生產(chǎn)率與效率分析lecture6-數(shù)據(jù)包絡(luò)分析1

Data

Envelopment

Analysis

(DEA)數(shù)據(jù)包絡(luò)分析Over

4000

published

articlesfrom

1978

to

2021DEAWhat

it

isConceptsFarrell

measures

of

EfficiencytechnicalallocativescaleRunning

DEADangers

of

DEA◆Data

Envelopment

Analysis

(DEA)This

method

is

a

special

but

importantapplication

of

linear

programming.DEA

is

widely

applied

to

evaluate

theperformance

of

Decision

Making

Units(DMUs)

such

as

private

firms,

publicagencies,

schools,

and

hospitals.Important

FeaturesEfficiency

(Productivity)

ManagementRelative

Measure

vs.

Absolute

Measure(Social

Science)

(Natural

Science)Multi-Dimensional

Analysisvs.

Cost

/

Benefit

Analysis(1)

Relative

Comparison(2)

Multiple

Inputs

and

Outputs(3)

Efficiency

Measurement

(0%-100%)

(4)

Avoid

the

Specification

Error

betweenInputs

and

Outputs(5)

Production/Cost

AnalysisImportant

FeaturesData

Envelopment

Analysis

(DEA)

A

mathematical

optimisation

techniquewhere

the

unknown

variablesaretheweightsWork

through

some

examples

to

explain

Start

with

a

simpleexample

of

8

similarorganisations

(decision

making

unitsDMUs)

producing

two

outputs

using

asingle

inputConcepts0

1

2

3

4

5

6

7

8

9Table

1.

Single

Input

and

Single

Output

Case654Sales3210EmployeeFigure

1.

Comparisons

of

Branch

StoresEfficiencyFrontierABCDFGEH0

1

2

3

4

5

6

7

8

9EfficiencyFrontierACB

DFGEHRegression

LineEmployeeFigure

2.

Regression

Line

vs.

Frontier

LineB:

the

best

storeThe

efficiency

of

the

other

stores

are

determin1

=

B

>

E

>

D>

C

>

H

>

A

=

G

>

F

=

0.4654Sales3210Table

2.

Efficiency0

12

3

4Employee43Sales210Figure

3.

Improvement

of

Store

AA1ABA2Table

3.

Two

Inputs

and

One

Output

CaseFigure

4.

Two

Inputs

and

One

Output

CaseProduction

Possibility

SetEfficient

FrontiersAEDCH

IFGB014553Area/Sales20

1

2

3

4Employee/SalesFigure

5.

Improvement

of

Store

AADEA1PTable

4.

One

Input

and

Two

Outputs

CaseProduction

Possibility

SetFigure

6.

One

Input

and

Two

Outputs

CaseDMUInputOut1Out2O1/IO2/I145450451012555281769.63.2375525525774601925763.29.6510010010001106552453304.5675516516533890450360549603603006510563641966.53.511611534152.56.812603301205.52誰最有效？W1=1

,

W2=0W1=1

,

W2=0W1=0

,

W2=1W1=0

,

W2=1W1=0.5

,

W2=0.5W1=0.5

,

W2=0.5W1=0.75

,

W2=0.25W1=0.75

,

W2=0.25W1=0.25

,

W2=0.75W1=0.25

,

W2=0.75樣本與實(shí)際Solid

lines

connectiDMUs

1,2,3,4,5Represents

anefficiency

frontierproduction

possibilicurve.Contrast

with“theoretical

fronti(broken

curve)

–

DEAefficiency

is

relatithe

sample

–

but

is

obest

estimateEfficiencies

for

all

DMUsmust

not

exceed

100%10.0

w1

+

1.0

w2

<=1

unit

19.6

w1

+

3.2

w2

<=

1

unit

27.0

w1

+

7.0

w2

<=

1

unit

33.2

w1

+

9.6

w2

<=

1

unit

41.0

w1

+

10.0

w2

<=

1

unit

54.5

w1

+

6.0

w2

<=

1

unit

63.2

w1

+

3.2

w2

<=

1

unit

75.0

w1

+

4.0

w2

<=

1

unit

86.0

w1

+

5.0

w2

<=1

unit

96.5

w1

+

3.5

w2

<=

1

unit

102.5

w1

+

6.8

w2

<=

1

unit

115.5

w1

+

1.5

w2

<=1

unit

12w1,

w2

>=

0生產(chǎn)可能集合的形成Max

10w1

+

1

w2St

10

w1

+

1

w2

=

1(only

active

constraintAnd

all

otherconstraints

inactiveA

solution:

w1

=0.1,

w2

=010w1

+

1w2

=

1.0(100%

efficient)DMU1DMU6Max

4.5

w1

+

6.0

w2

St

7.0

w1

+

7.0

w2

=

1

(activeconstraint

3)3.2

w1

+

9.6

w2

=

1

(activeconstraint

4)

And

all

other

constraintsinactiveUnique

solution:

w1=0.05803,

w2

=0.08484.5

w1

+

6.0

w2

=

0.77Peers

are

DMUs

3

and

4DMU6DEA

softwareFree

softwareEMS

–Holger

Scheel

DEA

---

Free

downloadDEAP

–

Tim

Coelli

–Free

downloadOther

softwareSaitech

–

Cooper,

Seiford

and

Tone

softwareFrontier

AnalysisWarwick

DEAEMQC

-

Fare

and

Grosskopf中文MaxDEALP

FormulationTable

5.

Hospital

CaseEfficiency

of

the

j

th

DMUEfficiency

of

the

j

th

DMUNow,

assuming

that

Hospital

(B)

is

evaluated

byDEA,

we

haveMaxs.t.Maxs.t.Efficiency

of

the

k

th

DMUMaxs.t.根本問題的一般表述DMU

DMU1

2DMUnx

11X21v1v2…………x12x22……x

1nX

2n……vmxm1xm2xmnu1u2……y11y12……y12y12……y1ny1n……usys1ys2ysn決策單元評(píng)估指標(biāo)……投入產(chǎn)出……………………………………CCR

modelR

:

A

Reference

SetPrimal

ProblemDual

ProblemSecond-stage:

SlackReference

Set:Table

1.6

:

2

inputs

–

1

output

CaseExample

ProblemPrimal

ProblemDual

ProblemDFE0Figure

1.7

:

Efficiency

of

DMU

AAA1CEfficiency

FrontierTable

1.7

:

Results

of

DEAReference

SetDMU

Efficiency0.167

0.167

0.8330.182

0.091

0.7270

0

00

0

00.200

0.2000.250

0.1250.5000100010001000A

0.833B

0.727C

1D

1E

1F

10

1

1

2.0

0

0Software:

EMS

ResultsBCC

modelVariable

Returns

to

ScaleCCR

modelBCC

modelBCC

model:

Dual

Problem0OutputInputFigure

2.1

:

Efficiency

Frontier

and

Production

Possibil(A)Efficiency

Frontier

of

CCR

modelEfficiency

Frontier

of

BCC

modeb(C)a(B)cdeCCR有效與BCC有效OutputInputFrontier

of

CCR

modelbaFrontier

of

BCC

modelcdPABRCost

Analysis0Efficiency

Frontier

of

CCR

modelEfficiency

Frontier

of

BCC

modeEdkgjihc

befPDual

ProblemAn

Economic

InterpretationLabourMin

combinationsrequired

(isoquant)Due

to

Michael

Farrell

(1957)Technical

efficiency

=

OB/OACapitalABOOther

thingsequal

=

outputAn

Economic

Interpretation(Output

maximisation

orientation)Output

1(e.g.

calls)Technical

efficiency

=

OB/OAOutput

2(e.g.

lines)DCOOther

thingsequal

=

inputsMax

combinations

achieveable(production

frontier)Economic

Interpretation

(3)

The

isoquant

and

production

frontier

are

not

knowndirectly,

but

might

be

estimated

from

known

data,

usingpiecewise

interpolationCapitalLabourMin

combinationsrequired

(isoquant)ABOEFGHJKLB

is

an

artificialobservation

-

acombination

ofF

and

GAllocative

efficiencyLabourMin

combinationsrequired

(isoquant)Depends

on

knowing

pricesAE

=

min

cost/actual

cost

=

OD/

OBCapitalABOCEfficientIsocostlineDInputPQR

AOutputMT.E.

=

PR/PAS.E.

=

PQ/PRT.&

S.E

=

PQ/PAOhis

is

input

orientation.

What

about

output

orientation?Running

DEAPurpose

-

built

softwareExcel/Solver

macrosOrganise

data

for

input

Identify

inputs,

outputs

andnoncontrollablesRunInterpretHow

reliable

is

DEA?

Depends

on

whether

frontier

can

bepopulated

by

efficient

firms:number