# A hybrid framework of hesitant fuzzy soft sets and rough sets for uncertainty modelling

## TL;DR

This paper introduces a new model combining hesitant fuzzy soft sets and rough sets to better handle uncertainty in decision-making processes.

## Contribution

The novel HFSRS model addresses simultaneous hesitation, indiscernibility, and parameterization with dynamic beta covers.

## Key findings

- HFSRS achieved 92% accuracy on photovoltaic fault detection, outperforming classical and fuzzy rough sets.
- The model reduced boundary regions by 35% and improved AUC to 0.97 compared to competing methods.
- HFSRS-TOPSIS offers computational tractability with complexity O(n × m × k) for datasets up to 10⁴ objects.

## Abstract

The process of decision making involves uncertainty due to lack of agreement among experts, inaccuracy in measurements and incomplete information. Current frameworks are inadequate in dealing with cases in which hesitation, indiscernibility, and parameterization may all take place simultaneously. The article proposes a new Hesitant Fuzzy Soft Rough Set (HFSRS) model that combines hesitant fuzzy soft sets and rough sets with dynamic \documentclass[12pt]{minimal}
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\begin{document}$$\:\boldsymbol{\beta\:}$$\end{document} covers that changes approximation boundaries in relation to hesitant membership levels. The suggested framework deals with severe constraints such as the impossibility to model parameter-dependent hesitation, duality violation of the classical fuzzy rough sets, and fixed thresholding processes that cannot be used in a noisy environment. The three fundamental properties provided by mathematical formalization: (a) duality preservation to provide logical consistency important for safety-critical applications, (b) monotonicity to provide predictable behavior important to explainable AI systems, and (c) topological consistency to provide hierarchical uncertainty modeling. HFSRS is empirically validated using synthetically generated datasets (500 photovoltaic modules with three fault indicators adjusted to IEC 61215-2:2021 standards) to achieve 92 per cent accuracy versus 85 per cent on classical rough sets, 86 per cent on fuzzy rough sets, 88 per cent on intuitionistic fuzzy rough sets, with 35 per cent reduction in boundary region and AUC of 0.97 versus 0.92 on competing methods running 30 times The best 0.65 -threshold of the beta value balances accuracy and coverage. The HFSRS-TOPSIS algorithm provides practitioners with strong decision support, computational tractability of \documentclass[12pt]{minimal}
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\begin{document}$$\:\left(O\right(n\:\times\:\:m\:\times\:\:k\left)\right)$$\end{document} on a dataset of up to \documentclass[12pt]{minimal}
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\begin{document}$$\:{10}^{4}$$\end{document} objects.