The efficiency decomposition and frontier projection of traditional two-stage network data envelopment analysis (DEA) model under variable returns to scale (VRS) are often not equivalent; which not only contradicts DEA theory, but also reduces the scientificity of the model. The main reason for this inequivalence is that there is a synergistic effect of variable scale return in two different stages. Therefore, this paper describes the production frontier of two-stage DEA under VRS for analyzing this synergistic effect, and then the efficiency evaluation pitfalls of two-stage DEA under VRS are identified. From the input orientation, output orientation, non-orientation perspectives, different two-stage network DEA models under VRS are respectively constructed to solve these evaluation pitfalls, and the equivalence relationships of their multiplier model and envelopment model are proved; and then the efficiency decomposition and frontier projection with equivalence relationship can be obtained to meet the different needs of decision-makers. Furthermore, variable intermediate element is discussed in the non-orientation model for achieving the Pareto optimality of two stages during the process of efficiency decomposition and frontier projection. By these models, the theoretical foundation of two-stage network DEA under VRS has been further improved. Finally, two examples are provided to illustrate the effectiveness of the new models.

中文翻译：

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变量尺度下两阶段网络 DEA 的效率分解与前沿投影


传统两阶段网络数据包络分析 （DEA） 模型在可变比例回报率 （VRS） 下的效率分解和前沿投影往往不等效;这不仅与 DEA 理论相矛盾，而且降低了模型的科学性。这种不等式的主要原因是可变规模回报在两个不同阶段存在协同效应。因此，本文描述了 VRS 下两阶段 DEA 的生产前沿，以分析这种协同效应，然后确定了 VRS 下两阶段 DEA 的效率评估陷阱。从输入取向、输出取向、非取向角度出发，分别构建了VRS下不同的两阶段网络DEA模型来解决这些评价陷阱，并证明了它们的乘数模型和包络模型的等效关系;进而得到具有等价关系的效率分解和前沿投影，以满足决策者的不同需求。此外，在非取向模型中讨论了可变中间元在效率分解和前沿投影过程中实现两个阶段的帕累托最优性。通过这些模型，进一步完善了 VRS 下两阶段网络 DEA 的理论基础。最后，提供了两个示例来说明新模型的有效性。