A framework for the fusion of non-exclusive and incomplete information on the basis of D number theory

Uncertainty is of great concern in information fusion and artificial intelligence. Dempster-Shafer theory is a popular tool to deal with uncertainty, but it cannot effectively represent and fuse uncertain information involving non-exclusiveness and incompleteness. In order to solve that problem, an idea of D number theory (DNT) has been proposed. In this paper, the basic theory of DNT for the fusion of non-exclusive and incomplete information is studied to strengthen its theoretical foundation, including concept formalization, uncertainty representation, information modelling and fusion. At first, the non-exclusiveness in DNT is defined formally and its basic properties are discussed. Secondly, new measures of belief and plausibility for D numbers are developed. Thirdly, the combination rule for D numbers is studied by extending the exclusive conflict redistribution rule. Fourthly, a method to combine information-incomplete D numbers is proposed. The proposed concepts, definitions, and methods are analyzed mathematically and exemplified through illustrative examples.