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Introduction to Data Envelopment Analysis

and Its Applications

Shinn Sun

Department of Management

Fo Guang University

6/10/2015

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Professor W. W. Cooper-Founder of DEA andShinn photoed at EURO XIV Conference on July 6, 1995

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V. Krivonozhko, J. C. Paradi, C. Chen, Rajiv Banker, Shinn, Hsihu Changphotoed at 5th International Symposium on DEA, January 6, 2007

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Lawrence M. Seiford

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Tsutsui, Tone, Fukuyama, Morita, Shinn, Hirotsu

DEA Symposium 2012, Feb 20-21

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Thanassoulis, Yu, Tone,

DEA Symposium 2012, Feb 20-21

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魏權齡（左三）與孫遜

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Joe Zhu

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Outline What is Data Envelopment Analysis (DEA) Efficiency Measures The Use of DEA DEA Linear Programming Model Example: Car Manufacturing DEA Models DEA Research 1996-2006 DEA Model Development Evolution of DEA Application Areas Future for DEA DEA Software

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Outline-continued

DEA Books Conclusions

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What is DEA Evaluating the productivity of Decision Making Units

(DMUs) Initially designed for non-profits where operating

ratios may not be appropriate schools public utilities vehicle maintenance of the Tactical Air Command (TAC)

Has been adopted for evaluating for-profit branches Airline, Banking, Health Care, Hotels, Service Industry,

Transportation, etc. Recently, Hi-Tech Industry

How can you compare various DMUs Determine appropriate inputs Determine appropriate outputs Measure relationships between these inputs and outputs

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Efficiency Measures

• However, with multiple inputs and outputs, it becomes more difficult to evaluate the efficiency of DMUs.

OutputEfficiency =

Input

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• Clearly, process A is more efficient than process B, but...

• A new assessment based on office space shows that process B is more efficient than process A, so…

Process Labor Cost ($/week)

Throughput (jobs/week)

Efficiency (jobs/$)

A 2,000 1,500 0.750B 1,500 1,100 0.733

Process Office Area

(ft2)

Throughput (jobs/week)

Efficiency

(jobs/ft2)A 10,000 1,500 0.15B 6,900 1,100 0.16

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The Use of DEA

Multiple inputs, multiple outputs. Measure efficiency relative to other DMUs. Linear Programming is used to determine

which DMUs are 100% efficient relative to the other units.

Determine relatively inefficient units. Provide ways of determining how to reduce

inefficiencies.

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DEA Linear Programming Model

Let Ek with k=1, 2, ... , K be the efficiency ratios of DMU k, where there are K total branch units.

Let uj, with j=1, 2, ... , M be the weight given for output j, where M is the total number of output types.

Let vi, with i=1, 2, ... , N be the weight given for input i, where N is the total number of input types.

Let Ojk be the number of observed units of output j generated by DMU k during one time period.

Let Iik be the number of actual units of input i used by DMU k during one time period

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DEA Efficiency Measure

Consider a single DMU B whose efficiency we want to measure.

Want to maximize its efficiency by choosing uj's and vi's.

However, in choosing, no other unit can exceed 100% efficiency. So we have the constraints

EB

u1O1B u2O2 B uMOMB

v1I1B v2I2 B vN INB

Ek

u1O1k u2O2k uMOMk

v1I1k v2I2 k vN INk

100%, k 1,2,K

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DEA Linear Program

Max EB u1O1B u2O2B uMOMB

v1I1B v2I2B vN INB 1

u1O1k u2O2k uMOMk v1I1k v2I2k vN INk 0

uj 0, j 1,2,, M

vi 0, i 1,2,, N

k 1,2,, K

Generally K ≥ 2(N+M)

subject to

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Example: Car Manufacturing Make-to-stock only Six units 3-door, 4-door, and 5-door cars only. Assume output 100 cars at each Inputs vary

Unit # Cars Labor Costs 1 100 2 200 2 100 4 150 3 100 4 100 4 100 6 100 5 100 8 80 6 100 10 50

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Productivity Frontier

0

50

100

150

200

0 2 4 6 8 10

Labor Hours

Mate

rial C

ost

s

For each DMU (unit) we need to solve a linear program to determine its efficiency.

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Unit #1

We see from its solutionthat it is 100% efficientrelative to the other units.

Final Reduced Objective Allowable AllowableCell Name Value Cost Coefficient Increase Decrease

$D$2 u1 0.01 0 100 1E+30 99.99143199$E$2 v1 0.166666667 0 0 0 3$F$2 v2 0.003333333 0 0 300 0

Constraints

Final Shadow Constraint Allowable AllowableCell Name Value Price R.H. Side Increase Decrease

$C$11 1 1 1 1E+30 1$C$5 S1 0 1 0 1 0.384615385$C$6 S2 -0.166666667 0 0 1E+30 0.166666667$C$7 S3 0 0 0 0.2 1$C$8 S4 -0.333333333 0 0 1E+30 0.333333333$C$9 S5 -0.6 0 0 1E+30 0.6$C$10 S6 -0.833333333 0 0 1E+30 0.833333333

Linear programMax 100 u1

subject to 100u1 - 2 v1 -200 v2 ≤ 0100u1 - 4 v1 -150 v2 ≤ 0100u1 - 4 v1 -100 v2 ≤ 0100u1 – 6 v1 -100 v2 ≤ 0100u1 - 8 v1 -80 v2 ≤ 0100u1 - 10 v1 -50 v2 ≤ 02v1 + 200 v2 = 1

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Consider Unit #4

• Here we find that unit 4 is relatively inefficient.• The shadow prices presented imply that the unit's

efficiency reference set are units 3 and 6. • Compare with the graph

Final Reduced Objective Allowable AllowableName Value Cost Coefficient Increase Decrease

u1 0.008888889 0 100 1E+30 100v1 0.055555556 0 0 2 7v2 0.006666667 0 0 116.6666667 33.33333333

Constraints

Final Shadow Constraint Allowable AllowableName Value Price R.H. Side Increase Decrease

1 0.888888889 1 1E+30 1S1 -0.555555556 0 0 1E+30 0.555555556S2 -0.333333333 0 0 1E+30 0.333333333S3 0 0.777777778 0 0.142857143 0.5S4 -0.111111111 0 0 1E+30 0.111111111S5 -0.088888889 0 0 1E+30 0.088888889S6 0 0.222222222 0 0.153846154 0.625

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Composite Reference Unit

• One efficient outcome can be obtained by combining the units in the efficiency set using the relative weight assigned to each in calculating the relative efficiency of unit 4.

• These weights turn out to be just the shadow prices on the efficiency constraints.

Calculation of composite unit and excess inputs usedReference

Set Composite ExcessOutputs and Reference Inputs

Inputs S3 S6 Unit C Unit 4 Used

Cars 0.7778 x 100 + 0.2222 x 100 = 100 100 0Labor Hours 0.7778 x 4 + 0.2222 x 10 = 5.3 6 0.7

Material Costs 0.7778 x 100 + 0.2222 x 50 = 88.9 100 11.1

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Alternate Efficient Changes The values for v1 and v2 measure the relative weight

given to the inputs labor-hours and material costs, respectively, in determining the efficiency.

For unit 4, each unit decrease in labor-hours, results in an efficiency increase of 5.55%.

An efficient firm is found by reducing labor-hours by

Also, for each unit decrease in material costs we increase efficiency by 0.67 % so unit 4 can become efficient by reducing costs by

.

100% 88.88%

5.55%2 hours

100% 88.88%

0.67%$16.67

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DEA Models

Traditional models: Charnes, Cooper and Rhodes Model (CCR) Banker, Charnes and Rhodes Model (BCC) Alternative models: Additive Model Slack-based Model Free Disposal Hull Multiplicative Model

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Cross Efficiency Model Window Analysis Models under weights restrictions Assurance Region Model Cone-Ratio Model Variable Models Non-controllable Model Categorical Model Bilateral Model

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Allocation Models

Profit Objective Model

Cost efficiency Model

Revenue Efficiency Model

Profit Efficiency Model

Revenue/Cost Efficiency Model

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DEA Research 1996-2006

A total of 1,030 journal articles is selected. (Theoretical Articles: 382, Applications: 648)

0

20

40

60

80

100

120

140

160

180

1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006

年代

篇數

整體文獻

理論文獻

應用文獻

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DEA Model Development方法論： 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 總計 百分比 排名

Parametric & Nonparametric 5 3 4 4 2 3 4 4 8 5 4 46 12.04% 1T

Restricting Multipliers 2 3 4 5 3 5 1 6 8 5 3 45 11.78% 2

RTS 2 2 1 5 4 2 3 3 10 3 3 38 9.95% 3

Ranking 2 2 0 2 2 2 4 4 6 7 5 36 9.42% 4

Stochastic & Alternate Frontiers 2 2 4 7 3 5 2 1 2 2 6 36 9.42% 4T

Discretionary & Nondiscretionary 3 2 5 0 1 4 3 2 2 5 5 32 8.38% 6

Allocative 1 0 0 2 4 1 4 1 4 4 3 24 6.28% 7

Sensitivity Analysis 2 0 3 4 0 3 1 1 3 3 2 22 5.76% 8

CCR 0 0 2 4 2 2 1 1 1 4 2 19 4.97% 9

Fuzzy Intervals 0 0 0 1 1 1 1 3 2 4 3 16 4.19% 10

Congestion 1 0 0 0 1 4 1 0 3 0 3 13 3.40% 11T

FDH 0 1 1 0 2 1 0 0 4 1 2 12 3.14% 12

Cone Ratio 0 0 0 2 1 1 1 2 0 0 1 8 2.09% 13T

Supper-efficiency 0 0 0 1 0 1 2 0 1 2 1 8 2.09% 13T

Malmquist Index 0 0 0 1 1 1 0 0 3 0 2 8 2.09% 13T

Discriminant Analysis 0 0 0 2 0 1 0 1 1 0 0 5 1.31% 16

Chance Constrained DEA 0 0 1 0 0 1 1 0 1 0 1 5 1.31% 16T

Noise 0 0 0 0 0 0 0 0 2 1 1 4 1.05% 18

RAM 0 0 0 1 0 2 0 0 0 0 0 3 0.79% 19

CFA 1 0 0 0 0 0 0 0 0 0 1 2 0.52% 20

總計 21 15 25 41 27 40 29 29 61 46 48 382 100.00%

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Evolution of DEA

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Application Areas

主題： 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 總計 百分比 排名

Banking 4 11 5 15 7 8 18 14 19 12 14 127 19.96% 1

Health Care 2 1 5 5 7 6 5 11 13 5 13 73 10.60% 2

Other 5 3 7 3 7 3 3 12 5 8 12 68 9.89% 3

Transportation 3 3 1 4 1 9 5 7 9 5 9 56 8.30% 4

Public Administration 3 2 3 1 5 2 6 3 5 7 3 40 6.54% 5

Energy 1 1 3 3 6 4 5 1 2 7 2 35 5.83% 6

Education 3 2 0 2 6 6 6 2 0 5 2 34 5.65% 7

Manufacturing 1 0 1 2 3 2 5 5 6 4 4 33 5.12% 8

Agriculture 0 2 3 2 4 0 1 1 8 3 5 29 4.24% 9

Telecommunication 2 2 2 1 3 4 1 1 5 2 1 24 4.06% 10

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# Seiford (1996) Gattoufi et al. (2004a) 本研究

1 農業

2 礦業

3 食品業

4 能源業

5 紡織業

6 製造業

7 營造業

8 鋼鐵業

9 銀行業

10 運輸業

11 通信業

12 電腦業

13 保險業

14 零售業

15 高科技業

16 國防

17 教育

18 環境

19 醫療保健

20 總體經濟

21 公共部門

22 公共財政

23 娛樂業

24 其他

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Future for DEA

Theoretical limitation Research Extended DEA models Information Science Statistics Stochastic DEA Dynamic DEA Network DEA Comparison of various DEA models Introduction to quality variables New Application area

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DEA Software

Commercial available software:

DEA Solver

Frontier Analyst

DEA Excel Solver

OnFront

Warwick DEA

MaxDEA, DEAOS Free Software: DEAP, EMS

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DEA Books

Charnes et al. (1994) Data Envelopment Analysis: Theory, Methodology and Applications.

Coelli et al. (1997) An Introduction to Efficiency and Productivity Analysis.

Cooper et al. (2000, 2007) Data Envelopment Analysis: A Comprehensive Text with Models, Application, References and DEA Solver.

孫遜 ( 民 93) 資料包絡分析法—理論與應用，揚智文化公司。

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Conclusions Can understate the inefficiency because

it is calculated by trying to put the inefficient DMU in the best light.

May correct by forcing one DMU, known to be efficient in general, to be explicitly efficient.

Much care must be taken in determining the input and output variables.

Can fail to give significant information if too few points available.

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Conclusions-continued

Serves as a tool for - productivity analysis; - performance measurement; - technology forecasting; - capacity planning; - process re-design; - R&D project evaluation; - strategy alliances selection; and - resources allocation.