NOVEL APPROACHES FOR THE TRANSFORMATION OF FUZZY MEMBERSHIP FUNCTION INTO BASIC PROBABILITY ASSIGNMENT BASED ON UNCERTAINTY OPTIMIZATION

Abstract

With the development of uncertainty reasoning and information fusion, there have emerged several theories including fuzzy set theory, Dempster-Shafer evidence theory, probability theory and rough set theory, etc., for representing and dealing with the uncertain information. When the fusion of the uncertain information originated from different sources is needed, how to construct a general framework for different theories of uncertainty or how to establish the connection between different theoretical frameworks has become a crucial problem. Particularly, to combine two kinds of information represented respectively by the BPA and the FMF, this paper proposes two transformations of an FMF into a BPA by solving a constrained maximization or minimization optimization problem. The objective function is the uncertainty degree of the body of evidence (BOE) and the corresponding constraints are established based on the given FMF. In fact the transformation of an FMF into a BPA is the transformation of fuzzy sets into random sets, which is currently accepted as a unified framework for several theories of uncertainty. Our proposed approaches have no predefinition of focal elements and they can be used as the general transformations of fuzzy sets into random sets. Some examples and analyses are provided to illustrate and justify the rationality and effectiveness of the proposed approaches.