Link between ultra-processed foods and drinks intake, gut microbiota and inflammation: an exploratory analysis in adult volunteers.

The concept of fuzzy sets, as introduced by Zadeh in 1965, brought about a paradigm shift and was used in various fields, especially in decision-making, medical diagnosis, pattern recognition, and fuzzy inference systems. In order to overcome the drawbacks of classical fuzzy sets, Atanassov proposed intuitionistic fuzzy sets. In 1986, IFS membership and non-membership degrees were used to express satisfaction (IFS): experiment and dissatisfaction, respectively. The total number of such degrees in IFS is not supposed to exceed 1, which limits the practice of decision-making when expert assessment does not comply with this norm.Yager proposed Pythagorean fuzzy sets (PyFS) to offer more expressiveness. The sum of the squares of membership and non-membership degrees is to be less than or equal to one. This loosening of admissible evaluations increases the range of possible evaluations. Germanizing this generation of sets, Yager introduced q-orthopair fuzzy sets (q-ROFS), in which the total of the q th powers () of the membership and non-membership degrees is not larger than unity. The rationality of q-ROFS lies in its parametric flexibility: by adjusting the value of , decision-makers can control the admissible evaluation space, thereby reducing information distortion and accommodating higher levels of uncertainty. Moreover, IFS and PyFS emerge as special cases of q-ROFS when and , respectively, demonstrating a clear hierarchical relationship among these models. As a flexible and inclusive model, q-ROFS has attracted significant academic interest and been widely applied to more sophisticated decision-making cases. Riaz and Hashmi observed that although IFSs, PyFSs, and q-ROFSs expand the permissible domain of evaluation, they remain fundamentally constrained by predefined algebraic conditions imposed solely on membership and non-membership degrees. In most real world problems uncertainty is not only fuzzy and also defined by structural, numerical or rule based relationship which cannot be sufficiently represented by power based constraints alone. In order to annul this shortcoming, they proposed linear Diophantine fuzzy sets (LDFS), which involve reference (control) parameters with membership and non-membership degrees. LDFS adds to the classical fuzzy structures an ordered integer pair, denoted by a pair of integers, which prove a linear Diophantine equation. This extra Diophantine structure enables the model to accommodate fuzzy uncertainty as well as underlying numerical constraints at the same time. In contrast to IFS, PyFS and q ROFS, which attain flexibility by nonlinear admissibility conditions of power, LDFS attains flexibility by reference parameters that govern the interplay between membership and non membership degrees. As a result, the space of evaluation is not only enlarged but the structure is under control as well, which increases logical consistency and interpretability in analyzing decisions. Ayub et al. also revealed that LDFS framework has a better flexibility and reliability compared to the available fuzzy paradigms. Regarding rationality, LDFS enhances the theory of fuzzy modeling by introducing arithmetic consistency to Diophantine constraints, such that the judgments are consistent even when there is contradictory or incomplete information. Also, LDFS introduces a conceptual development of q-ROFS through the change of power-based generalization into constraint-based generalization in order to deal with those cases in which structural dependencies or discrete regulatory conditions coexist with vagueness. Owing to these advantages, LDFS has attracted significant scholarly attention in both theoretical development and practical applications.