Hesitant Fuzzy $$\beta $$-Covering $$({\mathcal {I}},$$ $${\mathcal {O}})$$ Rough Set Models and Applications to Multi-attribute Decision-Making

The hesitant fuzzy $$\beta $$ -covering rough set offers stronger representational capabilities than earlier hesitant fuzzy rough sets. Its flexibility makes it more suitable for hesitant fuzzy multi-attribute decision-making (MADM). As a result, it has become a popular research focus in decision analysis and has drawn significant attention from scholars. However, the existing hesitant fuzzy $$\beta $$ -covering rough set based on t-norms cannot handle the overlap and correlation between hesitant information well. Addressing this problem, we propose the hesitant fuzzy overlap function and hesitant fuzzy $$\beta $$ -covering $$({\mathcal {I}},$$ $${\mathcal {O}})$$ rough set (HF $$\beta $$ CIORS) models based on the hesitant fuzzy overlap function. First, we establish the definition of the hesitant overlap function and representable hesitant fuzzy overlap function on a partial order relation. Based on proposed definitions, we provide examples of representable and unrepresentable hesitant fuzzy overlap functions and offer a detailed proof to explain the unrepresentable function. Second, we construct four types of HF $$\beta $$ CIORS models and prove some of its important properties. Thirdly, we integrate the HF $$\beta $$ CIORS models with the TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) method and apply them to solve MADM problems. The validity of the proposed method is demonstrated through a practical application, and its stability and effectiveness are confirmed via sensitivity and comparative analyses. Based on these validations, our method proves effective in addressing MADM problems, offering reliable decision-making support.