dea (data envelopment analysis) toshiyuki sueyoshi new mexico tech dept. of management
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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
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