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In this paper, we point out that Lin’s general neighborhood systems-based rough set model is an extension of Qian’s optimistic rough set model, and thus called optimistic general neighborhood systmes-based rough set model. Then we present a new rough set model based on general neighborhood systems, and prove that it is an extension of Qian’s pessimistic rough set model. Later, we study the basic properties of the proposed pessimistic rough sets, and define the serial, reflexive, symmetric, transitive and Euclidean conditions for general neighborhood systems, and explore the further properties of related rough sets. Furthermore, we apply the pessimistic general neighborhood systems-based rough set model in the research of incomplete information system, and build a three-way decision model based on it. A simple practical example to show the effectiveness of our model is also presented.

Attribute significance is very important in multiple-attribute decision-making (MADM) problems. In a MADM problem, the significance of attributes is often different. In order to overcome the shortcoming that attribute significance is usually given artificially. The purpose of this paper is to give attribute significance computation formulas based on inclusion degree. We note that in the real-world application, there is a lot of incomplete information due to the error of data measurement, the limitation of data understanding and data acquisition, etc. Firstly, we give a general description and the definition of incomplete information systems. We then establish the tolerance relation for incomplete linguistic information system, with the tolerance classes and inclusion degree, significance of attribute is proposed and the corresponding computation formula is obtained. Subsequently, for incomplete fuzzy information system and incomplete interval-valued fuzzy information system, the dominance relation and interval dominance relation is established, respectively. And the dominance class and interval dominance class of an element are got as well. With the help of inclusion degree, the computation formulas of attribute significance for incomplete fuzzy information system and incomplete interval-valued fuzzy information system are also obtained. At the same time, results show that the reduction of attribute set can be obtained by computing the significance of attributes in these incomplete information systems. Finally, as the applications of attribute significance, the attribute significance is viewed as attribute weights to solve MADM problems and the corresponding TOPSIS methods for three incomplete information systems are proposed. The numerical examples are also employed to illustrate the feasibility and effectiveness of the proposed approaches.