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In order to process the fuzzy and imprecise information in the evidential reasoning, the scholars have made many attempts to generalize belief and plausibility functions based on the Dempster-Shafer(D-S) evidence theory to fuzzy sets for many decades. A new method for defining the fuzzy closeness degree is put forward in this paper. Based on the closeness degree, another generalization of belief and plausibility functions to fuzzy sets is proposed which discards the max and min operators in foregoing generalizations according to the measure of fuzzy inclusion. We then make the comparisons of the proposed extension with some methods available. The results of the numerical experiments show the effectiveness of the proposed generalization, especially for being able to catch more information about the change of fuzzy focal elements.

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