Group Efficiency Evaluation Under Fixed-Sum Output Constraints: A Cross-EEF Approach with Application to Industrial Carbon Emissions in China

The existence of fixed-sum output constraints in real-world situations is widespread, such as market share and carbon dioxide emissions, etc. However, existing fixed-sum output data envelopment analysis (DEA) methods mostly focus on individual decision-making units (DMUs) and ignore the interactions between groups. Therefore, this study first establishes a systematic framework to quantify group performance by the average criterion, and constructs the equilibrium efficient frontier (EEF) to evaluate all groups on a common platform. To address the non-uniqueness issue of EEF, we further introduce the aggressive cross-efficiency mechanism, ultimately proposing a novel group cross-EEF methodology that explicitly accounts for competitive intergroup dynamics. The proposed method is applied in the assessment of carbon emission efficiency in the industrial sector for 30 provinces in China, and the validity of the method is verified. The result shows that (1) even though the average industrial carbon efficiency stands at 1.2015, half of the provinces exhibit values below 1; (2) significant regional heterogeneity is observed, with North China and East China exhibiting higher efficiency levels, while the Northeast and Northwest regions lag behind; (3) provinces such as Beijing, Guangdong, and Zhejiang demonstrate superior performance, in contrast to Ningxia, Hebei, and Qinghai, which remain at relatively low efficiency levels. This study provides theoretical and policy insights to support the advancement of low-carbon development in China’s industrial sector.