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Forest carbon sink project (FCSP) has received growing attention for its outstanding advantages in carbon emission reduction, environment protection, and achieving the carbon neutrality objective. Using a government-social capital cooperation model, FCSP helps to revitalize the ecological and economic benefits of the forest with standard and efficient management. However, the management and operation barriers of FCSP implementation are complex, multiple and indefinite in the long run, which requires a comprehensive barrier evaluation framework. Hence, this paper is devoted to constructing a barrier evaluation framework for FCSP implementation in China with an integrated multi-criteria decision-making (MCDM) method. To this end, critical barriers are identified and the criteria system is constructed with a two-stage systematic literature review and interaction with the experts. Later, an applicable and comprehensive barrier evaluation framework is built, wherein the interval type-2 fuzzy set (IT2FS) is used to process fuzziness and uncertain information, the best-worst method (BWM) is adopted to determine the criteria weight, and the Preference Ranking Organization Method for Enrichment Evaluations II (PROMETHEE II) is introduced for comprehensive barrier evaluation. The criteria weighting results indicate that FCSP implementation encounters challenges taking into account institutional and governance barriers (criteria weight: 0.3592), management barriers (0.2384), economic and market barriers (0.1637), knowledge barriers (0.0747), technical and infrastructure barriers (0.1639), C1 "Lack of clear leadership and policies" and C5 "Forest management performance" were determined the most significant criteria, A5 and A4 respectively performed the lowest and the highest barrier level. In sensitivity analysis, the ranking results were stable when the criteria changed, and Spearman's rank correlations of B2, B4, and B5 are significant. The results were validated by a comparative analysis with other methods with the help of Spearman's rank correlation coefficient, which reflects accepted high correlations (Spearman's pro values of BWM-IT2F-PROMETHEE II (Gaussian rule), BWM-IT2F-TOPSIS, and BWM-TFN-VIKOR are 1.00, 1.00, 1.00). The findings of the paper help the decision makers to develop more rational strategies and successfully implement the FCSP. [ABSTRACT FROM AUTHOR]