2.1 Engineering Example 1: Full Self-Driving (FSD) Systems

Autonomous vehicle systems are composed of sensors (LiDAR, radar, cameras), complex software algorithms, and machine learning models. FMEA struggles because: data from sensors is noisy and conflicting, failure modes are interdependent and nondeterministic, expert assessments (e.g., severity of misdetection) are subjective and since these systems are still in development stage, the experts do not have complete knowledge.

In this condition, Fuzzy Logic models degrees of severity and likelihood (e.g., “high” and “very high”) instead of crisp values; and Rough Sets can categorize failures into approximate boundaries when data is incomplete (e.g., unknown exact probability of failure but categorized as “likely” based on partial data). Since in FSD systems, a misclassification of a pedestrian by the AI can result in vastly different outcomes depending on speed, visibility, and traffic context, the improved prioritization of failure modes under data uncertainty, improves decision-making on which subsystems need fail-safes.

2.2 Engineering Example 2: Renewable Energy Production

Offshore wind turbines operate under uncertain weather and maintenance schedules. FMEA faces challenges due to: stochastic nature of environmental loads, difficulty in modeling degradation over time. In photovoltaic (PV) panels, the challenge becomes even more complex due to the significant influence of environmental parameters—such as synergetic impact of humidity, wind speed, and wind direction—that are often not explicitly captured in conventional physical or mathematical models. However, expert judgments tend to vary widely because these environmental factors introduce nonlinear and context-dependent effects. Over the past decade, AI techniques have been increasingly used to estimate and model the combined impact of these variables, further highlighting the limitations of traditional FMEA approaches with incomplete knowledge of the experts in handling such uncertainties.

In this domain, we consider scenario on wind turbines: blade erosion due to salt spray may accelerate over time depending on unpredictable microclimate effects. This causes difficulty in assigning static failure probabilities. In this condition, Fuzzy Logic allows considering of gradual degradation stages (e.g., “early erosion” and “moderate wear”), whereas Rough Sets can categorize failures into approximate boundaries when data is incomplete (e.g., unknown exact probability of failure but categorized as “likely” based on partial information).

2.3 Engineering Example 3: Safety-Critical Nature in Medical Device and Reliability

Wearables like continuous glucose monitors rely on physiological sensors whose performance depends on user behavior, skin condition, and environment. Sensor failures are difficult to pin down, diverse user conditions cause inconsistent readings, and safety-critical nature means errors must be minimized, even if data is partial.

For example a smart watch heart-rate sensor might falsely detect issue due to tattoo interference or motion.

In this condition, Fuzzy Logic captures subjective evaluations of risk levels; while Rough Sets can group diverse sensor performance outcomes into “safe”, “borderline”, and “alert” regions, even if not all failure mechanisms are fully known.

3.1 Theoretical Foundation of Triangular Fuzzy Set

A triangular fuzzy set is a simple and widely used representation of fuzzy sets for handling subjectivity [30], reducing possible defects of individual bias [31], and consistency in evaluation [31], which improves decision-making by accounting for uncertainty and partial truths. It is characterized by a triangular membership function (see Formula (1)) [32] shown in Table 3. The triangular function is defined by three parameters: $$$ a $$$ (lower bound), $$$ m $$$ (mode or peak value), and $$$ b $$$ (upper bound). The membership function, $$$ \mu (x) $$$, increases linearly from $$$ a $$$ to $$$ m $$$ and decreases linearly from $$$ m $$$ to $$$ b $$$. The membership degree ranges between 0 and 1, where the peak value $$$ m $$$ achieves a degree of 1 (see the representation in Figure 1).

Triangular fuzzy sets are particularly useful for representing vagueness in qualitative terms such as “low,” “medium,” and “high.” They are computationally efficient and suitable for applications in decision-making, risk assessment, and fuzzy logic-based models.

3.2 Theoretical Foundation of Rough Set

In emerging technologies, incomplete and partial information often arises due to systems' complexity and data collection limitations, posing challenges for accurate failure analysis. The rough set theory offers a powerful tool to address this by handling uncertainty and imprecision, making it ideal for such scenarios [33-35]. Figure 2 outlines the key steps in the rough set theory process [36]. It begins with data collection, followed by constructing the discernibility matrix ($$$ \Delta =\left\{\left(x,y\right)|x\ne y\right\} $$$) to identify differences between objects. Next, reducts are computed ($$$ \mathrm{RED}(B){=\bigcap}_{B\subseteq C}{\mathrm{POS}}_B $$$) to derive minimal attribute sets. Decision rules ($$$ R:\mathrm{IF}\kern0.3em A\kern0.3em \mathrm{THEN}\kern0.3em B $$$) are then generated based on these reducts. Finally, the model's accuracy $$$ \left(\mathrm{ACC}=\frac{\mathrm{Correct}\ \mathrm{Predictions}}{\mathrm{Total}\ \mathrm{Predictions}}\right) $$$ is evaluated before concluding the process.

Traditional FMEA relies heavily on crisp and subjective input data and identifying the values for severity (S), occurrence (O), and detectability (D), and multiplying the three values to obtain a risk priority number (RPN) [37]. However, RPNs can be inconsistent and ambiguous due to the inherent subjectivity of human judgment. Integrating fuzzy and rough set theories with FMEA, we aim to make the analysis more robust and less sensitive to uncertainty. Therefore, in the current study, we propose a variant of the fuzzy FMEA method integrated with rough set theory to address uncertainty and enhance the robustness of reliability analysis through FMEA.

3.3 Integration of Fuzzy and Rough Theories

Fuzzy and rough theories each have specific capabilities in dealing with uncertainty. Fuzzy theory captures the vagueness and ambiguity in expert reasoning during risk prioritization by representing their judgment through linguistic variables and membership functions. This is particularly crucial when implementing FMEA since it conventionally uses crisp values that may not reflect the true uncertainty in failure assessments [38, 39]. Rough theory, on the other hand, uses boundary regions instead of membership functions to represent uncertainty due to indiscernibility caused by lack of sufficient information [40]. In the implementation of FMEA, rough theory is particularly useful because it takes into account the decision maker's subjective judgment and preference and ensures the accuracy of FMEA ranking results [41].

Integrating fuzzy and rough theories with FMEA overcomes the limitations of traditional RPN by refining the calculation of the RPNs. Rough set theory complements fuzzy logic by managing uncertainty due to incomplete or imprecise information, leading to more nuanced modeling of uncertainty and improved decision-making accuracy [42]. This integration enhances the reliability of risk assessments by incorporating both fuzzy uncertainty (due to subjective expert opinions) and random uncertainty (due to variability in data) [43].

4.1 Step 1: Data Collection

Data collection is conducted through structured questionnaires. Experts are asked to evaluate failures based on FMEA risk factors: severity ($$$ S $$$), occurrence ($$$ O $$$), and detectability ($$$ D $$$). Each expert selected a linguistic term to describe the risk factors. The linguistic terms are then mapped to triangular fuzzy numbers (TFNs), as shown in Table 3, following the methodology in [44].

4.2 Step 2: Construct the Fuzzy Decision Matrices

4.3 Step 3: Generate FRNs

All collected TFNs are converted into Fuzzy Rough Numbers (FRNs). A TFN $$$ {\tilde{C}}_i=\left({C}_{il},{C}_{im},{C}_{iu}\right) $$$ is converted into an FRN through the following steps [45]:

4.4 Step 4: Obtain the Aggregate Fuzzy Rough Decision Matrix

4.5 Step 5: Calculating Fuzzy Rough RPNs

4.6 Step 6: Calculating the Mean Value of RPNs

To evaluate the overall importance of each failure, the mean value of $$$ FR{R}_i=\left(\left[ FR{R}_i^{lL}, FR{R}_i^{lU}\right]\right),\left(\left[ FR{R}_i^{mL}, FR{R}_i^{mU}\right]\right),\left(\left[ FR{R}_i^{uL}, FR{R}_i^{uU}\right]\right) $$$, indicated by $$$ RP{N}_i^{\Sigma} $$$ is calculated using Formula (11):

4.7 Step 7: Ranking

The mean values of RPNs are sorted in ascending order to generate the final ranking of failures.

5.1 Data Collection for Empirical Study

To calculate rough FMEA, rough-fuzzy FMEA, and fuzzy-rough FMEA, data needs to be collected from domain experts regarding the failure modes, S, O, and D ratings in Table 4. This data will serve as input to the respective FMEA models.

6.1 Sensitivity Analysis

For the sensitivity analysis of the proposed FR-FMEA, two different evaluations are considered. First, the TFNs used for the linguistic terms Table 3 are replaced with those of [59] and [60]. These two studies were selected because they also use a five-point TFN structure for FMEA risk factors' ranking. Second, the robustness of the proposed FR-FMEA is tested by changing the input of experts in two different scenarios: (1) replacing all inputs of one expert to very high (VH) and (2) replacing all inputs of one expert to very low (VL).

The outcome of the first evaluation is shown in Table 7. As shown in the ranking results, no significant change is seen in the ranking of most failures. The only failures whose ranking has changed drastically are F6 and F10. This shows the acceptable robustness of the results to changes in TFNs.

Table 8 shows outcomes of the second evaluation where the experts' inputs are changed. It is observed in the table that there have been some changes in the results, although the significant ones are seen only in four failures (F11, F16, F17, and F20). The relative stability of ranks with respect to drastic changes in one expert's input demonstrates the robustness of results to noise.

6.2 Computational Complexity

The time complexity values in Table 9 reflect the computational demands of each failure analysis method based on how they process data. For crisp (conventional) FMEA, the complexity is $$$ O(n) $$$, where $$$ n $$$ is the number of failure modes or components analyzed. This is because the method involves a simple calculation of the risk priority number (RPN) for each failure mode, using fixed numerical values for severity, occurrence, and detection. These are independent operations, requiring no iterations or comparisons between failure modes, resulting in linear complexity.

In the case of fuzzy FMEA, the complexity increases to $$$ O\left(n\cdotp r\right) $$$, where $$$ r $$$ is the number of fuzzy rules or membership functions. On the other hand, fuzzy sets typically rely on relation matrices, where every object is compared with every other object in the dataset. This results in a total of $$$ \left(\genfrac{}{}{0ex}{}{n}{2}\right)=\frac{n\left(n-1\right)}{2} $$$ pairwise comparisons, which leads to a time complexity of $$$ O\left({n}^2\right) $$$.

Each failure mode must be processed through a fuzzy inference system, which includes fuzzification of inputs, evaluation of fuzzy rules, and defuzzification of outputs. The rule base must be evaluated for each item, increasing the overall computation in proportion to both the number of items and the number of rules.

Fuzzy rough FMEA can have an even higher complexity, so typically $$$ O\left({n}^2\right) $$$ or worse. This is because the method involves computing pairwise similarity or indiscernibility relations between all failure modes, resulting in a relation matrix of size $$$ n\times n $$$. In addition to this, lower and upper approximations must be calculated for each object, often requiring multiple passes through the data. If attribute reduction or feature selection is also performed using fuzzy rough techniques, the complexity can grow further, depending on the number of attributes and the method used for searching attribute subsets.

The results of the comparison illustrate that while conventional FMEA is fastest, it lacks the ability to deal with uncertainty. Fuzzy FMEA adds uncertainty modeling with reasonable computational cost, and fuzzy rough FMEA offers the most powerful modeling for uncertainty and incomplete knowledge, at the cost of increased complexity.

6.3 Limitations and Future Research

This study recognizes several limitations. First, the proposed FMEA method holds the inherent limitation in all FMEA methods, that is, the reliance on experts' judgments, which may be influenced by different types of bias. Although the integration of rough and fuzzy theories with FMEA seeks to address the subjectivity in experts' judgments, the reliance on experts' inputs still remains. On the other hand, the number of experts may also influence the results. Moreover, this study is conducted on and validated by a specific use case and may not be generalized on more broad cases.

A valuable future research would be to compare FR-FMEA, RE-FMEA, Fuzzy FMEA, and conventional FMEA on the same case study, using the same team of interviewees, rather than data from different sources. This approach would allow for a more controlled comparison, highlighting which method offers greater granularity and effectiveness in complex systems. Additionally, exploring the sensitivity of these methods to input variations could provide deeper insights into their robustness and reliability under changing conditions.

Validating the method's adaptability and reliability through various tests will be essential before it can be confidently adopted as a standard tool in complex engineering risk management.