Due to the effectiveness and advantages of interval-valued intuitionistic fuzzy sets (IVIFSs) in evaluating uncertainty and risk, we introduce IVIFSs into loss functions of decision-theoretic rough sets (DTRSs) and propose an optimization-based approach to interval-valued intuitionistic fuzzy three-way decisions. First, based on the classical DTRSs and two previous optimization models, we construct a new concise linear programming model for simultaneously determining the threshold pair. Our model is mathematically equivalent to the DTRSs and the previous models under the Karush-Kuhn-Tucker (KKT) condition. Second, we extend the constructed model via the IVIFSs of loss functions and we discuss the relations between these loss functions based on a similarity measure function-based ranking method and a multiple score function-based ranking method for IVIFSs. Third, we develop our extended models via two ranking methods and we prove the existence and uniqueness of the optimal solution of the model. The optimization-based method, along with its algorithm for three-way decisions, is designed in an interval-valued intuitionistic fuzzy environment. Compared to the latest existing methods, our method has three advantages (see Advantages 1-3). Finally, an illustrative example is considered, and the advantages of our approach are demonstrated by this example.