Study On DEA Theory And Its Applications

Data envelopment analysis (DEA for short), originally formulated by A. Charnes, Cooper and Rhodes (CCR 1978), measures the relative efficiencies among the decision making units (DMUs) with multiple-input and multiple-output as a linear programming formulation. It is a new cross-subject that covers mathematics, operational research, economics as well as management science. It has been successfully employed to study the comparative performance of units by using mathematical programming (including linear programming, multiple objective programming, generalized optimization with cone-structure, Semi-infinite Multi-criteria Programming, chance programming, etc.).The performance of a DMU depends only on the identified efficient frontier characterized by the DMUs with a unity efficiency score. If the DMU is efficient then it lies on the efficient frontier. Using the data envelopment analysis, we can determine the structure, feature of the efficient frontier so as to know how to construct it. The nonparametric mathematical programming approach is one of the most popular techniques used in efficiency analysis. For its "natural" economic background, DEA method has attracted many researchers to study on it and they have done a lot of works during the course of data envelopment analysis being developed into a new research field. For 29 years, the founding works in DEA can be seen as follows:( i ) A lot of successful cases have been made which indicate the universality and applicability of DEA.(ii) Extension and perfectness of the DEA models. After the CCR model, some representative "classical models" appear such as BCC model, FG Model, ST Model, additive Model C~2GS~2; Semi-infinite Multi-criteria Programming C~2W with infinite DMUs , chance programming, DEA model with con-ratio C~2WH and generalized DEA model, Log-DEAmodel, chance DEA, DEA model in uncontrollable factors, inverse DEA model, etc.(iii) Research on the economic background and managing background of DEA model(iv) Research on mathematical theory including convex analysis, mathematical programming as well as the basic theory relevant to DEA ( v ) Computation and soft-developing are as important as the application of DEAThis dissertation constitutes the following parts:Firstly, a new two-objective DEA model is proposedIn the analysis of economic system, we always want to obtain the most production of output for the least input consumption because of the limited resources. So we proposed a new DEA model based on two objective programming in input-output oriented way , this DEA model is capable of maximizing the efficiency of individual units at the same time that total input on assumption is minimized and total output production is maximized. At last we give a new method for solving the model.Secondly, a ranking of DMUs in interval DEA model is givenThe classical DEA (input-oriented) model is to evaluate each DMU optimistically. Here we give an approach of determining the interval efficiency under output-riented DEA model. The DEA model is formulated with an interval efficiency which consists of efficiencies obtained from the optimistic and pessimistic viewpoints. Thus, two end points can construct the interval efficiency. With the same idea, we can deal with the interval inefficiency model which is inverse to interval efficiency. Finally, we compare the results under input DEA model and output DEA model respectively through an example.Thirdly, we study the problem of detecting influential problems in generalized DEA modelFor a given data set we deal with the problem of detecting influential problems in generalized DEA model. The technique we present here is intended to give a comprehensive description about the measurement of the influence of an extreme efficient DMU on the efficiency of an inefficient DMU from several aspects. Since the generalize DEA model contains all kinds of returns-to-scale (RTS) from constant RTS to increasing RTS to decreasing RTS. Then we extend the problem by introducing a new concept of gradient projection which under the BCC model (input-oriented) and a two-objective programming respectively to measure the influence of an extreme efficient DMU on the efficiency of an inefficient DMU. Accordingly, this chapter aims to give us a comprehensive understanding about the influential observations.The forth is about the allocation of limited resources—forecasting input in inverse DEA model (input-oriented)In this paper we show that the inverse data envelopment analysis model can be used to estimate inputs for a decision making unit (DMU): if among a group of decision making units, we increase certain outputs to a particular unit and assume that the DMU maintains its current efficiency level with respect to other units, how much more inputs should be provided to the unit? The problem is transformed into a multi-objective programming problem to solve. We use one example to illustrate our computation method and through the example we can also see the difference between the method presented in paper [38] and our method proposed here.Fifthly, we propose a method of determining super-most productive scale size (super-mpss)We introduce a new concept of super-most productive scale size (super-mpss) and present a procedure for determining it. Then the relationship between most productive scale size and super-most productive scale size is revealed. At last an example illustrates the feasibility of our method.At last, two applications of DEA are proposed.(1) Determine the optimal portfolio in DEAWe propose a new DEA method to obtain the optimal portfolio of n securities with multiple inputs and multiple outputs as well as the best optimal portfolio during different time periods. Here the return, expected return, etc. are regarded as inputs and the risk, loss, etc are regarded as outputs respectively. The main feature of the method is that it overcomes the difficulties that M. Markowits theory can not solve. At last an example illustrates the feasibility of the technique. (2) Forecast GDP in China and Efficient Input Interval We first give a new method of time series model to forecast GDP of China. The method proposed here aims to emphasis the importance of the impact of STEP-Affair on the GDP forecasting. The superiority of the method to ARMA model is illustrated in an example presented accordingly. Then in the whole economic system when the GDP forecasted above is given, how can we allocate the limited resources to make the economical behavior relative efficient. We use data envelopment analysis to show how to determine input interval. Each input among the input interval as well as the given output (GDP) constitute an efficient decision making unit (DMU).Main contributions of this dissertation are as follows:Theoretical aspects:1. Research on the innovation of DEA efficiency:We consider the problem of minimizing the total inputs and maximizing total outputs in traditional DEA model;Promote the development of the interval DEA model;Give a new comprehensive measure of detecting influential observations.2. Make a further study on the inverse DEA problem by presenting a method of forecasting input in data envelopment analysis.3. We give the concept of the super-most productive scale size and by determining the relationship between the most productive scale size and the super-most productive scale size. Practical aspects:1. We determine the optimal portfolio in DEA which can overcome the difficulties that Markwits theory can not solve.2. Show the application of DEA in state macro-economy by allocating the limited resources to make the economic activities efficient.