Covering-based Variable Precision And Grade Rough Fuzzy Set Models

As an extension of the classical rough set theory,the covering-based rough set,rough fuzzy set and fuzzy rough set have been researched by many scholars.On the other hand,covering rough fuzzy sets have been defined in four different ways according to the neighbourhood of the object x and the background of the prob-lem.However,for the follow-up research of this model,many researchers have only considered the minimum description neighbor domain of object x(i.e.nMd(x)).Moreover,rough fuzzy sets are mainly used to deal with real data,and the existence of errors in real data makes noise or disturbance more common.The model has not considered introducing a reasonable mechanism to deal with disturbances.Based on the above considerations,in this paper,we establish covering variable precision and grade rough fuzzy set models by extending the covering rough fuzzy set models.Firstly,considering the limitation of equivalence relation in practical application and the ubiquity of noise or disturbance in real data.Based on the discussion of the covering rough set models and the mininum description neighborhood of the elements for the relative error classification rate of a given fuzzy set,four kinds of covering variable preeision rough fuzzy set models are definel and established by means of the minimum description of neighbor domain,whole neighborhood,rule confidence and membership of the elements;Secondly,considering the quantitative information of overlap between sets and equivalents in approximation space,the grade rough set is introduced into the covering rough fuzzy set model,we propose four kinds of covering grade rough fuzzy set models.Finally,the covering variable precision rough fuzzy set models and the covering grade rough fuzzy set models are combined by logical operation,we propose four kinds of "logical operation" rough fuzzy set models of covering-based variable precision and grade.Covering variable precision rough fuzzy set models and covering grade rough fuzzy set models,based on four different covering rough fuzzy set models,using the mechanism that variable precision rough set and grade rough set can deal with disturbances,the covering rough fuzzy set models are further extended and have a certain fault tolerance and error correction ability;"logical operation" rough fuzzy set models of covering-based variable precision and grade makes the quantification of concepts more precise when combining the variable precision rough set and grade rough set by logical operation.At the same time,for every model,their proper-ties and relationships are investigated respectively;the relationships between the obtained and the existing models are discussed;the applicable examples are given respectively and the results obtained illustrate the rationality and superiority of the models.