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• Defining two IvIF granular structure distances. • Developing an extended partial order relation in IvIF granular space based on granular structure distance. • Proposing new uncertainty measures based on this partial order relation. • Presenting IvIF approximate granular structure for describing the target concept and a new attribute reduction algorithm. • Presenting a new method based on uncertainty measures for the linguistic large group decision-making problem. The interval-valued intuitionistic fuzzy (IvIF) set, a combination of intuitionistic fuzzy sets and interval-valued sets, has been widely employed for multi-attribute decision-making and group decision-making. However, two urgent problems remain to be solved. The first is to determine the coarseness/fineness relation in the IvIF granular space, and the other is to determine the difference between two IvIF granular structures with the same uncertainty. In this paper, a pair of IvIF granular structure distances is proposed to address the aforementioned issues. Then, a more generalized coarseness/fineness relation is defined for the IvIF granular space. Furthermore, two complementary uncertainty measures were formulated based on the coarseness/fineness relation. Finally, we investigate the IvIF approximate granular structure of a target concept and propose a new algorithm for attribute reduction from the perspective of distance. The experiments show that our algorithm possesses a shorter reduct than the other algorithms, ensuring relatively high classification accuracy. Furthermore, a case study is presented to demonstrate the feasibility of using the proposed method to solve a large linguistic group decision-making problem. [ABSTRACT FROM AUTHOR]