Fusing Constructions Of Rough Sets And Vague Sets And Their Three-Way Decision Models

Vague sets and rough sets are both mathematical tools dealing with uncertainty.The vague set is an extension of the fuzzy set and membership of every element in fuzzy sets is divided into supporting and opposing aspects,that is,true membership t and false membership f.Rough sets mainly rely on knowledge granularity for approximate representation and cognitive learning;they are convenient for data analysis,granularity calculation and approximate reasoning;they are associated with three-way of region division and three-way of decision making.Vague sets and rough sets are fused into vague rough sets and rough vague sets,and both of them deepen the expression ability of uncertain information.Specific research contents are provided as follows.(1)Step-average-vague sets have statistical equilibrium between vague sets.A new rough set model based on step-average-vague sets is constructed.The effective fusion of step-averagevague sets and rough sets strengthen knowledge representation and three-way approximation.The step-average-vague set is defined,the upper and lower approximations and three-way regions are determined,and the arithmetic properties of parallel complement of approximate operators in this model are studied.The corresponding fuzziness,accuracy,roughness,approximate accuracy and approximate quality are proposed,and the properties of domain division and hierarchical integration are obtained.Provide data examples,calculate concepts such as the upper and lower approximations,three-way regions,ambiguity,roughness,and verify related properties.The obtained vague rough set and its uncertainty measures are beneficial to further uncertainty processing.(2)For the rough vague set,its dual-approximate vague sets show singleness and extremality,and its integrated rough set model is worth constructing.In this paper,linear fused rough vague sets are proposed to improve rough vague sets,and the subsequent probabilistic linear fused rough vague sets adopt three-way decision modeling.The correlation uncertainty measures are also studied.Firstly,linear fused rough vague sets are constructed parametrically by fusion and extension of rough vague sets,and the ensemble algorithm and operator properties of linear fused rough vague sets are proposed.By three-way of similarity measures,the gransize cognitive approximation of vague sets to vague sets in linear fusion is optimized,and the optimal linear parameters are obtained theoretically and solved by discrete search algorithm.Based on this,a probabilistic linear fused rough vague set model is constructed by means of three linear fused rough vague sets,and its construction algorithm and arithmetic properties are given.Then for uncertainty measures,the accuracy,roughness and dependence are discussed,and their cognitive approximation and linear parameter optimization are given.Finally,the model,properties,measurements and algorithm results are fully verified by data examples and numerical experiments.In conclusion,rough models of vague sets are extended,balanced and improved systematically by effective fusion of vague sets and rough sets in this study,and vague sets are helpful for supporting,opposing and delaying decision making,and relevant uncertainty measurement and parameter optimization are beneficial for cognitive learning.