Best-worst method (BWM) has attracted increasing attention. It has been generalized to different fuzzy environments and applied to various real-life decision problems. This article develops a new intuitionistic fuzzy (IF) BWM (IFBWM) for multicriteria decision-making. When a decision maker (DM) makes comparisons, there may be some hesitancies. Thus, the reference comparisons are represented as intuitionistic fuzzy values (IFVs), the Best-to-Others vector and the Others-to-Worst vector are IF vectors. According to the multiplicative consistency of intuitionistic fuzzy preference relation, this article gives the consistency equations and views them as IF equations. The derivation of optimal IF weights of criteria is formulated as an IF decision-making problem. Thereby, a mathematical programming model is constructed to assure that the derived optimal IF weights of criteria is a normalized IF weight vector. Depending on the risk preference of DM, four linear programming models are presented to obtain the optimal IF weights based on the constructed mathematical programming model for the optimistic DM, the pessimistic DM, and the neutral DM, respectively. Furthermore, this article investigates the process of improving the consistency. Several examples are demonstrated to show the application and effectiveness of the proposed IFBWM.