Granular variable precision fuzzy rough sets with general fuzzy relations

The variable precision (θ,σ)(θ,σ)-fuzzy rough sets were proposed to remedy the defects of preexisting fuzzy rough set models. However, the variable precision (θ,σ)(θ,σ)-fuzzy rough sets were only defined and investigated on fuzzy ⁎-similarity relations. In this paper, the granular variable precision fuzzy rough sets with general fuzzy relations are proposed on arbitrary fuzzy relations. The equivalent expressions of the approximation operators are given with fuzzy (co)implications on arbitrary fuzzy relations, which can calculate efficiently the approximation operators. The granular variable precision fuzzy rough sets are characterized from the constructive approach, which are investigated on different fuzzy relations. The conclusions on the variable precision (θ,σ)(θ,σ)-fuzzy rough sets are generalized into the granular variable precision fuzzy rough sets.