Dynamic fuzzy sets model uncertainty over time in two test scenarios

A computational study has extended picture fuzzy sets with a discrete time dimension, producing 12 similarity measures for comparing evaluations that evolve across observations. The dynamic picture fuzzy set framework retains the original model’s four-part representation of positive, neutral, negative and refusal membership while allowing each value to change at different time points. The researchers tested the framework in two constructed scenarios: multi-criteria supplier selection over five quarters and type 2 diabetes risk assessment over five weeks.

These were mathematical and computational demonstrations, not a clinical trial or a live procurement study, and the paper reports no patient outcomes or operational business results. The approach addresses a limitation of static fuzzy models. A conventional fuzzy set assigns one membership degree to each element, while an intuitionistic fuzzy set adds a non-membership degree and combines the remaining uncertainty into a single hesitation degree.

A picture fuzzy set separates that uncertainty into neutral membership, representing hesitation or abstention, and refusal membership, representing an unjudged or non-committed portion. For every element and time point in a dynamic picture fuzzy set, the positive, neutral and negative degrees lie between zero and one and have a sum no greater than one; the refusal degree is one minus that sum. This structure can represent support, neutrality, opposition and non-commitment without collapsing the last two states into a single category.

Formally, the researchers define a dynamic picture fuzzy set over a non-empty universe and an ordered discrete set of time points. Its membership functions carry positive, neutral and negative degrees for every combination of an element and a time point. They extend inclusion, equality, union, intersection and complement from static picture fuzzy sets to each time node, then prove that the resulting operations satisfy idempotency, commutativity, associativity, distributivity, absorption, involution and De Morgan’s laws.

A dynamic picture fuzzy set with only one time point reduces to a conventional picture fuzzy set, establishing the proposed model as a temporal generalization rather than a separate representation of agreement alone. Similarity scores are produced by comparing corresponding attribute values over time and combining those differences with normalized time weights. The paper constructs Type-I and Type-II versions of six measures: Hamming, Euclidean, Minkowski, Chebyshev, Lambert and absolute sum-difference.

Type-I measures use positive, neutral and negative membership and therefore focus on explicit attitudes. Type-II measures add refusal membership, allowing differences in unjudged or indeterminate information to affect the score. In an illustrative comparison of user evaluations for two products over five weeks, every Type-II score was lower than its corresponding Type-I score.

The authors interpret this pattern as a consequence of penalizing differences in refusal rather than ignoring them. The weighting mechanism determines how strongly each observation contributes. The researchers describe an exponential decay factor between zero and one, with weights proportional to successive powers and then normalized to sum to one.

A smaller factor makes the weights decline more rapidly with distance from the most recent observation, whereas a factor of one gives equal weight; constant and linear strategies are also possible. Their theoretical analysis assumes positive time weights. The paper’s axiomatic results distinguish proper similarity measures, which are bounded, symmetric, identical for equal sets and monotonic for ordered sets, from quasi-similarity measures, which meet the first three properties but not the general monotonicity condition.

Under the stated assumptions, both Type-I and Type-II Hamming measures are proper similarity measures. The Type-I Euclidean, Minkowski, Chebyshev, Lambert and absolute sum-difference measures are also proper when the relevant regularity conditions hold. Most Type-II versions are classified as quasi-similarity measures because adding refusal can cause the monotonicity condition to fail.

The authors supply counterexamples demonstrating that an earlier proof approach incorrectly assumed that a refusal-degree difference between ordered sets always obeys a particular bound. The Lambert and absolute sum-difference formulas also contain denominators that can be zero. The paper therefore distinguishes regular data, for which those formulas produce usable results, from singular data that can generate anomalous values.

All six measure families have linear asymptotic time complexity, O(mn), where m is the number of time points and n the number of attributes. Hamming mainly uses absolute differences; Euclidean adds squaring and a square root; Minkowski generalizes this with p-th powers and roots. Chebyshev relies largely on maximum comparisons and has a low computational overhead, although the paper notes that focusing on the largest discrepancy can reduce discrimination.

Lambert performs division for individual attribute values, while absolute sum-difference performs divisions after aggregation and is therefore expected to be less costly in practice. Including refusal expands each time node from three to four dimensions, which the authors estimate adds about 33% to the calculation load without changing the O(mn) complexity class. In the supplier-selection demonstration, a manufacturing enterprise evaluated three suppliers over five quarters using product quality, unit price, delivery punctuality and after-sales service.

The ideal dynamic picture fuzzy set was constructed from the alternatives’ temporal values. For each benefit criterion, it used the maximum positive degree, minimum neutral degree and minimum negative degree across time; the cost criterion used the minimum positive degree, minimum neutral degree and maximum negative degree. With a decay factor of 0.95, the normalized weights rose from 0.180 for the oldest quarter to 0.221 for the most recent.

All 12 measures ranked the suppliers A1 above A2 above A3. Varying the decay factor from 0.65 to 1 changed numerical scores but not the supplier ranking. The medical demonstration used weekly data for three constructed patients and four type 2 diabetes indicators: fasting blood glucose, two-hour postprandial blood glucose, glycated haemoglobin and body-mass index.

A reference pattern encoded the study’s stated thresholds, including 7.0 mmol/L for fasting glucose, 11.1 mmol/L for two-hour postprandial glucose, 6.5% for glycated haemoglobin and 25 kg/m² for body-mass index. Similarity was classified as high risk at 0.85 or above, medium risk from 0.70 to below 0.85, and low risk below 0.70. Most measures ranked the persistently high-indicator patient first, the patient whose indicators moved toward the reference pattern second and the consistently low patient third.

Lambert and absolute sum-difference instead ranked the improving patient first, demonstrating that conclusions can depend on how relative changes are treated. The most conspicuous result came from the Type-II absolute sum-difference measure, which classified all three constructed patients as low risk, including the patient classified as high risk by the other 12 measures. The authors attribute this to structural sensitivity of its difference-to-sum normalization and describe the outcome as a potentially dangerous missed-diagnosis risk, not evidence that the patient was medically low risk.

Sensitivity analysis showed that classification could also change with temporal weighting for a changing trajectory: the Type-I Chebyshev score for the improving patient fell from 0.860 at α = 0.86 to 0.847 at α = 0.96, shifting that result from high to medium risk, while scores for the two stable patients changed little. The study’s central result is therefore a structured way to expose how recency, refusal information and formula selection influence decisions. Its limits include empirically selected decay factors, unresolved monotonicity for several Type-II measures, reliance on discrete time sets and constructed application data; the medical example does not establish diagnostic accuracy, clinical benefit or suitability for patient care.

Subject of Research: Time-aware modeling of changing uncertainty using dynamic picture fuzzy sets and similarity measures

Article Title: Dynamic Picture Fuzzy Sets and Its Applications

Article References: Shen, Y. (2026). Dynamic Picture Fuzzy Sets and Its Applications. Cognitive Computation, 18(1), 119. https://doi.org/10.1007/s12559-026-10656-w

Image Credits: AI Generated

DOI: 10.1007/s12559-026-10656-w

Keywords: dynamic picture fuzzy sets, uncertainty modeling, time-weighted similarity, fuzzy set theory, supplier selection, multi-criteria decision-making, type 2 diabetes, risk classification, computational complexity, symptom progression, Dynamic, Picture