数据包络分析法(DEA)-电子书.pdf
Chapter 1 DATA ENVELOPMENT ANALYSIS History,Models and Interpretations William W.Cooper1,Lawrence M.Seiford2 and Joe Zhu31 Red McCombs School of Business,University of Texas at Austin,Austin,TX 78712 USAemail:cooperwmail.utexas.edu 2 Department of Industrial and Operations Engineering,University of Michigan at Ann Arbor,Ann Arbor,MI 48102 USA email:seifordumich.edu 3 Department of Management,Worcester Polytechnic Institute,Worcester,MA 01609 USAemail:jzhuwpi.edu Abstract:In a relatively short period of time Data Envelopment Analysis(DEA)has grown into a powerful quantitative,analytical tool for measuring and evaluating performance.DEA has been successfully applied to a host of different types of entities engaged in a wide variety of activities in many contexts worldwide.This chapter discusses the fundamental DEA models and some of their extensions.Key words:Data envelopment analysis(DEA);Efficiency;Performance 1.INTRODUCTION Data Envelopment Analysis(DEA)is a relatively new“data oriented”approach for evaluating the performance of a set of peer entities called Decision Making Units(DMUs)which convert multiple inputs into multiple outputs.The definition of a DMU is generic and flexible.Recent years have seen a great variety of applications of DEA for use in evaluating the performances of many different kinds of entities engaged in many different activities in many different contexts in many different countries.These DEA 2 Chapter 1:Data Envelopment Analysis applications have used DMUs of various forms to evaluate the performance of entities,such as hospitals,US Air Force wings,universities,cities,courts,business firms,and others,including the performance of countries,regions,etc.Because it requires very few assumptions,DEA has also opened up possibilities for use in cases which have been resistant to other approaches because of the complex(often unknown)nature of the relations between the multiple inputs and multiple outputs involved in DMUs.As pointed out in Cooper,Seiford and Tone(2000),DEA has also been used to supply new insights into activities(and entities)that have previously been evaluated by other methods.For instance,studies of benchmarking practices with DEA have identified numerous sources of inefficiency in some of the most profitable firms-firms that had served as benchmarks by reference to this(profitability)criterion and this has provided a vehicle for identifying better benchmarks in many applied studies.Because of these possibilities,DEA studies of the efficiency of different legal organization forms such as stock vs.mutual insurance companies have shown that previous studies have fallen short in their attempts to evaluate the potentials of these different forms of organizations.Similarly,a use of DEA has suggested reconsideration of previous studies of the efficiency with which pre-and post-merger activities have been conducted in banks that were studied by DEA.Since DEA in its present form was first introduced in 1978,researchers in a number of fields have quickly recognized that it is an excellent and easily used methodology for modeling operational processes for performance evaluations.This has been accompanied by other developments.For instance,Zhu(2002)provides a number of DEA spreadsheet models that can be used in performance evaluation and benchmarking.DEAs empirical orientation and the absence of a need for the numerous a priori assumptions that accompany other approaches(such as standard forms of statistical regression analysis)have resulted in its use in a number of studies involving efficient frontier estimation in the governmental and nonprofit sector,in the regulated sector,and in the private sector.See,for instance,the use of DEA to guide removal of the Diet and other government agencies from Tokyo to locate a new capital in Japan,as described in Takamura and Tone(2003).In their originating study,Charnes,Cooper,and Rhodes(1978)described DEA as a mathematical programming model applied to observational data that provides a new way of obtaining empirical estimates of relations-such as the production functions and/or efficient production possibility surfaces that are cornerstones of modern economics.Formally,DEA is a methodology directed to frontiers rather than central tendencies.Instead of trying to fit a regression plane through the center of W.W.Cooper,L.M.Seiford and J.Zhu 3 the data as in statistical regression,for example,one floats a piecewise linear surface to rest on top of the observations.Because of this perspective,DEA proves particularly adept at uncovering relationships that remain hidden from other methodologies.For instance,consider what one wants to mean by“efficiency”,or more generally,what one wants to mean by saying that one DMU is more efficient than another DMU.This is accomplished in a straightforward manner by DEA without requiring explicitly formulated assumptions and variations with various types of models such as in linear and nonlinear regression models.Relative efficiency in DEA accords with the following definition,which has the advantage of avoiding the need for assigning a priori measures of relative importance to any input or output,Definition 1.1(Efficiency Extended Pareto-Koopmans Definition):Full(100%)efficiency is attained by any DMU if and only if none of its inputs or outputs can be improved without worsening some of its other inputs or outputs.In most management or social science applications the theoretically possible levels of efficiency will not be known.The preceding definition is therefore replaced by emphasizing its uses with only the information that is empirically available as in the following definition:Definition 1.2(Relative Efficiency):A DMU is to be rated as fully(100%)efficient on the basis of available evidence if and only if the performances of other DMUs does not show that some of its inputs or outputs can be improved without worsening some of its other inputs or outputs.Notice that this definition avoids the need for recourse to prices or other assumptions of weights which are supposed to reflect the relative importance of the different inputs or outputs.It also avoids the need for explicitly specifying the formal relations that are supposed to exist between inputs and outputs.This basic kind of efficiency,referred to as“technical efficiency”in economics can,however,be extended to other kinds of efficiency when data such as prices,unit costs,etc.,are available for use in DEA.In this chapter we discuss the mathematical programming approach of DEA that implements the above efficiency definition.Section 2 of this chapter provides a historical perspective on the origins of DEA.Section 3 provides a description of the original“CCR ratio model”of Charnes,Cooper,and Rhodes(1978)which relates the above efficiency definition to other definitions of efficiency such as the ones used in engineering and science,as well as in business and economics.Section 4 describes some 4 Chapter 1:Data Envelopment Analysis methodological extensions that have been proposed.Section 5 expands the development to concepts like“allocative”(or price)efficiency which can add additional power to DEA when unit prices and costs are available.This is done in section 5 and extended to profit efficiency in section 6 after which a conclusion section 7 is supplied.2.BACKGROUND AND HISTORY In an article which represents the inception of DEA,Farrell(1957)was motivated by the need for developing better methods and models for evaluating productivity.He argued that while attempts to solve the problem usually produced careful measurements,they were also very restrictive because they failed to combine the measurements of multiple inputs into any satisfactory overall measure of efficiency.Responding to these inadequacies of separate indices of labor productivity,capital productivity,etc.,Farrell proposed an activity analysis approach that could more adequately deal with the problem.His measures were intended to be applicable to any productive organization;in his words,from a workshop to a whole economy.In the process,he extended the concept of“productivity”to the more general concept of“efficiency”.Our focus in this chapter is on basic DEA models for measuring the efficiency of a DMU relative to similar DMUs in order to estimate a best practice frontier.The initial DEA model,as originally presented in Charnes,Cooper,and Rhodes(CCR)(1978),built on the earlier work of Farrell(1957).This work by Charnes,Cooper and Rhodes originated in the early 1970s in response to the thesis efforts of Edwardo Rhodes at Carnegie Mellon Universitys School of Urban&Public Affairs-now the H.J.Heinz III School of Public Policy and Management.Under the supervision of W.W.Cooper,this thesis was to be directed to evaluating educational programs for disadvantaged students(mainly black or Hispanic)in a series of large scale studies undertaken in U.S.public schools with support from the Federal government.Attention was finally centered on Program Follow Through-a huge attempt by the U.S.Office(now Department)of Education to apply principles from the statistical design of experiments to a set of matched schools in a nation-wide study.Rhodes secured access to the data being processed for that study by Abt Associates,a Boston based consulting film,under contract with the US Office of Education.The data base was sufficiently large so that issues of degrees of freedom,etc.,were not a serious problem despite the numerous input and output variables used in the study.Nevertheless,unsatisfactory and even absurd results were secured W.W.Cooper,L.M.Seiford and J.Zhu 5 from all of the statistical-econometric approaches that Rhodes attempted to use.While trying to respond to this situation,Rhodes called Coopers attention to M.J.Farrells seminal article The Measurement of Productive Efficiency,in the 1957 Journal of the Royal Statistical Society.In this article Farrell used activity analysis concepts to correct what he believed were deficiencies in commonly used index number approaches to productivity(and like)measurements.Cooper had previously worked with A.Charnes in order to give computationally implementable form to Tjalling Koopmans activity analysis concepts.So,taking Farrells statements at face value,Cooper and Rhodes formalized what was involved in the definitions that were given in section 1 of this chapter.These definitions then provided the guides that were used for their subsequent research.The name of Pareto is assigned to the first of these two definitions for the following reasons.In his Manual of Political Economy(1906)the Swiss-Italian economist,Vilfredo Pareto,established the basis of modern welfare economics,i.e.,the part of economics concerned with evaluating public policies,by noting that a social policy could be justified if it made some persons better off without making others worse off.In this way the need for making comparisons between the value of the gains to some and the losses to others could be avoided.This avoids the necessity of ascertaining the utility functions of the affected individuals and/or to weight the relative importance of each individuals gains and losses.This property,known as the“Pareto criterion”as used in welfare economics,was carried over,or adapted,in Activity Analysis of Production and Allocation,a book edited by Koopmans(1951).In this context,it was final goods which were accorded this property,in that they were all constrained so that no final good was allowed to be improved if this improvement resulted in worsening one or more other final goods.These final goods(=outputs)were to be satisfied in stipulated amounts while inputs were to be optimally determined in response to the prices and amounts exogenously fixed for each output(=final good).Special attention was then directed by Koopmans to efficiency prices which are the prices associated with efficient allocation of resources(=inputs)to satisfy the pre-assigned demands for final goods.For a succinct summary of the mechanisms involved in this activity analysis approach,see p.299 in Charnes and Cooper(1961).Pareto and Koopmans were concerned with analyses of entire economies.In such a context it is reasonable to allow input prices and quantities to be determined by reference to their ability to satisfy final demands.Farrell,however,extended the Pareto-Koopmans property to inputs as well as 6 Chapter 1:Data Envelopment Analysis outputs and explicitly eschewed any use of prices and/or related exchange mechanisms.Even more importantly,he used the performance of other DMUs to evaluate the behavior of each DMU relative to the outputs and the inputs they all used.This made it possible to proceed empirically to determine their relative efficiencies.The resulting measure which is referred to as the Farrell measure of efficiency,was regarded by Farrell as restricted to meaning technical efficiency or the amount of waste that can be eliminated without worsening any input or output.This was then distinguished by Farrell from allocative and scale efficiencies as adapted from the literature of economics.These additional efficiencies will be discussed later in this chapter where the extensions needed to deal with problems that were encountered in DEA attempts to use these concepts in actual applications will also be discussed.Here we want to note that Farrells approach to efficiency evaluations,as embodied in the Farrell measure,carries with it an assumption of equal access to inputs by all DMUs.This does not mean that all DMUs use the same input amounts,however,and,indeed,part of their efficiency evaluations will depend on the input amounts used by each DMU as well as the outputs which they produce.This equal access assumption is a mild one,at least as far as data availability is concerned.It is less demanding than the data and other requirements needed to deal with aspects of performance such as allocative or scope and scale efficiencies.Furthermore,as discussed below,this assumption can now be relaxed.For instance,one can introduce non-discretionary variables and constraints to deal with conditions beyond the control of a DMUs management-in the form of exogenously fixed resources which may differ for each DMU.One can also introduce categorical variables to insure that evaluations are effected by reference to DMUs which have similar characteristics,and still other extensions and relaxations are possible,as will be covered in the discussions that follow.To be sure,the definition of efficiency that we have referred to as Extended Pareto-Koopmans Efficiency and Relative Efficiency were formalized by Charnes,Cooper and Rhodes rather than Farrell.However,these definitions conform both to Farrells models and the way Farrell used them.In any case,these were the definitions that Charnes,Cooper and Rhodes used to guide the developments that we next describe.The Program Follow Through data with which Rhodes was concerned in his thesis rec
