Some complex probabilistic hesitant fuzzy aggregation operators and their applications to multi-attribute decision making.

As a generalization of complex fuzzy set (CFS), complex hesitation fuzzy set (CHFS) is a powerful tool to express two-dimensional fuzzy information, but it ignores the randomness that exists widely in real life. Considering the randomness and ambiguity often simultaneously occur in real-world problems, we come up with an innovative fuzzy set called complex probabilistic hesitant fuzzy set (CPHFS) in this article, which is a joint extension of CFS and probabilistic hesitant fuzzy set (PHFS). Moreover, we establish four basic operations for complex probabilistic hesitant fuzzy element(CPHFE) and a series of useful operational laws are also developed.