New roughness measures of the interval-valued fuzzy sets

In the paper (Gong, Sun, & Chen, 2008; Han, Chen, & Chen, 2008), by combining the rough set theory with the interval-valued fuzzy set theory, Gong and Han proposed the concept of the interval-valued rough fuzzy sets and studied a parameters-related roughness measure of the interval-valued fuzzy sets in an approximation space, respectively. Since the measure inherited from Banerjee’s about the fuzzy sets in Banerjee and Pal (1996), it inevitably holds some similar undesirable properties. In this paper, we first remark some defects about the above mentioned measure, and then give a new roughness measure–rough entropy of the interval-valued fuzzy sets in an approximation space by using the information entropy theory, and demonstrate the difference between the two measures by a simple illustration example. Furthermore, in respect that the two measures mentioned above both strongly depend on parameters, we propose a parameter-free roughness measure of the interval-valued fuzzy sets in an approximation space based on the notions of the average mass assignment and average membership function of an interval-valued fuzzy set, adhering to the idea of parameter-free roughness measure of the fuzzy sets proposed in Huynh and Nakamori (2005). Some related properties about the new measure are discussed. Finally, we discuss how use the new measure to evaluate the rough approximation quality of classification in an interval-valued fuzzy decision information system.