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