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The ranking of Atanassov's intuitionistic fuzzy values (A-IFVs) is nontrivial because there is no natural linear order among them, as opposed to fuzzy sets. In this paper, we tackle this difficult problem and develop a new novel technique for ranking A-IFVs based on amount of information using the geometrical representation of Atanassov's intuitionistic fuzzy sets (A-IFSs), with the aim of overcoming some drawbacks and ambiguities in existing methods and to build a new ranking model in the A-IFS context from the viewpoint of amount of information. We prove that the order on nonempty A-IFVs induced by our function is admissible in the A-IFS context. Thus, our developed method provides a total order that extends the usual partial order between A-IFVs. Moreover, we discuss the role that a decision maker's attitude can play in decision making under uncertainty. This allows us to introduce the idea of attitudinal-based extension of our developed technique, which is much more useful and flexible in real-world applications. By this, we also provide a new idea and explanation to evaluate two special A-IFVs: the completely known opposition (0,1,0) and the completely unknown information (0,0,1). Finally, an effort is made to extend the applicability of the developed methods by considering the situation in which Atanassov's interval-valued intuitionistic fuzzy sets are involved. Experimental studies show the good performance of the developed methods in the A-IF and A-IVIF situations, respectively.