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The problems of the <italic>A-IFS</italic> and <italic>A-IVIFS</italic> theories are analyzed and to solve them, their Dempster-Shafer theory based redefinition is developed and a new concept called Belief-Plausibility number (<italic>BPN</italic>) is introduced. The <italic>BPN</italic> can be applied beyond semantics of usual fuzzy sets and the <italic>A-IFS</italic>. Two main sources of uncertainty generating the <italic>A-IVIFS</italic> are considered: an uncertainty coursed by the membership and non-membership functions multiplicity and an uncertainty of inputs. For the fuzzy inputs, the definition of the fuzzy-valued <italic>BPN</italic> is proposed. The developed approach to treat interval-valued intuitionistic fuzzy objects and their Belief-Plausibility re-definitions allowed us to introduce a set of useful definitions. It is important that some of the them were not earlier mentioned in the literature. The corresponding new arithmetical operations with such objects are proposed. The practical usefulness of the introduced approach is illustrated by the solution of the real-world Fuzzy-Valued Belief-Plausibility multiple criteria problem of the raw material suppliers selection on the steel rolling plant. The obtained results allowed us to state that the introduced Belief-Plausibility approach can be successfully used in the analysis and applications independently, without the use of the <italic>A-IFS</italic> concepts. This approach owns visible advantages in comparison with the <italic>A-IFS</italic> based one.

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The New Definitions of Intuitionistic and Belief-Plausibility Based Local Criteria With Interval and Fuzzy Inputs Applied to the Multiple Criteria Problem of a Raw Material Supplier Selection