MISO hierarchical inference engine satisfying the law of importation with aggregation functions

Fuzzy inference engine, as one of the most important components of fuzzy systems, can obtain some meaningful outputs from fuzzy sets on input space and fuzzy rule base using fuzzy logic inference methods. In multi-input-single-output (MISO) fuzzy systems, in order to enhance the computational efficiency of fuzzy inference engine, this paper aims mainly to investigate three MISO fuzzy hierarchical inference engines employed fuzzy implications satisfying the law of importation with aggregation functions (LIA). We firstly find some aggregation functions for well-known fuzzy implications such that they satisfy LIA. The fuzzy implication satisfying LIA with respect to a given aggregation function is then characterized. Finally, three fuzzy hierarchical inference engines in MISO fuzzy systems are constructed according to aforementioned theoretical developments. Three examples are also provided to illustrate our theoretical arguments.