Basic concepts, definitions, and methods in D number theory
  As a generalization of Dempster-Shafer theory, D number theory (DNT) aims to
provide a framework to deal with uncertain information with non-exclusiveness
and incompleteness. Although there are some advances on DNT in previous
studies, however, they lack of systematicness, and many important issues have
not yet been solved. In this paper, several crucial aspects in constructing a
perfect and systematic framework of DNT are considered. At first the
non-exclusiveness in DNT is formally defined and discussed. Secondly, a method
to combine multiple D numbers is proposed by extending previous exclusive
conflict redistribution (ECR) rule. Thirdly, a new pair of belief and
plausibility measures for D numbers are defined and many desirable properties
are satisfied by the proposed measures. Fourthly, the combination of
information-incomplete D numbers is studied specially to show how to deal with
the incompleteness of information in DNT. In this paper, we mainly give
relative math definitions, properties, and theorems, concrete examples and
applications will be considered in the future study.
