New Divergence Measures Bring Sharper Uncertainty Handling to Fuzzy Decision Making

When experts rate a candidate, a product, or a medical diagnosis, their judgments are rarely crisp yes-or-no answers. A scout evaluating a cricketer might say the batting is probably strong, the fielding is somewhat uncertain, and the temperament is mildly worrying, all at the same time. Classical fuzzy set theory was invented to capture such graded judgments, but real evaluations often carry several conflicting dimensions of doubt at once. A new study published in the Journal of Big Data by Muhammad Jabir Khan of Nantong University, Ahmad N. Al-Kenani of King Abdulaziz University, and Kanikar Muangchoo and Sakulbuth Ekvittayaniphon of Rajamangala University of Technology Phra Nakhon tackles exactly this problem, introducing a family of divergence measures for T-spherical fuzzy sets that remain mathematically well-behaved in situations where earlier formulas break down.

T-spherical fuzzy sets are a generalization that has been gaining traction because they let modelers record three independent degrees of membership: a positive degree expressing support for an alternative, a neutral or abstinence degree expressing indecision, and a negative degree expressing opposition. Where an ordinary fuzzy set records a single number between zero and one, and intuitionistic or picture fuzzy sets record two or three tightly constrained degrees, the T-spherical framework adds a tunable parameter t that controls how many graded levels each membership degree can take. This makes the framework expressive enough to represent panel opinions in which several experts disagree in structured ways, but it also raises a subtle mathematical question: how do you measure how different two such rich, multi-dimensional judgments are?

That question is answered by divergence measures, the information-theoretic cousins of distance functions. A divergence quantifies the discrepancy between two uncertain descriptions, and it underpins everything from clustering algorithms to decision rules. The most famous example, the Kullback-Leibler divergence, compares probability distributions by averaging the logarithm of their likelihood ratios. Researchers had already extended the Kullback-Leibler idea to T-spherical fuzzy sets, but the new paper documents a serious practical flaw in that extension: whenever any of the membership, non-membership, or abstinence components approaches zero, the logarithm drives the divergence toward an undefined value. Since zero evaluations are common in real-world rating data, the existing measure can simply fail to produce an answer precisely when practitioners need one.

A second existing measure, developed earlier by Huang and colleagues, suffers from the opposite problem. Its formulation is built on exponential terms, which means it produces unbounded positive values. Even small differences between two T-spherical fuzzy values can yield enormous divergence numbers, which makes interpretation difficult and can distort downstream rankings. In other words, the two available tools for measuring disagreement in this framework were either undefined at boundary cases or numerically explosive, leaving a genuine gap in the toolbox for anyone working with T-spherical fuzzy data.

The authors close that gap by constructing new divergence measures and then rigorously verifying their axiomatic properties. In the fuzzy-set literature, a divergence measure is not just any formula that outputs a number; it must satisfy axioms such as being zero only when the two sets are identical, being symmetric in its arguments, and behaving monotonically as the sets become more different. The paper carries out a thorough mathematical investigation to show that the proposed measures meet these requirements, ensuring theoretical soundness rather than merely plausible behavior. This axiomatic grounding matters because downstream algorithms inherit the reliability, or the flaws, of the divergence they are built upon.

From the new divergence, the team builds two practical engines. The first is a divergence-based version of TOPSIS, a widely used multi-criteria decision-making technique whose name stands for Technique for Order of Preference by Similarity to Ideal Solution. Classical TOPSIS ranks alternatives by their geometric closeness to an ideal option and their remoteness from an anti-ideal one. The new formulation replaces the usual distance computations with the proposed divergence, so the ranking of alternatives reflects information-theoretic disagreement between fuzzy evaluations rather than simple Euclidean separation. Alongside it, the authors introduce a fresh method for determining the weights of criteria when the underlying data are spherical fuzzy, addressing the perennial problem that some evaluation criteria matter more than others and that those weights are themselves uncertain.

The second application pushes the divergence into machine learning territory: the authors extend a pattern classification algorithm so that it can operate on T-spherical fuzzy descriptions. Classification with fuzzy divergence works by measuring how far an unclassified object’s fuzzy feature profile diverges from the profiles of known classes, assigning the object to the class with the smallest divergence. Because the new measures stay defined across the full range of possible evaluations, the classifier does not stumble on boundary cases that would crash or distort a Kullback-Leibler-based approach.

To demonstrate that the framework survives contact with reality, the researchers validate it through a case study on selecting an all-rounder cricketer for a test-match competition under uncertain and imprecise evaluation conditions. The scenario is a natural fit for the method: cricket all-rounders must be judged simultaneously on batting, bowling, and fielding, and scouts’ opinions about each skill are naturally expressed with degrees of agreement, hesitation, and disagreement. The T-spherical fuzzy representation captures those layered opinions, and the divergence-based TOPSIS method converts them into a defensible ranking of candidates.

Crucially, the authors do not stop at a single worked example. They conduct sensitivity analyses to check whether small perturbations in the fuzzy evaluations or the criterion weights destabilize the ranking, comparative analyses to benchmark the new measures against the existing Kullback-Leibler-style and exponential-style alternatives, and numerical experiments to probe behavior across parameter settings. The reported results confirm that the proposed framework is stable, effective, and superior to prior approaches in the tested scenarios, and that it enhances both the interpretability and the reliability of decisions made under T-spherical fuzzy conditions. Interpretability is the quieter but arguably more important gain: because the divergence values remain bounded and defined, decision makers can actually read the numbers they are given rather than wrestling with infinities or wildly scaled outputs.

The significance of the work extends beyond cricket selection boards. Multi-criteria decision making under uncertainty appears in supplier selection, medical diagnosis, environmental assessment, and recommender systems, and pattern classification with fuzzy features is relevant wherever expert-labeled data carry hesitation and dissent. By supplying divergence measures that are axiomatically sound and numerically robust across the entire evaluation range, the study gives researchers and practitioners a dependable foundation for building such systems on T-spherical fuzzy data. The paper, published open access on 24 September 2026 in the Journal of Big Data, was supported by Rajamangala University of Technology Phra Nakhon, and its authors declare no competing interests. As fuzzy frameworks grow richer to match the messiness of human judgment, the mathematics for comparing those judgments must keep pace, and this contribution marks a substantial step in that direction.

Subject of Research: Divergence measures for T-spherical fuzzy sets applied to multi-criteria decision making and pattern classification

Article Title: Divergence measures for T-spherical fuzzy sets with applications to multi-criteria decision making and pattern classification

Article References: Khan, M. J., Al-Kenani, A. N., Muangchoo, K., & Ekvittayaniphon, S. (2026). Divergence measures for T-spherical fuzzy sets with applications to multi-criteria decision making and pattern classification. Journal of Big Data. https://doi.org/10.1186/s40537-026-01565-8

Image Credits: AI Generated

DOI: 10.1186/s40537-026-01565-8

Keywords: T-spherical fuzzy sets, divergence measures, multi-criteria decision making, TOPSIS, pattern classification, fuzzy set theory, uncertainty modeling, Kullback-Leibler divergence, criteria weights, sports analytics, cricket, Journal of Big Data