n-Dimensional (S,N)-implications

Abstract

The n -dimensional fuzzy logic ( n -DFL) has been contributed to overcome the insufficiency of traditional fuzzy logic in modeling imperfect and imprecise information, coming from different opinions of many experts by considering the possibility to model not only ordered but also repeated membership degrees. Thus, n -DFL provides a consolidated logical strategy for applied technologies since the ordered evaluations provided by decision makers impact not only by selecting the best solutions for a decision making problem, but also by enabling their comparisons. In such context, this paper studies the n -dimensional fuzzy implications ( n -DI) following distinct approaches: (i) analytical studies, presenting the most desirable properties as neutrality, ordering, (contra-)symmetry, exchange and identity principles, discussing their interrelations and exemplifications; (ii) algebraic aspects mainly related to left- and right-continuity of representable n -dimensional fuzzy t-conorms; and (iii) generating n -DI from existing fuzzy implications. As the most relevant contribution, the prospective studies in the class of n -dimensional interval (S,N)-implications include results obtained from t-representable n -dimensional conorms and involutive n -dimensional fuzzy negations. And, these theoretical results are applied to model approximate reasoning of inference schemes, dealing with based-rule in n -dimensional interval fuzzy systems. A synthetic case-study illustrates the solution for a decision-making problem in medical diagnoses.

Strategies on admissible total orders over typical hesitant fuzzy implications applied to decision making problems

“…By applying the Łukasiewicz implication, the example reported an ME‐MCDM problem in a CIM‐application, which could be analyzed from three distinct comparisons based on ‐operators. Ongoing works are focusing on other classes of implications, such as ‐implications, 43 and including the residuation principle related to R ‐implications and the left‐continuity of t ‐norms, in the context of admissible ‐orders. And, to show the advantage of the proposed method, further work extends case studies in cloud computing, 44 for hesitant fuzzy environments, based on the theoretical results achieved in this step of the research on admissible linear orders.…”

n-Dimensional Fuzzy Implications: Analytical, Algebraic and Applicational Approaches

“…• main results considering the extended study of n-DI, their main properties and classes contributing to explore other research fields taking advantages of logical structure of n-DFL: (i) formalizing IF-THEN rules of approximate reasoning; (ii) structuring new methods to measure similarity and consensus n-DFS; (iii) extracting parameters as correlation coefficient and entropy supporting the data analysis of applications. And, see in [Zanotelli et al 2020] partial relevant thesis results.…”

An approach for consensual analysis on Typical Hesitant Fuzzy Sets via extended aggregations and fuzzy implications based on admissible orders

“…Ongoing works are focusing on other classes of implications, such as (S, N )-implications [Zanotelli et al 2020] and including the residuation principle related to R-implications and the left-continuity of t-norms, in the context of admissible H, -orders. And, in order to show the advantage of the proposed method, further work extends case studies in cloud computing [Schneider et al 2020], for hesitant fuzzy environments, based on the theoretical results achieved in this step of the research on admissible linear orders.…”