A Generalized Belief Entropy With Nonspecificity and Structural Conflict

The Dempster–Shafer (D-S) evidence theory has been widely used in many fields from probabilistic inference, information fusion to decision analysis due to its superiority to formulate uncertain and incomplete information under a weaker condition than the Bayesian probability theory. To support the elicitation of basic probability assignment (BPA), numerous uncertainty measures have been proposed in the recent decades, which take into account nonspecificity and structural conflict (SC) in either the entropy-alike or nonentropy form. This article is focused primarily on developing a generalized belief entropy to measure and quantify the uncertainty of BPA in a consistent and comprehensive way. The definition of the proposed belief entropy involves three components. Specifically, the first component is Shannon’s definition of entropy of probability functions which can be interpreted as the measure of discord of the mass function among various focal elements. The second component is the measure of total nonspecificity which is the generalization of Dubois–Prade’s definition of nonspecificity measure. Furthermore, the critical component introduced in this article is the utilization of the SC coefficient within an evidence, which fully considers the intersection between each pair of focal elements of the BPA. The characteristics and properties of the new generalized belief entropy are analyzed systematically with theoretical proofs. Finally, the proposed belief entropy is embedded to a three-tuple approach to deal with a practical multiple attribute decision-making (MADM) problem to demonstrate its validity and strengths of measuring nonspecificity and SC in uncertain decision making.