Abstract: Data envelopment analysis (DEA), as a non-parametric statistical method dealing with evaluation problem that utilizes multiple inputs to produce multiple outputs, uses linear programming models to evaluate the relative efficiencies of a group of homogenous decision making units (DMUs). Due to the uncertainty of the decision-making environment and the limited knowledge of decision-makers in real applications, the input and output data of DMUs may be characterized by fuzzy numbers, such as interval numbers and triangular fuzzy numbers. At the same time, the subjective preference of decision-makers (DMs) under uncertain conditions has an important influence on the decision-making process. Aiming at the evaluation of the relative efficiencies of decision-making units in fuzzy environments, a variety of traditional fuzzy data envelopment analysis (FDEA) models are developed in the literature. However, these models may obtain unrealistic results and lack the discrimination power to distinguish efficient DMUs. Meanwhile, the currently published papers involving fuzzy cross-efficiency not only require considerable computational efforts to obtain fuzzy cross-efficiencies, but also adopt the arithmetic average method to aggregate cross-efficiencies and ignore the subjective preference of decision-makers such that the relative importance attached to the cross-efficiencies provided by the different decision-making units are neglected and a convincing result is hard to be attained. The significances of this paper can be summarized as the following two aspects: For one thing, this paper enriches the theoretical research of the cross efficiency method of fuzzy DEA. Both the fuzzy DEA model and the improved interval cross-efficiency model in this paper measure the efficiency of decision making units are based on the unified production frontier, so as to ensure the efficiency comparability between different decision making units and under different α values. For another, this paper expands the application of prospect theory to the efficiency evaluation problem in fuzzy environment. The interval reference point is defined instead of the traditional single reference point, which fully considers the influence of the change of decision-makers' psychological factors on decision-making behavior in the process of aggregating cross-efficiency.
To measure relative efficiency of DMUs in fuzzy environments, the fuzzy DEA cross-efficiency evaluation method based on prospect theory is proposed. Firstly, the α-level-based approach is applied to turn the inputs and outputs data represented by triangular fuzzy numbers into interval numbers, and then an improved interval cross-efficiency model is proposed, which considers all the optimal weights of decision-making units. Subsequently, the prospect theory is introduced to study the problem of interval cross-efficiency aggregation, and the interval reference point is defined to replace the traditional single reference point. Based on the principle of maximizing the prospect of cross-efficiency value of decision-making units, a model is constructed to solve the aggregation weight, and aggregate interval cross-efficiency value to replace the comprehensive prospect of cross-efficiency value. Finally, the preference degree approach is used to compare and rank the interval cross-efficiency values, which can provide a more comprehensive and reasonable result. The proposed method measures efficiencies of decision-making units based on a unified production frontier, ensuring that efficiencies of different decision-making units and under different α values are comparable. Moreover, the interval reference point takes the changes of the decision-maker's psychological factors into full consideration in fuzzy environments. Aggregating the interval cross-efficiency values of decision-making units rather than the comprehensive prospect value retains as much decision information as possible. At last, in order to illustrate the feasibility and validity of the proposed method by comparing it with different approaches, this paper adopts an example that evaluates the efficiency of ten decision making units. Each DMU has two inputs and two outputs, and all input and output data are characterized by triangular fuzzy numbers. Compared to interval DEA models and fuzzy cross-efficiency models proposed by other scholars, it is observed that the result of these methods is significantly different the proposed method. On the one hand, the interval DEA models solve the self-evaluation efficiency of DMUs, and each decision making unit evaluates from the perspective that is most beneficial to itself, resulting in overestimation of efficiency and other problems. On the other hand, fuzzy cross-efficiency models proposed by other scholars don't adopt a unified production frontier to measure the efficiencies of decision-making units. Most importantly, the use of the arithmetic average method has no way to take into consideration the DM's subjective preferences in the efficiency aggregation process. This paper provides an effective way to measure the performances of DMUs in fuzzy environments, and it can avoid the efficiency overestimation problem and obtain a unique ordering of the DMUs.
In this paper, the input and output data of DMUs are both characterized by triangular fuzzy numbers. In future research, we plan to study the approach to evaluate the efficiency of DMUs with different types of fuzzy numbers. Besides, measuring the performances of DMUs can be extended to a hesitant fuzzy environment and regret theory is utilized to replace the prospect theory.

Key words: cross-efficiency, fuzzy DEA, prospect theory

摘要： 针对模糊环境下决策单元的相对有效性评价问题,本文利用α-截集法将三角模糊数型的投入产出值转化为区间数,提出一种改进的区间交叉效率模型。随后,引入前景理论来研究区间交叉效率集结问题,定义区间参考点代替传统的单个参考点,以最大化所有决策单元的前景交叉效率为原则,构建最大化前景交叉效率模型求解集结权重。根据偏好度方法,比较区间交叉效率值。本文方法基于统一的生产前沿面来度量决策单元的效率,保证了不同决策单元之间以及不同α值下的效率可比;定义区间参考点充分考虑了决策者在模糊环境下的心理因素变化,集结决策单元的区间交叉效率值代替综合前景值,以保留尽可能多的决策信息。最后,通过例子验证方法的有效性。

关键词: 交叉效率, 模糊DEA, 前景理论