# DEA - Pàgines de la UAB

```TALLER DE EFICIENCIA
(noviembre de 2011)
Dr. Diego Prior
1
Plan de trabajo
Conceptos previos e introducci&oacute;n al
Data Envelopment Analysis (DEA)
Modelos DEA, extensiones y software
EMS
Aplicaciones con datos reales
2
Conceptos previos
Datos
Objetivos
organizaci&oacute;n
Dise&ntilde;o de la
evaluaci&oacute;n
Modelo
Conceptos previos
Datos
Objetivos organizaci&oacute;n
Modelo
outputs; precio outputs
inputs; precio inputs
Maximizaci&oacute;n de los
beneficios
Frontera de
beneficios
outputs; precio outputs
inputs
Maximizaci&oacute;n de los
ingresos
Frontera de
ingresos
outputs
inputs; precio inputs
Minimizaci&oacute;n de los costes
Frontera de
costes
outputs
inputs
Maximizaci&oacute;n de la
Frontera de
eficiencia
t&eacute;cnica
Introducci&oacute;n al Data Envelopment
Analysis (DEA)
1. &iquest;De qu&eacute; hablamos?
 Productividad: (Output/ Input) = (y/x)
 Eficiencia: max. (y/x)
 Si el input est&aacute; definido en unidades monetarias:
 Productividad: (y/x) = [1/(x/y)] =
 = (1/tasa de costes)
 PROBLEMA: &iquest;qu&eacute; hacer cuando tenemos m&uacute;ltiples
outputs e inputs?
Introducci&oacute;n al Data Envelopment
Analysis (DEA)
Introducci&oacute;n al Data Envelopment
Analysis (DEA)
Introducci&oacute;n al Data Envelopment
Analysis (DEA)
 DEA evaluates relative efficiencies of a homogenous
set of decision making units (DMUs) in the presence
of multiple input and output factors
 Efficiency is defined as the ratio of weighted sum of
outputs to weighted sum of inputs
 A DMU is considered efficient if it achieves a score of
1.00
 DEA identifies necessary improvements required in
making inefficient DMUs efficient
 DEA has extensively been applied in a variety of
business and decision making environments that
include banking, healthcare, transportation …
16
Some DEA Models and
Approaches
 CCR Model (Primal and Dual)
 BCC Model
 Super Efficiency Model
 DEA Models with Weight Restrictions
 Cross-Efficiency Models in DEA
 Benchmarking in DEA
 DEA Windows Analysis
 DEA with Ordinal and Cardinal Factors
17
CCR Ratio Model: Primal Form
s
v
k 1
m
max
u
j 1
k
j
y kp
s
k 1
x jp
s.t
k 1
m
s.t
u
j 1
j
u x
j 1
 1, i
x ji
v k , u j  0 k , j
k
m
s
 vk y ki
v
max
s
v
k 1
j
jp
y kp
1
m
k
y ki   u j x ji  0, i
j 1
v k , u j  0 k , j
where: p is the unit being evaluated; s represents the number of outputs; m represents the
number of inputs; yki is the amount of output k provided by unit i; xji is the amount of
input j used by unit i; vk and uj are the weights given to output k and input j, respectively.
18
CCR Ratio Model (Input-Oriented):
Dual Form
min 
 x
s.t
 x jp
i
ji
ki
 y kp
j
i
 y
i
k
i
i  0 i
where:  represents the efficiency score of unit p; s represent the dual variables that
identify the benchmarks for inefficient units.
19
CCR Ratio Model (OutputOriented): Dual Form
max 
 x
s.t
i
ji
 x jp
j
i
 y
i
ki
 y kp
k
i
i  0 i
20
CCR Model: Illustration
DMU
1
2
3
Input 1
5
8
7
Input 2
14
15
12
Output 1 Output 2 Output 3
9
4
16
5
7
10
4
9
13
Maximize 9v1  4v 2  16v 3
s.t.
5u1  14u 2  1
9v1  4v 2  16v3  5u1  14u 2  0
5v1  7v 2  10v3  8u1  15u 2  0
4v1  9v 2  13v3  7u1  12u 2  0
v1 , v 2 , v3 , u1 , u 2  0
21
Selection of Inputs, Outputs, and
units in DEA
 Inputs: resources (examples: workers,
machinery, operating expenses, budget, etc.)
 Outputs: actual number of products produced
to a host of performance and activity
measures (examples: quality levels,
 If there are m inputs and s outputs then
potentially ms DMUs can be efficient. Thus,
to achieve discrimination we need
substantially more units than ms
22
BCC Model
 CCR model considers constant returns
to scale (CRS) whereas the BCC model
considers variable returns to scale
(VRS)
min 
 x
s.t
 x jp
i
ji
ki
 y kp
i
 y
i
i

i
k
j
new constraint
(convexity)
1
i
i  0 i
23
Super Efficiency Model
Super efficiency model allows for
effective ranking of efficient DMUs
s
v
max
k 1
m
u
s.t
j 1
s
v
k 1
j
k
y kp
x jp  1
The DMU being evaluated
is removed from the constraint
set thereby allowing its efficiency
score to exceed a value of 1.00
m
k
y ki   u j x ji  0, i  p
j 1
v k , u j  0 k , j
24
DEA Model with Weight Restrictions
 Unrestricted weight flexibility in DEA can be
resolved through weight restrictions
 Weight restrictions also allow for the
incorporation of managerial input into DEA
models
25
Cross Efficiencies in DEA
 Cross efficiency in DEA allows for effective
discrimination between niche performers and
good overall performers
 Cross efficiency score of a DMU represents
how well the unit is performing with respect to
the optimal weights of another DMU
 A DMU that achieves high cross efficiency
scores is considered to be a good overall
performer
26
Cross Efficiency Matrix
DMU
1
2
3
N
1
Θ11
Θ21
Θ31
ΘN1
2
Θ12
Θ22
Θ32
ΘN2
3
Θ13
Θ23
Θ33
ΘN3
N
Θ1N
Θ2N
Θ3N
ΘNN
Efficiency score of DMU 2 when evaluated
with the optimal weights of DMU 1
27
Benchmarking in DEA
 We discussed traditional DEA benchmarking
in the illustrative example
 Benchmarks may not be inherently similar to
inefficient DMUs
 Virtual benchmarks do not exist in practice
 Benchmarking can also be performed based
on the cross efficiency matrix.
 Use of cluster analysis on cross efficiencies
28
Windows Analysis in DEA
Evaluating the performance of a DMU
over time by treating it as a different
entity in each period
A DMU is compared to itself over time
29
Questions
30
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