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Atanassov's intuitionistic fuzzy sets (IFSs) are a very convenient tool for describing alternatives/options while making decisions because they make it possible to naturally represent the pros, cons, and hesitation. The IFSs have attracted a significant interest and have been applied in various fields. Of course, their use poses some challenges. One of the main challenges is the ranking of alternatives/options described by the intuitionistic fuzzy sets, to be called for brevity the intuitionistic fuzzy alternatives. This is a crucial issue, notably for the applications, for instance, in decision making. We first present in detail and analyze the benefits of a method we introduced previously (cf. Szmidt and Kacprzyk [1]). For this method, we augment the original assumptions with an additional assumption, which is justified and inherently reasonable. As a result, we obtain formulas which are better justified than those previously used as they explicitly consider the arguments in favor (pro), against (con), and hesitance. Since the intuitionistic fuzzy alternatives can not be linearly ranked, then the additional assumptions during the ranking process are necessary. We address these issues and analyze examples to clarify our new approach. We examine some other methods discussed in the literature and analyze their results, and show that the new assumptions reconcile some misconceptions raised by those other papers. [ABSTRACT FROM AUTHOR]