Generalized belief function in complex evidence theory

The complex-value-based generalized Dempster-Shafer evidence theory, also called complex evidence theory is a useful methodology to handle uncertainty problems of decision-making on the framework of complex plane. In this paper, we propose a new concept of belief function in complex evidence theory. Furthermore, we analyze the axioms of the proposed belief function, then define a plausibility function in complex evidence theory. The newly defined belief and plausibility functions are the generalizations of the traditional ones in Dempster-Shafer (DS) evidence theory, respectively. In particular, when the complex basic belief assignments are degenerated from complex numbers to classical basic belief assignments (BBAs), the generalized belief and plausibility functions in complex evidence theory degenerate into the traditional belief and plausibility functions in DS evidence theory, respectively. Some special types of the generalized belief function are further discussed as well as their characteristics. In addition, an interval constructed by the generalized belief and plausibility functions can be utilized for fuzzy measure, which provides a promising way to express and model the uncertainty in decision theory.