Asymmetric models of intuitionistic fuzzy rough sets and their applications in decision-making

The variable precision rough set is one of the well-liked extensions of rough sets for misclassification and perturbations. The classical variable precision rough set and some of its extensions are based on equivalence relations. The applicability of those models are constrained by the too-strict equivalence relations. In this paper, we extend equivalence relations in an intuitionistic fuzzy environment and put forward some intuitionistic fuzzy neighborhoods. The classical variable precision rough set is based on two thresholds β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta$$\end{document} and 1-β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1-\beta$$\end{document} to construct the β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta$$\end{document}-lower and β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta$$\end{document}-upper approximations. The sum of two thresholds is 1. They are called the symmetrical thresholds. If the sum of two thresholds is not always 1, we call them asymmetrical thresholds. The model with symmetrical thresholds is called a symmetric model while the model with asymmetrical thresholds is called an asymmetric model. The thresholds play a key role in the classical variable precision rough set. However, the condition that the sum of two thresholds is 1, is too harsh to limit the applications of the classical variable precision rough set. Following the probabilistic rough sets, we weaken the conditions and propose three types of asymmetric models of intuitionistic fuzzy rough set models. These proposed models, which are the generalizations of the classical variable precision rough sets, are capable of successfully resolving uncertain problems with hesitant degrees, misclassifications, and perturbations in the intuitionistic fuzzy covering approximate space. The first type of our proposed model has greater accuracy than other two types of models. A new multi-attribute decision-making approach is proposed in order to effectively use our proposed models, combining the benefits of the traditional PROMETHEE and TOPSIS procedures. Two illustrated instances are also presented to test the decision-making process. The proposed method’s viability and efficacy are shown by comparison analysis, sensitivity analysis, and validity analysis.