Abstract Shadowed set provide a three-way approximation scheme for transforming a fuzzy set into three disjoint areas (elevated, reduced, and shadow areas). A fundamental issue in the construction of shadowed sets is the interpretation and determination of a pair of thresholds ( α , β ) . Several extended shadowed set models have been proposed to calculate ( α , β ) . However, the construction of a few of these models may have a large fuzzy entropy loss, and the determination of ( α , β ) involves artificial subjective parameters. Therefore, in this study, a novel shadowed set model is proposed, namely, mean-entropy-based shadowed sets (MESS). At first, based on the principle of uncertainty invariance, a novel framework of three-way approximations of fuzzy sets is proposed based on the mean of fuzzy entropy. Secondly, new decision rules are generated based on the fuzzy entropy loss, and ( α , β ) is obtained. Thirdly, the MESS model is optimized more reasonably using an iterative method, and thus the fuzzy entropy loss of the MESS model can be minimized. Finally, the validity and rationality of the proposed model are verified by instances and experimental analysis.

中文翻译：

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基于均值熵的阴影集：一种新的模糊集三向逼近

摘要 阴影集提供了一种将模糊集转换为三个不相交区域（升高区域、缩小区域和阴影区域）的三向近似方案。构建阴影集的一个基本问题是解释和确定一对阈值 (α, β)。已经提出了几个扩展的阴影集模型来计算 (α, β)。然而，其中一些模型的构建可能会有很大的模糊熵损失，并且（α，β）的确定涉及人为的主观参数。因此，在本研究中，提出了一种新的阴影集模型，即基于均值熵的阴影集（MESS）。首先，基于不确定性不变性原理，提出了一种基于模糊熵均值的模糊集三向逼近框架。第二，基于模糊熵损失生成新的决策规则，并获得（α，β）。第三，通过迭代方法对MESS模型进行更合理的优化，从而使MESS模型的模糊熵损失最小化。最后，通过实例和实验分析验证了所提出模型的有效性和合理性。