Generalized Data Envelopment Analysis Model Evaluated Based On Sample Units And ITS Properties

Data envelopment analysis(DEA) method is the intersection of operations research, management science and mathematical economics research. Traditional DEA method is an evaluation method based on self-assessment system. Suppose that the reference set of evaluation can be grouped into the decision-making unit set (DMU Set) and non-decision-making unit set (non-DMU Set), then traditional DEA methods can only give the information which have relation to DMU Set but not any non-DMU Set. It make traditional DEA method be restricted in many evaluation application, but gener-alized DEA method has certain advantages to evaluate this kind of problem, generalized DEA method can choose reference set freely and evaluate decision making units based on the reference set. In this paper, the main work is to further perfect the theory of generalized DEA method, study the output-oriented DEA model and input-oriented DEA model,output-oriented and input-oriented DEA model, the relationship between efficiency of decision making unit and moving of production frontier, efficiency rank-ing based on different sample production frontier, the choice of sample unit set and generalized stochastic DEA method are studied, the main conclusions are as follows:(a) The generalized DEA method based on C2R model, BC2 model, ST model and FG model and the concept of the generalized DEA efficiency are presented, including generalized DEA model for input-oriented and output-oriented situation, and generalized additive DEA model. The relationship between these models and the traditional DEA models is analyzed, the relationship between the generalized DEA efficiency and corresponding Pareto efficiency of multi-objective programming is explored, and the property of projection and the method of ranking decision making units are given.(b) The output-oriented generalized DEA method based on C2W model is presented, the input-oriented and output-oriented generalized DEA theory based on C2W model is perfected. The judgement conditions of generalized DEA efficiency, the relationship between generalized DEA method and traditional DEA method, the relationship between generalized DEA efficiency and Pareto efficiency of corresponding multiple objective programming, property of projection of decision making unit and ranking method of decision making unit by the d-moving of the sample production frontier were disscussed in the input-oriented and output-oriented generalized DEA based on C2W model.(c) DEA efficient and weak DEA efficient decision making units are located on the production frontier, DEA inefficient but not weak DEA efficient decision making units are in the production possibility set but not on the production frontier in the traditional DEA method. By introducing the idea of the moving of production possibility set and production frontier, prove that it exist reciprocal relationship between the optimal value base on the BC2 model with output (input) only and the moving value of production frontier, illustrate through the double output(input), and make clear the decision making units’location in production probability set.(d) Traditional DEA method evaluate decision making unit relative to all the decision making units, generalized DEA method evaluate decision making unit relative to all the sample units. Because the reference frame is different, the evaluation result for each decision making unit might be different. Explain this kind of situation through traditional and generalized DEA method with input (output) only based on BC2 model, and give the geometric features of relative efficient value in generalized DEA method by the moving of sample frontier.(e) Generalized DEA method can evaluate the efficiency of the decision making units through many different reference sets. But in terms of the rank in applying the generalized DEA method, the existing ranking method which uses the d moving of sample frontier method relative to outputs, can only rank the efficiency of the decision making units in the special region. So this paper gave definitions of the efficiency, which are based on the b moving of sample frontier relative to inputs and b+d moving of sample frontier relative to both inputs and outputs, and the corresponding evaluation model and judgment condition. The relationship between the efficiency of generalized DEA method and Pareto efficiency of multi-objective programming is also provided in this paper. The application of the above mentioned three methods can rank the decision making units in different regions.(f) Apply the DEA model with output only (D-BCO2) to evaluate the efficiency of de-cision making unit, the efficiency value maybe have difference with the qualitative analysis. For solving the problem, the generalized DEA model with output only (DG-BCO2) is given. Cluster analysis is applied to determine the sample unit set and (DGc) model is applied to evaluate the efficiency of the decision making unit. A virtual decision making unit, in accord with a certain condition, is constructed to do cluster analysis with all decision making units and take the group with virtual decision making unit as the sample unit. The sample unit can reflect the decision maker’s decision preference. Finally, through an example to show the method has a certain feasibility.(g) The generalized stochastic DEA method with random input and output of decision making units and sample units is presented. The relative efficiency evaluation and the efficiency ranking in the generalized stochastic DEA model are discussed based on expected value model and chance constrained programming. The evaluation efficiency is different based on the two methods, the effect of variance in the evalua-tion of decision making unit’s efficiency can be shown based on chance constrained programming.