Sometimes in the sense of intuitionist 2-tuple linguistic (I2TL) sets, experts may not be able to decide the most suitable criterion weight vector for multicriteria decision making (MCDM). To avoid this situation, the decision-makers (DMs) can use the Best-Worst method (BWM), in which DMs choose the best (most significant) criterion and the worst (least significant) criterion and then provide two preference vectors by comparing criteria best to other (BO) and other to worst (OW). In the Nonlinear Best-Worst Method (NBWM) it is more complicated to find the unique solution of the model. Therefore, the main goal of this study is to propose two approaches to BWM, namely, Linear Best-Worst Method (LBWM) and Euclidean Best-Worst method (EBWM) to achieve the best criteria priority vector for Multi-Criteria Group Decision Making (MCGDM) problems in the context of I2TL information. In the computational process of MCDM problems, we have to aggregate I2TL elements into a global one. Consequently, under certain critical properties, we are creating some operational laws for I2TL elements based on Hamacher operations. Also, the intuitionistic 2-tuple linguistic Hamacher weighted average (I2TLHWA), and the intuitionistic 2-tuple linguistic Hamacher weighted geometric (I2TLHWG) operators are introduced with the assistance of Hamacher operations and I2TL elements. Subsequently, we analyze some of the I2TLHWA operator’s related properties and we propose MCGDM framework under I2TL information. Finally we demonstrate the validity and efficiency of our method and operations.