Multi-granulation Fuzzy Decision-theoretic Rough Fuzzy Set Models

As an extension of the decision-theoretic rough set based on the equivalence relation on the universe, multigranulation decision-theoretic rough sets have been researched by many scholars. On the other hand, in the models, equivalence relation or partition is so strict that it may limit the application of generalized and the loss functions have uncertainty in some degree. Moreover, it is not taking into account, the fuzziness of decision states and approximation target on the universe and the fuzzy relations in fuzzy information system. Based on these considerations, in this paper, we establish multi-granulation fuzzy decision-theoretic rough fuzzy set models by extending multigranulation decision-theoretic rough sets.Firstly, loss functions are estimated by trapezoidal fuzzy numbers, we propose covering-based multi-granulation trapezoidal fuzzy decision-theoretic rough sets by combining multi-granulation decision-theoretic rough sets and covering probabilis-tic rough sets; Secondly, considering the fuzziness of decision states, we propose covering-based multi-granulation trapezoidal fuzzy decision-theoretic rough fuzzy sets by defining the conditional probability of fuzzy events. Finally, loss functions are estimated by interval-valued, considering the interval-valued fuzzy relations in the fuzzy information system, we propose multi-granulation interval-valued decision-theoretic rough interval-valued fuzzy sets based on interval-valued fuzzy probability measure.Covering-based multi-granulation trapezoidal fuzzy decision-theoretic rough sets and covering-based multi-granulation trapezoidal fuzzy decision-theoretic rough fuzzy sets are discussed by defining the magnitude of trapezoidal fuzzy number and using a ranking function to converting an trapezoidal fuzzy number into an real number. The probabilistic thresholds and the decision rules are derived by Bayes decision. Multi-granulation interval-valued decision-theoretic rough interval-valued fuzzy sets based on interval-valued fuzzy probability measure are investigated by defining the partial relation of interval and using the flexibility of interval, and we obtain the probabilistic thresholds and the decision rules. In the same, for every model, the average, optimistic and pessimistic situations are investigated respectively; the re-lationships between the obtained models and the existing models are discussed; the applicable examples are given respectively and the results obtained illustrate the applicability and generality of the models.