The petrochemical sector plays a vital role in supplying essential raw materials for downstream industries such as resin manufacturing, plastics, and coatings. Evaluating the performance of petrochemical suppliers has therefore become a strategic priority for ensuring operational efficiency, environmental compliance, and social responsibility throughout the supply chain. However, data uncertainty, market volatility, and the complex interdependencies between production stages make supplier evaluation a challenging task. To address these challenges, this study proposes an integrated two-stage fuzzy Data Envelopment Analysis framework that combines intuitionistic fuzzy entropy with cross-efficiency evaluation. The proposed model extends Wang et al.'s relative closeness approach to a fuzzy two-stage environment, enabling consistent and robust performance assessment under uncertainty. In the proposed framework, the lower and upper bounds of fuzzy efficiency are transformed into membership and non-membership degrees within an intuitionistic fuzzy set, while the Triangular Directional Distance Index is employed to obtain a reliable ranking of suppliers. The proposed framework is applied to evaluate 26 petrochemical suppliers within the supply chain of a resin manufacturing company. The results demonstrate that the proposed approach produces stable and interpretable supplier rankings under different possibility levels, providing decision-makers with a robust analytical tool for sustainable supplier evaluation.