Data Envelopment Analysis Methods Based On Fuzzy Set And Partial Ordered Set Theory

Data envelopment analysis (DEA) is the intersection of operations research, management science and mathematical economics research. This method is used to evaluate the efficiency of multiple input and multiple output system. The initial DEA model requires crisp input and output data, which makes it difficult to evaluate a model with fuzzy input and output data. Therefore, fuzzy data envelopment analysis method is proposed. However, because of the complexity of the fuzzy DEA approach, many problems still remain unresolved. Furthermore, some problems of DEA method itself have been limiting the development of fuzzy DEA method, such as the DEA has not yet been out of the plight of the randomized, and its applications lack of the theoretical basis in terms of not satisfying the axiom of economic system. To address the above problems, we conducted relative research, and10conclusions are concluded. Our research works are as follows:(1) We explained what make the efficiency value of DMU of the traditional fuzzy data envelopment analysis model greater than1, and propose a fuzzy sample data envelopment analysis model. The model also includes data envelopment analysis model with sample fuzzy target DMU and fuzzy data envelopment analysis model with sample target DMU. They are the key models to evaluate the fuzzy data envelopment analysis model.(2) For the simplicity of the selected special point of traditional fuzzy data envelopment analysis when evaluating the fuzzy DMUs, we propose a new selection method of special DMU, which includes not only the best decision making units and worst decision making unit of traditional method, but also the greatest decision making unit, the smallest decision making unit, the central decision making unit and the DMU in which the membership function obtain greatest value. Thus, this method extends the traditional fuzzy data envelopment analysis, and the application range of the fuzzy data envelopment analysis.(3) For the simplicity of the cut approach, we use different cuts for different input and output data.(4) For the simplicity of fuzzy number of the traditional fuzzy data envelopment analysis method, we extend the types of fuzzy numbers. This extension includes partial small type fuzzy number, partial large type fuzzy number and middle type fuzzy numbers.(5) For the simplicity of the evaluating methods of traditional fuzzy data envelopment analysis method, we propose a new evaluating method based on vector. The method can give excellences of the target sample fuzzy decision making unit compared with other fuzzy decision making units.(6) For the complexity computation of the efficiency values of decision making units in the traditional fuzzy data envelopment analysis method, a calculation method of the efficiency value of the fuzzy decision making unit based on Matlab is provided. The algorithm can not only calculate the efficiency of decision making units of the traditional fuzzy data envelopment analysis model, but also the efficiency value of the sample fuzzy decision making units.(7) For the limitation of the projection method of traditional data envelopment analysis and the uncertainty relationship between decision making units, the proper partial order relation is provided and we propose the data envelopment analysis method based on partial ordered set theory. Through the partial order relation, we establish the relationship between different decision making units, and provide a new projection method of decision making units. This projection method avoids the disadvantage of the projection method of traditional DEA method that must be clearly defined as input-oriented or output-oriented model.(8) To establish the relationship between the data envelopment analysis based on partial ordered set theory and the efficient decision making unit, weak efficient decision making unit and inefficient decision making unit of the traditional data envelopment analysis methods, we provide relative theorem. The results show that efficient decision making unit will be the maximal element of the related partial ordered set, but the maximal element is not necessarily effective. We still analyze the reasons why the maximal element is inefficient.(9) To visualize the partial order relation in complex decision problems, the algorithm is proposed to illustrate the partial order relation, and the partial ordered graph of decision making units is provided. According to the partial ordered graph, we can see clearly the distribution of the various decision making units and the partial relationship between decision making units, which provides an important reference for decision makers.Finally, further research and applications directions of data envelopment analysis method based on fuzzy sets and partially ordered set theory are provided, we also propose the research frontier of data envelopment analysis and its applications.