This article investigates group decision-making (GDM) problems, where the decision makers' (DMs) preference information is represented by incomplete interval-valued intuitionistic fuzzy preference relations (IVIFPRs). First, a multiplicative consistency property and an acceptably multiplicative consistency property for IVIFPRs are offered. Then, an optimization model to estimate the missing values in an incomplete IVIFPR is constructed. Subsequently, two optimization models are, respectively, established to derive a perfectly consistent IVIFPR and an acceptably consistent IVIFPR from a given inconsistent IVIFPR. Furthermore, a model is offered to gain the DMs' weights. Afterward, the consensus index is defined. When the consensus for IVIFPRs is unacceptable, a model is presented to reach the consensus requirement. Moreover, a novel GDM method for incomplete IVIFPRs is presented. Finally, the presented method is applied to an illustrative example that shows the feasibility of the offered method.