performance evaluation: network data envelopment analysis
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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. Contents 1. Efficiency 2. Data Envelopment Analysis 3. Mathematical Models 4. Network Models 5. Research Areas. 1. Efficiency. Definition Output point of view - PowerPoint PPT PresentationTRANSCRIPT
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