TI-fuzzy neighborhood measures and generalized Choquet integrals for granular structure reduction and decision making.

Granular structures are mathematical representations of knowledge used in granular computing. Any fuzzy information table can be seen as a fuzzy granular structure, which is called a fuzzy β-covering approximation space. Hence, corresponding granular computing methods are widely used in data analysis, granular reduction and decision making. Currently, these methods mainly use rough approximate operators to process data where attributes are independent of each other in rough approximate operators. However, attributes are often associated with each other in real-life problems. As a parametric nonlinear aggregation function, the Choquet integral (CI) with respect to a fuzzy measure (FM) solves the problem of attribute association well. In this paper, we present TI-fuzzy β-neighborhood measures, which are FMs and generalized CIs, to deal with granularity reduction and decision making in a fuzzy β-covering approximation space. Firstly, four pairs of TI-fuzzy β-neighborhood measures under the t-norm “T” and its residual implication “IT”, as FMs, are presented for use in granular computing instead of rough approximation operators. Then, a novel method with TI-fuzzy β-neighborhood measures is presented to deal with granularity reduction in the fuzzy β-covering approximation space. Thirdly, four pairs of generalized CIs based on the TI-fuzzy β-neighborhood measures are constructed. By combining the presented CIs with the fuzzy PROMETHEE method, we propose a new method to solve the problem of decision making. Finally, several numerical examples and UCI data sets are used to illustrate the feasibility and effectiveness of our proposed methods.