Fuzzy Math Gets Sharper: New Decision Tool Tames Uncertainty in Supplier Choices

Every major manufacturing decision is an exercise in judging the unknowable. When a company chooses a supplier, it weighs cost against reliability, delivery speed against environmental performance, and technical capability against price, all while the data itself is murky, incomplete, and contested by experts who disagree with one another. A new study published in the International Journal of Machine Learning and Cybernetics tackles this messiness head-on, introducing a mathematical framework that can absorb several layers of uncertainty at once and still deliver a defensible ranking of alternatives. The work, authored by Lin Yan of Tongji University in Shanghai, combines an extended form of fuzzy set theory with two established decision-analysis techniques to create what the author calls a PIVDHF-CRITIC-TOPSIS framework, and demonstrates it on a real-world problem: selecting digital suppliers for a Chinese new energy vehicle manufacturer.

To understand why this matters, it helps to trace the lineage of the underlying mathematics. Fuzzy sets, introduced by Lotfi Zadeh in 1965, replaced the rigid yes-or-no logic of classical set theory with degrees of membership, allowing an element to belong to a set partially. Hesitant fuzzy sets, proposed by Vicenç Torra in 2010, went a step further by permitting several possible membership values for a single element, capturing the situation where a panel of experts offers conflicting assessments rather than a single consensus figure. Dual hesitant fuzzy sets then added a second dimension: alongside multiple possible membership degrees, they allow multiple possible non-membership degrees, so an evaluator can simultaneously express hesitation about how well an option performs and hesitation about how badly it fails.

The new paper builds on two prominent extensions of that dual structure. Probabilistic dual hesitant fuzzy sets attach probability weights to each possible membership and non-membership value, acknowledging that some expert opinions are more likely or better supported than others. Interval-valued dual hesitant fuzzy sets, by contrast, replace single numbers with intervals, which is useful when an assessor can only bound a judgment rather than pin it down. Each extension captures something the other misses. Probabilities encode the relative credibility of individual assessments; intervals encode the inherent imprecision of the assessments themselves. Yan’s contribution is the probabilistic interval-valued dual hesitant fuzzy set, or PIVDHFS, which fuses both properties into a single representation. In a PIVDHFS, each element carries a set of interval-valued membership degrees and a set of interval-valued non-membership degrees, and each of those interval values is associated with a probability. The result is a data structure that can model disagreement among experts, imprecision within each expert’s judgment, and the unequal reliability of different judgments, all simultaneously.

Representing information this way is only useful if you can compute with it, and that requires a way to measure how different two PIVDHFSs are. Distance measures are the workhorses of fuzzy decision theory: they underpin ranking methods, clustering, pattern recognition, and medical diagnosis applications. The paper defines the operational laws of PIVDHFSs and then derives a family of distance formulas. The probabilistic interval-valued dual hesitant fuzzy weighted distance treats each attribute with a fixed importance weight. The ordered weighted distance instead assigns weights to positions in an ordered sequence, so that the largest deviations can be emphasized or downplayed depending on the decision-maker’s risk posture. The hybrid weighted distance combines both ideas, weighting attributes and positions at the same time, and the generalized hybrid weighted distance adds a tunable parameter that lets the analyst adjust how aggressively large differences are penalized. Together these measures give decision-makers a flexible toolkit for comparing alternatives described in this rich uncertain format.

A second, often overlooked problem in multi-attribute decision-making is determining how much each criterion should matter in the first place. When attribute weights are unknown, subjective assignment invites bias, so objective weighting methods infer importance directly from the data. The study enhances one such method, CRITIC, which stands for criteria importance through intercriteria correlation. The original CRITIC, introduced by Diakoulaki and colleagues in 1995, assigns higher weights to criteria that carry more information, measured by standard deviation, and that are less redundant with other criteria, measured by correlation. Yan’s version incorporates the Spearman correlation coefficient, a rank-based measure of association, within the PIVDHFS context. Rank correlation is more robust than the Pearson coefficient when the underlying data are ordinal or non-linear, which is precisely the character of hesitant fuzzy assessments. The enhanced CRITIC therefore extracts objective weights from the structure of the probabilistic interval-valued data itself, reducing the influence of arbitrary human judgment on the weighting stage.

With distances and weights in hand, the framework turns to ranking. TOPSIS, the technique for order of preference by similarity to ideal solution, is one of the most widely used methods in operations research. Its logic is intuitive: the best alternative is the one that is simultaneously closest to a hypothetical positive ideal solution and farthest from a negative ideal one. The paper extends TOPSIS to operate directly on PIVDHFSs, using the newly developed distance measures to compute each alternative’s separation from the ideal and anti-ideal points. The extended PIVDHF-CRITIC-TOPSIS pipeline thus proceeds in three stages: encode expert evaluations as PIVDHFSs, derive objective attribute weights with the Spearman-enhanced CRITIC method, and rank alternatives by their relative closeness computed through the new distance family.

The framework was put to the test on a problem with genuine industrial stakes: digital supplier selection for a new energy vehicle manufacturer in China. Digital suppliers provide the software platforms, data services, and connected-technology capabilities that modern electric vehicle production increasingly depends on, and choosing among them involves criteria that are hard to quantify precisely, such as technological maturity, data security posture, service responsiveness, and long-term partnership potential. Evaluations from multiple experts naturally conflict, and confidence in each assessment varies. The PIVDHFS format captured this layered uncertainty, and the extended framework produced a complete ranking of candidate suppliers. The author validated the result in two ways. A sensitivity analysis varied key parameters, including the generalized distance parameter and the derived attribute weights, to check whether the ranking was an artifact of particular settings. A comparative study then applied alternative decision methods to the same data to confirm that the proposed approach yields consistent and scientifically rational outcomes.

What makes this work notable in the broader landscape of decision science is its position at the confluence of several research threads that had previously advanced separately. Probabilistic dual hesitant fuzzy sets have been applied to risk evaluation, sustainable supplier selection, and rainfall analysis, while interval-valued dual hesitant fuzzy sets have supported emergency response evaluation and aggregation-based decision models. Distance measures for dual hesitant structures have found uses ranging from image segmentation to medical diagnosis. By unifying the probabilistic and interval-valued extensions and wiring them into an objective weighting scheme and a classical ranking method, the new framework offers a template that other researchers can adapt. The same architecture could, in principle, support green supplier selection, renewable energy assessment, medical diagnosis, or any domain where expert judgment is hesitant, imprecise, and unevenly reliable.

The practical implications extend to the fast-growing field of digital supply chain management. Recent literature on supplier selection in digital manufacturing has emphasized resilience, cyber risk, and data-driven simulation, and machine learning approaches have been proposed for modeling supplier performance under disruption. Yet the front end of those pipelines, the structured elicitation of expert judgment about candidates, still relies heavily on decision frameworks that assume crisp numbers or single-layer fuzziness. A representation that preserves probabilities, intervals, and dual hesitation through every stage of weighting and ranking could feed cleaner, more honest inputs into downstream analytics. For industries such as electric vehicles, where supply chains span continents and technology cycles are measured in months, the difference between a robust ranking and a fragile one can translate into millions in procurement outcomes.

There are, of course, limits worth noting. The framework’s sophistication comes at the cost of computational and elicitation overhead: asking experts to supply interval-valued memberships, non-memberships, and probabilities for every attribute of every alternative is demanding, and the quality of the output depends on the quality of those inputs. The study itself reports validation through a single real-world case, supplemented by sensitivity and comparative analyses, which is a solid but not exhaustive evidence base. Still, the paper represents a careful piece of mathematical engineering, defining operational laws, deriving a graded family of distance measures, enhancing an objective weighting method with a rank correlation coefficient, and demonstrating the whole pipeline on a decision that manufacturers actually face. As uncertain, expert-mediated judgments continue to pervade high-stakes industrial and environmental choices, tools that quantify hesitation rather than papering over it are likely to become standard equipment in the decision scientist’s toolkit.

Subject of Research: Multi-attribute decision-making using probabilistic interval-valued dual hesitant fuzzy sets with distance measures and an extended TOPSIS framework

Article Title: Multi-attribute decision-making based on probabilistic interval-valued dual hesitant fuzzy sets and extended TOPSIS

Article References: Yan, L. (2026). Multi-attribute decision-making based on probabilistic interval-valued dual hesitant fuzzy sets and extended TOPSIS. International Journal of Machine Learning and Cybernetics, 17(10), Article 486. https://doi.org/10.1007/s13042-026-03294-z

Image Credits: AI Generated

DOI: 10.1007/s13042-026-03294-z

Keywords: fuzzy sets, multi-attribute decision-making, TOPSIS, CRITIC method, distance measures, dual hesitant fuzzy sets, supplier selection, uncertainty modeling, operations research, new energy vehicles, Spearman correlation, decision science