关键词: 数据包络分析, 乐观效率, 悲观效率, 区间效率

Abstract: Data envelopment analysis (DEA) is an effective tool to evaluate the performance of decision-making units (DMU) based on multiple inputs and outputs. In 1978, CHARNES et al. measured the efficiency (CCR efficiency) by the ratio of total weighted outputs to total weighted inputs on condition that the similar ratios of each DMU do not exceed the value of 1. Therefore, the CCR ratio model is identified as the best relative efficiency or optimistic efficiency. The traditional DEA models are usually constructed from the optimistic perspective to achieve the best DMU performance. But DMU efficiency can also be measured from the pessimistic perspective, by maximizing the ratio of its weighted sum of outputs to the weighted sum of inputs, on condition that the efficiency of each DMU is no less than 1. Hence, applying both perspectives can comprehensively evaluate the two extreme performances of every DMU. At present, there are not enough studies on how to evaluate DEA and optimize performance measurement. Therefore, this study proposes a simpler and more effective way from both optimistic and pessimistic perspectives to measure DMU performance and efficiency within intervals. On such a basis, new DEA adjustment coefficient models are built to identify the range of interval efficiency. In doing so, the pessimistic efficiencies of DMU are adjusted to the lower bounds of efficiencies, so the best and worst relative efficiencies form an interval to comprehensively measure DMU performance. This not only expands the research scope of DEA, but also is more in line with reality, which is convenient for DM to provide more comprehensive and in-depth decision-making reference.
Since optimistic efficiency and pessimistic efficiency are efficiency values obtained by DEA models in different ranges, they cannot be directly compared. Theoretically, optimistic efficiency and pessimistic efficiency should form an efficiency interval. Therefore, it is necessary to adjust the pessimistic efficiency so that the adjusted pessimistic efficiency of each DMU and its optimistic efficiency form an efficiency interval. In order to measure the efficiency interval of each DMU reasonably, we introduce the adjustment coefficient, because the efficiency interval of each DMU will be affected by the value of the adjustment coefficient. Therefore, the optimization models are constructed from optimistic and pessimistic perspectives to determine the adjustment coefficient, and the adjustment coefficient is used to adjust the pessimistic efficiency of DMU to the lower bound of the efficiency interval, so that it and optimistic efficiency constitute the efficiency interval. After the efficiency interval of DMU is determined, since the final efficiency of each DMU is expressed by the number of intervals, a simple and practical sorting method is needed to compare and sort them. In order to facilitate the comparison with other interval efficiency ranking methods, we choose the Harwicz criterion method as the method for comparing and ranking interval efficiency. Finally, in order to illustrate the feasibility and effectiveness of the proposed method by comparing it with other methods, the research performance of 12 key science and engineering universities in China is evaluated. The data comes from the survey report of Science and Technology work in Chinese universities in 2016. Each DMU has two inputs and four outputs, and all input and output data are represented by exact numbers. The example analysis shows that the proposed method can not only obtain reasonable efficiency interval of DMU, but also identify efficient and inefficient DMU. At the same time, all DMU are completely sorted by combining optimistic and pessimistic efficiency. All DEA efficient DMU together form an efficiency frontier, while all DEA inefficient DMU together form an inefficiency frontier. For DMU that are not specified, they are always surrounded by efficiency and inefficiency frontier. At the same time, some DMU may belong to both DEA efficient and DEA inefficient, and these DMU have the widest efficiency interval, so they contain the greatest uncertainty in the actual evaluation.
Compared with current methods, the method proposed in this study makes the following contributions. First, it can identify both DEA efficient and inefficient DMU; the former constitutes the efficiency frontier, and the latter the inefficiency frontier, covering all the DEA unspecified DMU. Second, the adjustment coefficient only needs to be solved once to adjust the pessimistic efficiency of each DMU, so as to obtain the lower bound of efficiency interval—simpler than other interval DEA models. Third, the efficiency intervals are in line with the optimistic and pessimistic efficiencies with consistent ranking orders. Given interval efficiencies offer a more comprehensive assessment of DMU' performance than traditional DEA efficiency, they hold significant potential for applications. It is worth noting that the input-oriented DEA adjustment coefficient model developed in this study can be easily adapted to other scenarios, such as output-oriented, BCC (Banker-Charnes-Cooper) and additive DEA models. Furthermore, it can also be applied to model interval input and output data, which would be a major focus of future study.

Key words: data envelopment analysis, optimistic efficiency, pessimistic efficiency, interval efficiency