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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.

The main purpose of this paper is to present a general framework for the study of fuzzy rough approximation operators based on pseudo-operations. In this paper, using the pseudo-operation, the pseudo-generalized fuzzy rough sets are presented and some properties of the pseudo fuzzy rough approximation operators are investigated. Meanwhile, connections between the proposed pseudo fuzzy rough approximation operators and the existing fuzzy rough approximation operators are also discussed.

According to the membership and non-membership functions of intuitionistic fuzzy sets, we show that a triangular norm can induce an inclusion degree, and using this triangular norm and the induced inclusion degree, a similarity measure of intuitionistic fuzzy sets can be obtained. At the same time, we illustrate the proposed method with a numerical example.

The soft set theory, proposed by Molodtsov, can be used as a general mathematical tool for dealing with uncertainty. By analyzing the characters of the interval-valued intuitionistic fuzzy set and soft set models, the purpose of this article is to introduce the concept of the interval-valued intuitionistic fuzzy soft set and then employ its corresponding knowledge to solve multiple parameter group decision making. Finally, an illustrative investment decision problem is employed to demonstrate how to apply the proposed procedure.

The TOPSIS (technique for order preference by similarity to ideal solution) method has been widely used to rank the preference order of alternatives and determine the optimal choice. In this paper, we firstly give the significant degree of covering in covering rough sets model, and then propose the TOPSIS method based on covering rough sets by combining the significant degree of covering. At the same time, we illustrate the proposed method with a numerical example.

In this paper, we show that the similarity between covering rough sets can be measured by means of a triangular norm. At the same time, we illustrate the proposed method with a concrete example.