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Uncertainty and imprecision characterize human cognitive and reasoning processes. Fuzzy cognitive maps (FCMs) are computationally simple yet effective structures to approximately model and simulate such processes. A limitation of current FCMs is that they are unable to model the hesitancy introduced into a complex system due to imperfect facts, missing information, and indecision. To cope with this issue, we propose a novel extension of the FCM model which is based on the theory of intuitionistic fuzzy sets. This intuitionistic FCM (iFCM) model, which is denoted as iFCM-II, inherently exploits the mathematical framework of intuitionistic fuzzy sets for the definition of the concepts constituting the cognitive map and their interrelations, as well as for reasoning. Furthermore, unlike the previous iFCM model, which is denoted as iFCM-I, it enables an intuitionistic estimation of hesitancy at the output concepts, thus offering a natural mechanism to assess the quality of its output. The advantages of the proposed iFCM model over the current FCM and iFCM models are demonstrated with reproducible numeric examples for process control and decision support applications.