Fuzzy Analytic Hierarchy Process (FAHP)

• FAHP is an extension of AHP that uses fuzzy numbers, typically triangular fuzzy numbers, to model the uncertainty and vagueness inherent in expert pairwise comparisons.
• It employs methodologies like Chang’s extent analysis and Buckley’s fuzzy geometric mean to extract robust priority weights from fuzzy comparison matrices.
• Applications span diverse fields such as software effort estimation, AI model evaluation, supplier selection, and environmental assessment, demonstrating its practical impact.

Fuzzy Analytic Hierarchy Process (FAHP) is an extension of the classical Analytic Hierarchy Process designed to address the imprecision, vagueness, and subjectivity that arise when eliciting pairwise comparison judgments from experts or decision makers. Instead of crisp numbers, FAHP encodes these comparisons as fuzzy numbers—typically triangular fuzzy numbers (TFNs)—which allows the decision framework to propagate uncertainty from initial judgments through to final priority weights, producing more nuanced, robust, and context-aware multi-criteria decision analysis. FAHP has been applied in domains including software effort estimation, LLM evaluation, IoT-adoption risk, supplier selection, ethical risk in AI, network selection, and environmental assessment, among others (Sehra et al., 2013, He et al., 4 Apr 2026, Shahriar et al., 2023, El-Hawy, 2024, Dyoub et al., 28 Jul 2025, Lahby et al., 2012, Ayhan, 2013, Ashek-Al-Aziz et al., 2020, Inzmam et al., 20 Mar 2026, Zhang et al., 2023, Afolayan et al., 2021).

1. FAHP: Principles, Rationale, and Mathematical Foundations

FAHP generalizes Saaty’s original AHP by allowing all pairwise comparison values $a_{ij}$ to be represented as fuzzy numbers $\tilde a_{ij}$, usually TFNs $(l_{ij}, m_{ij}, u_{ij})$. This models the range of possible importance ratios, from a minimum plausible (pessimistic), through the most likely (modal), to a maximum plausible (optimistic) value, aligning with the inherently linguistic or uncertain way in which experts express relative preferences. The membership function for a TFN is: $$\mu_{\tilde a}(x) = \begin{cases} \dfrac{x-l}{m-l}, & l \le x \le m \ \dfrac{u-x}{u-m}, & m \le x \le u \ 0, & \text{otherwise} \end{cases}$$ Classical AHP requires crisp $1$–$9$ (Saaty) scores for each pair of criteria or alternatives. In contrast, FAHP accommodates expressions such as “Criterion A is between moderately and strongly more important than B,” encoded as a TFN such as $(3, 5, 7)$ (Sehra et al., 2013, Ashek-Al-Aziz et al., 2020).

The principal motivations for FAHP are:

• To explicitly embed epistemic and linguistic uncertainty into all stages of multi-criteria decision making;
• To allow optimistic, moderate, or pessimistic decision-maker attitudes to be reflected in the range of comparisons;
• To propagate vagueness in expert judgment through aggregation, weight extraction, and final ranking, yielding priorities more faithful to the input consensus (Sehra et al., 2013, He et al., 4 Apr 2026).

2. FAHP Workflow and Core Algorithms

Most FAHP implementations follow a sequence of hierarchical problem decomposition and fuzzy computation:

1. Problem Structuring: Construct a hierarchical model (Goal → Criteria → Alternatives), as in classical AHP, but explicitly define at each level whether comparisons will be crisp or fuzzy (Afolayan et al., 2021, Shahriar et al., 2023).
2. Fuzzy Pairwise Judgment Elicitation:

For each criterion or alternative pair $(i, j)$, experts specify a linguistic judgment (e.g., "strongly more important"), mapped onto a TFN using a predefined scale—e.g., - Equal: $(1, 1, 1)$ - Weak: $(1, 3, 5)$ - Strong: $\tilde a_{ij}$0 - Very Strong: $\tilde a_{ij}$1 - Extreme: $\tilde a_{ij}$2 Reciprocals are taken: $\tilde a_{ij}$3 (Sehra et al., 2013, Ayhan, 2013, Lahby et al., 2012).

1. Fuzzy Comparison Matrix Construction: Assemble the $\tilde a_{ij}$4 fuzzy reciprocal matrix $\tilde a_{ij}$5 for $\tilde a_{ij}$6 criteria, enforcing $\tilde a_{ij}$7 and $\tilde a_{ij}$8 (Sehra et al., 2013, Shahriar et al., 2023).
2. Fuzzy Weight Extraction:

The two most common algorithms are: - Chang’s Extent Analysis (Sehra et al., 2013, Ashek-Al-Aziz et al., 2020, Lahby et al., 2012):

For each criterion $\tilde a_{ij}$9, compute the fuzzy synthetic extent:

$(l_{ij}, m_{ij}, u_{ij})$0

For two TFNs $(l_{ij}, m_{ij}, u_{ij})$1, $(l_{ij}, m_{ij}, u_{ij})$2, compute the degree of possibility $(l_{ij}, m_{ij}, u_{ij})$3.

$(l_{ij}, m_{ij}, u_{ij})$4

Raw support $(l_{ij}, m_{ij}, u_{ij})$5; normalized to weights $(l_{ij}, m_{ij}, u_{ij})$6.

• Buckley’s Fuzzy Geometric Mean (Shahriar et al., 2023, Ayhan, 2013, Afolayan et al., 2021):

  Each row $(l_{ij}, m_{ij}, u_{ij})$7:

  $(l_{ij}, m_{ij}, u_{ij})$8

  Normalize TFNs to derive $(l_{ij}, m_{ij}, u_{ij})$9; defuzzify by centroid: $$\mu_{\tilde a}(x) = \begin{cases} \dfrac{x-l}{m-l}, & l \le x \le m \ \dfrac{u-x}{u-m}, & m \le x \le u \ 0, & \text{otherwise} \end{cases}$$0; normalize.

1. Aggregation and Final Ranking: Local (criterion) weights are multiplied by alternative weights and summed to get global priorities (He et al., 4 Apr 2026, Sehra et al., 2013, El-Hawy, 2024).
2. Consistency Verification: The consistency ratio (CR) is calculated on the matrix of TFN modal (midpoint) values.

$$\mu_{\tilde a}(x) = \begin{cases} \dfrac{x-l}{m-l}, & l \le x \le m \ \dfrac{u-x}{u-m}, & m \le x \le u \ 0, & \text{otherwise} \end{cases}$$1

Acceptable if $$\mu_{\tilde a}(x) = \begin{cases} \dfrac{x-l}{m-l}, & l \le x \le m \ \dfrac{u-x}{u-m}, & m \le x \le u \ 0, & \text{otherwise} \end{cases}$$2 or $$\mu_{\tilde a}(x) = \begin{cases} \dfrac{x-l}{m-l}, & l \le x \le m \ \dfrac{u-x}{u-m}, & m \le x \le u \ 0, & \text{otherwise} \end{cases}$$3, otherwise revision is required (Afolayan et al., 2021, Sehra et al., 2013).

3. Extensions: Type-2, Shadowed, and Multi-Granular FAHP

Recent work has expanded the FAHP formalism to handle richer types of uncertainty and granular input:

• Interval Type-2 FAHP (IT2-FAHP):

Pairwise comparisons are represented as trapezoidal interval type-2 fuzzy numbers (IT2FNs) with upper and lower membership functions, allowing modeling of both epistemic fuzziness and ambiguity due to class boundaries. This is particularly relevant for air quality assessment, where pollutant weights are uncertain near class breakpoints (Inzmam et al., 20 Mar 2026).

• Shadowed FAHP:

Shadowed fuzzy numbers (SFNs) generalize TFNs and other fuzzy numbers to four-point representations that retain both fuzziness and “non-specificity.” They provide a unified approach for multi-granular, heterogeneous preference data, combining scalars, intervals, TFNs, and even intuitionistic fuzzy inputs into a single calculus (El-Hawy, 2024).

• Confidence-Aware and Data-Fused FAHP:

Some approaches (notably for LLM evaluation) modulate the width of TFNs by explicit decision-maker or model-generated confidence scores, reducing interval width as confidence increases. Other work combines subjective, FAHP-derived weights with data-driven objective weights (e.g., from fuzzy clustering) in a convex combination to leverage both expert judgments and empirical sensitivity (He et al., 4 Apr 2026, Zhang et al., 2023).

4. Applications Across Domains

FAHP has been implemented in a variety of multi-criteria decision contexts:

• Software Effort Estimation:

FAHP captures the uncertainty in ranking such criteria as reliability, MMRE, prediction accuracy, and confidence, reversing the output ranking compared to classical AHP in at least one benchmark case (Sehra et al., 2013).

• LLM and AI Model Evaluation:

Incorporating LLM-generated confidence into the widths of comparison TFNs yields more calibrated and stable model assessments, improving over both direct scoring and crisp AHP (He et al., 4 Apr 2026).

• Supplier Selection and Contractor Appraisal:

FAHP addresses the ambiguity in evaluating supplier criteria (quality, delivery, after-sales, etc.) and is implemented with group feedback and web-based support for iterative consistency correction (Ayhan, 2013, Afolayan et al., 2021, El-Hawy, 2024).

• Ethical Risk in AI Systems:

The ff4ERA framework computes ethical risk scores as the product of FAHP-derived risk weights, certainty factors, and risk-level, producing interpretable and context-sensitive output (Dyoub et al., 28 Jul 2025).

• Network Selection and IoT Adoption:

FAHP is used to synthesize weights for network performance and adoption barriers, supporting robust prioritization of deployment factors in uncertain environments (Lahby et al., 2012, Shahriar et al., 2023).

• Environmental Quality Assessment:

IT2-FAHP assigns robust, uncertainty-aware importance weights to pollutants for air quality index computation, directly capturing hesitation and class-boundary fuzziness in policy and sensor-driven monitoring (Inzmam et al., 20 Mar 2026).

5. Empirical Results and Interpretive Context

Empirical application of FAHP robustly demonstrates:

• Rankings and priority weights may change substantially compared to crisp AHP. For instance, in software effort model selection, FAHP ranked COCOMO (algorithmic) models above fuzzy neural network models, inverting the crisp AHP output (Sehra et al., 2013).
• In LLM judgment tasks, FAHP consistently improves accuracy—by up to 1 percentage point over crisp AHP and more over direct scoring—especially when comparisons are ambiguous or exhibit low confidence (He et al., 4 Apr 2026).
• For group decision support (e.g., contractor selection, Delphi-FAHP for IoT risk), FAHP enables aggregation of anonymous, distributed, and multi-stage feedback, while preserving transparency and traceability of the full fuzzy-judgment chain (Shahriar et al., 2023, Afolayan et al., 2021).
• In settings with multi-granular, interval-valued, or intuitionistic fuzzy data, SFN-based “Shadowed AHP” or IT2-FAHP approaches allow fully uncertainty-preserving aggregation and provide stability near critical threshold boundaries (El-Hawy, 2024, Inzmam et al., 20 Mar 2026).
• Empirical comparisons between FAHP and classical AHP show that while the absolute weights may differ (with FAHP often spreading weights more widely), trend-wise fluctuations in ranking are often similar for ≈50% of cases. However, FAHP increases the robustness to inconsistent or linguistically vague data (Ashek-Al-Aziz et al., 2020).

6. Strengths, Limitations, and Consensus-Building

Strengths

• Captures Vagueness and Hesitation: Enables the full propagation of decision-maker uncertainty through the decision process.
• Supports Interval and Qualitative Judgments: Accommodates multi-granular and heterogeneous inputs (e.g., TFNs, intervals, IFNs).
• Improved Robustness: Reduces arbitrariness, dampens overconfidence, and permits explicit modeling of optimistic/pessimistic scenarios.
• Supports Feedback and Consensus: Well-suited for iterative, group-oriented, Delphi-style consensus-building, with built-in mechanisms for consistency checking and revision (Shahriar et al., 2023, Afolayan et al., 2021).

Limitations

• Computational Complexity: Higher computational cost in fuzzy arithmetic and possibility calculations; additional structural choices over membership function shapes and aggregation rules.
• Reliance on Judgment Consistency: Propagation of inconsistency in input can affect the reliability of outcomes; consistency verification is still essential, often on the defuzzified (modal) matrices (Sehra et al., 2013, Afolayan et al., 2021).
• Interpretation: The meaning of fuzzy weights may differ from crisp priorities, and the selection of defuzzification methods and ranking indices can vary between implementations.
• No Standard for Group/Aggregate Consistency in Fuzzy Domain: Generalization of CR measures and group feedback loops remains a topic of research (El-Hawy, 2024).

7. Future Directions and Research Challenges

Key research directions include:

• Higher-Order Fuzziness: Deployment of interval type-2, shadowed, and intuitionistic FAHP frameworks to further increase the expressiveness and reliability in contexts where both inter- and intra-expert uncertainty is significant (El-Hawy, 2024, Inzmam et al., 20 Mar 2026).
• Automated and Confidence-Modulated Fuzzification: Embedding confidence-driven interval shrinkage strategies for model-based or crowd-sourced group decision scenarios (He et al., 4 Apr 2026).
• Hybrid Aggregation and Data Fusion: Integrating subjective FAHP weights with empirical, data-driven or clustering weights, as done in VSRQ for vehicle risk, to enhance objectivity and adaptivity (Zhang et al., 2023).
• Fuzzy Consistency Indices: Research into direct, fuzzy-domain consistency metrics and automated advice/repair mechanisms to strengthen the reliability and interpretability of FAHP group consensus (Afolayan et al., 2021, El-Hawy, 2024).
• Sensitivity and Robustness Analysis: Systematic studies of output sensitivity to variation in TFN endpoints, defuzzification schemes, and confidence-interval choices remain crucial for robust real-world deployment (Dyoub et al., 28 Jul 2025).

FAHP thus represents a rigorously justified extension of the AHP paradigm, equipping decision-makers and systems with the critical capacity to reason under uncertainty, propagate subjective imprecision, and generate priorities that are maximally faithful to the complexity of human judgment and heterogeneous information sources across contemporary multi-criteria environments.