This paper studies a multigranulation extension of $ \mathrm{S} $-approximation spaces based on an optimistic fusion principle. Instead of using a single knowledge mapping, we consider a family of knowledge mappings defined on the same universe and evaluated by a common decision mapping. Given this setting, we introduce optimistic lower and upper approximation operators and derive the corresponding acceptance, rejection, and deferment regions from the viewpoint of three-way decisions. Several basic properties of the proposed operators are established, including duality and monotonicity-type relations. We also examine how the new model is connected with the underlying single-granularity $ \mathrm{S} $-approximation spaces. In addition, when the involved granularities satisfy a non-contradictory condition, we prove that optimistic combination preserves decisions' consistency and yields extremal regional behavior, namely, the largest acceptance region and the smallest deferment region among admissible combination strategies. The obtained results extend the theory of $ \mathrm{S} $-approximation spaces to a multigranulation setting and provide a formal basis for three-way decision analysis with multi-source knowledge. A semi-realistic multi-source assessment case study and a quantitative comparison with classical optimistic multigranulation rough set, majority voting, and averaging-based fusion strategies are also presented to illustrate the applicability and decisiveness of the proposed model.
Citation: Jing Huang. Optimistic multigranulation $ \mathrm{S} $-approximation spaces and three-way decisions[J]. AIMS Mathematics, 2026, 11(8): 25355-25389. doi: 10.3934/math.20261018
This paper studies a multigranulation extension of $ \mathrm{S} $-approximation spaces based on an optimistic fusion principle. Instead of using a single knowledge mapping, we consider a family of knowledge mappings defined on the same universe and evaluated by a common decision mapping. Given this setting, we introduce optimistic lower and upper approximation operators and derive the corresponding acceptance, rejection, and deferment regions from the viewpoint of three-way decisions. Several basic properties of the proposed operators are established, including duality and monotonicity-type relations. We also examine how the new model is connected with the underlying single-granularity $ \mathrm{S} $-approximation spaces. In addition, when the involved granularities satisfy a non-contradictory condition, we prove that optimistic combination preserves decisions' consistency and yields extremal regional behavior, namely, the largest acceptance region and the smallest deferment region among admissible combination strategies. The obtained results extend the theory of $ \mathrm{S} $-approximation spaces to a multigranulation setting and provide a formal basis for three-way decision analysis with multi-source knowledge. A semi-realistic multi-source assessment case study and a quantitative comparison with classical optimistic multigranulation rough set, majority voting, and averaging-based fusion strategies are also presented to illustrate the applicability and decisiveness of the proposed model.
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Jing Huang. Optimistic multigranulation $ \mathrm{S} $-approximation spaces and three-way decisions[J]. AIMS Mathematics, 2026, 11(8): 25355-25389. doi: 10.3934/math.20261018
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