Chapter 7 Additional Topics on Data Envelopment Analysis AN INTRODUCTION TO EFFICIENCY AND PRODUCTIVITY ANALYSIS Prof. Ming-Miin Yu ( 游明敏 ) National Taiwan

Chapter 7 Additional Topics on Data Envelopment Analysis

AN INTRODUCTION TO EFFICIENCY AND PRODUCTIVITY ANALYSIS

Prof. Ming-Miin Yu ( 游明敏 )National Taiwan Ocean University

slide 2CHAPTER 7 Additional Topics on Data Envelopment Analysis

Term paper requirement Oral presentation and defense on 24 Jun. Deadline for Final report at AM:12:00 on

30 Jun. Final report format （ at least 10 pages ）

– Introduction– Problem Description– Methodology– Data– Empirical Result– Conclusion– Reference


7.1 Introduction

Some popular extensions of the basic DEA models (CRS, VRS) ︰Allocative efficiencyEnvironmental variablesNon-discretionary variablesThe treatment of slacks (slack-based measure)Congestion efficiencyUndesirable outputWeights restrictions Super efficiencyCross-efficiencyMeta-FrontierNetwork DEA


7.2 Price Information and Allocative Efficiency

If price information is available and a behavioral objective, such as cost minimisation or revenue or profit maximisation, is appropriate, then it is possible to measure allocative efficiency.

Two sets of linear programs are required.

one to measure technical efficiency (TE)

the other one to measure economic efficiency (EE)


7.2.1 Cost Minimization

For the case of VRS cost minimization

,0

1'1

,0

,0

,min

*

*'

, *

N

Xx

Yyst

xw

i

i

iixi

)1.7(


Recall: TE under VRS

running a DEA problem with variable returns to scale (VRS) imposed by N1’ λ=1

,0

1'1

,0

,0

,min ,

N

Xx

Yyst

i

i )12.6(


7.2.1 Cost Minimization

Note that this procedure implicitly includes any slacks into the allocative efficiency measure

iiii xwxwEEEfficiencyEconomic '*' /

TEEEAEEfficiencyAllocative /


7.2.2 Revenue Maximization

For the case of VRS revenue maximization

*'' / iiii ypypEEEfficiencyEconomic

,0

1'1

,0

,0

,max

*

*'

, *

N

Xx

Yyst

yp

i

i

iiyi

)2.7(

TEEEAEEfficiencyAllocative /


Recall: Output-Orientated DEA under VRS

The output-orientated VRS DEA model:

Where 1 ≦Φ<∞, and Φ–1 is the proportional increase in outputs that could be achieved by the i-firm, with input quantities held constant.

Note that 1/Φ defines a TE score which varies between 0 and 1 .

,0

1'1

,0

,0

,max ,

N

Xx

Yyst

i

i

)14.6(


Profit Efficiency

Cost minimization and revenue maximization together imply profit maximization

Measurement of profit efficiency using DEA methods:

Directional Distance Functions

(Fare, Grosskopf and Weber, 1997)



Recall: A Simple Numerical example

firm y x1 x2 x1/y x2/y

1 1 2 5 2 5

2 2 2 4 1 2

3 3 6 6 2 2

4 1 3 2 3 2

5 2 6 2 3 1

Table 6.1 Example data for CRS DEA example (EG1.DTA)


Listing of Data File, EG3.DTA

Y X1 X2 W1 W2

1 2 5 1 3

2 2 4 1 3

3 6 6 1 3

1 3 2 1 3

2 6 2 1 3

Note: All firms face the same prices, which are 1 and 3 for inputs 1 and 2, respectively


Recall: A Simple Numerical Example

x1/y

x2/y

0

2

5

3’3

1

1’4

4’

Figure 6.6

1 2 3 4 5

1

2

3

4

5

FRONTIER


7.2.3 A CRS Cost Efficiency DEA Example

x2/y

0

2

5

3’3

1

1’4

4’

Figure 7.1

1 2 3 4 5

1

2

3

4

5

FRONTIER

6

3’’

ISOCOST LINE

x1/y


Listing of Instruction File, EG3.INS

eg3.dta DATA FILE NAME

eg3.out OUTPUT FILE NAME

5 NUMBER OF FIRMS

1 NUMBER OF TIME PERIODS

1 NUMBER OF OUTPUTS

2 NUMBER OF INPUTS

0 0=INPUT AND 1=OUTPUT ORIENTATED

0 0=CRS AND 1=VRS

1 0=DEA(MULTI-STAGE), 1=COST-DEA, 2=MALMQUIST-DEA, 3=DEA(1-

STAGE),

4=DEA(2-STAGE)



Firm Technical efficiency

Allocative efficiency

Cost efficiency

1 0.5 0.706 0.353

2 1.0 0.857 0.857

3 0.833 0.900 0.750

4 0.714 0.933 0.667

5 1.0 1.000 1.000

Mean 0.810 0.879 0.725

Table 7.1 CRS Cost Efficiency DEA Result


Exercises:

Add another firm (6-th firm) with y = 2, x1 = 12, and x2 = 4

Check: Cost efficiency = TE x AE Cost efficiency, Overall efficiency,

Economic efficiency Note that all of the optimal input

quantities are used in the same ratio (3:1) because each firm faces the same relative input prices (see EG3.OUT)



Y X1 X2 W1 W2

1 2 5 1 2

2 2 4 1 2

3 6 6 1 2

1 3 2 1 2

2 6 2 1 2

Note: 1.All firms face the same prices, which are

1 and 2 for inputs 1 and 2, respectively. 2. W2=3 更改為 W2=2



Table 7.1 CRS Cost Efficiency DEA Result

Firm Technical efficiency

Allocative efficiency

Cost efficiency

1 0.5 0.833 0.417

2 1.0 1.000 1.000

3 0.833 1.000 0.833

4 0.714 1.000 0.714

5 1.000 1.000 1.000

Mean 0.810 0.967 0.793



x2/y

x1/y0

1 2 3 4 5 6

1

2

3

4

5

ISOCOST LINE

FRONTIER

1

1’

1’’2 3 4

5

3’

4’


Solutions to Exercises:

Add another firm (6-th firm) with y = 2, x1 = 12, and x2 = 4

Check: Cost efficiency = TE x AE Cost efficiency, Overall efficiency,

Economic efficiency Note that all of the optimal input

quantities are used in the same ratio (3:1) because each firm faces the same relative input prices (see EG3.OUT)



Y X1 X2 W1 W2

1 2 5 1 3

2 2 4 1 3

3 6 6 1 3

1 3 2 1 3

2 6 2 1 3

2 12 4 1 3




Y X1 X2 W1 W2

1 2 5 1 2

2 2 4 1 2

3 6 6 1 2

1 3 2 1 2

2 6 2 1 2

2 12 4 1 2



7.3 Non-Discretionary Variables




7.4 Non-Discretionary Variables In some instance, the manager may not be

able to alter all inputs or outputs (Short-Run).

In above DEA problem the θ-parameter is only associated with the discretionary inputs and hence the problem only seeks radial reduction in this subset of the inputs (Discretionary subset).

)6.7(

,0

1'1

0

,0

,0

,min,

N

Xx

Xx

Yyst

NDND

i

DD

i

i


Efficiency measure with Non-Discretionary Inputs

S’

S

Labor(Discretionary input)

Capital(Non-Discretionary input)

0

C

D

B’

B

Figure 6.5


7.4 Adjusting for Environment

We use the term environment to describe factors which could influence the firm, where such factors are not traditional inputs and are assumed not under the control of the manager.

1. Ownership differences

2. Location characteristics

3. Labor union power

4. Government regulations


Method 1

Assumption: if the values of the environment variable can be ordered from the least to the most detrimental effect on efficiency ( 假設可以排序 )

In the approach the efficiency of the i-firm is compared with those firms in sample which have a value of environmental variable which is less than or equal to that of the i-firm. ( 例如區位 [ 鄉村、郊區、市區 ] 影響漢堡餐廳經營效率的排序 )



Method 2

If there is no natural ordering of environmental variable (e.g., public versus private ownership), Charnes, Cooper and Rhodes (1981) propose a method which involves three stages:

1. Divide the sample into public/private sub- samples and solve DEAs for each sub-sample.

2. Project all observed data points onto their respective frontiers

3. Solve a single DEA using the projected points and assess any difference in the mean efficiency of the two sub-sample.


Methods 1 and 2 的缺點 one problem with methods 1 and 2 is that

the comparison set can be greatly reduced, resulting in many firms being found to efficient and thus reducing the discriminating power of the analysis.

Another problem is that only one environmental variable can be considered by these two method.

Method 2 requires categorical variable, Method 1 requires the direction of the influence of the environmental variable (upon efficiency) be known a priori.



Method 3

The method is to include the environmental variable(s) directly into the LP formulation (e.g., input or output?).



,0

1'1

0

,0

,0

,min ,

N

Zz

Xx

Yyst

i

i

i


One advantage of this approach is that i-firm is only compared with a theoretical firm which has an environment which is no better than that of the i-firm.

One disadvantage is that the method requires that the direction of the influence of the environmental variable be known in advance which is not always the case.



Neutral variable

If one is unsure as to the direction of the influence of the environmental variables, then the variables can be included in the LP problem in an equality form.

,0

1'1

0

,0

,0

,min ,

N

Zz

Xx

Yyst

i

i

i

)8.7(


Neutral variable

This formulation ensures the i-th firm is only compared to a (theoretical) frontier firm which has the same environment.

The restriction can greatly reduce the reference set and hence inflate the efficiency scores.

In some cases it may be unacceptable (unfair?) to compare the i-firm with another firm which has a more favorable environment.



non-discretionary input

First disadvantage of methods 3a to 3c is the environmental variable must be continuous variables, they cannot be categorical. (mixed integer LP models)

The second disadvantage of methods 3b and 3c is that one must decide a priori whether the environmental variable has a positive or negative influence upon efficiency.


Method 4: the two-stage method

The two-stage method involves solving a DEA problem in a first-stage analysis, involving only the traditional input and output.

In the second stage, the efficiency scores from the first stage are regressed upon the environmental variables.

Direction of influence

Hypothesis Testing

Tobit regression method to account for truncated data


Method 4: the two-stage method

One disadvantage is that if the variables used in first stage are highly correlated with the second stage variables then the result are likely to be biased.

It only considers radial inefficiency and ignores the slacks.

Possible solution: Three-stage approach

Decomposition into 3 factors: environmental variables, stochastic noise, and managerial inefficiency


Recommendations for the two-stage method

It can accommodate more than one variable;

It can accommodate both continuous and categorical variables;

It does not make prior assumptions regarding the direction of the influence of the categorical variable;


Recommendations for the two-stage method

One can conduct hypothesis tests to see if the variables have a significant influence upon efficiencies;

It is easy to calculate; The method is simple and

transparent. The method can be used to assess

the influence of various management factors (e.g., age, experience, education and training of the manager) upon efficiency.


7.5 Input Congestion

“Backward-bending” isoquants due to congestion in the use of an input, to the extent that it begins to have a negative marginal product (i.e., declining part of the TP curve)

The excess input use is due to constraints which are not under the control of the firm

Examples: labor unions preventing a reduction in stuff, or government controls setting the levels of various inputs


7.5 Input Congestion

strong disposability ( 強可拋 ) in inputs (and outputs) ： firms can always costlessly dispose of unwanted input (and outputs)

weak disposability ( 弱可拋 ) in inputs (and outputs)


Input Congestion

We introduce a δ parameter in the input restrictions.

,0

1'1

,0

,0

,min ,

N

Xx

Yyst

i

i

)7.7(


Input Congestion

We solve a strong disposability and a weak disposability DEA and identify input congestion efficiency (CE) from the difference in the TE scores from previous VRS model above. An input congestion efficiency measure is expressed as

CE=OPs/OPw

OPs/OP=(OPs/OPw) ×(OPw/OP)

TEs=CE×TEw


Efficiency measurement and Input Disposability (Congestion)

S

Ss

x1

x2

0

Ps

Pw

PA

Sw

Figure 7.2

CE=OPs/OPwOPs/OP=(OPs/OPw) ×(OPw/OP)TEs=CE×TEw



Efficiency measurement and Input Disposability (Congestion)

Note that the technical efficiency scores obtained from this weak disposability VRS DEA are greater than or equal to the strong disposability VRS DEA scores . This is because the effects of congestion inefficiency have been removed from the technical measure.


Input Congestion

The technical efficiency scores calculated from a CRS DEA can be decomposed into (1) congestion inefficiency,

(2) scale inefficiency,

(3) “pure” technical efficiency

by solving three DEA models: (DEAP ?)

(1) a CRS assuming strong disposability;

(2) a VRS assuming strong disposability;

(3) a VRS assuming weak disposability.


Weak disposability in outputs

Existence of a positively sloped portion of the production possibility curve, implying a negative shadow price for a particular output.

Unwanted outputs or undesirable outputs: pollutants, noise, bad loans

Goods v.s. Bads



7.6 Treatment of Slacks

S’

S

x1/y

x2/y

0

C

D

B’

B

A

A’

Figure 6.5


Treatment of Slacks

the use of a second-stage linear programming problem to ensure the identification of an efficient frontier point by maximizing the sum of slacks

0,0,0

,0

,0

)'1'1(min ,,

ISOS

ISXx

OSYyst

ISKOSM

i

i

ISOS

)8.7(


Treatment of Slacks

There are two major problems associated with second-stage LP. The first and most obvious problem is that the sum of the slacks is maximized rather than minimized. Hence, it identifies not the nearest efficient point but the furthest efficient point.

The second major problem is that it is not invariant to units of measurement.


Treatment of Slacks

Coelli suggests using a multi-stage DEA method to avoid the problems inherent in the two-stage method.

DEAP Software:

One-stage DEA (slacks are calculated as residuals)

Two-stage DEA

Multi-stage DEA


Listing of Instruction File, EG1.INS

eg1.dta DATA FILE NAME

eg1.out OUTPUT FILE NAME

5 NUMBER OF FIRMS

1 NUMBER OF TIME PERIODS

1 NUMBER OF OUTPUTS

2 NUMBER OF INPUTS

0 0=INPUT AND 1=OUTPUT ORIENTATED

0 0=CRS AND 1=VRS

0 0=DEA(MULTI-STAGE), 1=COST-DEA, 2=MALMQUIST-DEA, 3=DEA(1-

STAGE),

4=DEA(2-STAGE)


Importance of Slacks can be overstated

Slacks may be viewed as being an artefact( 人工製品 ; 加工品 ) 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 frontier, the slack issue would disappear.

Ferrier and Lovell (1990) argued that slacks may essentially be viewed as allocative inefficiency.


7.7 Additional Methods The treatment of slacks (slack-based

measure)

Congestion efficiency

Undesirable output

Weights restrictions

Cross-efficiency

Meta-Frontier

Network DEA

7.5 超效率 (Super-Efficiency) DEA 模式

DEA 法的缺點： (1) 同時會有多個 DMU 均被評估為具有效率，其效率分數均為一，尤其是在 DMU 個數，相對於投入與產出數目和並未超過很多時，這種結果易於成立。 (2) 一個“特殊”的 DMU 可能只因為其具有某種獨特的投入產出組合，即使那獨特的組合可能不重要，仍被列為有效率。

超效率 DEA 模式：由 Andersen and Petersen (1993) 所提出，其目的在區分及排列這些 DEA 估計結果為有效率的該組 DMU 之順位。




利用標準 DEA 模式，進行 DMU(C) 之變動規模報酬效率之評估時，可藉由 BCC 線性規劃模式來求解，


超效率 DEA 模式之線性規劃限制式，旨在去除受評 DMU 的投入產出組合。評估 DMU(C) 所用之超效率模式的 BCC 線性規劃

式：


7.6 交叉效率 DEA 模式交叉效率法之應用，係在計算效率時包括了同儕評估 (peer evaluations) 之考量。交叉效率之構想，來自 Sexton, Silkman and Hogan

(1986) ，其係利用效率矩陣來表達自我與同儕評估。 DMU(i) 之相對效率，可藉下列極大化之線性規劃模式求得，

7.6 交叉效率 DEA 模式若將自己選擇之權數應用在其它 DMU 上，則限制式之 Ein 值，可解釋為將第 i 個 DMU 之權數，應用於第 n 個 DMU 之效率評估，

因此，此 Ein 之公式，在表達其他 DMU 利用 DMU(i) 權數的交叉評估，亦即 DMU(i) 對其它 DMU 之同儕評估。

7.6 交叉效率 DEA 模式矩陣 E 之第 i 列效率值，為所有 DMU 應用第 i 個 DMU 的產出與投入權數下之效率評估值，亦即 DMU(i) 對其它 DMU 之同儕評估。 Eii 值，為第 i 個 DMU 自我評估或標準 DEA 效率值。矩陣 E 中第 j 欄之交叉效率，為所有 DMU(i = 1,

2, …, N) 應用他們的權數對第 j 個 DMU 之同儕效率評估值，亦即每個 DMU 對 DMU(j) 。

7.6 交叉效率 DEA 模式










Super Efficiency



7.8 Empirical Application: Australian Universities

A DEA study (Coelli, 1996)—the relative performance of Australia’s 36 universities

Input-orientated DEA models

Construction of three separate models:

one for the administration sectors ( 本例 )

one for the academic sectors

one for universities as a whole



Inputs:

1.expenditure on administrative staff.

2.other administrative costs.

Outputs:

1.total number of students.

2.total number of staff.



University VRS TE CRS TE Scale eff.

Australian Catholic University 0.757 0.806 0.938 irs

Australian National University

1.000 1.000 1.000 -

Central Queensland University 0.499 0.557 0.896 irs

Charles Sturt University 1.000 1.000 1.000 -

Curtin University of Technology

0.700 0.702 0.997 drs

Deakin University 0.786 0.800 0.982 drs

Edith Cowan University 0.784 0.861 0.911 drs

Flinders University of South Australia

1.000 1.000 1.000 -

Griffith University 0.720 0.738 0.975 drs

James Cook University 0.725 0.757 0.958 irs

Table 7.3




La Trobe University 0.930 0.947 0.982 drs

Macquarie University 0.770 0.778 0.990 irs

Monash University 0.728 1.000 0.728 drs

Murdoch University 0.779 0.824 0.946 irs

Northern Territory University 0.662 0.980 0.676 irs

Queensland University of Technology

0.978 1.000 0.978 drs

Royal Melbourne Institute of Tech.

0.739 0.786 0.939 drs

Southern Cross University 0.950 1.000 0.950 irs

Swinburne University of Technology

0.876 0.925 0.947 irs

University of Adelaide 0.653 0.665 0.982 drs

Table 7.3




University of Ballarat 0.867 1.000 0.867 irs

University of Canberra 1.000 1.000 1.000 -

University of Melbourne 0.838 1.000 0.838 drs

University of New England 0.707 0.713 0.991 irs

University of New south Wales 0.745 0.930 0.801 drs

University of Newcastle 0.404 0.404 1.000 -

University of Queensland 1.000 1.000 1.000 -

University of South Australia 1.000 1.000 1.000 -

University of Southern Queensland

0.798 0.826 0.967 irs

University of Sydney 0.765 1.000 0.765 drs

Table 7.3




University of Tasmania 1.000 1.000 1.000 -

University of Technology, Sydney

1.000 1.000 1.000 -

University of Western Australia

0.865 0.870 0.995 irs

University of Western Sydney 0.622 0.625 0.995 drs

University of Wollongong 0.882 0.892 0.989 irs

Victoria University of Technology

0.904 0.918 0.985 irs

Mean 0.818 0.870 0.944

Table 7.3

slide 100

CHAPTER 7 Additional Topics on Data Envelopment Analysis

7.8 Further Reading

Fare, R., S. Grosskopf and C.A. K. Lovell (1985), The Measurement of Efficiency of Production, Boston: Kluwer Academic Publishers. [HB241.F335]

Cooper, W.W., L. M. Seiford and K. Tone (1999), Data Envelopment Analysis: A Comprehensive Text with Models, Applications, References and DEA-Solver Software, Boston: Kluwer Academic Publishers. [HA31.38.C66]

Fare, R., S. Grosskopf and C.A. K. Lovell (1994), Production Frontiers, Cambridge: Cambridge University Press. [HB241.F336]

slide 101


Conclusions

An analysis of technical efficiency can reasonably concentrate upon the radial Farrell efficiency score provided in the first stage DEA LP.

However, if one insists on identifying Koopmans-efficient projected points then we would strongly recommend the use of the multi-stage method.

slide 102


Thank

You ！

Documents

Chapter 7 Additional Topics on Data Envelopment Analysis AN INTRODUCTION TO EFFICIENCY AND PRODUCTIVITY ANALYSIS Prof. Ming-Miin Yu ( 游明敏 ) National Taiwan