deap使用说明中英对照版

新英格兰大学

效率与生产力分析中心(CEPA)

工作报告

DEAP 2.1版本指南:数据包络分析

（计算）程序

Coelli T.J

No.8/96

澳大利亚，NSW2351，阿米代尔

新英格兰大学，计量经济学系

CEPA工作报告

https://www.360docs.net/doc/8f16180109.html,.au/econometrics/cepawp.htm ISSN1327-435X

ISBN 1 86389 4969

目录

摘要 (3)

1.INTRODUCTION说明 (4)

2.EFFICIENCY MEASUREMENT CONCEPTS效率测度的概念。 (5)

2.1 Input-Orientated Measures投入主导型的测度 (6)

2.2 Output-Orientated Measures 产出主导型测量方法 (9)

3.Data Envelopment Analysis(DEA)数据包络分析 (12)

3.1 The Constant Returns to Scale Model(CRS) 规模报酬不变模型 (13)

Slacks 松弛变量 (15)

Example 1 例子1 (19)

3.2 The Variable Returns to Scale Model (VRS)and Scale Efficiencies 规模

收益变化模型（VRS）和规模效率 (22)

Calculation of Scale Efficiencies 规模效率的计算 (23)

Example 2例子2 (25)

3.3 Input and Output Orientations 投入和产出主导型 (26)

3.4 Price Information and Allocative Efficiency 价格信息和配置效率 (29)

Example 3 例子3 (30)

3.5 Panel Data,DEA and the Malmquist Index 面板数据，DEA和Malmquist

指数 (31)

Scale Efficiency 规模效率 (34)

Example 4 例子4 (34)

4.The DEAP Computer Program DEAP计算机程序 (35)

Data file 数据文件 (36)

Instruction file 向导文件 (37)

Output file 输出文件 (38)

5.Examples 例子 (38)

5.1 Example 1:An Input-orientated CRS DEA Example 例子1：一个CRS投

入主导型DEA例子 (39)

5.2 Example 2:An Input-orientated VRS DEA Example 例子2：一个VRS投入

主导型的DEA模型例子 (43)

5.3 Example 3:A Cost Efficiency DEA Example 例子3：成本效率DEA例子

(47)

5.4 Example 4:A Malmquist DEA Example 例子4：Malmquist DEA例子 49

6.Concluding Comments 结束语 (53)

REFERENCES 参考文献： (53)

APPENDIX 附录 (55)

Tips on using DEAP in File Manager in Windows 3.1: 在Windows 3.1的File Manager使用DEAP的小贴士： (56)

DEAP 2.1版本指南:

数据包络分析(计算)程序

澳大利亚，威尔士，2351，阿米代尔

新英格兰大学，计量经济学系

效率和生产力分析中心

Tim Coelli

Email:tcoelli@https://www.360docs.net/doc/8f16180109.html,.au

Web:https://www.360docs.net/doc/8f16180109.html,.au/econometrics/cepa.htm

CEPA 工作报告96/08

摘要

This paper describes a computer program which has been written to conduct data envelopment analyses(DEA)for the purpose of calculating efficiencies in production. The methods implemented in the program are based upon the work of Rolf Fare, Shawna Grosskopf and their associates. Three principal options are available in the computer program. The first involves the standard CRS and VRS DEA models(that involve the calculation of technical and scale efficiencies)which are outlined in Fare, Grosskopf and Lovell(1994). The second option considers the extension of these models to account for cost and allocative efficiencies.These methods are also outlined in Fare et al(1994). The third option considers the application of Malmquist DEA methods to panel

data to calculate indices of total factor productivity (TFP) change; technological change;technical efficiency change and scale efficiency change. These latter methods are discussed in Fare,Grosskopf,Norris and Zhang(1994). All methods are available in either an input or an output orientation(with the exception of the cost efficiencies option).

这篇论文描述了一个程序，这个程序是用来实施数据包络分析（DEA），以此来计算生产中的效率。程序中实施的方法是基于Rolf Fare，Shawna Grosskopf 和他的同事的工作。计算程序中有三个主要的选择。第一个包括标准的CRS和VRS两个DEA模型（这包括了技术效率和规模效率的计算），这两个模型是Fare, Grosskopf and Lovell(1994)设计的。第二个选择考虑了这些模型的延伸，即说明成本和配置效率。这些模型也是Fare et al(1994)设计的。第三个选择考虑到了Malmquist DEA模型的使用，这个模型是用面板数据来计算全要素生产率变化（TFP）、技术进步、技术效率变化和规模效率变化的指数。后面所说的模型Fare,Grosskopf,Norris and Zhang(1994)曾经讨论过。所有的模型无论在投入主导型还是产出主导型（除了成本效率）都能够获得。

1.INTRODUCTION说明

This guide describes a computer program which has been written to conduct data envelopment analyses(DEA). DEA involves the use of linear programming methods to construct a non-parametric piecewise surface(or frontier)over the data,so as to be able to calculate efficiencies relative to this surface. The computer program can consider a variety of models.The three principal options are:

这个指南描述了一个实施数据包络技术的计算机程序。DEA模型是运用线性规划的方法通过建造一个非参数分段的面（前沿），然后相对这个面计算效率。计算机程序能够考虑模型的多样性。三个主要的选择是：

1.Standard CRS and VRS DEA models that involve the calculation of

technical and scale efficiencies(where applicable).These methods are outlined in Fare, Grosskopf and Lovell(1994).

2.The extension of the above models to account for cost and allocative

efficiencies. These methods are also outlined in Fare et al(1994).

3.The application of Malmquist DEA methods to panel data to calculate

indices of total factor productivity(TFP)change;technological

change;technical efficiency change and scale efficiency change.These methods are discussed in Fare, Grosskopf,Norris and Zhang(1994).

1.标准CRS 和VRS的数据包络分析模型，它们包括技术效率和规模效率的

计算（当规模效率适合）。这些模型是Fare,Grosskopf and Lovell(1994)构造的。

2.上述模型的延伸，考虑了成本和配置效率。这些模型也是Fare et al(1994)

构造的。

3. 第三个选择考虑到了Malmquist DEA模型的使用，这个模型是用面板数

据来计算全要素生产率变化（TFP）、技术进步、技术效率变化和规模效率

变化的指数。这些方法Fare,Grosskopf,Norris and Zhang(1994)曾经讨论过。

All methods are available in either an input or an output orientation(with the exception of the cost efficiencies option). The output from the program includes,where applicable,technical,scale,allocative and cost efficiency estimates;residual slacks;peers;TFP and technological change indices.

这些方法无论是在投入主导型还是在产出主导型（除了成本效率）都是能够获得的。当合适的时候，程序的结果可以包括技术、规模、配置、成本效率的估计值；还有松弛变量的残值；对应点；全要素生产率和技术进步变化指数。

The paper is divided into sections.Section 2 provides a brief introduction to efficiency measurement concepts developed by Farrell(1957);Fare,Grosskopf and Lovell(1985, 1994)and others. Section 3 outlines how these ideas may be empirically implemented using linear programming methods(DEA). Section 4 describes the computer program,DEAP,and section 5 provides some illustrations of how to use the program.Final concluding points are made in Section 6. An appendix is added which summarises important technical aspects of program use

文章可以分为以下几章。第二章给出了Farrell(1957);Fare,Grosskopf and Lovell(1985,1994)等关于效率测度概念的简短的介绍。第三章概述了这些思想是怎样通过DEA方法实现实证研究的。第四部分描述了计算程序，DEAP。第五章给了一些运用程序的例子。最后，在第六章，总结了要点。附录里面包含了程序所用的重要技术方面的概述。

2.EFFICIENCY MEASUREMENT CONCEPTS效率测度的概念。

The primary purpose of this section is to outline a number of commonly used efficiency measures and to discuss how they may be calculated relative to an efficient technology, which is generally represented by some form of frontier function. Frontiers have been estimated using many different methods over the past 40 years.The two principal methods are:

这章的主要目的是概述一些常用的测度效率的方法，并且讨论了相对于技术有效（通常由某种形式的前沿机构所代表）怎样计算出他们。过去的40年里，我们运用了很多不同的方法估计前沿效率。两个有代表性的方法是：

1.data envelopment analysis(DEA)and

2.stochastic frontiers,

1.数据包络分析

2.随机前沿法

which involve mathematical programming and econometric methods, respectively.This paper and the DEAP computer program are concerned with

the use of DEA methods. The computer program FRONTIER can be used to estimate frontiers using stochastic frontier methods.For more information on FRONTIER see Coelli(1992, 1994).

他们分别包含了数学规划和计量经济学的方法。这篇论文和DEAP计算机程序是关于DEA 方法的。计算机程序FRONTIER可以通过随即前沿法估计前沿效率，想得到更多关于FRONTIER请看Coelli(1992,1994)的研究。

The discussion in this section provides a very brief introduction to modern efficiency measurement. A more detailed treatment is provided by

Fare,Grosskopf and Lovell (1985,1994)and Lovell(1993). Modern efficiency measurement begins with Farrell (1957)who drew upon the work of

Debreu(1951)and Koopmans(1951)to define a simple measure of firm efficiency which could account for multiple inputs. He proposed that the efficiency of a firm consists of two components:technical efficiency,which reflects the ability of a firm to obtain maximal output from a given set of inputs,and allocative efficiency,which reflects the ability of a firm to use the inputs in optimal proportions,given their respective prices. These two measures are then combined to provide a measure of total economic efficiency.1

这章对现代的效率测度给了一个简短的介绍。更详细的方法请参考Fare, Grosskopf and Lovell (1985,1994)and Lovell(1993)的研究。现代效率测量是与Farrell (1957)，他在Debreu(1951)and Koopmans(1951)的工作的基础上定一个一个简单的测度公司效率的方法，这个方法考虑了多投入。他认为公司的效率可以分为两个部分：技术效率，反映了公司在给定投入的情况下所能获得的最大的产出的能力。另一个是配置效率。它反映了给定各自投入价格的情况下使用最优比例的能力。两个测量结合起来给出了总经济效率的测度。

The following discussion begins with Farrell’s original ideas which were illustrated in input/input space and hence had an input-reducing focus.These are usually termed input-orientated measures.

接下来的讨论从Farrell最初的想法开始，这个想法产生于投入，然后关注于投入的减少。这通常被称为投入主导型。

2.1 Input-Orientated Measures投入主导型的测度

Farrell illustrated his ideas using a simple example involving firms which use two inputs (x1 and x2)to produce a single output(y),under the assumption of constant returns to scale.2 Knowledge of the unit isoquant of the fully efficient 1Some of Farrell’s terminology differed from that which is used here. He used the term price efficiency instead of allocative efficiency and the term overall efficiency instead of economic efficiency.The terminology used in the present document conforms with that which has been used

most often in recent literature.

一些Farrell的经济效率和这里使用是不同的。他使用价格效率代替这里的配置效率，使用总体效率代替这里的经济效率。这篇文献里面的术语同最近大部分文献的术语是一致的。

2The constant returns to scale assumption allows one to represent the technology using a unit

firm3, represented by SS′in Figure 1,permits the measurement of technical efficiency. If a given firm uses quantities of inputs,defined by the point P,to produce a unit of output,the technical inefficiency of that firm could be represented by the distance QP,which is the amount by which all inputs could be proportionally reduced without a reduction in output .This is usually expressed in percentage terms by the ratio QP/0P,which represents the percentage by which all inputs could be reduced. The technical efficiency(TE) of a firm is most commonly measured by the ratio

TEI=0Q/0P, (1) which is equal to one minus QP/0P. It will take a value between zero and one,and hence provides an indicator of the degree of technical inefficiency of the firm. A value of one indicates the firm is fully technically efficient.For example,the point Q is technically efficient because it lies on the efficient isoquant.

Farrell通过一个简单的例子阐述了他的观点。这个例子假设一个公司的规模报酬不变，使用了两个投入（x1和x2）生产单一的产出（y）已知全效率公司的单位等产量曲线，由数据1的SS′代表，允许技术效率的度量。如果给定的公司用一定数量的投入，由P点所定义，去生产一单位的产出，公司的技术无效可以有QP的距离所表示，它就是在不减少产出的情况下，所有的投入按比例减少的数量。这通常由百分数QP/OP的比率表示，它代表了所有投入减少的百分比。

公司的技术效率通常由

TE I=OQ/OP （1）也就是1减去QP/OP4。它取值0~1，因此提供了公司技术无效的指标。1代表完全的技术有效，例如，Q点就是技术有效的，因为他在效率曲线上。

数据 1

技术和配置效率

isoquant.Furthermore,Farrell also discussed the extension of his method so as to accommodate more

than two inputs,multiple outputs,and non-constant returns to scale.

规模报酬不变的假设允许我们使用单位等产量曲线代表效率。进一步讲，Farrell也讨论了他方法的进一步延伸，以使他的方法适应比两个个多的投入和多产出以及规模报酬变化的情形。

3The production function of the fully efficient firm is not known in practice,and thus must be estimated from observations on a sample of firms in the industry concerned.In this paper we use DEA to estimate this frontier.

在现实中，全效率公司是不知道的，所以在工业中必须在相关的一些公司观测样本里面估计它。在这篇论文里，我们用DEA方法估计它。

4The subscript“I”is used on the TE measure to show that it is an input-orientated

measure.Output-

orientated measures will be defined shortly.

脚注“I”用于技术效率测量，表示是投入主导型方法。产出主导型定义的就比较简洁。

If the input price ratio,represented by the line AA′in Figure 1,is also known, allocative efficiency may also be calculated.The allocative efficiency(AE)of the firm

operating at P is defined to be the ratio

AEI=0R/0Q, (2) since the distance RQ represents the reduction in production costs that would occur if production were to occur at the allocatively(and technically)efficient point Q′,instead of at the technically efficient,but allocatively

inefficient,point Q.5

如果投入的价格的比率由数据1的AA’表示，那么配置效率也就算出来了。P点运作的公司的配置效率由下面的比率给出定义。

AE I=OR/OQ （2）因为RQ的距离代表如果生产发生在配置有效的点Q’,而不是技术有效但是配置无效的Q点，那么就可以降低生产成本。

The total economic efficiency(EE)is defined to be the ratio

EEI=0R/0P, (3) where the distance RP can also be interpreted in terms of a cost

reduction.Note that the product of technical and allocative efficiency provides the overall economic efficiency

TEI×AEI=(0Q/0P)×(0R/0Q)=(0R/0P)=EEI. (4) Note that all three measures are bounded by zero and one.

总经济效率（EE）可以由下面的比率

EE I=OR/OP （3）RP的距离可以解释为成本降低。注意，技术和配置的效率给出了总的经济效率，TE I*AE I=（OQ/OP）*（OR/OQ）=（OR/OP）=EE I（4）

数据 2

分段线性等量曲线

（Piecewise Linear Convex Isoquant）

5One could illustrate this by drawing two isocost lines through Q and Q′.Irrespective of the slope of

these two parallel lines(which is determined by the input price ratio)the ratio RQ/0Q would represent

the percentage reduction in costs associated with movement from Q to Q′.

我们可以通过画一条通过点Q和点Q’的等成本曲线来阐述这个问题。不考虑两条平行线（由投入的价格比率决定）的斜率，RQ/0Q的比率就代表了从Q点到Q’点成本下降的百分比。

These efficiency measures assume the production function of the fully efficient firm is known.In practice this is not the case,and the efficient isoquant must be estimated from the sample data. Farrell suggested the use of either(a)a

non-parametric piecewise-linear convex isoquant constructed such that no observed point should lie to the left or below it(refer to Figure 2),or(b)a parametric function,such as the Cobb- Douglas form,fitted to the data,again such that no observed point should lie to the left or below it.Farrell provided an illustration of his methods using agricultural data for the 48 continental states of the US.

这些效率测量假设完全有效率的公司是已知的。在实践中，这并不是问题，效率等量曲线是从样本数据中估计出的。Farrell认为无论是使用非参数分段线性等量曲线（没有观测的点在它的左边或下边）（参考数据2），还是参数形式，比如柯布道格拉斯形式，相对应的数据也都没有在它左边或者下边的。Farrell用美国48个洲的农业数据给出了他的方法的一个例子。

2.2 Output-Orientated Measures 产出主导型测量方法

The above input-orientated technical efficiency measure addresses the question:“By how much can input quantities be proportionally reduced without changing the output quantities produced?”. One could alternatively ask the question“:“By ho w much can output quantities be proportionally expanded without altering the input quantities used?”. This is an

output-orientated measure as opposed to the input-oriented measure discussed above. The difference between the output-and input-orientated measures can be illustrated using a simple example involving one input and one output. This is depicted in Figure 3(a)where we have a decreasing returns to scale technology represented by f(x),and an inefficient firm operating at the point P. The Farrell input-orientated measure of TE would be equal to the ratio AB/AP,while the output-orientated measure of TE would be CP/CD. The output-and input-orientated measures will only provide equivalent measures of technical efficiency when constant returns to scale exist,but will be unequal when increasing or decreasing returns to scale are present(Fare and Lovell

1978). The constant returns to scale case is depicted in Figure 3(b)where we observe that AB/AP=CP/CD,for any inefficient point P we care to choose.

上述投入主导型技术效率测量方法提出了一个问题：在不减少产出数量的同时，能按比例减少多少的投入？人们也可以问这样的问题：不改变投入数量的同时，产出数量可以按比例增加多少呢？这是一个与上面阐述的投入主导型测量方法

相反的产出主导型测量方法。投入主导型测度方法和产出主导型测度方法可以用仅包含一个投入还让一个产出的简单例子进行阐述。数据3（a）描述了这一问题。我们有一个规模收益递减的函数f（x），和一个在P点运行的无效率的公司。Farrell的投入主导型测量方法中的TE与图中的AB/AP是相同的，然而产出主导型测度方法的TE是CP/CD。仅当规模收益不变的时候，产出主导型和投入主导型测度方法的技术效率是相等的。但是存在规模报酬递增还是递减的时候就不相等。数据3（b）描述了规模报酬不变。我们可以看到AB/AP=CP/CD，对于任意无效率的P,我们都可以选择。

One can consider output-orientated measures further by considering the case where production involves two outputs(y1 and y2)and a single input(x1). Again , if we assume constant returns to scale,we can represent the technology by a unit production possibility curve in two dimensions. This example is depicted in Figure 4 where the line ZZ′is the unit production possibility curve and the point A corresponds to an inefficient firm. Note that the inefficient point,A,lies below the curve in this case because ZZ′represents the upper bound of production possibilities.

我们可已通过考虑包括两产出（y1和y2）和一投入（x1）来进一步考虑产出主导型测度方法。再一次，我们假设规模报酬不变，我们可以用一单位产品可能曲线（产品等量曲线）的两个方面来表示技术。数据4描述了这个例子，ZZ’就是单位产品可能曲线，点A是无效率的公司。注意，无效率的点A，在曲线下方，因为ZZ’代表最高的生产可能线。

数据3

投入和产出主导型技术效率

测度方法和规模收益

Input-and Output-Orientated Technical Efficiency Measures

and Returns to Scale

数据 4

产出主导型的技术和配置效率

The Farrell output-orientated efficiency measures would be defined as follows.In Figure 4 the distance AB represents technical inefficiency. That is,the amount by which outputs could be increased without requiring extra inputs.Hence a measure of output-orientated technical efficiency is the ratio TEO=0A/0B. (7) Farrell的产出主导型测度模型可以如下定义。在数据4中，AB的距离代表技术无效。这就是，在不增加额外的投入的基础上，可以增加多少产出。因此，产出主导型的技术效率就是这个比率。

TE0=OA/OB （7）

If we have price information then we can draw the isorevenue line DD′,and define the allocative efficiency to be

AEO=0B/0C (8)

如果我们有价格的信息，我们就可以画等产量曲线DD’，并定义配置效率为

AE0=OB/OC （8）which has a revenue increasing interpretation(similar to the cost reducing interpretation of allocative inefficiency in the input-orientated case). Furthermore,one can define overall economic efficiency as the product of these two measures

EEO=(0A/0C)=(0A/0B)×(0B/0C)=TEO×AEO. (9)

他有一个使收益增加的定义（类似于投入主导型的减少成本的配置无效的定义）。进一步，我们可以定义总体经济效率为两个测度的综合：

EE0=（OA/OC）=（OA/OB）*（OB/OC）=TE0*AE0 （9）Again,all of these three measures are bounded by zero and one.

这三个测量也都在0~1之间。

Before we conclude this section,two quick points should be made regarding the six efficiency measures that we have defined:

在总结这章之前，关于我们定义的六个效率测度的两点我们需要说明。

1)All of them are measured along a ray from the origin to the observed production point.Hence they hold the relative proportions of inputs(or

outputs)constant. One advantage of these radial efficiency measures is that they are units invariant.That is,changing the units of measurement (e.g. measuring quantity of labour in person hours instead of person years)will not change the value of the efficiency measure. A non-radial measure,such as the shortest distance from the production point to the production surface,may be argued for,but this measure will not be invariant to the units of measurement chosen. Changing the units of measurement in this case could result in the identification of a dif ferent“nearest”point. This issue will be discussed further when we come to consider the treatment of slacks in DEA.

2)The Farrell input-and output-orientated technical efficiency measures can be shown to be equal to the input and output distance functions discussed in Shepherd(1970). For more on this see Lovell(1993,p10).This observation becomes important when we discuss the use of DEA methods in calculating Malmquist indices of TFP change.

1）所有的测度都是从原点到观察到的生产点的射线。因此，他们都有相对恒定的投入（产出）比例。这种射线的效率测度的优点是他们都是单位不变的。

这就是变化度量的单位（例如：测量劳动数量用人时而不是人年）将不会改变效率值。非射线的测度，比如较短的生产点到生产面的距离，可能是被要求的，但是测度将相对选择的测度单位而不是恒定的。在这种情况下，改变测度单位可能导致发现一个不同的更近的点。当我们考虑DEA的松弛变量的时候，在我们将对这一问题进行深入的研究。

2）Farrell的投入主导和产出主导技术效率的测量与Shepherd(1970)的投入产出距离模型可以看做是等价的。想知道更多请看Lovell(1993,p10)。当我们运用DEA计算全要素变化的Malmquist指标的时候，这项观察变得重要。

3.Data Envelopment Analysis(DEA)数据包络分析

Data envelopment analysis(DEA)is the non-parametric mathematical programming approach to frontier estimation. The discussion of DEA models presented here is brief,with relatively little technical detail. More detailed reviews of the methodology are presented by Seiford and Thrall(1990), Lovell (1993),Ali and Seiford(1993),Lovell (1994),Charnes et al(1995)and Seiford(1996). 数据包络分析是前沿估计的非参数数学规划方法。这里讨论的DEA模型是很简短的，只有相对很少的技术细节。想知道更多方法论的细节请参考Seiford and Thrall(1990),Lovell(1993),Ali and Seiford(1993),Lovell (1994),Charnes et

al(1995)and Seiford(1996)的研究。

The piecewise-linear convex hull approach to frontier estimation,proposed by Farrell (1957),was considered by only a handful of authors in the two decades following Farrell’s paper. Authors such as Boles(1966)and Afriat(1972) suggested mathematical programming methods which could achieve the task,but the method did not receive wide attention until a the paper by Charnes,Cooper and Rhodes(1978) which coined the term data envelopment analysis(DEA). There has since been a large number of papers which have

extended and applied the DEA methodology.

Farrell(1957)的关于前沿估计的分段线性规划方法仅仅被Farrell论文后的20年里少数的几个作者所推崇。作者，比如Boles(1966)and Afriat(1972)，所推崇的数学规划方法可以完成任务，但是这种方法并没有引起人们的广泛关注，直到Charnes,Cooper and Rhodes(1978)发表了论文，并且创造了数据包络分析这个词。从那以后，涌现了大量的拓展和运用DEA方法的文献。

Charnes,Cooper and Rhodes(1978)proposed a model which had an input orientation and assumed constant returns to scale(CRS).6Subsequent papers have considered alternative sets of assumptions,such as Banker,Charnes and Cooper(1984)who proposed a variable returns to scale(VRS) model. The following discussion of DEA begins with a description of the input-orientated CRS model in section 3.1,because this model was the first to be widely applied.

Charnes,Cooper and Rhodes(1978)提出了一个模型，这个模型是投入主导型并假设规模报酬不变（CRS）。接下来的论文提出了相反的假设，比如Banker , Charnes and Cooper(1984)提出了规模报酬变化的模型（VRS）。接下来DEA 的讨论从3.1的投入主导型的CRS模型开始，因为这个模型是最应该被广泛应用的。

3.1 The Constant Returns to Scale Model(CRS) 规模报酬不变模型

We shall begin by defining some notation.Assume there is data on K inputs and M outputs on each of N firms or DMU’s as they tend to be called in the DEA literature.7For the i-th DMU these are represented by the vectors xi and yi,respectively. The K×N input matrix,X,and the M×N output matrix,Y,represent the data of all N DMU’s. The purpose of DEA is to construct a non-parametric envelopment frontier over the data points such that all observed points lie on or below the production frontier. For the simple example of an industry where one output is produced using two inputs, it can be visualised as a number of intersecting planes forming a tight fitting cover over a scatter of points in three-dimensional space. Given the CRS assumption,this can also be represented by a unit isoquant in input/input space(refer to Figure 2)The best way to introduce DEA is via the ratio form.For each DMU we would like to

6At this point we will begin to use CRS to refer to constant returns to scale rather than CRTS.Most economics texts use the latter,while most DEA papers use the former.

在这点我们开始使用规模收益不变的CRS模型而不是CRTS。很多经济文章用的是后者，但是在大多数的DEA论文里用的都是前者。

7DMU stands for“decision making unit”.It is a more appropriate term than“firm”when,for example,a bank is studying the performance of its branches or an education district is studying the performance of its schools.

DMU代表“决策单元”。这是一个更恰当的术语，比如：银行研究他的分支机构的行能，教育区研究学校的性能。

obtain a measure of the ratio of all outputs over all inputs,such as u′yi/v′xi, where u is an M×1 vector of output weights and v is a K×1 vector of input weights. To select optimal weights we specify the mathematical programming problem:

我们从定义一些记号开始。假设有N个公司或者像DEA文献里面叫的DMU。每个公司有K个投入和M个产出的数据。对于第i个DMU，他们分别由xi和

yi来代表，K×N的投入矩阵为X，M×N的产出矩阵为Y，代表了所有N个DMU 的所有数据。DEA的目的就是在数据点的基础上构造一个非参数的包络前沿，使所有的观测的数据都在生产前沿的上面或者下面。比如工业的一产出，两投入的简单例子，可以看做是一些相交平面，形成了一个涵盖三维空间的散点的紧紧的盖子。给出了规模报酬不变的假设，这可以由投入的单位等产量曲线代表。（参考数据介绍DEA的最好的办法是通过比率的形式。对于每个DMU，我们都得到所有产出关于所有投入的比率的测量，比如：u’y i/v’x i，其中u是M×1的输出权重矩阵，v是K×1的的投入权重矩阵。选择最优的权重就是数学规划要解决的问题。

max u,v(u’y i/v’x i),

st u’y j/v’x j≤1,j=1,2,...,N,

u,v≥0 （10）This involves finding values for u and v, such that the efficiency measure of the i-th DMU is maximised, subject to the constraint that all efficiency measures must be less than or equal to one. One problem with this particular ratio formulation is that it has an infinite number of solutions.8To avoid this one can impose the constraint v′xi=1,which provides:

这就包括了寻找u和v的过程，这样第i个DMU的效率测度就被最大化了，并且由于约束，所有的效率都是小于等于1.一个特殊的比率的问题就是他有无限个解决办法。为了避免这个问题，我们就可以加入这样的一个假设v’x i=1,这就提出了：

maxμ,ν(μ’y i),

st ν’x i=1,

μ’y j-ν’x j≤0, j=1,2,...,N,

μ,ν≥0, （11）where the notation change from u and v to μ and ν reflects the transformation.This form is known as the multiplier form of the linear programming problem.

这里，符号由u 和v 变为μ和ν正反映了这种转变。这种形式在线性规划里面被称为乘数形式。

Using the duality in linear programming,one can derive an equivalent envelopment form of this problem:

使用线性规划的二元形式，我们可以得到这个问题的相等的形式。

8That is,if(u*,v*)is a solution,then(αu*,αv*)is another solution,etc.

那就是，如果(u*,v*)是解，那么(αu*,αv*)就是另外一个解。

minθ,λθ,

st -y i+Yλ≥0,

θx i-Xλ≥0,

λ≥0, （12）where θ is a scalar and λ is a N×1 vector of constants.This envelopment form involves fewer constraints than the multiplier form(K+M

其中θ是一个标量而λ是个N×1的常数矢量，这个包络形式比乘数形式少了很多的约束（K+M

Slacks 松弛变量

The piecewise linear form of the non-parametric frontier in DEA can cause a few difficulties in efficiency measurement. The problem arises because of the sections of the piecewise linear frontier which run parallel to the axes (refer Figure 2 )which do not occur in most parametric functions (refer Figure 1). To illustrate the problem , refer to Figure 5 where the DMU’s using input combinations C and D are the two efficient DMU’s which define the frontier, and DMU’s A and B are inefficient DMU’s. The Farrell(1957)measure of technical efficiency gives the effici ency of DMU’s A and B as OA′/OA and OB′/OB, respectively. However, it is questionable as to whether the point A′is an efficient point since one could reduce the amount of input x2 used (by the amount CA′)and still produce the same output. This is known as input slack in the literature.10Once one considers a case involving more inputs

and/or multiple outputs, the diagrams are no longer as simple, and the possibility of the related concept of output slack also occurs.11Thus it could be 9The forms defined by equations 10 and 11 are introduced here for expository purposes.They are no used again in the remainder of this paper.The multiplier form has,however,been estimated in a number of studies.The μ and ν weights can be interpreted asnumber of

studie s.Theμandνweights can be interpreted as normalised shadow prices.

在这里引入等式10和11定义的形式是为了解释目的。在这篇文章的其他部分是没有用处的。然而，乘数形式在很多论文里被估计。μ和ν的权重可以被解释为正常化的影子价格。

10Some authors use the term input excess.

一些作者使用术语：额外的投入。

11Output slack is illustrated later in these notes(see Figure 4.8).

产出松弛变量接下来在这些文件里阐述（见数据4.8）

argued that both the Farrell measure of technical efficiency(θ)and any

non-zero input or output slacks should be reported to provide an accurate indication of technical efficiency of a DMU in a DEA analysis.12Note that for the i-th DMU the output slacks will be equal to zero only if Yλ-yi=0, while the input slacks will be equal to zero only if θxi-Xλ=0(for the given optimal values of θ and λ).

DEA的非参数前沿分段线性形式会产生效率测度的一些不同的地方。问题的产生是因为分段前沿函数的与坐标轴平行的部分。（参考数据2）这在大多数的参数模型里面是不存在的（参考数据1）。为了阐述这一问题，参考数据5，其中DMU的投入包括C和D是两个有效率的DMU，他代表了前沿。DMU的A和B是无效率的DMU。根据Farrell(1957)的技术效率测度，DMU A和B的技术效率分别为OA’/OA and OB’/OB。然而，问题是A’点是否是效率点呢。因为我们可以在得到同样产出的情况下减少投入的数量x2，（通过CA’）。这在文献里称做投入松弛变量。当我们考虑更多投入和更多产出的情况时，图示就不再简单了，并且相关的概念产出松弛也是可能发生的。因此，在DEA的分析中，提供Farrell的技术效率测度（θ）和非零的投入或产出松弛变量，以此来提供准确的DMU的技术效率指标，这件事是值得争论的。注意，对于第i个DMU的产出松弛变量仅仅当Yλ-y i=0的时候才等于0，投入松弛变量也仅仅当θx i-Xλ=0的时候才等于0。（对于给定的θ和λ）。

数据 5

效率测度和投入松弛变量

Efficiency Measurement and Input Slacks

In Figure 5 the input slack associated with the point A′is CA′of input x2. In cases when there are more inputs and outputs than considered in this simple example,the identification of the “nearest” efficient frontier point (such as C), and hence the subsequent calculation of slacks, is not a trivial task. Some authors (see Ali and Seiford 1993) have suggested the solution of a

12Koopman’s(1951)definition of technical efficiency was stricter than the Farrell(1957)definition. The former is equivalent to stating that a firm is only technically efficient if it operates on the frontier and furthermore that all associated slacks are zero.

Koopman’s(1951)的技术效率的定义比Farrell(1957)的定义要严格。前者等价于说公司如果在效率前沿运行，那么他仅仅是技术有效的，进一步所有的相关的松弛变量都是0。

second-stage linear programming problem to move to an efficient frontier point by MAXIMISING the sum of slacks required to move from an inefficient frontier point(such as A′in Figure 5)to an efficient frontier point(such as point C). This second stage linear programming problem may be defined by:

在数据5中，与A’点相关的投入松弛变量就是投入x2的CA’。当比简单的例子里面有更多的投入和产出的时候，我们就可以发现更近的效率前沿点（比如C 点）然后，接下来的松弛变量的计算就不是没有意义的。一些作者建议用两阶段线性规划的方法去移动效率前沿点，通过最大化需要的松弛变量的总和，把无效的的前沿点（比如数据5的A’点）移动到有效率的点（如C点）。两阶段线性规划问题可以如下定义：

minλ,OS,IS-(M1’OS+K1’IS),

st -y i+Yλ-OS=0,

θx i-Xλ-IS=0,

λ≥0,OS≥0,IS≥0, （13）where OS is an M×1 vector of output slacks,IS is a K×1 vector of input slacks, and M1 and K1 are M×1 and K×1 vectors of ones,respectively. Note that in this second-stage linear program,θ is not a variable,its value is taken from the

first-stage results. Furthermore,note that this second-stage linear program must also be solved for each of the N DMU’s involved.13

其中OS是M×1的产出松弛变量矩阵，IS是K×1的投入松弛变量矩阵，M1和K1分别是他们的M×1和K×1 的矩阵。注意，在两阶段线性规划中，θ不是变量，他的取值来于第一阶段。更进一步，我们要注意两阶段线性规划的问题对于N个DMU来说每个都要解决。

There are two major problems associated with this second stage LP.The first and most obvious problem is that the sum of slacks is MAXIMISED rather than MINIMISED. Hence it will identify not the NEAREST efficient point but the FURTHEST efficient point.The second major problem associated with the above second-stage approach is that it is not invariant to units of measurement. The alteration of the units of measurement,say for a fertiliser input from kilograms to tonnes(while leaving other units of measurement unchanged),could result in the identification of different efficient boundary points and hence different slack and lambda measures.14

有两个主要问题是关于两阶段线性规划的。第一个也是最显而易见的是松弛变量的总和是最大化而不是最小化。因此，我们找到的不是最近的效率点而是最远的

13This method is used by all the popular DEA software such as Warwick DEA and IDEAS.

这个方法被所有的流行的DEA软件所使用，比如Warwick DEA 和IDEAS。

14Charnes,Cooper,Rousseau and Semple(1987)suggest a units invariant model where the unit worth of a slack is made inversely proportional to the quantity of that input or output used by the i-th firm. This does solve the immediate problem,but does create another,in that there is no obvious reason for the slacks to be weighted in this way.

Charnes,Cooper,Rousseau 和Semple(1987)设计了一个单位不变的模型，在这个模型里松弛变量的单位价值和第i个公司的使用的投入或者产出的数量成反比。这能解决直接问题，但是由创造了另一个问题，因为这种方法没有足够的理由计算松弛变量的权重。

效率点。第二个关于两阶段方法的主要问题是对于计量单位来说他不是不变的。计量单位的改变，比如说更多的投入，从千克到吨（在其他计量单位不变的前提下），可以导致发现不同的效率前沿点和不同的松弛变量和更多的测度方法。

Note,however,that these two issues are not a problem in the simple example presented in Figure 5 because there is only one efficient point to choose from on the vertical facet. However, if slack occurs in 2 or more dimensions (which it often does) then the above mentioned problems can come into play.

然而，我们也要注意，在数据5中的简单的例子里有两点并不是问题所在。因为在垂直面上仅有一个效率点。然而，如果松弛变量发生在两个或者更多维的结构中（这是经常发生的）上述问题就会发生了。

As a result of this problem,many studies simply solve the first-stage linear program (equation 12)for the values of the Farrell radial technical efficiency measures(θ)for each DMU and ignore the slacks completely,or they report both the radial Farrell technical efficiency score(θ)and the residual slacks, which may be calculated as OS=-yi+Yλand IS=θxi-Xλ. However,this approach is not without problems either because these residual slacks may not always provide all (Koopmans) slacks (e.g.,when a number of observations appear on the vertical section of the frontier in Figure 5.5)and hence may not always identify the nearest(Koopmans)efficient point for each DMU.

这个问题的结果是，许多研究仅仅解决第一阶段的线性规划问题（等式12），从而得到Farrell的每个DMU的辐射的技术效率值（θ）。并且完全忽略松弛变量，或者他们记录辐射的Farrell的技术效率值（θ）和残余松弛变量，这个变量是这样计算的OS=-y i+Yλand IS=θx i-Xλ.然而，这个方法并非能解决所有的问题，可能是因为残余松弛变量不一定提供所有的松弛（Koopmans）变量（例如，数据5.5中有一些观测点在前沿的垂直面部分上时），或者是可能不总是能找出每个DMU的最近的效率点。

In the DEAP software we give the user three choices regarding the treatment of slacks.

在DEAP软件中，我们关于松弛变量给使用者三种选择。

These are:

1.One-stage DEA,in which we conduct the LP in equation 12 and

calculate slacks residually;

2.Two-stage DEA,where we conduct the LP’s in equations 12 and 13;and

3.Multi-stage DEA,where we conduct a sequence of radial LP’s to identify

the efficient projected point.

他们是：

1.一阶段DEA，我们在等式12构造线性规划模型并计算松弛变量残值。

2.两阶段DEA，我们用等式12和13构造线性规划模型。

3.多阶段DEA，我们构造一系列的辐射的线性规划模型以此来识别有效预测点。

The multi-stage DEA method is more computationally demanding that the other two methods(see Coelli 1997 for details). However,the benefits of the approach are that it identifies efficient projected points which have input and output mixes which are as similar as possible to those of the inefficient points,and that it is also invariant to units of measurement.Hence we would recommend the use of the multi-stage method over the other two alternatives.

同其他两种方法相比，多阶段DEA方法相比计算复杂。然而，这种方法的优点是他能识别投入和产出混合的效率预测点，这些点与非有效点十分的相似，并且识别出的效率预测点相对于测度单位来说是不变的。因此，同另外两个方法相比我们推举使用多阶段的DEA。

Having devoted a number of pages of this manual to the issue of slacks we would like to conclude by observing that the importance of slacks can be overstated. Slacks may be viewed as being an artefact of the frontier construction method chosen (DEA)and the use of finite sample sizes. If an infinite sample size were available and/or if an alternative frontier construction method was used,which involved a smooth function surface,the slack issue would disappear. In addition to this observation it also seems quite reasonable to accept the arguments of Ferrier and Lovell(1990)that slacks may essentially be viewed as allocative inefficiency. Hence we believe that an analysis of technical efficiency can reasonably concentrate upon the radial efficiency score provided in the first stage DEA LP(refer to equation 12). However if one insists on identifying Koopmans-efficient projected points then we would strongly recommend the use of the multi-stage method in preference to the two-stage method for the reasons outlined above.15

我们在这个指南中说了很多关于松弛变量的问题，现在我们总结出我们也许夸大了松弛变量的作用。松弛变量可以看做是用DEA方法来得到前沿结构和使用有限样本的人工品。如果我们能够得到DEA的无限的样本，或者使用另外一个估计前沿结构的方法，这种方法有一个光滑的结构表面，那么松弛变量的问题就消失了。另外对于这个观察，我们接受Ferrier and Lovell(1990)的观点是合理的。他们认为松弛变量可以看做是配置无效率。因此，我们相信技术效率分析可以合理的集中于在一阶段DEA线性规划（参考等式12）中得到的辐射效率指数。

然而，如果我们坚持想得到Koopmans效率预测点，我们就强力建议使用多阶段的方法而不是两阶段的方法，原因如上所述。

Example 1 例子1

We will illustrate CRS input-orientated DEA using a simple example involving

15However we have also included the 2-stage option in our software because it is the method used in other popular DEA software packages such as Warwick DEA and IDEAS.

然而，我们的软件也包括了两阶段的选择。因为在其他流行的DEA软件包里面使用，比如Warwick DEA 和IDEAS。

five observations on DMU’s(firms)which use two inputs to produce a single output.The data is as follows:

我们用一个包括五个DMU（公司）的简单的例子阐述规模报酬不变投入主导型的数据包络分析。每个DMU都是两投入一产出，数据如下：

表 1

规模报酬不变的DEA例子数据

Example Data for CRS DEA

The input/output ratios for this example are plotted in Figure 6,along with the DEA frontier corresponding to equation 12. You should keep in mind, however, that this DEA frontier is the result of running five linear programming

problems-one for each of the five DMU’s. For example, for DMU 3 we could rewrite equation 12 as

这个例子的投入产出比率在数据6中绘出，同时还汇出了DEA的同等式12对应的前沿。我们可以记在心里，然而，这个DEA前沿是对总的5个DMU每个都计算一次线性规划的结果。例如，对于DMU3我们可以这样重新书写等式12.

minθ,λθ,

st -y3+(y1λ1+y2λ2+y3λ3+y4λ4+y5λ5)≥0,

θx13-(x11λ1+x12λ2+x13λ3+x14λ4+x15λ5)≥0,

θx23-(x21λ1+x22λ2+x23λ3+x24λ4+x25λ5)≥0,

λ≥0, （14）whereλ=(λ1,λ2,λ3,λ4,λ5)′.

The values of θ and λ which provide a minimum value for θ are listed in row 3 of Table 2. We note that the TEI of DMU 3 is 0.833. That is,DMU 1 should be able to reduce the consumption of all inputs by 16.7% without reducing output. This implies production at the point denoted 3′in Figure 6.This projected point, 3′,lies on a line joining points 2 and 5.DMU 2 and DMU 5 are therefore usually referred to as the peers of DMU 3. They define where the relevant part of the frontier is(i.e.relevant to DMU 3)and hence define efficient production for DMU 3. Point 3′is a linear combination of points 2 and 5,where the weights in this linear combination are the λ‘ s in row 3 of Table 2.

θ和λ的值在表2第三行中提供了最小的θ值。我们注意到DMU3的技术效率值是0.833。DMU1可以在不减少产出的情况下将投入降低16.7%。这就意味着应该在数据6的3’点生产。这个估计点3’在DMU2和DMU5的连线上，它被认为是点3的对应点。他们定义了前沿相关部分的所在（例如与DMU3相关的）