Choonjoolee(DEA)10Boston Conf

An Efficient Data Envelopment Analysis with a large data set in Stata

15-16 July, 2010

Boston10 Stata Conference

Choonjoo Lee, Kyoung-Rok Lee

[email protected], [email protected]

Korea National Defense University

ContentsPart I. A Large Data Set in Stata/DEA

Large Data Set in DEA?

Computational Aspects of Large Data Set

The Scope of this Study

Efficiency Matters in Stata/DEA/Linear Programming

Tasks to be covered

Part II. Malmquist Index Analysis with the Panel DataBasic Concept of Malmquist Index

The User Written Command “malmq”

Part I. A Large Data Set in Stata/DEALarge Data Set in DEA?



Efficiency Matters in Stata/DEA/Linear Programming

Tasks to be covered

Large Data Set in DEA?• Graphical illustration of DEA concept

Large Data Set in DEA?

• Variables and Observation Constraints by the Features of DEA Domain Programs(Language)– Statistical Package based DEA Programs– Spreadsheet based DEA Programs– Language based DEA Codes

• Performance of Linear Program(LP): Efficiency and Accuracy– LP is the Critical Component of DEA Program– Approaches to Solve LP: Simplex, Interior Point Methods(IPMs)

☞ Numerous Variants of the Basic LP Approach

• DEA Report Format(User Interface Design)– Results(input, output)– Graphical Display– Log


• Matrix Size for the Data Set in Matrix Format– # of rows and columns(variables and observations) allowed by the Program– The storage limit of the computer memory upgrade of computer technology, the way to access the data in the memory

• Matrix Density– # of nonzeros of the matrix– How many zero elements in the matrix?

• A Computationally Demanding Procedure of DEA due to the LP– The number of iterations needed to solve a problem grows exponentionally as a

function of variables and observations

• Numerical Difficulties– Inaccuracy and inefficiency due to the Floating Point Arithmetic with finite

precision– Numerical Precision due to the binary representation of number


• Performance of DEA code– Linear Program/Simplex Method

– Computational Technique

– Illustration

• Panel Data in DEA–Malmquist Index Analysis

Efficiency Matters in Stata/DEA/LP

• DEA program demands heavy computation– Computation time heavily depends on the number of

observations(DMUs), variables(inputs, outputs), LP

process, etc.

• Stata uses RAM(memory) to store data– The memory size matters for the large data set


ModelComputation(sec)

MemoryMajor Areas Revised

5-2-2-V1 ~20 1G

5-2-2-V2 (released) <2 <300MBasic feasible solution

5-5-5-V3 <1 <300MRevised Simplex Method

365-1-5-V1 ? 6G

365-1-5-V2* ~14600 6G Two-stage LP

365-1-5-V3* (under development)

20 <300M Mata, Tolerance

• The performance of Input Oriented DEA models

※ Stata SE

– if the number of observations(n) becomes significantly

larger than the number of variables(m)?

Method

Operation Pivoting Pricing Total

Tableau

Simplex

Multiplication,Division

(m+1)(n-m+1)

m(n-m)+n+1

Addition,Subtraction

m(n-m+1) m(n-m+1)

Revised

Simplex

Multiplication,Division

(m+1)2 m(n-m) m(n-m)+(m+1)2

Addition,Subtraction

m(m+1) m(n-m) m(n+1)

Efficiency Matters in Stata/DEA/LP• Understanding the difference of computation

– Data

Source: Cooper et al.(2006), table3-7

StoreInput Data Output Data

Employee Area Sales ProfitA 10 20 70 6B 15 15 100 3C 20 30 80 5D 25 15 100 2E 12 9 90 8


• Tableau and Revised Simplex in DEA/LP

– For DMU A

StoreInput Data Output Data

Employee Area Sales ProfitA 10 20 70 6

OrientatioOrientationn

Constant Return to ScaleConstant Return to Scale Variable Returns to ScaleVariable Returns to Scale

Input Input OrientedOriented

Min Min θs.t. s.t. θxxAA - X - Xλ ≥ 0YYλ -y -yAA ≥ 0

λ ≥ ≥ 00

Min θMin θs.t. θxs.t. θxAA - Xλ ≥ 0 - Xλ ≥ 0

Yλ -yYλ -yAA ≥ 0≥ 0

eλ=1eλ=1 λ λ ≥ ≥ 00

Output Output OrientedOriented

Max ηMax ηs.t. xs.t. xAA - Xμ ≥ 0 - Xμ ≥ 0

ηyηyAA -yμ ≤ 0 -yμ ≤ 0

μ μ ≥ ≥ 00

Max ηMax ηs.t. xs.t. xAA - Xμ ≥ 0 - Xμ ≥ 0

ηyηyAA -yμ ≤ 0 -yμ ≤ 0

eλ=1 eλ=1 μ μ ≥ ≥ 00


• Tableau and Revised Simplex in DEA/LP

– The Basic DEA Models

Input &Output

Variablesdata file

DEA Options: Basic, Variants

Data conversion:Scaling, Tolerance

DEA result Report

LinearProgramming: Simplex Method

Files ofEfficiency

& Lambdas

Basic Solution Generating

DEA Loop: RTS, ORT

DATA Stata/DEA RESULT


• Program Structure

dea ivars = ovars [if] [in] [, rts(crs | vrs | drs | irs) ort(in | out) stage(1 | 2) trace saving(filename)]

– rts(crs | vrs | drs | irs) specifies the returns to scale. The default, rts(crs), specifies constant returns to scale.

– ort(in | out) specifies the orientation. The default is ort(in), meaning input-oriented DEA.

– stage(1 | 2) specifies the way to identify all efficiency slacks. The default is stage(2), meaning two-stage DEA.

– trace specifies to save all the sequences displayed in the Results window in the dea.log file. The default is to save the final results in the dea.log file.

– saving(filename) specifies that the results be saved in filename.dta.


• Program Syntax

– Canonical form

– Standard form

Min θ

s.t. 10θ - 10λA - 15λB - 20λC - 25λD - 12λE ≥ 0

20θ - 20λA - 15λB - 30λC - 15λD - 9λE ≥ 0

70λA+ 100λB + 80λC + 100λD + 90λE ≥ 70

6λA +3λB + 5λC + 2λD + 8λE ≥ 6

Min θ

s.t. 10θ - 10λA - 15λB - 20λC - 25λD - 12λE - S1- + x1 = 0

20θ - 20λA - 15λB - 30λC - 15λD - 9λE - S2- + x2 = 0

70λA + 100λB + 80λC + 100λD + 90λE - S1+ + x3 = 70

6λA + 3λB + 5λC + 2λD + 8λE -S2 + +x4 = 6

Efficiency Matters in Stata/DEA/LP• Develop the Basic Data Bank(input oriented CRS)

• Model V1: Tableau DEAX θ λA λB λC λD λE S1

- S2- S1

+ S2+ x1 x2 x3 x4 RHS MRT

1 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 0x1 0 10 -10 -15 -20 -25 -12 -1 0 0 0 1 0 0 0 0x2 0 20 -20 -15 -30 -15 -9 0 -1 0 0 0 1 0 0 0x3 0 0 70 100 80 100 90 0 0 -1 0 0 0 1 0 70x4 0 0 6 3 5 2 8 0 0 0 -1 0 0 0 1 6

1 30 46 73 35 62 77 -1 -1 -1 -1 0 0 0 0 76x1 0 10 -10 -15 -20 -25 -12 -1 0 0 0 1 0 0 0 0 ×x2 0 20 -20 -15 -30 -15 -9 0 -1 0 0 0 1 0 0 0 ×x3 0 0 70 100 80 100 90 0 0 -1 0 0 0 1 0 70 70/90x4 0 0 6 3 5 2 8 0 0 0 -1 0 0 0 1 6 6/8

Ⅰ 1 30-

47/4353/

8

-105/

8

171/4

0 -1 -1 -1 69/8 0 0 0-

77/873/4

x1 0 10 -1-

21/2-

25/2-22 0 -1 0 0 -3/2 1 0 0 3/2 9 ×

x2 0 20-

53/4-

93/8

-195/

8

-51/4

0 0 -1 0 -9/8 0 1 0 9/8 27/4 ×

x3 0 0 5/2265/

495/4

155/2

0 0 0 -1 45/4 0 0 1-

45/45/2

10/265

λE 0 0 6/8 3/8 5/8 2/8 1 0 0 0 -1/8 0 0 0 1/8 6/8 1/2


• Model V1: Tableau DEA

– Efficiency score(θ) of DMU A is 14/15

Z θ λA λB λC λD λE S1- S2

- S1+ S2

+ RHS MRTⅤ 1 0 0 -11/70 -32/35 -89/70 0 -39/350 1/175 -1/70 0 1λA 0 0 1 1/7 6/21 -33/21 0 -6/35 3/35 -1/70 0 1 35/3

θ 0 1 0 -11/70 -32/35-

267/210

0 -39/350 1/175 -1/70 0 1 175/1

S2+ 0 0 0 41/7 43/21 152/21 0 4/105 -2/105

-159/18

551 0 ×

λE 0 0 0 49/8 59/24 182/21 1 1/6 -1/12-

159/2120

0 0 ×

Ⅵ 1 0 -1/15 -1/6 -14/15 -7/6 0 -1/10 0 -1/75 0 14/15S2

- 0 0 35/3 5/3 10/3 -55/3 0 -2 1 -1/6 0 35/3θ 0 1 -1/15 -1/6 -14/15 -7/6 0 -1/10 0 -1/15 0 14/15

S2+ 0 0 2/9 53/9 19/9 62/9 0 0 0 -4/45 1 2/9

λE 0 0 35/36 451/72 177/72 257/36 1 0 0 -4/45 0 35/36


• Model V3: Revised DEAc 0

A I b

0

cB cN

B N b

0

0 cN-cBB-1N

I B-1N B-1b

cBB-1b


• Model V3: Revised DEA

– Step1: Set up the initial tableau factors.

– Step2: Find entering variable.

– Step3: Find leaving variable.

– Step4: Update the tableau. (Update the basis.)

X θ λA λB λC λD λE S1- S2

- S1+ S2

+ x1 x2 x3 x4 RHS

1 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 0

x1 0 10 -10 -15 -20 -25 -12 -1 0 0 0 1 0 0 0 0

x2 0 20 -20 -15 -30 -15 -9 0 -1 0 0 0 1 0 0 0

x3 0 0 70 100 80 100 90 0 0 -1 0 0 0 1 0 70

x4 0 0 6 3 5 2 8 0 0 0 -1 0 0 0 1 6

cB

BN

cN

b


- 1st step: The initial tableau factors.

B= xB= CB= CBB-1=

θ λA λB λC λD λE S1- S2

- S1+ S2

+

30 46 73 35 62 77 -1 -1 -1 -1

Max

- 2nd step: Finding entering variablecN -cBB-1N: Max value is selected as a entering variable

- 3rd step: Finding leaving variableB-1N = Min{xB/(B-1N)} ={×, ×, 70/90, 6/8} = 6/8 (←x4)



- 4th step: Update the tableauX θ λA λB λC λD λE S1

- S2- S1

+ S2+ x1 x2 x3 x4 RHS

1 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 0

x1 0 10 -10 -15 -20 -25 -12 -1 0 0 0 1 0 0 0 0

x2 0 20 -20 -15 -30 -15 -9 0 -1 0 0 0 1 0 0 0

x3 0 0 70 100 80 100 90 0 0 -1 0 0 0 1 0 70

x4 0 0 6 3 5 2 8 0 0 0 -1 0 0 0 1 6

cB

BN

cN

b

X θ λA λB λC λD x4 S1- S2

- S1+ S2

+ x1 x2 x3 x4 RHS

1 0 0 0 0 0 -1 0 0 0 0 -1 -1 -1 0 0

x1 0 10 -10 -15 -20 -25 0 -1 0 0 0 1 0 0 -12 0

x2 0 20 -20 -15 -30 -15 0 0 -1 0 0 0 1 0 -9 0

x3 0 0 70 100 80 100 0 0 0 -1 0 0 0 1 90 70

λE 0 0 6 3 5 2 1 0 0 0 -1 0 0 0 8 6



1 1.341099143-

61.133949280.4455321 1.883781314

2.587946653

0 0 0 0.0588235 0 0

0 0.116421975-

6.672515869-0.110761 0.495342732

-0.09713860

6

0-

0.172319263-

19.71403694-0.262333

-0.074690066

1.54739666

0-

0.046367686-

4.060891628-0.082268

-0.009800959

0.25169459

0 0.105886854 4.651313305 0.1136269-

0.0158843140.03722914

3

Tasks to be covered• Computational Accuracy– Example: Obtaining Inverse Matrix• Matrix D

Tasks to be covered

• Computational Accuracy– Example: Obtaining Inverse Matrix• Inverse matrix D by Stata/Mata “luinv (D)”

1 162470623.2 -4.022811871-

81235306487411816.6 81235289.98

0 -147760451.4 -0.087162294 73880208-

443281245.5-73880196.74

0 3410527.559 0.007873073 -1705264 10231581.38 1705263.517

0 16.99999999 -2.96E-17 -2.77E-08 1.66E-07 2.77E-08

0 86785601.44 2.18378179-

43392792260356746.7 43392788.04

0 31184842.39 0.196004759-

1559241893554511.28 15592419.02

6 15592419.02 5 43392788.04 4 2.76977e-08 3 1705263.517 2 -73880196.74 1 81235289.98 6 6 0 31184842.39 .1960047586 -15592418.13 93554511.28 5 0 86785601.44 2.18378179 -43392791.54 260356746.7 4 0 16.99999999 -2.95716e-17 -2.76977e-08 1.66186e-07 3 0 3410527.559 .0078730725 -1705263.586 10231581.38 2 0 -147760451.4 -.0871622935 73880208.39 -443281245.5 1 1 162470623.2 -4.022811871 -81235305.55 487411816.6 1 2 3 4 5: b

: b=luinv(X)

: st_view(X=.,.,(" a1"," a2"," a3"," a4"," a5","a6")) mata (type end to exit) . mata

Tasks to be covered• Computational Accuracy– Example: Obtaining Inverse Matrix• Inverse matrix D by Stata/Mata “luinv (D)”

Tasks to be covered

• Computational Accuracy– Example: Obtaining Inverse Matrix• D*D-1 in Stata/Mata(default tolerance)

1 5.96E-08 2.36E-08 -3.73E-08 5.96E-08 -7.45E-08

0 1.000000003 -1.74E-18 -1.63E-09 9.78E-09 1.63E-09

0 4.66E-10 1 -1.63E-09 -2.98E-08 -3.96E-09

0 -1.49E-08 1.81E-09 1 0 -7.45E-09

0 -2.79E-09 2.95E-10 4.66E-100.99999998

9-1.40E-09

0 4.66E-09 3.84E-11 -1.28E-09 7.45E-09 1.000000001

Should it be Identity Matrix?

Tasks to be covered

• Computational Accuracy– Example: Obtaining Inverse Matrix• D*D-1 in Excel

Where the computational inaccuracy comes from?

1 5.96046E-08-7.77156E-

167.45058E-09-5.96046E-08

-1.49012E-08

00.999999999 2.72414E-17 0 7.31257E-09 0

0 4.19095E-09 1 6.98492E-10 1.49012E-08 7.21775E-09

0 1.49012E-08 00.999999996 0 0

0 9.31323E-10-3.46945E-

17-4.65661E-10 0.999999996

-9.31323E-10

0-4.88944E-09 4.85723E-17 4.19095E-09-2.42144E-08 1

Tasks to be covered

• Computational Accuracy– One of the possible reasons: Decimal and Binary numbers

17(decimal number)

• 17 / 2 = 1

• 8 / 2 = 0

• 4 / 2 = 0

• 2 / 2 = 0

• 1 / 2 = 1

= 10001(binary number)

0.7(decimal) 0.101100110011(binary)

0.75(decimal) 0.11(binary)

0.05(decimal) 0.000011001100(binary)

0.10(decimal) 0.000110011001(binary)

0.6(decimal) 0.100110011001(binary)

How computer saves a=0.75, b=0.7+0.05, c=0.6+0.1+0.05?

Tasks to be covered

• Accuracy– Tolerance• to set upper or lower limit on the number of iterations.

• to stop an unattended run if the algorithm falls into a cycle

– Preprocessing: Scaling• to improve the numerical gap and get a safe solution.

Ex) Rank(D)

Part II. Malmquist Index Analysis with the Panel Data

Basic Concept of Malmquist Index

The User Written Command “malmq”


• Malmquist Productivity Index(MPI) measures

the productivity changes along with time

variations and can be decomposed into changes

in efficiency and technology.

The input oriented MPI can be expressed in terms of input oriented CRS efficiency as Equation 1 and 2 using the observations at time t and t+1.


Basic Concept of Malmquist IndexThe input oriented geometric mean of MPI can be decomposed using the concept of input oriented technical change and input oriented efficiency change as given in equation 4.

malmq ivars = ovars [if] [in] [, ort(in | out) period(varname) trace saving(filename)]

– ort(in | out) specifies the orientation. The default is ort(in), meaning input-oriented DEA.

– period(varname) identifies the time variable.– trace specifies to save all the sequences displayed in the Results

window in the malmq.log file. The default is to save the final results in the malmq.log file.

– saving(filename) specifies that the results be saved in filename.dta.

The User written command “malmq”

• Program Syntax


• Example

– Data

– Result


• Example

Notes

• The data and code related to the presentation will

be available from the Conference website.

References

• Cooper, W. W., Seiford, L. M., & Tone, A. (2006). Introduction

to Data Envelopment Analysis and Its Uses, Springer

Science+Business Media.

• Ji, Y., & Lee, C. (2010). “Data Envelopment Analysis”, The

Stata Journal, 10(no.2), pp.267-280.

• Lee, C., & Ji, Y. (2009). “Data Envelopment Analysis in Stata”,

DC09 Stata Conference.

• Maros, Istvan. (2003). Computational techniques of the simplex

method, Kluwer Academic Publishers.

Choonjoolee(DEA)10Boston Conf

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