Abstract Network data envelopment analysis (NDEA) is used to evaluate the performance of a set of multi-stage decision-making units (DMUs) (or multi-stage production processes) taking into account the internal structure of these DMUs. Finding efficiency stability regions (ESRs) for efficient DMUs is one of the most important issues in DEA because it can provide useful information for their managers (or decision-makers) from the managerial and economic viewpoints. ESR of an efficient DMU is the region that this DMU will remain efficient if and only if after changing its inputs and its outputs it stays in this region. Hitherto, there exist many DEA approaches in the context of finding the ESRs of DMUs however, none of them take into account the internal structures of DMUs to find their ESRs. They only consider the inputs and the (final) outputs of DMUs to find their ESRs. This problem is the drawback of all existing approaches in this context. Hence, this study contributes to network DEA by proposing a novel approach to tackle this problem. To do this, we first define the three concepts of ‘network-efficient’, ‘extreme network-efficient’, and ‘ESR’ in network DEA and then introduce a linear DEA model to specify the extreme network-efficient two-stage production processes. Second, we propose a DEA approach to find the ESRs of these two-stage production processes. The proposed approach finds the ESR of an extreme network-efficient two-stage production process when its inputs increase, its intermediate products and (final) outputs decrease, and the data of the other two-stage production processes remain unchanged. This research also clarifies the managerial and economic implications of finding the ESRs. Finally, three numerical examples and an empirical application are given to show the use of the proposed approach. Proofs of the theorems are also presented in the appendix.

中文翻译：

---

具有中间产品的两阶段生产过程的效率稳定区

摘要 网络数据包络分析（NDEA）用于评估一组多阶段决策单元（DMU）（或多阶段生产过程）的性能，同时考虑这些 DMU 的内部结构。为有效的 DMU 寻找效率稳定区 (ESR) 是 DEA 中最重要的问题之一，因为它可以从管理和经济的角度为其管理者（或决策者）提供有用的信息。有效 DMU 的 ESR 是该 DMU 将保持有效的区域，当且仅当在更改其输入和输出后它仍停留在该区域中。迄今为止，在寻找 DMU 的 ESR 的背景下存在许多 DEA 方法，然而，它们都没有考虑到 DMU 的内部结构来寻找其 ESR。他们只考虑 DMU 的输入和（最终）输出来找到它们的 ESR。这个问题是在这种情况下所有现有方法的缺点。因此，本研究通过提出一种新的方法来解决这个问题，为网络 DEA 做出了贡献。为此，我们首先在网络 DEA 中定义了“网络高效”、“极端网络高效”和“ESR”三个概念，然后引入线性 DEA 模型来指定极端网络高效的两阶段生产过程. 其次，我们提出了一种 DEA 方法来找到这些两阶段生产过程的 ESR。当输入增加，中间产品和（最终）输出减少，而其他两阶段生产过程的数据保持不变时，所提出的方法找到了一个极端网络高效的两阶段生产过程的 ESR。这项研究还阐明了寻找 ESR 的管理和经济影响。最后，给出了三个数值例子和一个经验应用来说明所提出方法的使用。定理的证明也在附录中给出。