dea (data envelopment analysis)
DESCRIPTION
DEA (Data Envelopment Analysis). Toshiyuki Sueyoshi New Mexico Tech Dept. of Management. Data Envelopment Analysis. (1) Relative Comparison (2) Multiple Inputs and Outputs (3) Efficiency Measurement (0%-100%) (4) Avoid the Specification Error between Inputs and Outputs - PowerPoint PPT PresentationTRANSCRIPT
DEA (Data Envelopment Analysis)
Toshiyuki SueyoshiNew Mexico Tech
Dept. of Management
Data Envelopment Analysis
• (1) Relative Comparison
• (2) Multiple Inputs and Outputs
• (3) Efficiency Measurement (0%-100%)
• (4) Avoid the Specification Error between
• Inputs and Outputs
• (5) Production/Cost Analysis
Table 1.1 : 1 input – 1 output Case
Company A B C D E F G HEmployees 4 3 3 2 8 6 5 5Output 3 2 3 1 5 3 4 2Output/Employee 0.75 0.667 1 0.5 0.625 0.5 0.8 0.4
Case : 1 input – 1 output
0
Out
put
Employees
D
B
A
G
H
F
E
C
Efficiency Frontier
Figure 1.1:Comparison of efficiencies in 1 input–1 output case
0
Out
put
Employees
C
Efficiency Frontier
Figure 1.2 : Regression Line and Efficiency Frontier
Regression Line
D
B
A
G
H
F
E
Table 1.2 : Efficiency
Company A B C D E F G HEfficiency 0.75 0.667 1 0.5 0.625 0.5 0.8 0.4
1 of employeeper Sales
another of employeeper Sales0
C
1 = C > G > A> B > E > D = F > H = 0.4
0
Out
put
Employees
D
C
Efficiency Frontier
Figure 1.3 : Improvement of Company D
D2
D1
Table 1.3 : 2 inputs – 1 output Case
Company A B C D E F G H IEmployees 4 4 2 6 7 7 3 8 5Offices 3 2 4 2 3 4 4 1 3Sales 1 1 1 1 1 1 1 1 1
Case : 2 inputs – 1 output
0
Off
ices
/Sal
es
Employees/Sales
DB
A
G
H
F
E
C
Efficiency Frontier
Figure 1.4 : 2 inputs – 1 output Case
I
Production Possibility Set
0
Off
ices
/Sal
es
Employees/Sales
B
C
Figure 1.5 : Improvement of Company A
AA1
A2
C and B :A of set referenceR
OAOAA of efficiency The 2
Table 1.4 : 1 input – 2 outputs Case
Company A B C D E F GOffices 1 1 1 1 1 1 1Customers 1 2 4 4 5 6 7Sales 6 7 6 5 2 4 2
Case : 1 input – 2 outputs
0
Off
ices
/Sal
es
Customers/Offices
B
F
C
Figure 1.6 : 1 input – 2 outputs Case
G
A
A1
D
E1
E
Efficiency Frontier
Production Possibility Set
1
1
OEOEE of efficiency The
OAOAA of efficiency The
Table 1.5 : Example of Multiple inputs–Multiple outputs Case
Company A B C D E F G H I J K LEmployees 10 26 40 35 30 33 37 50 31 12 20 45Offices 8 10 15 28 21 10 12 22 15 10 12 26Customers 23 37 80 76 23 38 78 68 48 16 64 72Sales 21 32 68 60 20 41 65 77 33 36 23 35
Case : Multiple inputs – Multiple outputs
1.1,
21
22221
11211
21
22221
11211
snss
n
n
mnmm
n
n
yyy
yyy
yyy
Y
xxx
xxx
xxx
X
nsnm
sjj2j1
ij
mjj2j1
ij
j
y,,y,y
DMU th j theofinput th i theofamount The :y
x,,x,x
DMU th j theofinput th i theofamount The :x
n), 2, 1,j ( UnitMakingDecision th j The : DMU
0,,,,0,,,
2.1,,2,11subject to
Maximize
2121
11
11
2211
2211
sm
mjmj
sjsj
mkmkk
skskk
uuuvvv
njxvxv
yuyu
xvxvxv
yuyuyu
CCR model
s,,rru
m,,iiv
r
i
21output th the toassigned weight The :
21 input th the toassigned weight The :
0u,,u,u,0v,,v,v
xvxvyuyu
1xvxvsubject to
yuyuMaximize
s21m21
mjmj11sjsj11
mkmk11
sksk11
n,,1j,xvyu:jRm
1iij
*i
s
1rrj
*rk
*** θ,u,v :Solution OptimalAn
R : A Reference Set
0u and 0v
1xv
0yuxvsubject to
yuMaximize
ri
m
1iiki
s
1rrjr
m
1iiji
s
1rrkr
Primal Problem
edunrestrict: and 0
yy
0xxsubject to
Minimize
j
rkn
1jjrj
ikn
1jjij
Dual Problem
0d and 0d ,0
ydy
xdxsubject to
ddMaximize
yyd and xxd
yr
xij
rkyi
n
1jjrj
ik*x
i
n
1jjij
s
1r
yr
m
1i
xi
rkn
1jjrj
yr
n
1jjijik
xi
Slack
*yr
kRjrj
*jrk
*xi
kRjij
*jik
* dyy and dxx
*y
rik
*xiik
**xiik
*ikik
dy
dx1dxxx
*yrrkrkrkrk
*xiik
*ikikik
dyyyy
dxxxx
n,,1j,0jR *jk Reference Set:
Table 1.6 : 2 inputs – 1 output Case
1x2xy
DMU A B C D E FInput 4 4 4 3 2 6
2 3 1 2 4 1Output 1 1 1 1 1 1
Example Problem
D,CR,833.0u,167.0v,167.0v
0u,0v,0v
1v2v4
F0uvv6
E0uv4v2
D0uv2v3
C0uvv4
B0uv3v4
A0uv2v4subject to
uMaximize
A**
2*1
21
21
21
21
21
21
21
21
Primal Problem
D ofOutput 667.0C ofOutput 333.0A ofOutput
D ofInput 677.0C ofInput 333.0A ofInput 833.0
833.0,0,667.0,333.0,0
:,,,,0
1
024232
04623444subject to
Minimize
*******
FEDCBA
j
FEDCBA
FEDCBA
FEDCBA
edunrestrictFBAj
Dual Problem
0
D
F
E
Figure 1.7 : Efficiency of DMU A
A
A1
C
yx2
yx1
Efficiency Frontier
0,667.0,333.0
,0,0ddd,0,667.0
,333.0,0,0ddd,833.0
0d,0d,0d,F,,B,Aj0
1d
833.02d4232
833.04d623444subject to
dddMaximize
*F
*E
*D
*C
*B
*A
*y1
*x2
*x1
*F
*E
*D
*C
*B
*A
*y1
*x2
*x1
*
y1
x2
x1j
y1FEDCBA
x2FEDCBA
x1FEDCBA
y1
x2
x1
*1v
667.0,333.0 ** DC DC 091.0,909.0 ** ED ED 1* CC 1* DD 1* EE 1* CC
*2v
*u *1xd *
2xd *
1yd
Table 1.7 : Results of DEADMU Efficiency Refference Set
A 0.833 0.167 0.167 0.833 0 0 0B 0.727 0.182 0.091 0.727 0 0 0C 1 0.200 0.200 1 0 0 0D 1 0.250 0.125 1 0 0 0E 1 0.500 0 1 0 0 0F 1 0 1 1 2.000 0 0
BCC model
Variable Returns to Scale
edunrestrict: and 0
yy
0xxsubject to
Minimize
j
rkn
1jjrj
ikn
1jjij
CCR model
edunrestrict: and 0
1
yy
0xxsubject to
Minimize
j
n
1jj
rkn
1jjrj
ikn
1jjij
BCC model
edunrestrict: and 0u,0v
1xv
0yuxvsubject to
yuMaximize
ri
m
1iiki
s
1rrjr
m
1iiji
s
1rrkr
Dual ProblemBCC model:
0
Out
put
Input
b
a
c
Efficiency Frontier of CCR model
Figure 2.1 : Efficiency Frontier and Production Possibility Set
d
(A)
(C)
(B)Efficiency Frontier of BCC model
0x and 0
1
yy
0xxsubject to
xpMinimize
ij
n
1jj
rkn
1jjrj
in
1jjij
m
1iiik
m
1iikik
m
1i
*iik
k
*k
*
xp
xp
C
C
Cost Actual
Cost Minimized
Cost Analysis
0
Efficiency Frontier of CCR model
Efficiency Frontier of BCC modelyx2
yx1
g
j
E
ihbc
de
k
g
jih
b
f
P
2p
1p
'2p
'1p
l
'l
edunrestrict: and 0u ,0v
pv
0yuxvsubject to
yuMaximize
rj
iki
s
1rrjr
m
1iiji
s
1rrkr
Dual Problem