Suppliers significantly affect the effectiveness of sustainable supply chain management. Hence, it is extremely important to evaluate and select suppliers scientifically and objectively. Based on the theory of triple bottom line (economic, social, and environmental dimension) and a balanced scorecard, a measureable supplier evaluation framework in a sustainable supply chain is first formulated. Second, to reduce the defects of the single weight method, the subjective and objective weights of evaluation indicators are determined by combining the fuzzy best-worst method (BWM) and the entropy method, and then the combination weights are obtained through linear weighting. Third, the grey relational technique for order performance by similarity to ideal solution (TOPSIS) method is further adopted to evaluate and rank the suppliers. Finally, a case study illustrates and demonstrates the availability of the proposed supplier evaluation index system and evaluation method. Subsequently, some suggestions are proposed according to the results.