda \in \Delta I_{p-1}}[{(w_{\lambda })}_{\alpha }^U-{(w_{\lambda })}_{\alpha }^L]\ln {(R_{\lambda })}_{\alpha }^L}}{({\gamma }_{L(p-1)}+{\displaystyle \sum _{\lambda \in \Delta I_{p-1}}[{(w_{\lambda })}_{\alpha }^U-{(w_{\lambda })}_{\alpha }^L]})}\end{array}.$$

(12)

Step 5: The benchmark value ${\rho }_q$ can be regulated to calculate the upper limits of the alpha-level sets of the three fuzzy risk factors and their relative weights by Equation (13), with $J_q=\left\{\lambda \in J_{q-1}|\ln {(R_{\lambda })}_{\alpha }^U>{\rho }_q\right\}$, $\Delta J_q=J_{q-1}\backslash J_q$. If $\Delta J_q=\varnothing$, the optimal testing is satisfied, and ${(FRPN)}_{\alpha }^U=\exp ({\rho }_q)$. Otherwise, the benchmark value needs to be re–determined:

$$\{\begin{array}{l}{\rho }_{q=1}=\frac{{\beta }_{U1}}{{\gamma }_{U1}}=\frac{{\beta }_{U0}+{\displaystyle \sum _{\lambda \in J_0}[{(w_{\lambda })}_{\alpha }^U-{(w_{\lambda })}_{\alpha }^L]\ln {(R_{\lambda })}_{\alpha }^U}}{{\gamma }_{U0}+{\displaystyle \sum _{\lambda \in J_0}[{(w_{\lambda })}_{\alpha }^U-{(w_{\lambda })}_{\alpha }^L]}}\\ {\rho }_{q\ge 2}=\frac{{\beta }_{Up}}{{\gamma }_{Up}}=\frac{{\beta }_{U(p-1)}-{\displaystyle \sum _{\lambda \in \Delta J_{q-1}}[{(w_{\lambda })}_{\alpha }^U-{(w_{\lambda })}_{\alpha }^L]\ln {(R_{\lambda })}_{\alpha }^U}}{{\gamma }_{U(p-1)}+{\displaystyle \sum _{\lambda \in \Delta J_{q-1}}[{(w_{\lambda })}_{\alpha }^U-{(w_{\lambda })}_{\alpha }^L]}}\end{array}$$

(13)

Step 6: The alpha-level set $[{(\mathrm{FRPN})}_{\alpha }^L,{(\mathrm{FRPN})}_{\alpha }^U]$ of the fuzzy RPNs can be obtained at confidence level $\alpha$ by Equation (9). Let $\alpha ={\alpha }_1,\dots ,{\alpha }_N$ repetitively, and the alpha-level set $\left[{(\mathrm{FRPN})}_{{\alpha }_l}^L,{(\mathrm{FRPN})}_{{\alpha }_l}^U\right]$ ($l=1,\dots ,N$) of the fuzzy RPNs can be ascertained at each confidence level with the repeated steps 2–5. Ultimately, the fuzzy RPNs can be determined according to Equation (8).

3.4. Defuzzification of Fuzzy RPNs

The centroid defuzzification method is applied to defuzzify the fuzzy RPNs, which can be expressed as

$$x_0(\mathrm{FRPN})={\displaystyle \int x.{\mu }_{\mathrm{FRPN}}(x)dx}/{\displaystyle \int {\mu }_{\mathrm{FRPN}}(x)dx}$$

(14)

where $x_0(FRPN)$ and ${\mu }_{\mathrm{FRPN}}(x)$ are the defuzzified values and fuzzy membership functions of the fuzzy RPNs, respectively.

In light of the situation that only the alpha-level sets are available at different confidence levels, and the precise membership functions are not known. In this study, we assume that the precise membership functions can be approximated using piecewise linear functions based on alpha-level set theory [37]. The formulas of the centroid defuzzification method are as follows:

$$\left\{\begin{array}{c}\int {\mu }_{\mathrm{FRPN}}\left(x\right)dx=\frac{1}{2N}[({\left(\mathrm{FRPN}\right)}_{{\alpha }_N}^U-{\left(\mathrm{FRPN}\right)}_{{\alpha }_N}^L)+({\left(\mathrm{FRPN}\right)}_{{\alpha }_0}^U-{\left(\mathrm{FRPN}\right)}_{{\alpha }_0}^L)+2\sum _{l=1}^{N-1}({\left(\mathrm{FRPN}\right)}_{{\alpha }_l}^U-{\left(\mathrm{FRPN}\right)}_{{\alpha }_l}^L)]\hfill \\ \int x.{\mu }_{\mathrm{FRPN}}\left(x\right)dx=\frac{1}{6N}[({\left(\mathrm{FRPN}\right)}_{{\alpha }_N}^{2U}-{\left(\mathrm{FRPN}\right)}_{{\alpha }_N}^{2L})+({\left(\mathrm{FRPN}\right)}_{{\alpha }_0}^{2U}-{\left(\mathrm{FRPN}\right)}_{{\alpha }_0}^{2L})+2\sum _{l=1}^{N-1}({\left(\mathrm{FRPN}\right)}_{{\alpha }_l}^{2U}-{\left(\mathrm{FRPN}\right)}_{{\alpha }_l}^{2L})]\hfill \\ \hfill +\frac{1}{6N}\sum _{l=0}^{N-1}[{\left(\mathrm{FRPN}\right)}_{{\alpha }_l}^U{\left(\mathrm{FRPN}\right)}_{{\alpha }_{l+1}}^U-{(FRPN)}_{{\alpha }_l}^L{\left(\mathrm{FRPN}\right)}_{{\alpha }_{l+1}}^L]\hfill \end{array}\right..$$

(15)

Obviously, ${\displaystyle \int {\mu }_{\mathrm{FRPN}}(x)dx}$ and ${\displaystyle \int x.{\mu }_{\mathrm{FRPN}}(x)dx}$ are only determined by the alpha-level set $\left[{(\mathrm{FRPN})}_{{\alpha }_l}^L,{(\mathrm{FRPN})}_{{\alpha }_l}^U\right]$, which was acquired in Section 3.3. Therefore, the defuzzified values of the fuzzy RPNs can be determined with Equations (14) and (15).

3.5. DEMATEL Technique

In a complicated system like a shipboard IEPS, there are often ambiguous correspondences between the failure modes and their causes. In other words, each failure mode may be induced by multiple causes, and each failure cause may induce multiple failure modes. In light of the common cause failure problems and complex failure mechanisms, the DEMATEL technique is applied to a shipboard medium-voltage DC IEPS to handle the correlation effects between the failure modes and causes. The implementation details of the DEMATEL technique are described as follows.

Step 1: The initial direct–relation matrix $D\in R^{(m+n)\times (m+n)}$ is constructed between the failure modes and causes, where $D_{11}\in R^{m\times m}$, $D_{12}\in R^{m\times n}$, $D_{21}\in R^{n\times m}$, and $D_{22}\in R^{n\times n}$ are block matrixes. $D_{11}\in R^{m\times m}$ indicates the direct impacts of failure causes on the failure causes, and $D_{12}\in R^{m\times n}$ represents the direct impacts of the failure causes on the failure modes. $D_{21}\in R^{n\times m}$ denotes the direct impacts of failure modes on the failure causes, and $D_{22}\in R^{n\times n}$ means the direct impacts of the failure modes on the failure modes. It should be noted that only the correlation effects between the failure modes and causes are considered. Consequently, $D_{11}$, $D_{21}$, and $D_{22}$ are all zero matrices, while $D_{12}$ is a nonzero matrix, where the elements are derived from the defuzzified values of the fuzzy RPNs. For example, the defuzzified value of the fuzzy RPN for the second failure mode induced by the third cause is the element in the third row and second column of matrix $D_{12}$:

$$D={\left[d_{ij}\right]}_{(m+n)\times (m+n)}=\left[\begin{array}{cc}{D}_{11}& D_{12}\\ D_{21}& D_{22}\end{array}\right]$$

(16)

Step 2: The initial direct–relation matrix is normalized, and the relative direct–relation matrix $\overline{D}\in R^{(m+n)\times (m+n)}$ is obtained as follows:

$$\{\begin{array}{l}{\overline{d}}_{ij}=d_{ij}/\underset{1\le i\le m+n}{\max }({\displaystyle \sum _{j=1}^{m+n}{d}_{ij}})\\ \overline{D}={\left[{\overline{d}}_{ij}\right]}_{(m+n)\times (m+n)}\end{array}$$

(17)

Step 3: The total–relation matrix $C\in R^{(m+n)\times (m+n)}$ is calculated from the following formula, where $E$ is an identity matrix, and element $c_{ij}$ shows the total impact of failure mode or cause $i$ on failure mode or cause $j$. Specifically, if $i\le m$ and $j\le m$, $c_{ij}$ is the total impact of failure cause $i$ on cause $j$; if $i\le m$ and $j>m$, $c_{ij}$ is the total impact of failure cause $i$ on mode $j$; if $i>m$ and $j\le m$, $c_{ij}$ is the total impact of failure mode $i$ on cause $j$; otherwise, $c_{ij}$ is the total impact of failure mode $i$ on mode $j$:

$$C={\displaystyle \sum _{\kappa =1}^{\infty }{\overline{D}}^{\kappa }}=\overline{D}{(E-\overline{D})}^{-1}={\left[c_{ij}\right]}_{(m+n)\times (m+n)}$$

(18)

Step 4: In order to measure the degree of interactions between a certain failure mode or cause and all other failure modes and causes, the correlation effect matrices $C_r\in R^{m+n}$ and $C_u\in R^{m+n}$ are introduced as follows. In the formula, $C_r(i)$ is the influence degree given by the failure mode or cause $i$ to all other failure modes and causes. Meanwhile, $C_u(j)$ is the influence degree received by failure mode or cause $j$ from all other failure modes and causes:

$$\{\begin{array}{l}{C}_r(i)={\displaystyle \sum _{j=1}^{m+n}{c}_{ij}}\\ C_u(j)={\displaystyle \sum _{i=1}^{m+n}{c}_{ij}}\end{array}$$

(19)

Step 5: The causal degree of each failure mode and cause is given by

$$Rea=C_r-C_u=\left[\begin{array}{ccc}Rea(1)& \begin{array}{ccc}\dots & Rea(i)& \dots \end{array}& Rea(m+n)\end{array}\right]$$

(20)

where $Rea(i)$ is the causal degree of failure mode or cause $i$. Specifically, $Rea(i)>0$ indicates a net failure cause. Meanwhile, $Rea(i)<0$ represents a net failure mode clustered into the FMEA process.

Finally, the risk level of each known or potential failure mode is prioritized according to its causal degree. In summary, the flowchart of the proposed improved risk assessment method using fuzzy logic and the DEMATEL technique is presented, as shown in Figure 3.

Figure 3. Flowchart of the proposed improved assessment method. RPN, risk priority number.

4. Case Application and Results Analysis

To ensure that the integrated all-electric ship does not lose its propulsion power arbitrarily, the propulsion subsystem is not allowed to install circuit breaker protection devices or implement conventional overload relay protection schemes. Therefore, the known or potential failure modes of the propulsion subsystem will seriously affect the propeller propulsion’s performance and the ship’s maneuvering characteristics. Taking the typical failure modes of the propulsion subsystem for the described shipboard medium-voltage DC IEPS as a case study, the effectiveness of the risk assessment method based on fuzzy logic and DEMATEL theory is verified.

4.1. Failure Mode Analysis of the Propulsion Subsystem

The propulsion subsystem mainly includes a PWM (pulse width modulation) inverter, a permanent magnet synchronous motor, speed controller, propeller, shafting, and other key equipment. Through fault simulations, expert experience knowledge bases, and equipment instruction, typical failure modes of the propulsion subsystem, failure causes, detection methods, and their local and final effects are collected and summarized in Table 5.

Table 5. Failure mode and effects analysis (FMEA) of the propulsion subsystem for a shipboard IEPS.

Equipment	Failure Mode (FM)	Failure Cause (FC)	Detection Method	Local Effect	Final Effect
Inverter	DC bus fault (FM1)	Overcurrent, overvoltage, undervoltage (FC1)	Alarm, software monitoring	Power supply instability, capacitor damage	Propulsion performance degradation, even loss of function
	Overheat fault (FM2)	Insulation damage, thermal switch failure, cooling fan damage (FC2)	Alarm, software monitoring	Inverter damage	Propulsion performance degradation
	Overload fault (FM3)	Motor overload (FC3)	Alarm, software monitoring	Inverter damage	Propulsion performance degradation
		Power device short or open circuit (FC4)
Permanent magnet synchronous motor	Internal material or mechanical structure damage (FM4)	Permanent magnet loss of magnetism, bearing fatigue wear, improper processing and assembly (FC5)	Regular maintenance	Motor loss of working ability	Propulsion function loss
	Winding and core overheating (FM5)	Motor overload (FC3)	Alarm, software monitoring	Motor damage	Propulsion performance degradation
		Winding short circuit (FC6)
		Excessive grid voltage, poor ventilation (FC7)
	Bearing oil leakage (FM6)	Poor sealing of bearing, blockage of oil outlet pipe (FC8)	Visual inspection	Motor performance affected	Propulsion performance affected
	Winding line fault (FM7)	Interturn or phase–to–phase short circuit (FC6)	Software monitoring	Motor damage, output performance degradation	Propulsion performance degradation
		Stator winding open circuit (FC9)
	Large vibration and noise (FM8)	Unbalanced load, loose core, bearing wear, too large bearing bush clearance (FC10)	Audible inspection	Instable motion, shell vibration	Propulsion performance affected
Speed controller	Communication fault (FM9)	Control circuit damage, wiring fault (FC11)	Software monitoring	Loss of speed control function	Loss of propulsion function
	Software error (FM10)	Human error, design error (FC12)	Software monitoring	Loss of speed control function	Loss of propulsion
	IO failure (FM11)	Sensor failure, interface failure (FC13)	Software monitoring	Loss of speed control function	Loss of propulsion
Propeller	Blade damage (FM12)	Beep or cavitation, deformation or fracture (FC14)	Regular maintenance, software monitoring	Propeller performance degradation	Propulsion performance degradation
	Jamming failure (FM13)	Foreign matter entanglement like fishing net (FC15)	Alarm, software monitoring	Loss of propeller function	Loss of propulsion function
Shaft	Fatigue Wear (FM14)	Rusting, Connection Key Failure (FC16)	Regular Inspection	Transmission efficiency and sealing affected	Propulsion performance affected
	Shaft breakage (FM15)	Stress concentration factor is not eliminated in the manufacturing process (FC17)	Alarm, software monitoring	Loss of transmission function Loss	Propulsion function loss

4.2. Calculation Results and Comparative Analysis

Suppose that the FMEA assessment team has four expert members. According to the assessment environment, membership qualifications, and system familiarity, the expert credibility weights obtained by the qualitative analysis can be determined as $\chi =\left[\begin{array}{cccc}0.2& 0.2& 0.3& 0.3\end{array}\right]$. In addition, the scores for expert members on work experience, technical field, and professional level are listed in Table 6. Accordingly, the expert entropy weights are $\theta =\left[\begin{array}{cccc}0.2438& 0.2517& 0.2526& 0.2519\end{array}\right]$. The comprehensive weights of expert credibility based on the combination of information entropy and qualitative analysis are written as $h=\left[\begin{array}{cccc}0.2219& 0.2258& 0.2763& 0.2760\end{array}\right]$.

Table 6. Scores of expert members for three performance indicators.

Expert Member	Work Experience	Technical Field	Professional Level
Expert 1	2	3	4
Expert 2	3	4	4
Expert 3	4	4	4
Expert 4	3	5	3

Using expert knowledge experience combined with fuzzy linguistic terms and evaluation criteria, the fuzzy evaluation information for the three risk factors (and their relative weights) for the permanent magnet synchronous motor of a propulsion subsystem) is presented, as shown in Table 7. The fuzzy evaluations of other key equipment, such as the PWM inverter, speed controller, propeller, and shafting, can be obtained in the same way. However, for the sake of simplicity, they are not listed here.

Table 7. Fuzzy evaluation of the risk factors and relative weights for a permanent magnet synchronous motor. Severity (S), occurrence (O), and detectability (D).

FM	FC	$\mathbf{Expert}1({\mathit{h}}_1=0.2219)$			$\mathbf{Expert}2({\mathit{h}}_2=0.2258)$			$\mathbf{Expert}3({\mathit{h}}_3=0.2763)$			$\mathbf{Expert}4({\mathit{h}}_4=0.2760)$
		S	O	D	S	O	D	S	O	D	S	O	D
FM4	FC5	L	R	MH	M	R	M	M	R	MH	M	R	MH
FM5	FC3	M	M	H	H	ML	H	M	ML	H	H	ML	H
FM5	FC6	M	ML	MH	H	L	H	M	L	MH	H	L	MH
FM5	FC7	M	ML	H	H	ML	H	M	L	H	H	L	H
FM6	FC8	VL	VL	H	VL	VL	H	VL	VL	VH	VL	L	VH
FM7	FC6	M	L	MH	M	VL	H	M	L	MH	M	VL	H
FM7	FC9	M	VL	H	M	VL	VH	M	VL	H	M	VL	VH
FM8	FC10	MR	M	VH	VL	M	VH	VL	M	VH	VL	M	VH
Relative weight		MH	MH	M	VH	MH	ML	VH	MH	M	VH	VH	M

Because of the fuzzy evaluation information and the defined fuzzy numbers, the overall fuzzy number evaluation results for the risk factors and their relative weights can be determined. The confidence level is set as $A=\left\{0,0.2,0.4,0.6,0.8,1.0\right\}$. With the designed benchmark adjustment search algorithm, the alpha-level sets of fuzzy RPNs for the typical failure modes of a permanent magnet synchronous motor can be computed and arranged (as shown in Table 8).

Table 8. Alpha-level sets of fuzzy RPNs for a permanent magnet synchronous motor.

FM	FC	${\mathit{\alpha }}_0=0$	${\mathit{\alpha }}_1=0.2$	${\mathit{\alpha }}_2=0.4$	${\mathit{\alpha }}_3=0.6$	${\mathit{\alpha }}_4=0.8$	${\mathit{\alpha }}_5=1.0$
FM4	FC5	[1.3603, 4.9520]	[1.7378, 4.7110]	[2.0945, 4.4638]	[2.4361, 4.2076]	[2.7660, 3.9409]	[3.0861, 3.6594]
FM5	FC3	[3.0015, 6.1343]	[3.3201, 5.9233]	[3.6317, 5.7110]	[3.9377, 5.4970]	[4.2380, 5.2809]	[4.5335, 5.0612]
FM5	FC6	[3.1648, 5.9269]	[3.3464, 5.8090]	[3.6089, 5.5918]	[3.8733, 5.3720]	[4.1404, 5.1505]	[4.4101, 4.9259]
FM5	FC7	[2.8002, 5.8667]	[3.0997, 5.6514]	[3.3885, 5.4347]	[3.6726, 5.2164]	[3.9582, 4.9958]	[4.2465, 4.7731]
FM6	FC8	[1.6066, 4.3693]	[1.8584, 4.1645]	[2.1054, 3.9586]	[2.3488, 3.7524]	[2.5883, 3.5452]	[2.8264, 3.3371]
FM7	FC6	[2.5348, 5.4909]	[2.7941, 5.2725]	[3.0557, 5.0521]	[3.3198, 4.8293]	[3.5859, 4.6034]	[3.8547, 4.3741]
FM7	FC9	[1.8369, 4.9938]	[2.1255, 4.7693]	[2.4155, 4.5417]	[2.7072, 4.3116]	[3.0015, 4.0776]	[3.2986, 3.8397]
FM8	FC10	[1.6382, 5.1008]	[2.0127, 4.8671]	[2.3636, 4.6297]	[2.6982, 4.3886]	[3.0210, 4.1421]	[3.3344, 3.8896]

The defuzzified values of the fuzzy RPNs can be determined by employing the centroid defuzzification method and alpha-level set theory. Then, the causal degrees of the failure modes and causes are acquired based on the above DEMATEL technique. In addition, in order to verify the feasibility and superiority of the proposed risk assessment method, the traditional RPN and fuzzy RPN (FRPN) are compared and analyzed. Based on the fuzzy information given in Table 7, and exact ratings in the evaluation criteria, a traditional RPN is derived. The fuzzy RPN is defined as the defuzzified values of the fuzzy RPNs obtained in the above section. The risk calculation results of the three evaluation methods for the typical failure modes of a propulsion subsystem in a shipboard medium-voltage DC IEPS are depicted in Table 9 and Figure 4.

Table 9. Results comparison of the three evaluation methods. FRPN, fuzzy RPN.

FM	FC	RPN	Ranking 1	FRPN	Ranking 2	FRPN + DEMATEL	Ranking 3
FM1	FC1	43.75	16	4.6581	12	0.3878	12
FM2	FC2	73.5	7	5.4885	6	0.4569	7
√ FM3	FC3	78.75	5	5.7936	4	1.0000	1
√ FM3	FC4	63	9	5.3597	8	0.4462	8
FM4	FC5	49	13	4.3515	16	0.3622	16
√ FM5 +	FC3	102	2	6.2190	1	1.0000	1
√ FM5 +	FC6	103	1	6.1305	2	0.9602	3
√ FM5 +	FC7	87.75	4	5.8755	3	0.4891	5
FM6 ※	FC8	32	18	4.0385	18	0.3362	18
√ FM7	FC6	75	6	5.4035	7	0.9602	3
√ FM7 ○	FC9	45	14	4.6349	15	0.3858	15
FM8 ※○	FC10	45	14	4.6434	14	0.3865	14
FM9	FC11	53.25	11	4.6477	13	0.3869	13
FM10	FC12	89.25	3	5.6741	5	0.4723	6
FM11	FC13	49.5	12	4.8024	11	0.3998	11
FM12	FC14	67	8	5.1530	9	0.4290	9
FM13※	FC15	33	17	4.1676	17	0.3469	17
FM14※	FC16	53.5	10	4.8735	10	0.4057	10
FM15 ※	FC17	22.5	19	3.5368	19	0.2944	19

Figure 4. Risk ranking of failure modes for the propulsion subsystem.

As stated in Table 9, the results of the comparative analysis show that failure modes FM6, FM8, FM13, FM14, and FM15, marked with ※ in the table, have the same risk prioritization in the three assessment methods. Furthermore, the rankings of failure mode FM5 (permanent magnet synchronous motor windings and iron core overheating), marked with + in the table (induced by different causes FC3, FC6, and FC7, respectively), are in the top five in terms of risk priority, indicating the effectiveness of the proposed method. The assessment results are consistent with the practical engineering failure cases, because the failures of propulsion motors often lead to a loss of the propulsion system’s functions, which can seriously affect the ship’s maneuvering characteristics and the crew’s safety onboard. Therefore, in the design and operation stage, it is necessary to fully consider the typical failure mode impacts of propulsion motors and develop preventive and compensatory recommendations, such as condition monitoring, maintenance management, and risk control, to enhance the system’s safety and reliability.

Compared with the other two methods, in the situation that different failure modes have an identical value for RPN, it is difficult to determine the risk implication and prioritize the risk level in a traditional RPN. For instance, the failure mode FM7, induced by FC9, and failure mode FM8, induced by FC10 (marked with ○ in the table), have the same RPN value of 45. However, the risk factor values of both these events are 6, 3, 2.5, and 3.75, 6, 2, respectively. Due to the introduction of risk factor weights into the fuzzy RPN and the proposed improved assessment method, the probabilities of severity, occurrence, and detectability are given different levels of attention to avoid the loss of risk information. Moreover, in the fuzzy RPN and the proposed method, expert credibility weights are adopted to express the relative importance of different expert opinions and decrease the subjectivity of expert knowledge. Furthermore, each risk factor, and its relative importance, is evaluated by linguistic terms and fuzzy ratings to reduce the dependence of accurate ratings. Improvements in these aspects conform to the realistic requirements of safety-critical systems like the shipboard IEPS and contribute to avoiding assessment defects with the same RPN values.

As shown in Table 9 and Figure 4, the failure modes and causes are not a one-to-one relationship (as indicated in the table) For instance, a failure mode FM5 can be induced by various failure causes, FC3, FC6, and FC7, respectively. Meanwhile, the failure cause FC3 can induce multiple failure modes (FM3 and FM5). Compared with fuzzy RPN, the fuzzy logic and DEMATEL based risk assessment method can take into account the correlation effects between failure modes and causes and give higher risk priority to common cause failure modes. In other words, if multiple failure modes are induced by the same cause, a higher risk priority is achieved, such as FM3 and FM5 induced by FC3, as well as FM5 and FM7 induced by FC6. This situation is in line with the applications of practical engineering, because, when multiple failure modes induced by common causes are given higher risk priority, more attention is paid to detect the failure’s causes and maintain the relevant equipment, thus eliminating the effects of different failures simultaneously. Consequently, the proposed assessment method can produce more reasonable risk prioritization results in the FMEA process. With the results of risk priority decision-making, preventive and compensatory recommendations can be designed purposefully to optimize maintenance resources optimization and prevent disastrous accidents.

5. Conclusions

Because of its complex structures and a poor environment, shipboard medium voltage DC IEPSs suffer from system failures and security threats. This paper proposes an improved risk assessment method integrating fuzzy logic and the DEMATEL technique to improve the system’s safety and reliability. Using this method, the loss of risk information can be avoided by the introduction of risk factor weights. Further, each risk factor and its relative importance are evaluated by linguistic terms and fuzzy ratings to reduce the dependence of accurate ratings. Additionally, expert credibility weights are adopted to decrease the subjectivity of expert knowledge and experience. By considering the correlation effects between the failure modes and their causes, the common failure modes are given higher priority to make the results more realistic and represent a flexible flection of real situations. With the results of risk priority decision-making, preventive and compensatory recommendations can be designed purposefully to optimize maintenance resources and prevent disastrous accidents.

The main contributions of this study can be stated as follows. Information entropy and qualitative analysis are synthesized to determine the credibility weights of domain experts, which can overcome the implicit subjectivity and inaccuracy of expert opinions on system failure analysis. The three risk factors and their relative importance are evaluated in the form of fuzzy linguistic terms and fuzzy ratings rather than by exact numerical values, thus making the results more realistic and objective. Since fuzzy RPNs are not trapezoidal or triangular fuzzy numbers, and their precise membership functions are unknown, a benchmark adjustment search algorithm is designed to obtain alpha-level sets of fuzzy RPNs for the corresponding defuzzification calculations. Additionally, the DEMATEL technique is employed to measure the correlation effects between the failure modes and causes, which have not been investigated in previous studies on shipboard medium-voltage DC IEPSs.

The typical failure modes of propulsion subsystems are analyzed and assessed for verification. The calculation results show that the inverter failure FM3, induced by cause FC3, and the permanent magnet synchronous motor failures FM5 and FM7, induced by causes FC3, FC6, and FC7, respectively, have higher risk levels and should be given more attention. Therefore, in the design and operation stage, it is essential to give full consideration to the failure effects of inverters and propulsion motors, to ensure effective maintenance, improvements, and control actions. In addition, it should be noted that the improved risk assessment method proposed in this paper is not only applicable to ship IEPSs, but also to the reliability analysis of safety-critical systems, such as aerospace and nuclear power plants. Considering the coupling effects of failure modes and failure causes, establishing a more comprehensive risk assessment model establishment for the FMEA process is planned for future research.