In this paper, we will propose the novel notion of neighborhood rough sets on a universe set and study some of their basic properties. Then, the relationships between the neighborhood rough sets and covering rough sets are established. Further, the several related notions of probabilistic neighborhood rough sets are investigated and their basic theoretical are discussed. In addition, the notion of neighborhood rough sets over two different universes is defined, and interesting in their properties are explained. Depend on the neighborhood rough sets over two different universes, two algorithms are designed to solve the rough decision-making problems and clarify their applicability by two illustrative examples, respectively. Finally, a comparison between Liu et al.’s approach and our approach is given.

中文翻译：

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两个不同域上基于邻域算子的分级粗糙集及其在决策问题中的应用

在本文中，我们将在宇宙集上提出邻域粗糙集的新概念，并研究它们的一些基本性质。然后，建立邻域粗糙集和覆盖粗糙集之间的关系。此外，研究了概率邻域粗糙集的几个相关概念并讨论了它们的基本理论。此外，定义了两个不同宇宙的邻域粗糙集的概念，并解释了它们有趣的性质。依赖于两个不同域上的邻域粗糙集，设计了两种算法来解决粗糙决策问题，并分别通过两个说明性示例阐明其适用性。最后，给出了 Liu 等人的方法与我们的方法之间的比较。