Sustainable supply chain management (SSCM) plays a great part in enterprise production operation management due to strict government regulations and increased public awareness. Selecting the best sustainable supplier is essential for companies to promote SSCM, which, as indicated in many studies, is a multi-criteria decision making (MCDM) problem. Besides, decision makers tend to utilize linguistic terms for expressing their evaluations owing to their fuzzy knowledge. This paper reports an innovative MCDM model for sustainable supplier selection by integrating best-worst method (BWM) and alternative queuing method (AQM) within the interval-valued intuitionistic uncertain linguistic setting. The novel approach allows to capture the uncertainty and vagueness of decision makers’ judgements with the aid of interval-valued intuitionistic uncertain linguistic sets. Furthermore, the BWM method can obtain the optimal weights of criteria by constructing a nonlinear programing model. The AQM is reliable and intuitive to generate the ranking of candidate suppliers. Finally, a watch manufacturer is used as an example for illustrating the practicability and effectiveness of the proposed sustainable supplier selection model.