Uncertainty analysis of hierarchical granular structures for multi-granulation typical hesitant fuzzy approximation space

Hierarchical structures and uncertainty measures are two main aspects in granular computing, approximate reasoning and cognitive process. Typical hesitant fuzzy sets, as a prime extension of fuzzy sets, are more flexible to reflect the hesitance and ambiguity in knowledge representation and decision making. In this paper, we mainly investigate the hierarchical structures and uncertainty measures in typical hesitant fuzzy backgrounds. Firstly, we propose the parameterized scalar cardinalities of typical hesitant fuzzy elements, typical hesitant fuzzy sets and typical hesitant fuzzy relations based on a more reasonable partial orders with a disjunctive semantic meaning, respectively, where the parameters represent the decision makers' risk preferences. Secondly, we present four ordered relations for typical hesitant fuzzy space and four uncertainty measures to characterize the ambiguity in typical hesitant fuzzy approximation space and discuss their relationships. Thirdly, the hierarchical structures of a multi-granulation typical hesitant fuzzy space are analyzed by various multi-granulation typical hesitant fuzzy knowledge bases. In addition, we construct the framework of multi-granulation typical hesitant fuzzy rough sets in terms of optimistic and pessimistic attitudes. Finally, we study the uncertainty measures for the multi-granulation typical hesitant fuzzy approximation space based on the maximal and minimal knowledge bases, respectively.