Quadri-Polar Fuzzy Fantastic Ideals in BCI-Algebras: A TOPSIS Framework and Application

Abstract

A quadri-polar fuzzy ($q\mathcal{P}$-$\mathcal{F}$) set is an extension of a traditional fuzzy set that uses four degrees of membership to represent different aspects of belonging to provide a more detailed framework for handling uncertainty and vagueness. In this paper, we propose the notion of quadri-polar-$(\varpi,\vartheta)$-fuzzy fantastic ideals ($q\mathcal{P}$-$(\varpi,\vartheta)$-$\mathcal{FFI}(s)$) in BCI-algebras based on $q\mathcal{P}$-$\mathcal{F}$ set. Also, the notion of quadri-polar-$(\in_{\tilde{\sigma}}, \in_{\tilde{\sigma}} \vee q_{\tilde{\tau}})$-fuzzy fantastic ideals ($q\mathcal{P}$-$(\in_{\tilde{\sigma}}, \in_{\tilde{\sigma}} \vee q_{\tilde{\tau}})$-$\mathcal{FFI}(s)$) is introduced, and the characterizations for an $\in_{\tilde{\sigma}}$-$q\mathcal{P}$-$\mathcal{F}$ set and $q_{\tilde{\tau}}$-$q\mathcal{P}$-$\mathcal{F}$ set to be  quadri-polar fuzzy ideals ($q\mathcal{P}$-$\mathcal{FI}$) in BCI-algebras are established. Furthermore, we present the $q\mathcal{P}$-$\mathcal{F}$ TOPSIS technique for multi-criteria Group decision-making (MCGDM), which is a natural extension of the TOPSIS method and used to rank and choose the best alternatives under $q\mathcal{P}$-$\mathcal{F}$ positive and negative ideal solutions. Finally, practical examples interpreting the applicability of our proposed $q\mathcal{P}$-$\mathcal{F}$-TOPSIS are solved.