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Bipolar Linguistic Z-Number Framework for FMEA Integrating BWM, CRITIC, and MABAC

This paper introduces a Bipolar Linguistic Z-Number (BLZ-Number) framework that unifies positive/negative assessments with linguistic reliability to enhance Failure Mode and Effects Analysis (FMEA) through a novel integration of BLZ-BWM, BLZ-CRITIC, and MABAC methods for more expressive and reliable risk evaluation.

Original authors: Rıdvan Şahin, Elif Başkan, Ahmet Emir Köse

Published 2026-07-13

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CC BY 4.0

Original authors: Rıdvan Şahin, Elif Başkan, Ahmet Emir Köse

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

=== DRAFT ===
Imagine you are trying to rank the dangers of a giant, clanking crane. In the old days, experts would just say, "That part is very dangerous," or "That part is somewhat dangerous." But real life is messy. An expert might think, "Well, this part could crush a worker (that's a high severity rating!), but it's also usually very sturdy (that's a low severity rating!)." Plus, they might be 90% sure about their guess, or only 50% sure.

The paper by Rıdvan Şahin, Elif Başkan, and Ahmet Emir Köse argues that the old way of thinking is like trying to describe a storm using only a single word like "wet." It misses the wind, the thunder, and how hard the rain is actually falling.

The New Tool: The "Bipolar Linguistic Z-Number"

The authors invented a new kind of information package called a Bipolar Linguistic Z-Number (BLZ-Number). Think of this as a "three-part secret code" for every risk:

The Primary Assessment: The expert's main evaluation of the risk (e.g., "It is highly severe!").
The Counter-Assessment: The expert's opposing or negative evaluation of that same risk (e.g., "But it is also somewhat stable!").
The Confidence Meter: How sure is the expert giving the report? (e.g., "I'm pretty sure," or "I'm just guessing.")

In the old models, you could have the confidence meter, but you couldn't easily hold the "primary view" and the "opposing view" at the same time without them canceling each other out. This new model keeps them side-by-side, like a tug-of-war where you can see exactly how hard both sides are pulling, and how strong the referee (the expert) is.

The Three-Step Detective Game

To figure out which crane parts are the most dangerous, the authors didn't just guess. They built a three-step machine to process these "three-part codes":

Step 1: The Subjective Detective (BLZ-BWM)
First, they asked experts to compare the importance of different risk factors: Severity (how bad the injury is), Occurrence (how often it happens), and Detectability (how easy it is to spot).

The Twist: Instead of just saying "Severity is 10 times worse than Detectability," the experts gave their answers in the new three-part code.
The Result: The machine calculated that Severity is the most important factor, followed by Occurrence, then Detectability. But it didn't just give a simple number; it kept track of the experts' internal conflicts and how sure they were.

Step 2: The Objective Detective (BLZ-CRITIC)
Next, the machine looked at the actual data without asking anyone for their opinion. It asked: "Which factor changes the most between different parts? Which factor is the most unique?"

The Twist: This step measures how much "noise" and "conflict" is in the data. It found that Detectability was actually the most informative factor because the data varied so wildly on that point.
The Result: This gave a different ranking than the experts did.

Step 3: The Final Showdown (BLZ-MABAC)
Finally, the machine combined the expert opinion (Step 1) and the data facts (Step 2) into a single score. It used a method called MABAC, which is like checking if a failure mode is standing above or below a "border line" of safety.

The Result: The machine ranked 13 different hazards. The winner (the most dangerous) was H11 (related to molten metal and burning). The loser (the least dangerous) was H12 (related to smoke).

What the Paper Says (and What It Doesn't)

The authors ran this whole process on a real-world crane case study (using data from a previous study by Akram et al.). They found that their new method produced a ranking that was positively correlated with other methods, but not identical.

The Correlation: When they compared their ranking to other famous methods, the agreement was moderate. For example, compared to a method by Krohling et al., the agreement score was 0.4670. Compared to the original source study (Z-FMEA), the agreement was lower at 0.1923.
The Takeaway: The paper suggests this isn't a mistake. Because their method looks at "positive vs. negative" assessments and "confidence" simultaneously, it naturally sorts the middle-ranked items differently than methods that only look at a single "danger score." The top danger (H11) and the bottom danger (H12) stayed the same, but the middle ones shuffled around.

Testing the Engine

To see if their machine was sturdy, they played with a "balance knob" (called β).

If they turned the knob all the way to 0, they used only the data (Objective).
If they turned it all the way to 1, they used only the experts (Subjective).
They found that if they went to the extremes, the rankings got weird (like a low-risk part suddenly jumping to the top). But when they kept the knob in the middle (0.5), the results were stable and made the most sense.

What This Means

The paper doesn't claim to have "solved" all risk analysis forever. Instead, it suggests that by using this new Bipolar Linguistic Z-Number framework, we can capture the messy, two-sided nature of human thinking better than before. It allows us to say, "This part is risky, but we have conflicting evidence, and we are only 70% sure," all in one package.

The authors admit this is a simulation based on existing data, not a new experiment with real people. They suggest that in the future, this tool could be tested on bigger problems like healthcare or supply chains, but for now, it's a solid proof-of-concept that shows a more nuanced way to handle the uncertainty of the real world.

Technical Summary: Bipolar Linguistic Z-Number Framework for FMEA Integrating BWM, CRITIC, and MABAC

Problem Statement
Real-world decision-making in Failure Mode and Effects Analysis (FMEA) involves complex uncertainties, including vagueness, reliability uncertainty, and the coexistence of positive and negative assessments. While classical FMEA relies on the Risk Priority Number (RPN), it often fails to capture linguistic uncertainty and assumes equal criterion importance. Existing extensions using Z-numbers address reliability by pairing an assessment with a confidence level, and bipolar linguistic models address the duality of expert judgment (supportive vs. opposing evidence). However, these approaches have largely developed separately. Current models typically capture either reliability or polarity, but rarely both within a single information object. This creates a representational gap where expert judgments containing positive content, negative content, and varying confidence cannot be systematically retained in a unified framework. Furthermore, existing FMEA studies often lack a unified structure where representation, weighting, and ranking follow the same semantic logic.

Methodology
The study proposes a unified framework based on the Bipolar Linguistic Z-Number (BLZ-Number), which integrates three distinct dimensions: a positive linguistic assessment, a negative linguistic assessment, and a linguistic reliability term. The methodology is structured into three main phases:

Theoretical Foundation (BLZ-Number):
- Definition: A BLZ-Number is defined as $\tilde{Z}_{abc} = \langle (s_a^A, s_b^A), s_c^R \rangle$ , where $(s_a^A, s_b^A)$ is a bipolar linguistic pair (positive and negative terms) and $s_c^R$ is the reliability term.
- Operations: The framework establishes linguistic scale functions, score, accuracy, and contradiction functions to compare BLZ-Numbers. It defines algebraic operations (addition, multiplication, scalar multiplication) and aggregation operators (BLZWA and BLZWG) that preserve the bipolar and reliability structure.
- Distance Measures: Generalized distance measures (Hamming and Euclidean) are defined in the semantic space to quantify the deviation between BLZ-Numbers.
Hybrid Weighting Mechanism:
- Subjective Weighting (BLZ-BWM): The Best-Worst Method is extended to the BLZ environment. It utilizes two parallel optimization channels: a positive channel (standard BWM logic) and a negative channel (reversed BWM logic where the worst criterion acts as the reference-best for negative concern). This yields latent BLZ-valued weights containing positive importance, negative importance, and reliability, which are later defuzzified into crisp weights.
- Objective Weighting (BLZ-CRITIC): The CRITIC method is adapted to incorporate reliability and internal contradiction. It calculates criterion importance based on three components: contrast intensity (standard deviation of exact values), inter-criterion conflict (correlation), and a new local BLZ credibility coefficient. This coefficient measures the "reliable clarity" of an evaluation by penalizing high reliability when internal contradiction (simultaneous high positive and negative evidence) is present.
- Integration: Subjective and objective weights are combined using a multiplicative integration rule controlled by a balance parameter $\beta$ .
Ranking Mechanism (BLZ-MABAC):
- The Multi-Attributive Border Approximation Area Comparison (MABAC) method is adapted to the BLZ environment. Instead of comparing alternatives to an ideal solution, the method calculates the distance of each weighted BLZ evaluation from a BLZ border approximation value (derived via geometric aggregation).
- A signed border deviation is calculated, indicating whether a failure mode lies above or below the border region. The final risk index is the sum of these signed deviations.

Numerical Application and Results
The framework was applied to a crane-operation FMEA case study adapted from a previous Z-number-based study (Akram et al.). The problem involved 13 failure modes and 3 risk factors (Severity, Occurrence, Detectability) evaluated by three experts.

Weighting Results: The subjective BLZ-BWM identified Severity as the most important criterion. The objective BLZ-CRITIC identified Detectability as the most informative criterion due to its high dispersion and conflict contribution. The integrated weights balanced these perspectives, with Severity remaining dominant but Occurrence and Detectability becoming nearly equal.
Ranking Results: The final ranking identified failure mode H11 (molten metal ground) as the highest risk, followed by H3 and H8.
Sensitivity Analysis: Varying the integration parameter $\beta$ showed that while the top-ranked hazard (H11) remained stable in the balanced region ( $\beta \in [0.25, 0.50]$ ), extreme settings (purely subjective or objective) led to significant shifts in the ranking of middle-tier hazards.
Comparative Analysis: The proposed ranking showed positive but moderate Spearman rank correlations with benchmark methods (including the source Z-FMEA, TOPSIS, and VIKOR-based approaches). The correlation was highest with ZE-BWM-MABAC and lowest with the source Z-FMEA, indicating that the proposed model generates a distinct prioritization pattern due to its bipolar and contradiction-aware logic.

Key Contributions

BLZ-Number Definition: The introduction of a new information object that unifies bipolar linguistic assessment (positive and negative evidence) with linguistic reliability, allowing for the simultaneous representation of support, opposition, and confidence.
Methodological Extensions: The development of BLZ-BWM (dual-channel optimization for subjective weights) and BLZ-CRITIC (incorporating reliability-aware clarity and contradiction into objective weights).
Integrated Framework: The construction of a complete FMEA architecture where representation, weighting, and ranking are semantically aligned under the BLZ logic, moving beyond simple Z-number extensions.
Contradiction Sensitivity: The ability of the model to distinguish between clear dominant judgments and highly conflicting evaluations, a feature absent in standard unipolar Z-number models.

Significance and Claims
The authors claim that the proposed framework addresses a critical representational gap in risk analysis by preserving the "directional duality" of expert judgment alongside reliability. Unlike previous models that reduce expert opinion to a single-axis assessment or treat reliability and polarity separately, this framework maintains all three dimensions (positive, negative, reliability) throughout the decision process.

The study posits that the moderate rank correlations with existing methods are not a sign of instability but a natural consequence of fundamentally changing the information structure, weighting logic, and ranking mechanism simultaneously. The framework is presented as a more expressive and interpretable alternative for complex risk analysis, particularly in scenarios where expert judgments involve internal conflict or contradictory evidence. The authors conclude that the model offers a semantically aligned decision architecture for reliability-aware bipolar risk prioritization, though they acknowledge the need for careful implementation regarding scale functions and parameter selection.