Performance Evaluation: Network Data Envelopment Analysis

高 強國立成功大學工業與資訊管理學系

於中山大學企業管理系

100 年 11 月 5 日

Contents1. Efficiency

2. Data Envelopment Analysis

3. Mathematical Models

4. Network Models

5. Research Areas

1. Efficiency

DefinitionOutput point of view(Actual output produced)/(Maximal output can be produced)

Input point of view(Minimal input required)/(Actual input used)

Technically Efficient ProductionT. Koopmans ： A feasible input/output vector where it is technolo

gically impossible to increase any output (and/or reduce any input) without simultaneously reducing another output (and/or increasing any other input).

=> Pareto optimality

Measurement

Parametric approachRegression analysis (Aigner-Chu)

Nonparametric approachData envelopment analysis (Charnes-Cooper-Rhodes)

I

A

A *

●

Output Eff. ＝ A/A*

( )平均產量 迴歸Ave. production

投入

產出

o

oo

oo

o

o

o

o

input

output

( )最大產量 資料包絡 Max. production

I*

Input Eff. ＝ I*/I

Parametric approach

Production function

X2

Y

X1

Two-input single-output

Y0

I1* I1

A

A *I2*

I2●

Input Eff. ＝ OA*/OA

Input efficiency

o

o

o

X1

X2

O

Dominated region

●

●

Isoquant (Y0)

Single-input two-output

Output Eff.=OA/OA*

O1*O1

A

A*O2

*

O2

●

● ●

●

Y1

Y2

O

Dominated region

o

oo

Product transformation curve (X0)

Example:

)lnln'(ln .min 12 n

i iii KcLbaY

' ,0 , acb

niYKcLba iii ,...,1 ,lnlnln' s.t.

Production function: cbKaLY

unrestricted in sign.

2. Data Envelopment Analysis

OA●

●

●

●

●

B

C

D

E

Y

X

E*

Productionfunction

Single-input single-output output side

DMU X Y

A 10 5

B 20 20

C 30 30

D 40 35

E 36 22

Eff.

1

1

1

1

2/3

Non-parametric approach

等產量線Isoquant

O

機器

A

工人

●

●

●

●

●B

C

D

E

X2

X1

E *

Input side

DMU X1 X2 Y

A 20 5 100

B 30 3 100

C 50 2 100

D 40 4 100

E 48 3 100

Eff.

1

1

1

4/5

5/6

E *

O

Y 2

A●

●

●

●

●

B

C

D

E

Y 1

產品轉換曲線Production transformation curve

Output side

DMU X Y1 Y2

A 100 20 50

B 100 40 40

C 100 50 20

D 100 30 30

E 100 40 12

Eff.

1

1

1

3/4

4/5

Emrouznejad et al. (2008) Socio-economic Planning Science 42, 151-157

3. Mathematical Models

Ratio form Input i and output r of DMU j: (Xij , Yrj)

m

i iki

s

r rkrk

Xv

YuE

1

1 .max

njXv

Yum

i iji

s

r rjr ..., ,1 ,1 s.t.

1

1

misrvu ir ,...,1 ; ..., ,1 , ,

DMU k chooses most favorable multipliers ur ,vi to calculate Ek

Linear transformation

.max1

s

r rkrk YuE

1 s.t.1

m

i iki Xv

misrvu ir ,...,1 ; ..., ,1 , ,

m

i iji

s

r rjr njXvYu11

,..., 1 ,0

Envelopment form(Dual of the ratio form)

θ .min

miXX iknj ijj ..., ,1 , s.t. 1

njj ..., ,1 ,0

srYY rknj rjj ..., ,1 ,1

● is the target on the frontier.） ,（1 1

**

n

j rj

n

j jijj YX

A●

A *

A 0

●

●

●

●

X

Y

變動規模報酬

固定規模報酬

v 0

v 0

v 0

Constant RTS

Variable RTS

Variable returns-to-scale

Technical Eff. ＝ A/A ＊ ,Scale Eff. ＝ A ＊ /A0,Aggregate Eff.=A/A0＝ (A/A ＊ )×(A*/A0)

m

i iki

s

r rkrk

Xvv

YuE

10

1 .max

njXvv

Yum

i iji

s

r rjr ..., ,1 ,1 s.t.

10

1

; ..., ,1 , , srvu ir

signin edunrestrict0 v

mi ,...,1

4. Network Models

X1k

X2k

Xmk

.

.

.

Y1k

Y2k

Ysk

.

.

.

DMU k

Conventional black box concept

CCR Ratio model

m

iiji

s

rrjr njXvYu

11

,...,1 ,0

m

iiki Xv

1

1 s.t.

s

rrkr

CCRk YuE

1

max.

misrvu ir ,...,1 ,,...,1 ,,

Envelopment model

n

jikiijj miXsX

1

,...,1 , s.t.

srminjss rij ,...,1 ,,...,1 ,,...,1 ,0,,

m

i

s

rri

CCRk ssθE

1 1)( min.

n

jrkrrjj srYsY

1

,...,1 ,

θ unrestricted in sign

Two-stage series system

Zpj: Intermediate product p of DMU j

Process 1

X1k

X2k

Xmk

...

DMU k

Process 2

Y1k

Y2k

Ysk

...

Z1k

Z2k

Zqk

...

System

s

rrkrk YuE

1 .max

m

iiki Xv

1

1 s.t.

m

iiji

s

rrjr njXvYu

11

,...,1,0

m

iiji

q

ppjp njXvZw

11

,...,1,0

q

ppjp

s

rrjr njZwYu

11

,...,1,0

qpmisrwvu pir ,...,1;,...,1;,...,1,,,

m

iiki

s

rrkrk XvYuE

1

*

1

* /

m

iiki

q

ppkpk XvZwE

1

*

1

*)1( /

q

ppkp

s

rrkrk ZwYuE

1

*

1

*)2( /

)2()1(kkk EEE

Ratio model

)( .min 11 1

s

r rmi

qp pik sssE

n

j ikiijj miXsX1 ,...,1 , s.t.

rpijsss rpijj ,,, ,0,,,,

n

j rkrrjj srYsY1 ,...,1 ,

Envelopment model

nj p

nj pjjpjj qpsZZ1 1 ,...,1 ,0

… hlZp

(l)

p=1,…,q

… tZp

(t)

p=1,…,q

Xi

i=1,…,m

Yr

r=1,…,s

General case

System efficiency is the product ofthe h process efficiencies.

More general case

s

rrkrk YuE

1 .max

1 s.t.1

m

iiki Xv

njXvYum

iiji

s

rrjr ,...,1 ,0

11

,0)()()1()()()(

)1()()()(

pppp MlIiMlOr

pljl

piji

pljl

prjr ZwXvZwYu

njqp ,...,1 ;,...,1

tlmisrwvu lir ,...,1 ;,...,1 ;,...,1,,,

Ratio model

.min

q

p

n

j ikp

ijp

j miXX1 1

)()( ,...,1 , s.t.

q

p

n

j rkp

rjp

j srYY1 1

)()( ,...,1 ,

qpnjpj ,...,1 ;,...,1 ,0)(

)0()0(

1

)1( ,0 MlZ lj

n

j j

)()(

1

)( ,0 qqlj

n

j

qj MlZ

qpMlZZ pplj

n

j

pj

plj

n

j

pj ,...,1, ,0 )()(

1

)1()(

1

)(

signin edunrestrict θ

Envelopment model

Parallel system

s

rrkrk YuE

1

.max

m

iiki Xv

1

1 s.t.

m

iiji

s

rrjr njXvYu

11

,...,1 ,0

)()(

,...,1 ,,...,1 ,0)()(

pp Ii

piji

Or

prjr njqpXvYu

misrvu ir ,...,1 ,,...,1 ,,

Ratio model

m

liki

s

rrkrk XvYuE

1

*

1

* /

q

p

pkk

Ii

piki

Or

prkr

pk ssqpXvYuE

pp 1

)()(*)(*)( ,,...,1 ,/)()(

1 , Define1

)(

1*

)(*)( )(

q

p

pm

i iki

Iip

ikip wXv

Xvw

p

q

p Iip

iki

Orp

rkr

m

i iki

Iip

iki

p

pp

Xv

Yu

Xv

Xv

1)(*

)(*

1*

)(*

)(

)()(

q

p

pk

p Ew1

)()(

q

pm

i iki

Orp

rkr

Xv

Yup

11

*

)(*)(

m

i iki

s

r rkr

Xv

Yu

1*

1*

A network system

Y1, Y2, Y3

1

2

X1, X2

3

,X )1(1

)1(2X

)3(2

)3(1 X ,X

,X )2(1

)2(2X

)I(2Y

)O(2Y

)O(1Y

)I(1Y

Model

kkk YuYuYu 33)O(

22)O(

11 .max

1 s.t. 2211 kk XvXv

njXvXvYuYuYu jjjjj ,...,1 ,0)()( 221133)O(

22)O(

11

njXvXvYu jjj ,...,1 ,0)( )1(22

)1(1111

njXvXvYu jjj ,...,1 ,0)( )2(22

)2(1122

njYuYuXvXvYu jjjjj ,...,1 ,0)( )I(22

)I(11

)3(22

)3(1133

21321 ,,,, vvuuu

Efficiencies

)/( )I(2

*2

)I(1

*1

)3(2

*2

)3(1

*13

*3

)3(kkkkkk YuYuXvXvYuE

)*3()*2()*1(*kkkk ssss

*1 kk sE

)/( )1(2

*2

)1(1

*11

*1

)1(kkkk XvXvYuE )/(1 )1(

2*2

)1(1

*1

)*1(kkk XvXvs

)/(1)/( )2(2

*2

)2(1

*1

)*2()2(2

*2

)2(1

*12

*2

)2(kkkkkkk XvXvsXvXvYuE

)/(1 )I(2

*2

)I(1

*1

)3(2

*2

)3(1

*1

)*2(kkkkk YuYuXvXvs

5. Research Areas

Models

I: Increasing marginal product II: Decreasing marginal productIII: Negative marginal product- Congestion

I IIIII

MultipliersStrictly positive

0,0, irir vuvu

ε ： non-Archimedean number ， 10-5

Absolute range

, / 1orr

or LuuU

Relative range (Assurance region, cone ratio)

/ 1Iii

Ii LvvU

,

orr

or LuU

Iii

Ii LvU

Data type

Traditional data

Undesirable data

Ordinal data

Qualitative data

Interval data

Stochastic data

Fuzzy data

ApplicationsNovel application A new area

A new journal

Implications

Special data type

Derivation of multiplier restrictions

References Chiang Kao and Shiuh-Nan Hwang, 2008, Efficiency decomposition in two-stage data env

elopment analysis: An application to non-life insurance companies in Taiwan. European J. Operational Research 185, 418-429.

Chiang Kao, 2009, Efficiency decomposition in network data envelopment analysis: A relational model. European J. Operational Research 192, 949-962.

Chiang Kao, 2009, Efficiency measurement for parallel production systems. European J. Operational Research 196, 1107-1112.

Chiang Kao and Shiuh-Nan Hwang, 2010, Efficiency measurement for network systems: IT impact on firm performance. Decision Support Systems 48, 437-446.

Chiang Kao and Shiuh-Nan Hwang, 2011, Decomposition of technical and scale efficien-cies in two-stage production systems. European J. Operational Research 211, 515-519.

Chiang Kao, 2011, Efficiency decomposition for parallel production systems. J. Operational Research Society (accepted) (SCI) doi:10.1057/jors.2011.16.

Chiang Kao, 2008, A linear formulation of the two-level DEA model. Omega, Int. J. Management Science 36, 958-962.

Chiang Kao and Shiang-Tai Liu, 2004, Predicting bank performance with financial forecasts: A case of Taiwan commercial banks. J. Banking & Finance 28, 2353-2368.

Thank You