Abstract: The data envelopment analysis (DEA) was proposed in 1978 as an efficiency evaluation method with a variety of inputs and outputs. It is a “black box” evaluation method using objective data with the most favorable weight for the decision-making unit (DMU) itself. Traditional DEA models generally have two problems: First, the “black box” evaluation can’t detect the impact of internal processes on efficiency; Second, the influence of other DMUs is not taken into account, and the phenomenon of partial generalization is likely to occur. Network DEA is an effective means to solve the first problem, by opening the “black box” for evaluation, and mining the impact of internal factors on efficiency. The cross efficiency evaluation method is useful to solve the second problem, which fully considers the role of all DMUs in efficiency evaluation. However, none of them take into account the influence of decision makers’ subjective preferences, and scholars have made many efforts to improve DEA, but the research on combining cross efficiency and network DEA is still in its infancy. Therefore, this paper combines them and constructs two-stage additive efficiency models according to the most basic two-stage chain network structure. This not only expands the research scope of DEA, but also is more in line with reality, which is convenient for DMs to provide more comprehensive and in-depth decision-making reference.
Cross-efficiency evaluation is mainly a “benevolent” and “aggressive” model constructed by auxiliary goal optimization. The “benevolent” model maximizes the efficiency of other DMUs as much as possible under the premise of ensuring its own highest efficiency, so the objective function is to find the maximum. The “aggressive” model, on the contrary, takes the minimum value and the other constraints remain unchanged. The network DEA considers the most basic chain structure, the output of the first stage is also the input of the second stage, and the overall efficiency of the system is equal to the product of the two-stage efficiency. This paper draws on the idea of additive efficiency decomposition, combines network DEA and cross efficiency method, and considers the preferences of decision-makers to construct a two-stage additive cross efficiency model by setting priorities for different processes. By calculating the proportion of sub-stage input in the overall input, the variables of optimization decision-making are obtained, and three new models in different situations are proposed, namely overall priority, stage 1 priority and stage 2 priority model. Finally, the overall efficiency, stage 1 efficiency and stage 2 efficiency of the evaluated unit are calculated by arithmetic average. There are two points to note.One is that when the model is a nonlinear programming, the Charnes-Cooper transformation should be used to solve the linear programming model; Second, each model is calculated using the weight that is most favorable to its own process, which is not necessarily optimal for other processes, so there may be a situation in the cross efficiency matrix where the mutual evaluation efficiency is greater than the self-assessment efficiency.
This paper takes banks, insurance, and supply chains as examples to analyze the Stage 1 priority, Stage 2 priority, and overall priority models respectively. In the case of banks, the 1st stage is the deposit absorption process, which is invested in the net value of fixed assets and the number of employees. The 2nd stage is the profit process, and the output is the book profit, which verifies the feasibility of the priority model of phase 1, and the efficiency at this time mainly depends on the efficiency level of phase 1. In the case of an insurance, stage 1 is the marketing process, with operating expenses and insurance expenses as inputs, and output as direct written premiums and reinsurance expenses, and stage 2 inputs. Phase 2 is the investment process of the insurance company, and the output is underwriting profit and investment profit, which verifies the feasibility of the Phase 2 priority model, and the efficiency at this time mainly depends on the efficiency level of Phase 2. In the supply chain, stage 1 is selling, with manpower, operating costs, and transportation costs as inputs, output as the number of products shipped, and stage 2 inputs. The 2nd stage is buying, and output is sales and profit. It can be seen that the overall efficiency and ranking are in the middle or near the efficiency and ranking of the sub-stages, and the overall efficiency priority model is feasible.
The model can also be applied to environmental efficiency assessment, enterprise performance evaluation, and other fields, and to various industries such as the hotel industry and pharmaceutical industry. At the macro level, the method can be applied to the efficiency evaluation of regional or international production activities; At the medium level, this method can be used to analyze the internal development factors of different industries in the same region or the same industry in different regions; At the micro level, it can be used for organizations such as businesses, banks or hospitals to find the causes of inefficiencies.

Key words: two-stage DEA, cross efficiency, additive efficiency decomposition

摘要： 数据包络分析是利用对决策单元自身最有利的权重进行“黑箱”评价的方法,无法观察到系统内部因素的影响,仅从自评角度进行评价易导致以偏概全。网络DEA和交叉评价方法是近年来热门的研究话题,一方面打开了“黑箱”进行效率评价,另一方面考虑了其他决策单元对被评价单元的影响。本文借鉴加性效率分解思想,把网络DEA和交叉效率评价方法相融合,根据决策者偏好设定不同过程的优先级,从而构建在两阶段结构下的加性交叉效率模型,具体分为三种情况:整体优先、阶段一优先和阶段二优先,被评价单元的最终效率由所有交叉效率的算术平均值计算得到。最后,结合不同行业的例子说明新模型的合理性和适用性,考虑了主观因素更符合现实环境下的决策行为,结合效率分解结果探究了导致低效率的原因,为决策者提供更全面更深入的决策意见参考。

关键词: 两阶段DEA, 交叉效率, 加性效率分解