Theory of Extended Fuzzy Discrete-Event Systems for Handling Ranges of Knowledge Uncertainties and Subjectivity

In 2001, we originated a theory of fuzzy discrete-event systems (FDESs) that generalized the conventional/crisp discrete-event systems (DESs). Vagueness and imprecision concerning states and event transitions of DESs were represented by membership grades and computed via fuzzy logic. Our application of the FDES theory to computerized human immunodeficiency virus/acquired immune deficiency syndrome treatment regimen selection, although preliminarily successful, suggests that a more comprehensive FDES theory is needed to address two general issues critically important not only to biomedical applications, but also to real-world problems in other industries. First, domain experts should have means other than point estimates and type-1 fuzzy sets mandated in the current framework to describe uncertainties, subjectivity, and imprecision in their (complex) knowledge and experience. Second, when a group of experts with distinct opinions is involved, they should not be forced to reach consensus for the sake of system development. This is because collective consensus may not be achievable, which is often the case in medicine, where individual expertspsila opinions should be equally respected since the underlying ground truth is unknown most of the time. The theory of extended FDES presented in this paper addresses both the problems and contains the FDES theory as a special case. Experts are now allowed to use interval numbers and type-1 and type-2 fuzzy sets to intuitively and quantitatively express their diverse knowledge and experience, which will then be processed by the new theory to form fuzzy state vectors and fuzzy event transition matrices. Accordingly, we have established mathematical operations that cover the computations of fuzzy states, fuzzy event transitions, and parallel composition. Numerical examples are provided.