The operations on interval-valued intuitionistic fuzzy values in the framework of Dempster–Shafer theory

• Operations on interval-valued intuitionistic fuzzy values.
• Dempster–Shafer theory of evidence.
• Decision making.

This work was motivated by the revealed limitations and drawbacks of the classical definition of interval-valued intuitionistic fuzzy set (IVIFS) proposed by Atanassov and Gargov and operations on interval-valued intuitionistic fuzzy values (IVIFV). It is shown that the classical definition of Atanassov’s IVIFS may lead to controversial results. Therefore, a new more constructive definition of IVIFS is proposed. It is shown that this new definition makes it possible to present IVIFVs in the framework of Dempster–Shafer theory (DST) as belief intervals with bounds presented by belief intervals. A new set of operations on IVIFVs in the framework of DST is proposed. It is proved that these operations provide IVIFVs in the sense of new definition of interval-valued intuitionistic fuzzy set. A critical analysis of commonly used operations for comparison of intuitionistic fuzzy values and IVIFVs is made. As a result, a new more justified operation for comparison of IVIFVs in the framework of interval extended Dempster–Shafer theory is proposed. It is proved that introduced set of operations on IVIFVs is free of revealed limitations and drawbacks of commonly used operations based on the classical definition of Atanassov’s IVIFS. The corresponding methods for aggregation of local criteria presented by interval-valued intuitionistic fuzzy values in the framework of interval-extended Dempster–Shafer theory are proposed and analysed. The proposed approach makes it possible to introduce the new interval-valued intuitionistic weighted geometric operators for the cases when the weighs are presented by intuitionistic and IVIFVs. These operators are not defined in the framework of classical approach.