Introduction

Intelligent risk assessment plays a crucial role in the era of big data and machine learning [1–3]. Among the various techniques, trapezoidal fuzzy fault tree analysis has emerged as a powerful tool by integrating traditional fault tree logic with fuzzy set theory to handle system uncertainties [4,5]. However, significant challenges persist in its application, particularly concerning model uncertainties that can undermine the reliability of assessment results [6,7].

Uncertainties in trapezoidal fuzzy fault tree assessment generally fall into three categories: data uncertainty [8–10], stemming from insufficient data and limited expert knowledge; relational uncertainty [11,12], arising from complex dependencies among basic events; and model uncertainty [13], resulting from computational approximations that may introduce information distortion. These uncertainties propagate upward through the fault tree hierarchy, from basic events to the top event, ultimately compromising the accuracy of risk assessment outcomes [14,15].

To mitigate data uncertainty, researchers have developed a range of methods centered on extending fuzzy mathematics, integrating expert knowledge, and constructing hybrid models. For instance, Liu et al. [16] incorporated intuitionistic trapezoidal fuzzy numbers into fault tree analysis and introduced a new ranking method based on expected values and compromise possibilities. Although this approach helped quantify the membership and non-membership degrees of basic event probabilities, it did not fully address multi-level uncertainty propagation. Kaushik and Kumar [17] later developed intuitionistic fuzzy importance measures using α-cut operations to process qualitative linguistic data, improving the reliability of criticality analysis, though computational complexity limited its use in large-scale systems. Yu et al. [18] applied the weakest t-norm to trapezoidal fuzzy number aggregation, reducing distortion and bringing results closer to statistical values compared to conventional linear weighting. Komal [19] extended this idea by combining Tω operators with fuzzy reliability theory in a generalized lambda-tau methodology for repairable systems, though its adaptability in dynamic environments remains to be fully tested. Badida et al. [20] applied a fuzzy fault tree framework to pipeline failure estimation, enabling risk assessment without historical data via Fussel-Vesely importance measures, albeit with reliance on subjective expert weighting. To address this, Lu et al. [21] introduced an improved SAM method using a butterfly optimization algorithm to adaptively determine the relaxation factor β, removing the need for manual tuning. Zhang et al. [22] combined regret theory with squared confidence Pythagorean fuzzy hybrid weighted operators in a multi-criteria decision model, improving semantic interpretability and reliability in information aggregation. Sun et al. [23] further proposed a Pythagorean fuzzy Bayesian network model using enhanced trapezoidal fuzzy Einstein geometric operators to integrate subjective and objective weights, effectively reducing prior probability uncertainty in submarine pipeline risk assessment. Kaushik and Kumar [24] built a hybrid intuitionistic fuzzy fault tree-Bayesian network model, using conditional probability tables to quantify system failure risks under imprecise data conditions. Finally, Tan et al. [25] integrated fuzzy fault trees with probabilistic risk assessment into a dynamic framework for battery energy storage safety, allowing real-time tracking of risk variations due to design parameter changes.

To address relational uncertainty, several integrated approaches have been proposed. Ding et al. [26] combined Bayesian networks with trapezoidal fuzzy fault trees to develop a dynamic risk assessment framework for residual heat removal systems, modifying conditional probability tables to represent event polymorphism and non-deterministic logic. Separately, Bi et al. [27] used particle swarm optimization to tune the parameters of a backpropagation neural network coupled with trapezoidal fault trees, capturing nonlinear data relationships and achieving high-accuracy risk prediction with a maximum error of 0.0006. Similarly, Sakar and Sokukcu [28] integrated fault trees with Bayesian networks to model conditional dependencies in pilot transfer risks, quantifying interactions between equipment failures and human operations through conditional probability tables. While effective, these methods mainly addressed static uncertainties. To overcome this, Meng et al. [29] introduced a hybrid event tree–fault tree and dynamic Bayesian network model capable of capturing interdependencies among risk factors while accounting for both stochastic and fuzzy uncertainties. Extending this effort, Ahmed et al. [30] developed a type-2 fuzzy Bayesian method to adaptively update linguistic variables in dynamic environments. Singh et al. [31] employed a hybrid model with bidirectional probabilistic updating to reveal nonlinear propagation patterns of fuzzy uncertainties in dependency networks. In a related innovation, Djemai et al. [32] converted Bowtie models into dynamic Bayesian networks, creating a bridge between system prediction and conditional dependency analysis.

To further address these challenges, particularly in integrating multi-source information and quantifying complex human factors, hybrid fault tree models have been increasingly adopted. For example, Saraliolu et al. [33] integrated the Human Factors Analysis and Classification System (HFACS) with fuzzy FTA to analyze ship engine room fires. Their model uses HFACS to structurally classify human factors and employs fuzzy FTA for quantitative probability calculation, revealing the fire mechanisms in ships older than 20 years caused by mechanical fatigue and insulation aging. Similarly, the combination of D–S evidence theory and the Human Error Assessment and Reduction Technique (HEART) offers a new approach for assessing cargo tank crack risks in chemical tankers, using Bayesian networks to handle uncertainty in expert opinions while quantifying human error probability [34]. Research indicates that D–S evidence theory can provide a unified framework for modeling uncertain, incomplete, or even unknown information, effectively overcoming the limitations of traditional probabilistic reasoning [35]. In container transportation, the integration of Type-2 fuzzy sets with the Success Likelihood Index Method (SLIM) has further improved risk prediction accuracy, with its multi-layer fuzzy logic effectively addressing the quantification of subjective judgments [36].

Although significant progress has been made in handling data and relational uncertainties, current approaches still largely rely on methods such as the SAM or Bayesian networks [37–40]. As a result, model uncertainty—stemming from approximation calculations—remains a persistent challenge. A key limitation is the widespread dependence on defuzzification techniques to obtain crisp probability values [41–43]. A common defuzzification formula employed in the literature [39,44–46] is the Fuzzy Possibility Score (FPS) = [(c + d)² - cd – (a + b)² + ab]/ [3*(c + d-a-b)], where a, b, c, d are the four parameters of a trapezoidal fuzzy number. However, this method has an inherent flaw: it discards the shape information of the fuzzy number itself. For instance, four distinct trapezoidal fuzzy numbers—(0.2,0.4,0.6,1.0), (0.25,0.45,0.668,1.0), (0.15,0.35,0.535,1.0), and (0.3,0.5,0.738,1.0)—all yield an identical FPS value of 0.56. This result is clearly inconsistent with reality, demonstrating that traditional defuzzification causes distortion through information loss. The root cause of this problem is twofold: first, it maps distinct fuzzy numbers to the same crisp value, failing to preserve their essential differences; second, it reduces the nonlinear arithmetic of trapezoidal fuzzy numbers to linear operations, thereby misrepresenting the true mechanism of uncertainty propagation.

This practice often leads to the premature simplification of basic event probabilities, causing information distortion and reducing the reliability of outcomes [47–49]. The issue is particularly pronounced during nonlinear multiplication operations involving trapezoidal fuzzy numbers, where model uncertainty is further amplified [50,51].

This systematic oversight represents a critical gap in current trapezoidal fuzzy fault tree risk assessments. To overcome these limitations, we propose a novel algorithm that specifically targets model uncertainty by preserving the integrity of fuzzy information throughout the analysis. Unlike conventional approaches, our method avoids defuzzification entirely by building on the cut-set theorem and the representation theorem of fuzzy numbers. This allows the original trapezoidal fuzzy shape to be maintained across all calculation stages, thereby minimizing information distortion and curbing the propagation of uncertainty. Key innovations of the algorithm include: (1) high-precision computation that reduces approximation errors, (2) full retention of fuzzy information for more reliable outcomes, and (3) broad applicability across different fault tree configurations.

The main contributions of this work are twofold. First, we introduce an uncertainty-aware algorithm that maintains fuzzy information across the assessment process, offering a new perspective on model uncertainty reduction. Second, we demonstrate the algorithm’s practical effectiveness through case studies, underscoring its potential to improve the accuracy and reliability of risk assessments in complex engineering systems. The structure of the paper is as follows: the Preliminaries section introduces the necessary theoretical background, including trapezoidal fuzzy numbers, the cut-set theorem, and the representation theorem. The Proposed Algorithm section presents the derivation and implementation of the proposed algorithm. The Case Studies and Sensitivity Analysis section validates the method using real and literature-based case studies, and the Conclusions section summarizes the findings and suggests directions for future research.

Preliminaries

Trapezoidal fuzzy numbers

Fuzzy set theory, introduced by Zadeh [52,53], provides a mathematical framework for handling data uncertainty. A fuzzy set on a universe X is defined by a membership function : X→ [0,1], which quantifies the degree to which an element belongs to . This function can be represented using various types of fuzzy numbers, among which trapezoidal fuzzy numbers are widely used due to their practicality and versatility [54–56]. A trapezoidal fuzzy number is denoted as , and its membership function is defined by Eq. (1) [57]:

(1)

1. (1). Arithmetic operations of trapezoidal fuzzy numbers

Let and be two trapezoidal fuzzy numbers. Their arithmetic operations are defined as follows [45]:

(2)(3)(4)

1. (2). Traditional Calculation of Top Event Probability

Consider a basic event i, whose failure probability is represented by a trapezoidal fuzzy number . The conventional approach for calculating the top event probability in a trapezoidal fuzzy fault tree is as follows [58]:

For an AND gate:

(5)

For an OR gate:

(6)

cut-set theorem

For a fuzzy number and a parameter , the λ-cut set is defined as:

(7)

Here, λ is referred to as the cut level, threshold, or belief level. By varying the value λ, a fuzzy set can be transformed into a classical set.

Representation theorem of fuzzy numbers

Consider a mapping: , that satisfies:

(8)

Then, is called a nest of sets on .

Let , with , satisfy

Then, the fuzzy number can be represented as:

(9)(10)(11)

where

(12)(13)(14)(15)

A novel algorithm for precise trapezoidal fuzzy fault tree analysis

Building on the cut-set theorem and the representation theorem of trapezoidal fuzzy numbers, we propose a novel computational algorithm designed to significantly reduce model uncertainty in risk assessment. As illustrated in S1 Fig, the algorithm consists of five key steps, guiding the entire process from fuzzy number input to final risk assessment. This workflow preserves the integrity of fuzzy information while effectively controlling uncertainty propagation through precise computation.

Step 1: Fault Tree Construction

Construct the fault tree to identify basic events and their logical relationships leading to the top event. The probabilities of basic events are determined using the expert scoring method [59] and expressed as trapezoidal fuzzy numbers.

Step 2: Logic Gate Classification

As the types of logic gates directly influence the computational approach, the fault tree is classified into one of three categories prior to calculation:

Fault trees containing only AND gates (proceed to Step 3, Branch 1)

Fault trees containing only OR gates (proceed to Step 3, Branch 2)

Fault trees containing both AND and OR gates (proceed to Step 3, Branch 3)

Step 3: Precise Computational Algorithm

The following exact computation methods are applied according to the logic gate type:

Branch 1: Fault trees with only AND gates

If the top event consists of n basic events connected by an AND gate, and the failure probability of the i-th basic event is represented by a trapezoidal fuzzy number, then the λ-cut interval for the i-th basic event is given by Equation (16):

(16)

Then the λ-cut interval for the product of n basic events is:

(17)

According to the representation theorem given in Equation (11), we obtain:

(18)

From Equation (7), it follows that:

(19)

(20)

Based on Equation (10), the values of and are derived as:

(21)

(22)

From Equation (12) and (13), the values of and are derived as:

(23)

(24)

Using Equation (14) and (15), the values of and are derived as:

(25)

(26)

Within the range of (0, 1), all values of and are calculated. Since is a monotonically increasing function of , for any given , the maximum value of that is less than or equal to can be determined. The corresponding value of is denoted as , which allows the coordinates on the left half of the precise membership curve of the trapezoidal fuzzy number to be determined. Subsequently, the precise membership curve of can be derived.

Similarly, since is a monotonically decreasing function of for any given , the minimum value of that is greater than or equal to can be determined. The corresponding value of is denoted as , which allows the coordinates on the right half of the precise membership curve of the trapezoidal fuzzy number to be determined. Subsequently, the precise membership curve of can be derived.

monotonicity proof is detailed in S1 Appendix

monotonicity proof is detailed in S2 Appendix

Branch 2: Designed for fault trees consisting exclusively of OR gates.

If n basic events of a fault tree are connected through an OR gate, the failure probability of the i-th basic event not occurring can be expressed as a trapezoidal fuzzy number (, according to the analysis of Branch 1. Based on Formula (7), the λ -cut interval of the i-th basic event is:

(27)

The λ -cut interval for the simultaneous non-occurrence of all n basic events is:

(28)

Similar to the analysis of Branch 1, we can obtain:

(29)

(30)

(31)

(32)

(33)

(34)

(35)

(36)

(37)

For any given , the maximum value of satisfying can be determined, and its corresponding can be represented by Thus, the coordinate values () on the left half of the precise membership curve of the trapezoidal fuzzy number can be determined, and the precise membership curve of can be derived.

For any given , the minimum value of satisfying can be determined, and its corresponding can be represented by . Thus, the coordinate values ) on the right half of the precise membership curve of the trapezoidal fuzzy number can be determined, and the precise membership curve of can be derived.

monotonicity proof is detailed in S3 Appendix

monotonicity proof is detailed in S4 Appendix

Branch 3: Applicable to fault trees featuring a combination of AND gates and OR gates.

If the fault tree consists of n basic events and includes both AND gates and OR gates, the λ-cut interval for the failure probability of the top event occurring can be expressed as follows, based on Formula (7):

(38)

According to the representation theorem (11), the following can be derived:

(39)

(40)

(41)

Using Formula (10), the values of parametersandcan be obtained:

(42)

(43)

Similarly, the values of parameters and can be derived from Formula (12) and (13):

(44)

(45)

Furthermore, the values of parameters and can be determined using Formula (14) and (15):

(46)

(47)

For the Left Half of the Membership Curve:

Within the interval (0,1), all values of = and are computed. Given that increases monotonically with , for any specified , the largest satisfying can be determined. The corresponding is denoted as , which enables the identification of the coordinates on the left half of the precise membership curve for the trapezoidal fuzzy number. From these coordinates, the precise membership curve of can be constructed.

For the Right Half of the Membership Curve:

On the other hand, since decreases monotonically with , for any given , the smallest satisfying can be found. The corresponding is denoted as , which facilitates the determination of the coordinates on the right half of the precise membership curve for the trapezoidal fuzzy number. Using these coordinates, the precise membership curve of can be derived.

monotonicity proof is detailed in S5 Appendix

monotonicity proof is detailed in S6 Appendix

Step 4: Membership Function Curve Plotting

Based on the precise computational results, the membership function curves for the trapezoidal fuzzy fault tree are plotted, providing a visual representation of the fuzzy distribution of the top event failure probability.

1. (1). By connecting the points , , , …, , , , , , …, , and , the precise membership curve of the trapezoidal fuzzy fault tree composed of AND gates can be plotted.
2. (2). By linking the points (), (), (), …, (), (,1), (,1), ), (), …, (), and (), the precise membership curve of the trapezoidal fuzzy fault tree composed of OR gates can be constructed.
3. (3). By joining the points (), (), (), …, (), , (), ), , …, , and (), the precise membership curve of the trapezoidal fuzzy fault tree composed of both AND and OR gates can be generated.

Here, the values of parameters e, f, g, ℎ, u, v are determined by the required analysis precision.

Step 5: Risk Assessment

The precise membership functions obtained in Step 4 provide a basis for predicting the top event’s failure probability and conducting importance analyses. These results can also be integrated with other risk assessment methods for further refinement.

Comprehensive case studies and sensitivity analysis

Validation and comparative analysis against benchmark literature data

To validate the reliability of the proposed algorithm and perform a comparative analysis against established methods, we replicated the electrostatic spark explosion case study from Zhang et al. [60]. This fault tree, containing both OR and AND gates across nine basic events, provides a robust benchmark for evaluating both accuracy and the ability to control model uncertainty.

To ensure a fair comparison, the normal fuzzy numbers from the reference study [60] were converted into trapezoidal fuzzy numbers based on the equal-area and centroid-equivalence principle [61]. The conversion results, summarized in Table 1 and visualized in S2 Fig, ensure the input data are equivalent while allowing us to apply our trapezoidal fuzzy-based algorithm.

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Table 1. Conversion of Normal Fuzzy Numbers to Trapezoidal Fuzzy Numbers.

https://doi.org/10.1371/journal.pone.0335759.t001

The proposed algorithm was then applied to compute the top event probability and the fuzzy importance measures of basic events. Critically, to highlight the issue of model uncertainty, we compared our precise results against those generated by the traditional approximate method. The comparative results are detailed in Tables 2 and 3.

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Table 2. Calculation results of top event failure probability.

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Table 3. Degree of Uncertainty Reduction in Trapezoidal Fuzzy Importance.

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The data reveals a key finding: while the central values (e.g., fuzzy medians) between methods are close, the model uncertainty inherent in the traditional approximate method is substantial. Our algorithm consistently reduces this uncertainty, as quantified in Table 3, with reductions reaching up to 100% for some events and averaging 45.25%. This performance notably surpasses the 36.65% reduction reported in the benchmark study [60], underscoring the advantage of our approach in preserving information integrity.

This reduction effect is further visualized in S3 Fig to S12 Fig, which contrast the precise, well-defined trapezoidal outputs of our method against the distorted shapes produced by the traditional approximation.

The most comprehensive comparison is presented in S13 Fig. It demonstrates that the precise computational results from our method are in near-perfect agreement (99.40% consistency) with the precise results from the benchmark study. Conversely, the approximate results from both methods deviate significantly from their respective precise curves.

This divergence is not a discrepancy but a validation of our core thesis: model uncertainty introduced by approximation is a systematic error. The high agreement between the two precise methods confirms the correctness of our algorithm, while the larger uncertainty gap (blue shaded region) in the benchmark study underscores the superior ability of our method to mitigate this error. Therefore, the proposed algorithm not only validates successfully against a known benchmark but also provides a demonstrably more reliable and informative output for risk assessment.

OREDA-based fault tree analysis: demonstrating robustness to system configuration changes

Building on the benchmark validation, this section employs real-world data from the OREDA handbook [62] to evaluate the algorithm’s performance under realistic conditions. The primary objective is to assess its robustness—specifically, how stable the algorithm’s output remains when key system parameters are varied. We begin by establishing a baseline using industrial valve failure data, a critical scenario in petrochemical industries.

Baseline analysis with OREDA data.

To ground the analysis in practical engineering, valve failure was selected as the top event, with nine associated basic events identified from the OREDA handbook. Table 4 presents the raw data, where each basic event’s failure probability is represented as a trapezoidal fuzzy number, forming the foundation for all subsequent sensitivity tests.

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Table 4. Raw Data of Valve Failures from the OREDA Handbook [62].

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Robustness to changes in system scale and complexity.

A key indicator of a robust algorithm is its ability to handle systems of varying scale without unpredictable performance degradation. To test this, we systematically increased the complexity of the fault tree by varying the number of basic events from 4 to 9. For each configuration, the precise result from the proposed algorithm was compared against the traditional approximate method.

Results: Stable and Predictable Performance Enhancement

The data in Table 5 reveals a crucial pattern: as system complexity increases, the proposed algorithm not only maintains stability but also delivers a progressively greater reduction in model uncertainty (from 1.20% to 7.49%). This trend is visualized in S14 Fig.

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Table 5. Algorithm Performance with Increasing System Complexity (Number of Basic Events).

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This demonstrates a critical aspect of robustness: graceful scaling. The algorithm consistently outperforms the traditional method, and the benefit of using the precise method becomes more pronounced for larger, more complex systems. This predictable enhancement—where increased complexity leads to greater relative accuracy—provides engineers with high confidence when applying the algorithm to real-world systems of unknown or variable scale. The algorithm effectively controls uncertainty propagation, ensuring reliable risk assessment even as the system model grows in complexity.

Robustness to variations in input fuzzy parameters.

To further interrogate the algorithm’s robustness, we tested its sensitivity to changes in the fundamental input parameters—the trapezoidal fuzzy numbers themselves. This assesses whether slight variations or uncertainties in defining the failure probabilities of basic events would lead to unstable or erratic outputs. We introduced an additional AND gate event, Eor0, into the valve failure fault tree (Structure: .) and systematically varied the centroid value of Eor0’s failure probability across two orders of magnitude.

Results: Predictable Response and Consistent Superiority

The results, detailed in Tables 6 and 7, demonstrate that the algorithm’s performance exhibits a highly stable and predictable pattern in response to input variations.

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Table 6. Algorithm Performance with Varying AND Event Probability (Centroid 0.02 to 0.08).

https://doi.org/10.1371/journal.pone.0335759.t006

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Table 7. Algorithm Performance with Varying AND Event Probability (Centroid 0.003 to 0.008).

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As shown in S15 Fig and S16 Fig, the algorithm consistently reduces model uncertainty across the entire tested range. A key finding is the smooth, monotonic response of the output to input changes. While the absolute value of the top event probability reasonably increases with the AND event’s probability, the uncertainty reduction effect provided by our algorithm remains substantial and stable.

Based on the data presented in Table 7, a graph illustrating the reduction in uncertainty is plotted, as shown in S16 Fig.

Critically, the data reveals that the algorithm’s advantage over the traditional method is preserved regardless of the input probability’s magnitude. The difference in results between the precise and traditional methods follows a consistent pattern, and the percentage of uncertainty reduction remains significant. This demonstrates that the algorithm is not overly sensitive to the specific values of input parameters. It performs reliably whether the additional event has a high or low probability, confirming its robust stability and suitability for assessing systems with diverse risk levels.

Robustness to Combined Variations in System Scale and Input Parameters

The most rigorous test of an algorithm’s robustness is its performance under combined variations of multiple parameters. This section investigates the algorithm’s response to the simultaneous increase in system complexity (by adding AND gates) and variation in input characteristics (by changing the centroid values of the added AND events from 0.2 to 0.8). This two-dimensional sensitivity analysis probes the algorithm’s stability boundaries.

The results, detailed across Tables 8–14, reveal a remarkable finding: the algorithm’s effectiveness in reducing model uncertainty remains highly stable and consistently superior despite the concurrent changes in both scale and input parameters.

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Table 8. Algorithm Performance with Increasing AND Gates and Varying Centroid Values (centroid value of 0.2).

https://doi.org/10.1371/journal.pone.0335759.t008

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Table 9. Algorithm Performance with Increasing AND Gates and Varying Centroid Values (centroid value of 0.3).

https://doi.org/10.1371/journal.pone.0335759.t009

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Table 10. Algorithm Performance with Increasing AND Gates and Varying Centroid Values (centroid value of 0.4).

https://doi.org/10.1371/journal.pone.0335759.t010

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Table 11. Algorithm Performance with Increasing AND Gates and Varying Centroid Values (centroid value of 0.5).

https://doi.org/10.1371/journal.pone.0335759.t011

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Table 12. Algorithm Performance with Increasing AND Gates and Varying Centroid Values (centroid value of 0.6).

https://doi.org/10.1371/journal.pone.0335759.t012

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Table 13. Algorithm Performance with Increasing AND Gates and Varying Centroid Values (centroid value of 0.7).

https://doi.org/10.1371/journal.pone.0335759.t013

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Table 14. Algorithm Performance with Increasing AND Gates and Varying Centroid Values (centroid value of 0.8).

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The data is synthesized in S17 Fig, which visually consolidates the results from all tested scenarios.

The most significant conclusion from this comprehensive analysis is the narrow range of performance variation. As the system scales up in complexity and the input parameters vary widely, the model uncertainty reduction effect remains tightly bounded between 9.23% and 11.33%. This indicates that the algorithm’s advantage is not contingent on a specific system configuration or input value. It delivers reliable, high-quality results across a broad spectrum of conditions.

This exceptional stability demonstrates that whether dealing with a few or many AND events, and whether those events have high or moderate probability, the algorithm consistently controls uncertainty propagation. This makes it particularly valuable for modeling large, complex systems where the exact number of components and their failure probabilities may be subject to uncertainty, providing “robust theoretical support” for safety-critical decision-making.

Robustness verification under input uncertainties

Methodology: Probability perturbation approach.

To evaluate robustness, we selected representative events from the OREDA dataset with high, medium, and low failure probabilities: E4 (central value: 0.672, high probability), E5 (central value: 0.503, medium probability), and E7 (central value: 0.008, low probability).

For each event, random perturbations of ±5%, ± 10%, and ±15% were applied to its central probability value while preserving the trapezoidal fuzzy number structure. At each perturbation level, 20 independent trials were conducted using different random seeds. The Uncertainty Reduction (UR) metric was recorded for each trial, and statistical analysis (mean, standard deviation, range) was performed to quantify robustness.

The complete dataset comprising 60 perturbation tests, including all input values and corresponding UR results for events E4, E5, and E7, is provided in S1, S2, S3, S4, S5, S6, S7, S8, S9 Tables to ensure transparency and reproducibility.

Results for high-probability events (E4).

The uncertainty reduction performance for the high-probability event E4 under different perturbation levels is presented in Table 15.

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Table 15. Uncertainty reduction performance under probability perturbations for event E4.

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The algorithm demonstrated stability under mild perturbations (±5%). As the perturbation magnitude increased, the variability in results also increased, yet remained within acceptable limits. The algorithm exhibited adaptability: output fluctuations grew with larger input perturbations, but without performance collapse. A positive correlation exists between perturbation level and output variability. Notably, even under ±15% perturbation, the standard deviation remained relatively low.

Results for medium-probability events (E5).

Table 16 presents the uncertainty reduction performance for the medium-probability event E5, which demonstrated high stability across all perturbation levels.

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Table 16. Uncertainty reduction performance under probability perturbations for event E5.

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The algorithm showed consistent stability across all perturbation levels for event E5, indicating strong robustness for medium-probability events. It adapted well to perturbations, with output variability increasing only slightly as the perturbation magnitude grew, remaining within a small range. The positive correlation between perturbation level and output variability was observed, but the standard deviation stayed low (e.g., 0.0835% at ±15%). While the distribution of results became slightly more dispersed with higher perturbations, it maintained good consistency and symmetry. These results underscore the algorithm’s excellent capability in handling medium-probability events, which are often the most common fault type in practical engineering applications.

Results for low-probability events (E7).

The results for the low-probability event E7 (Table 17) reveal the algorithm’s exceptional robustness.

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Table 17. Uncertainty reduction performance under probability perturbations for event E7.

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For the low-probability event E7, the algorithm exhibited exceptional stability. Under input perturbations ranging from ±5% to ±15%, the mean UR values remained tightly clustered between 5.230% and 5.234%, with extremely low standard deviations (< 0.00011). This indicates that the algorithm provides highly reliable and consistent results even for low-probability events, which are often challenging to measure precisely, demonstrating very strong robustness.

Comparative stability analysis.

The algorithm is not entirely insensitive to input perturbations, but its response is moderate and largely linear. The fluctuation in output results is significantly smaller than the input perturbation (e.g., a ± 15% input perturbation resulted in less than ±1% output fluctuation). This demonstrates the algorithm’s effective inherent capability to suppress the propagation of input uncertainty, meeting the robustness requirements for industrial applications.

The algorithm’s robustness is not uniform across all inputs; instead, it intelligently adjusts its sensitivity based on the event’s intrinsic importance within the system, as reflected by its base probability. This characteristic is a significant advantage for practical engineering applications. It implies that careful data collection remains crucial for high-probability components, while the algorithm can tolerate greater data uncertainty for numerous low-probability components. This effectively reduces the data acquisition and precision requirements, thereby lowering the overall cost of system modeling.

Conclusions

This study has introduced a novel computational algorithm designed to tackle the persistent challenge of model uncertainty in trapezoidal fuzzy fault tree analysis. By leveraging the cut-set theorem and operating directly on trapezoidal fuzzy numbers without intermediate defuzzification, the proposed method effectively preserves information integrity and minimizes distortion throughout the uncertainty propagation process. The algorithm provides a generalized framework applicable to fault trees with OR-gates, AND-gates, and mixed logics.

The primary contribution of this work is the development of a precise computational framework that addresses a critical gap in existing methodologies. Comprehensive validation demonstrates the algorithm’s significant advantages. In the benchmark case study, it achieved a superior model uncertainty reduction of 45.25% compared to 36.65% from existing methods, while maintaining a 99.40% consistency with established precise results, confirming its accuracy. Furthermore, extensive simulations based on OREDA data robustly verified its stability and scalability. The algorithm exhibits a “graceful scaling” property, where the benefit of uncertainty reduction becomes more pronounced as system complexity increases (e.g., from 2.40% to 7.49% with growing basic events in OR-gate trees).

A pivotal finding is the algorithm’s exceptional robustness under input uncertainties. The dedicated perturbation analysis (60 tests across events with high, medium, and low probabilities) revealed that output variations remain remarkably small (<±1%) even under significant input perturbations (±15%). This controlled sensitivity, which is intelligently aligned with an event’s base probability (higher robustness for low-probability events), is a key practical advantage. It implies that rigorous data collection can be focused on critical high-probability components, while the algorithm reliably tolerates greater uncertainty for numerous low-probability ones, thereby reducing the overall cost and effort of system risk modeling.

In conclusion, this algorithm offers not only a theoretical advancement in managing model uncertainty but also a practical, robust, and efficient tool for risk assessment in complex engineering systems. Future work will focus on optimizing the computational efficiency for large-scale fault trees and exploring integrations with dynamic models like Bayesian networks to handle time-evolving uncertainties.

Supporting information

Acknowledgments

The authors are grateful to Prof. Z for insightful discussions on specific method.