Researchers have introduced a new mathematical framework designed to help large industrial facilities rank safety response strategies when the information available about hazards is incomplete, ambiguous, or imprecise. The approach, described in a study published on 3 November 2026 in the open-access journal Discover Analytics, combines a newly defined type of fuzzy set with a well-established multi-criteria decision-making method to evaluate dozens or even hundreds of candidate response plans against multiple, often conflicting, safety criteria.
The core of the work is the Circular Complex Fermatean Fuzzy Set, or CrC-FFS, a structure the authors say extends existing complex Fermatean fuzzy sets and circular Fermatean fuzzy sets. Fuzzy sets, first introduced by Zadeh, provide a way to model uncertainty and imprecision mathematically, and successive extensions such as intuitionistic, Pythagorean, and Fermatean fuzzy sets have progressively increased the flexibility of these models. The new framework goes further by simultaneously characterizing membership, abstinence, and non-membership grades through complex-valued representations while adding a circular radius that captures an additional degree of uncertainty. The structure satisfies the Fermatean constraint, meaning that the sum of the powers of the real and imaginary parts of the membership, abstinence, and non-membership grades does not exceed the unit interval.
According to the study, industrial safety management has become increasingly challenging in modern large-scale systems because of the growing complexity of manufacturing processes, interconnected operational environments, diverse hazard sources, and the presence of uncertain, incomplete, ambiguous, and imprecise information. Effective safety planning and emergency response require frameworks capable of simultaneously evaluating multiple conflicting criteria, including hazard severity, operational reliability, response effectiveness, worker safety, environmental impact, resource allocation, system resilience, and mitigation efficiency. Conventional safety management approaches often depend on deterministic risk assessments and precise numerical evaluations, which the authors argue may not adequately capture ambiguous hazard levels and uncertain emergency conditions.
To manipulate information within the new framework, the researchers established refined algebraic operational laws, including direct sum, direct product, and scalar multiplication based on generalized t-norms and t-conorms. Building on these laws, they developed four aggregation operators: the Circular Complex Fermatean Fuzzy Weighted Arithmetic Mean (CrC-FFWAM), the ordered weighted arithmetic mean (CrC-FFOWAM), the weighted geometric mean (CrC-FFWGM), and the ordered weighted geometric mean (CrC-FFOWGM). These operators fuse uncertain multi-criteria safety information and expert judgments into single values that can be compared. The study reports that the fundamental mathematical properties of the operators, including boundedness, idempotency, and monotonicity, were rigorously established to demonstrate their validity, consistency, and suitability for information aggregation.
The aggregation operators are then incorporated into a multigranulation-based multi-criteria decision-making framework built around the Weighted Aggregated Sum Product Assessment (WASPAS) method. WASPAS, proposed by Zavadskas and colleagues, combines the advantages of the Weighted Sum Model and the Weighted Product Model, and the authors note that integrating additive and multiplicative aggregation strategies generally provides higher ranking accuracy and greater stability than either method alone. In the proposed procedure, experts provide assessments of each alternative against each criterion, cost criteria are converted into benefit criteria during normalization, expert opinions are aggregated using the arithmetic and geometric operators, and a balancing coefficient beta determines the relative contributions of the two aggregation models to the final utility value. The alternative with the highest utility value is considered the optimal choice.
To validate the methodology, the team applied it to a practical case study involving large-scale industrial hazard-risk assessment and safety response optimization. In the first problem, seven evaluation criteria were used to rank 27 industrial safety response strategies, with the resulting ordering reported as C7, C4, C2, C3, C1, C5, and C6 in descending order of preference. In a second, much larger problem, 200 evaluation criteria were applied to 100 decision alternatives, with computations performed using MATLAB code that evaluates the weighted sum and weighted product components systematically and combines them into final utility values. Under one parameter setting with beta equal to 0.1 and lambda equal to 1, the largest utility value was 0.3087, occurring at case C471, while under another setting with beta equal to 0.1 and lambda equal to 2, the largest value was 0.6502 at case C166. The complete dataset used in the study is publicly available in the Zenodo repository, which the authors say ensures transparency and reproducibility of the experimental results.
Comparative analyses against existing approaches, including Fermatean fuzzy sets proposed by Senapati and Yager, complex Fermatean fuzzy sets from Akram and colleagues, a circular Fermatean fuzzy framework from Luo, and related methods, showed that although the best alternative remained identical across all competing approaches, the obtained score values and ranking patterns differed considerably. The authors attribute this to the fact that existing operators primarily aggregate information using only membership and non-membership grades, ignoring the additional uncertainty represented by the circular radius and leading to partial information loss during aggregation. By simultaneously incorporating membership, non-membership, complex-valued information, and radius parameters, the proposed framework preserves a richer representation of uncertainty and produces more discriminative rankings while maintaining consistency in identifying the optimal solution.
Sensitivity investigations examined how the results respond to different values of the decision parameters lambda and beta. The study reports that the score values of the alternatives change moderately with different combinations of the two parameters, reflecting the flexibility of the parameterized aggregation mechanism, but that the overall ranking pattern remains highly stable. In particular, the highest-ranked alternatives consistently preserve their leading positions across all tested parameter settings. The authors argue that this stability confirms the framework is robust against parameter variations and that the inclusion of lambda and beta gives decision-makers additional flexibility to adjust aggregation behavior according to practical requirements without compromising the credibility of the final ranking.
The study also addresses computational scalability, noting that each stage of the procedure involves only elementary arithmetic and algebraic operations on the fuzzy components, so the overall computational effort grows polynomially with the number of alternatives, evaluation criteria, and decision-makers. Because the aggregation, score computation, and ranking procedures for different alternatives are mutually independent, the authors state that the framework is naturally suitable for parallel and distributed computing environments, enabling efficient implementation for large industrial safety datasets. They add that the richer mathematical representation introduces only a modest computational overhead without significantly affecting overall efficiency.
The authors conclude that the proposed framework offers a flexible, robust, intelligent, and reliable decision-support tool for industrial safety management and hazard-response optimization under uncertain environments, contributing to resilient industrial operations, improved risk mitigation, and informed strategic decision-making. They suggest future extensions to large-scale group decision-making with dynamic and heterogeneous environments, integration with objective and hybrid weighting techniques such as CRITIC, Entropy, FUCOM, and machine learning-based weighting schemes, combination with artificial intelligence, big data analytics, and digital twins for real-time decision support, and generalization to other decision-making models including COPRAS, MARCOS, CoCoSo, EDAS, CODAS, TODIM, PROMETHEE, and VIKOR.
Subject of Research: A new circular complex Fermatean fuzzy decision-making framework for optimizing industrial safety response strategies under uncertain hazard risk conditions
Article Title: A robust multigranulation-based circular complex Fermatean fuzzy framework for large-scale optimization of industrial safety response strategies under uncertain hazard risk conditions
Article References: Ullah, K., Rehman, N., Movaheedi, Z., & Ali, A. (2026). A robust multigranulation-based circular complex Fermatean fuzzy framework for large-scale optimization of industrial safety response strategies under uncertain hazard risk conditions. Discover Analytics, 4(1), Article 47. https://doi.org/10.1007/s44257-026-00096-z
Image Credits: AI Generated
DOI: 10.1007/s44257-026-00096-z
Keywords: industrial safety, fuzzy sets, Fermatean fuzzy, multi-criteria decision-making, WASPAS, hazard risk, uncertainty modeling, aggregation operators, emergency response, risk assessment, Discover Analytics, decision support
News Source: Denise Maddox. (November 5, 2026). New Fuzzy Math Framework Ranks Industrial Safety Strategies Under Uncertain Hazard Risks. Scienmag.
