HyperRough Number and SuperHyperRough Number with Applications

Abstract

Rough set theory provides a systematic framework for modeling uncertainty by approximating subsets through lower and upper approximations induced by an equivalence relation. The rough number, defined as the interval between the mean attribute values of these approximations, offers a numerical measure of uncertainty. HyperRough Sets extend this framework by incorporating multiple attributes into the approximation process, while SuperHyperRough Sets further generalize it via iterated powerset constructions. However, numerical counterparts of these extended structures—the HyperRough Number and the SuperHyperRough Number—have not yet been formally introduced. In this paper, we define HyperRough Numbers and SuperHyperRough Numbers, establish their fundamental properties, and illustrate their usefulness in multi-attribute decision-making problems. The author contends that these numerical constructs provide a basis for future research on sophisticated applications across a wide range of applied sciences.