Data envelopment analysis

Hakim SabzevariUniversity
Data Envelopment Analysis
Mohmmad Mahdi Sahebi
mahdisahebi@hotmail.com
February 27, 2018

Table of Contents I
Introduction
Assessment
Production Function
Curve Fitting
Cobb Douglas
Efficiency
Absolute Efficiency
Relative Efficiency
Introduction
What is DMU?
Numerical Example
Using Graphical Analysis
DEA CCR
M. Sahebi ·DEA ·February 27, 2018 2 / 49

Table of Contents II
DEA Variable Returns to Scale

Introduction Assessment
Assessment
Introduction
How do you manage proﬁtability of a network of hundreds or
thousands of bank branches disbursed over several states and
countries?

Assessment
Introduction
How can a managed care organization manage the quality and costs
of the thousands of physicians providing health services to millions
of plan members?

Assessment
Introduction
What methods would enable a government to ensure that the
multiple ofﬁces serving citizens across a country are operating at low
cost while meeting the required service quality?

Assessment
Introduction
Schools: S1, S2, S3, . . .
University: U1, U2, U3, . . .
University Department D1, D2, D3, . . .
Airports: A1, A2, A3, . . .
Hospitals: H1, H2, H3, . . .
Banks : B1, B2, B3, . . .
Bank Branches: BB1, BB2, BB3, . . .

Assessment
Which school is efﬁcient?
School %Pass %Pass
Final Exam Entrance Exam
A 100% 90%
B 40% 30%
Table: Percentage of passing exams

Assessment
Which airport is efﬁcient?
Avg Service/Day
Airplan1 60
Airplan2 60
Airplan3 20
Airplan4 10
Airplan5 0.5
Table: Flight numbers

The decision making based on the output parameters is not correct.

Introduction Production Function
Production Function
The basic relationship between the factors of production and the
output is referred to as a Production Function.
Q = f (V1, V2, . . . , Vm, U1, U2, . . . , Un)
Where:
Q = output
V1, . . . , Vm = Controllable inputs
U1, . . . , Un = Uncontrollable inputs

Production Function
Why we need production functions?
Choose best Inputs
Maximize Outputs
Benchmarking Decision Making Units
...

Production Function
Curve Fitting
Min L1
Min
n∑
i=1
|yi − αxi − β|
Min L2 (LSS1)
Min
n∑
i=1
(yi − αxi − β)2
Min L∞
Min{ max
i=1,...,n
(|yi − αxi − β|)}
1
Least Square EstimationM. Sahebi ·DEA ·February 27, 2018 11 / 49

Production Function
Cobb Douglas
Q = ALβ
Kα
QGeneral = x0Ax1
1 Ax2
2 . . . Axn
n
Where:
Q is total production2
L is Labor input 3
K is Captial input 4
A is total factor productivity
2
The real value of all goods produced is a year
3
The total number of person-hours worked in a year
4
The real value of all machinary, equipment, and building

Figure: Image source:
https://economicpoint.com/production-function/cobb-douglas

Introduction Efficiency
Efficiency
Figure: Image source: https://www.tapinfluence.com

Efﬁciency
EfﬁciencyAbsolut =
Y
Y∗

Disadvantages/Advantages
Disadvantages
Global index not exist for any type of decision making units 5
In developing countries similar to Iran, calculating the Absolute
Efﬁciency is not beneﬁcial
Advantage
Benchmark with global indexes is good when we want to
introduce the future path
5
DMU

Efﬁciency
EfﬁciencyRelativek =
Yk
Xk
max{
Yj
Xj
, j = 1, 2, . . . , n}

Frontier Analysis
Stochastic Frontier Analysis
Data Envelopment Analysis6
6
DEA

Data Envelopment Analysis Introduction
Data envelopment analysis (DEA) is a data oriented, no-parametric
method to evaluate relative efﬁciency. based on pre-selected iputs
and outputs.7 DEA is an increasingly popular management tool.
DEA can handle multiple input and multiple output models.
It doesn’t require an assumption of a functional form relating
inputs to outputs.
Inputs and outputs can have very different units. For example,
X1 could be in units of lives saved and X2 could be in units of
dollars without requiring an a priori tradeoff between the two.
7
Journal ot the Operat, ioiis Itesearch, Society of Japan 2009, VoL 52,
No. 2,163-173

Data Envelopment Analysis What is DMU?
DMU
x1
x2
x3
xm
DMUj
y1
y2
y3
yn
DMU = Decision Making Unit(s)

Data Envelopment Analysis Numerical Example
Numerical Example
Personal Number
Branch Transactions of Staff
Tehran 125 18
Sabzevar 44 16
Tabriz 80 17
Mashhad 23 11
Table: Bank Branches
How can we compare these branches and measure their performance
using this data?

Ratio
Personal Number Personal transactions
Branch Transactions of Staff per staff member
Tehran 125 18 6.94
Sabzevar 44 16 2.75
Tabriz 80 17 4.71
Mashhad 23 11 2.09

0 1 2 3 4 5 6 7
Tehran 6.94
Sabzevar 2.75
Tabriz 4.71
Mashhad 2.09

Personal transactions
Branch per staff member Relative Efﬁciency
Tehran 6.94 100(6.94/6.94) = 100%
Sabzevar 2.75 100(2.75/6.94) = 40%
Tabriz 4.71 100(4.71/6.94) = 68%
Mashhad 2.09 100(2.09/6.94) = 30%

Extending the example
Personal Business Number of
Branch transactions transactions staff
Tehran 125 50 18
Sabzevar 44 20 16
Tabriz 80 55 17
Mashhad 23 12 11

Extending the example Ratio
Personal transaction Business transactions
Branch per staff per staff
Tehran 6.94 2.78
Sabzevar 2.75 1.25
Tabriz 4.71 3.24
Mashhad 2.09 1.09

Extending the example Ratio
Personal transaction Business transactions
Branch per staff per staff
Tehran 6.94 2.78
Sabzevar 2.75 1.25
Tabriz 4.71 3.24
Mashhad 2.09 1.09
A 1.23 2.92
B 4.43 2.23
C 3.32 2.81
D 3.70 2.68
E 3.34 2.96

Data Envelopment Analysis Using Graphical Analysis
Business transactions per staff
Tehran
Tabriz
Sabzevar
Mashhad
0 1 2 3 4 5
0
1
2
3
4
5
6
7

Data Envelopment Analysis DEA CCR
TECCR:
Min θ
Subject to:
ΣλiXi ≤ θX0 i = 1, . . . , m
ΣλiXi ≥ Y0 i = 1, . . . , m
λ ≥ 0

Data Envelopment Analysis DEA Variable Returns to Scale
TEVariableReturnstoScale:
Min θ
Subject to:
ΣλiXi ≤ θX0 i = 1, . . . , m
ΣλiXi ≥ Y0 i = 1, . . . , m
Σλi = 1
λ ≥ 0

Example 1
Personal Business Number of
Branch transactions transactions staff
Tehran 125 50 18
Sabzevar 44 20 16
Tabriz 80 55 17
Mashhad 23 12 11

Problem:
Min θTehran
Subject to:
18L1 + 16L2 + 17L3 + 11L4 ≤ 18θ
125L1 + 44L2 + 80L3 + 23L4 ≥ 125
50L1 + 20L2 + 55L3 + 12L4 ≥ 50
L1 , L2 , L3 , L4 ≥ 0

Problem:
Min θSabzevar
Subject to:
18L1 + 16L2 + 17L3 + 11L4 ≤ 16θ
125L1 + 44L2 + 80L3 + 23L4 ≥ 44
50L1 + 20L2 + 55L3 + 12L4 ≥ 20
L1 , L2 , L3 , L4 ≥ 0

Problem:
Min θTabriz
Subject to:
18L1 + 16L2 + 17L3 + 11L4 ≤ 17θ
125L1 + 44L2 + 80L3 + 23L4 ≥ 80
50L1 + 20L2 + 55L3 + 12L4 ≥ 55
L1 , L2 , L3 , L4 ≥ 0

Note that DMUs Tehran and Tabriz are overall efficient and DMUs
Sabzevar and Mahshad are ineficient with an efficiency rating of
xxxxxxx.
Efficient level of InputsSabzevar:
0.285217391
[
18
]
+ 0.104347826
[
17
]
=
[
6.907826087
]
Efficient levels of OutputsSabzevar:
0.285217391
[
125
50
]
+ 0.104347826
[
80
55
]
=
[
44
20
]

Tehran
Tabriz
Sabzevar
Mashhad
0 1 2 3 4 5
0
1
2
3
4
5
6
7
SabzevarEfﬁcient

Problem:
Min θMashhad
Subject to:
18L1 + 16L2 + 17L3 + 11L4 ≤ 11θ
125L1 + 44L2 + 80L3 + 23L4 ≥ 23
50L1 + 20L2 + 55L3 + 12L4 ≥ 12
L1 , L2 , L3 , L4 ≥ 0

Example 2
DMU Inputs Outputs
1 5 14 9 4 16
2 8 15 5 7 10
3 7 12 4 9 13
Table: 3 DMUs with 2 inputs and 3 outputs

LP Program for DMU1
Problem:
Min θ
Subject to:
5L1 + 8L2 + 7L3 ≤ 5θ
14L1 + 15L2 + 12L3 ≤ 14θ
9L1 + 5L2 + 4L3 ≥ 9
4L1 + 7L2 + 9L3 ≥ 4
16L1 + 10L2 + 13L3 ≥ 16
L1, L2, L3 ≥ 0

LP Program for DMU2
Problem:
Min θ
Subject to:
5L1 + 8L2 + 7L3 ≤ 8θ
14L1 + 15L2 + 12L3 ≤ 15θ
9L1 + 5L2 + 4L3 ≥ 5
4L1 + 7L2 + 9L3 ≥ 7
16L1 + 10L2 + 13L3 ≥ 10
L1, L2, L3 ≥ 0

LP Program for DMU3
Problem:
Min θ
Subject to:
5L1 + 8L2 + 7L3 ≤ 7θ
14L1 + 15L2 + 12L3 ≤ 12θ
9L1 + 5L2 + 4L3 ≥ 4
4L1 + 7L2 + 9L3 ≥ 9
16L1 + 10L2 + 13L3 ≥ 13
L1, L2, L3 ≥ 0

Figure: DMU1 LP Solved

Note that DMUs 1 and 3 are overall efficient and DMU 2 is ineficient
with an efficiency rating of 0.733333.
Efficient levels of Inputs:
0.261538
[
5
14
]
+ 0.661538
[
7
12
]
=
[
5.935
11.6
]
Efficient levels of Outputs:
0.261538


9
4
16

 + 0.661538


4
9
13

 =


5
7
12.785



@articlemurillo2004economic, title=Economic efﬁciency and frontier
techniques, author=Murillo-Zamorano, Luis R, journal=Journal of
Economic surveys, volume=18, number=1, pages=33–77, year=2004,
publisher=Wiley Online Library

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Data envelopment analysis

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