Series on Advanced Economic Issues Faculty of Economics, VŠB‐TU Ostrava
www.ekf.vsb.cz/saei [email protected]
Series on Advanced Economic Issues
Faculty of Economics, VŠB‐TU Ostrava
Mehdi Toloo
DATA ENVELOPMENT ANALYSIS
WITH SELECTED MODELS AND
APPLICATIONS
Ostrava, 2014
Mehdi Toloo
Department of Business Administration
Faculty of Economics
VŠB‐Technical University of Ostrava,
Sokolská 33
701 21 Ostrava, CZ
Reviews
Adel Hatami‐Marbini, Université catholique de Louvain, Belgium
Sahand Daneshvar, Eastern Mediterranean University, Turkey
Reza Farzipour Saen, Islamic Azad University, Iran
This publication has been elaborated in the framework of the project Support
research and development in the Moravian‐Silesian Region 2013 DT 1 – International
research teams (02613/2013/RRC). Financed from the budget of the Moravian‐
Silesian Region. It has been also supported by the Czech Science Foundation
(GACR project 14‐31593S) and through European Social Fund (OPVK project
CZ.1.07/2.3.00/20.0296).
The text should be cited as follows: Toloo, M., (2014). Data Envelopment Analysis
with Selected Models and Applications, SAEI, Vol. 30. Ostrava: VŠB‐TU Ostrava.
© VŠB‐TU Ostrava 2014
Printed in Grafico, s.r.o.
Cover design by MD communications, s.r.o.
ISBN 978‐80‐248‐3738‐3
Preface
If you cannot measure it, you cannot improve it.
Lord William Thomson Kelvin
In microeconomics a production function is a mathematical function that
transforms all combinations of inputs of an entity, firm or organization into the
output. Given the set of all technically feasible combinations of outputs and inputs,
only the combinations encompassing a maximum output for a specified set of
inputs would constitute the production function. Data Envelopment Analysis
(DEA), which has initially been originated by Charnes, Cooper and Rhodes in 1978,
is a well‐known non‐parametric mathematical method with the aim of estimating
the production function. In fact, DEA evaluates the relative performance of a set of
homogeneous decision making units with multiple inputs and multiple outputs.
This book covers some basic DEA models and disregards more complicated ones,
such as network DEA, and mainly stresses on the importance of weights in DEA
and some of their applications. As a result, this book mostly considers the
multiplier form of DEA models to extend some new approaches, however the
envelopment forms are introduced in some possible approaches. This book also
aims at dealing with some innovative uses of binary variables in extended DEA
model formulations. The auxiliary variables enable us formulate Mixed Integer
Programming (MIP) DEA models for addressing the problem of finding a single
efficient and ranking efficient DMUs. In some cases, the status of input(s) or
output(s) measure is unknown and binary variables are utilized to accommodate
these flexible measures. Furthermore, the binary variables approach tackles the
problem of selecting input or output measures.
The book also stresses the mathematical aspects of selected DEA models and their
extensions so as to illustrate their potential uses with applications to different
contexts, such as banking industry in the Czech Republic, financing decision
problem, technology selection problem, facility layout design problem, and
selecting the best tennis player. In addition, the majority of the extended models in
this book can be extended to some other DEA models, such as slacks‐based
measures, hybrids, non‐discretionary and fuzzy DEA which are applicable on
some other contexts.
VI Preface
This research‐based book contains six chapters as follows:
The first chapter (General Discussion) starts with a simple numerical example to
explain the concept of relative efficiency and to clarify the importance of input and
output weights in measuring the efficiency score. Then these basic concepts are
extended to some more complex cases. Efficient frontiers and projection points are
illustrated by means of some constructive and insightful graphs.
The second chapter (Basic DEA Models) presents both envelopment and multiplier
forms of the DEA models in the presence of multiple inputs and multiple outputs.
However, this book mainly focuses on multiplier form of DEA models. In addition,
this chapter illustrates the role of each axiom to construct the production possibility
set (PPS). It is also concerned with some DEA models to deal with pure input data
as long as with pure output data set. Apart from basic input‐ and output‐oriented
DEA models with different returns to scale, the chapter includes a model that
combines both orientations. Three different case studies involving banking
industry, technology selection, and asset financing are provided in this section.
In chapter 3 (GAMS Software), we briefly introduce General Algebraic Modeling
System (GAMS) software, a modeling system for linear, nonlinear and mixed
integer optimization problems for solving DEA models.
Chapter 4 (Weights in DEA) treats the weights in DEA and their importance along
with various weight restrictions and common set of weights (CSW) approaches.
The chapter includes Assurance Region (AR) and Assurance Region Global (ARG)
methods to restrict weight flexibility in DEA. Two DEA models with different
types of efficiency, i.e. minsum and minimax, with their integrated versions are
introduced in this chapter. The evaluation of facility layout design problem is
addressed as a numerical example.
Chapter 5 (Best Efficient Unit) considers CSW and binary variable approaches as
the main tool for developing models that have the capability to find the most
efficient DMU and also rank DMUs. We cover WEI/WEO data sets along with
multiple input and multiple output data set. Some epsilon‐free DEA models are
introduced to overcome the problem of finding a set of positive weights. The
problem of finding the most cost efficient under uncertain input prices are also
discussed. Two real data set involving professional tennis players and Turkish
automotive company are rendered to validate the approaches in this chapter.
Chapter 6 (Data Selection in DEA) closes the book by considering data selection
problem in DEA and presenting some modifications of the standard DEA models
to accommodate flexible and selective measures. To deal with these problems, two
multiplier and envelopment DEA models are developed where each model
contains two alternative approaches: individual and integrated models. Individual
approach classifies flexible measures and identifies selective measures for each
DMU, and aggregate approach accommodates these measures using integrated
Preface VII
DEA models. We present three case studies to examine and validate the
approaches in this chapter.
Evidently, my deepest gratitude and love go to my family, Laleh and Arad, for
supporting me in writing this book. Ronak Azizi saved me a lot of trouble by
tackling all formatting issues in Microsoft. Last, but certainly not least, I would like
to extend my thanks to my friend, Dr. Adel Hatami‐Marbini, for helping me with
editing the book and invaluable ideas and comments.
This publication has been elaborated in the framework of the project Support
research and development in the Moravian‐Silesian Region 2013 DT 1 – International
research teams (02613/2013/RRC). Financed from the budget of the Moravian‐
Silesian Region. It has been also supported by the Czech Science Foundation
(GACR project 14‐31593S) and through European Social Fund (OPVK project
CZ.1.07/2.3.00/20.0296).
Mehdi Toloo, Ph.D.
Contents
Preface ............................................................................................................ V
Contents ........................................................................................................... IX
List of Abbreviations .................................................................................. XIII
Glossary of Symbols .................................................................................... XV
Chapter 1 General Discusion .......................................................................... 1
1. 1 A simple case (single input and single output) ............................................ 1 1. 2 Two inputs and one output ............................................................................. 6 1. 3 One input and two outputs ........................................................................... 10 1. 4 Two inputs and two outputs ......................................................................... 12
Chapter 2 Basic DEA Models ........................................................................ 17
2.1 The CCR model ............................................................................................... 17 2.2 Numerical Example ........................................................................................ 22 2.3 The Output‐oriented CCR model ................................................................. 25 2.4 Production Possible Set ................................................................................. 27 2.5 Envelopment form of the CCR model ......................................................... 33 2.6 Application (Bank industry) ......................................................................... 36 2.7 The BCC model ............................................................................................... 39 2.8 The output‐oriented BCC model .................................................................. 41 2.9 Without explicit inputs/outputs DEA models ............................................ 42 2.9.1 WEI‐DEA models .................................................................................. 43 2.9.2 Industrial robot evaluation problem ................................................... 45 2.9.3 WEO‐DEA models ................................................................................ 46 2.9.4 Financing decision problem ................................................................. 49
2.10 The additive model ........................................................................................ 50
Chapter 3 GAMS Software ........................................................................... 53
3.1 The GAMS software ....................................................................................... 54 3.2 GAMS IDE ....................................................................................................... 54 3.3 Structure of GAMS codes .............................................................................. 55 3.4 Running a model ............................................................................................ 62 3.5 Navigating the listing file .............................................................................. 62
X Contents
3.6 Listing Window .............................................................................................. 63 3.7 Compilation ..................................................................................................... 64 3.8 Equation listing ............................................................................................... 67 3.9 Column listing ................................................................................................ 68 3.10 Model Statistic ................................................................................................. 69 3.11 Solution report ................................................................................................ 69 3.12 $Include option ............................................................................................... 71 3.13 The Put Writing Facility ................................................................................ 72 3.14 GAMS Data Exchange (GDX) ....................................................................... 77
Chapter 4 Weights in DEA ............................................................................ 81
4.1 Weight restrictions ......................................................................................... 85 4.2 Some other approaches .................................................................................. 88 4.3 Common set of weights ................................................................................. 92 4.3.1 Integrated minsum approach .............................................................. 92 4.3.2 Integrated minimax approach ............................................................. 97 4.3.3 Facility layout design problem ............................................................ 97
Chapter 5 Best Efficient Unit ...................................................................... 101
5.1 Multi‐input and multi‐output case ............................................................ 102 5.2 DEA‐WEI approach...................................................................................... 113 5.2.1 Penalty function approach ................................................................. 114 5.2.2 Minimax approach .............................................................................. 115 5.2.3 Professional tennis players ................................................................. 117 5.2.4 New minimax method ........................................................................ 119
5.3 DEA‐WEO approach .................................................................................... 121 5.4 Epsilon‐free approaches .............................................................................. 124 5.5 Most cost efficient DMU .............................................................................. 130 5.5.1 CE with input price uncertainly ........................................................ 130 5.5.2 Turkish automotive company ............................................................ 133
Chapter 6 Data Selection in DEA .............................................................. 135
6.1 Flexible measures ......................................................................................... 135 6.1.1 Multiplier approach ............................................................................ 136 6.1.2 University evaluation .......................................................................... 139 6.1.3 Envelopment approach ....................................................................... 143
6.2 Selective measures ........................................................................................ 146 6.2.1 The rule of thumb in DEA .................................................................. 146 6.2.2 Multiplier form of selecting model ................................................... 148 6.2.3 Envelopment form of selecting model .............................................. 151 6.2.4 Aggregate approach ............................................................................ 154 6.2.5 Banking industry applications ........................................................... 156
Contents XI
Chapter 7 Conclusion ................................................................................... 161
Appendix ........................................................................................................ 163
References ...................................................................................................... 183
List of Tables ................................................................................................. 191
List of Figures ................................................................................................ 193
Index ................................................................................................................ 195
Summary ........................................................................................................ 201
List of Abbreviations
AHP Analytic Hierarchy Process
AMT Advanced Manufacturing Technology
ATP Association of Tennis Professionals
BCC Banker, Charnes, Cooper’s model in 1984
CCR Charnes, Cooper, Rhodes’s model in 1978
CE Cost Efficiency
CRS Constant Returns to Scale
CSW Common Set of Weights
DEA Data Envelopment Analysis
DMU Decision Making Unit
FLD Facility Layout Design
FMS Flexible Manufacturing System
GAMS General Algebraic Modeling System, an optimization software
KKT Karush‐Kuhn‐Tucker
LP Linear Programming
MCDM Multiple Criteria Decision Making
MIP Mixed Integer Programming
MINLP Mixed Integer Non‐Linear Programming
VRS Variable Returns to Scale
WCM World Class Manufacturing
Glossary of symbols
, , lowercases in italic denote scalars or variables
, lowercases in boldface denote vectors
, uppercases in boldface denote matrices
, open interval
, closed interval
origin in , i.e. 0,… ,0 ∈
1,… ,1 ∈
jth unit vector; i.e. , presents 1,0,
∗, ∗ Asterisk denotes the optimal value of a vector/variable
This Book is dedicated to my dear and loving wife, Laleh, and my curious son, Arad, who helped me with patience and kindness.
Thank you
1
CHAPTER 1
General Discussion
Limitation of resources is an undeniable fact, which is dealt with by many
organizations such as business firms, banks, hospitals, universities, etc. Hence,
improving the performance of resource utilization for organizations is one of the
most important concerns of managers. As a result, a manager must evaluate
frequently the performance of the organization. One essential step to improve the
performance of organizations is to measure the efficiency. In general, the efficiency
score of an organization with multiple inputs and multiple outputs is defined as
the ratio of the weighted sum of outputs to the weighted sum of inputs. In this
definition, the efficiency score is closely related to the determination of weights
while different weights leads to different efficiency scores. Experts and managers
usually, based on their experiments, assign the values to these weights. In contrast
to this method, data envelopment analysis (DEA) finds the optimal weights with
the aim of maximizing the efficiency score for each organization. These weights are
obtained precisely from the data set (inputs and outputs) and may differ from one
organization to another. In the rest of this chapter, the concept of input and output
weights in evaluation performance is discussed.
1. 1 A simple case (single input and single output)
Suppose there are 12 hospitals that are labeled H1 to H12 at the head of each row in
Table 1–1. The second and third columns are the number of staff and the number
of patients (measured in 100 persons/month), respectively. Let us start our
discussion with comparing H1 and H5. Both hospitals have the same amount of
staff, whereas more patients are admitted to H5 compared to H1; hence H5 works
better than H1 and is relatively more efficient. In this case, we say that H5 dominates
H1 (or equivalently H1 is dominated by H5). In a similar manner, consider H8 and
H12: with the same quantity of admitted patients, the number of employed staff in
the former hospital is less than the latter one which means H8 dominates H12 and
hence is relatively more efficient.
2 Chapter 1
2014 M. Toloo
Table 1–1 Hospital case with single input and single output
Hospitals Staff Patients
H1 173 222 1.283 0.659
H2 280 240 0.857 0.440
H3 225 212 0.942 0.484
H4 246 297 1.207 0.620
H5 173 248 1.434 0.736
H6 323 316 0.978 0.502
H7 253 342 1.352 0.694
H8 212 362 1.708 0.877
H9 197 353 1.792 0.920
H10 190 370 1.947 1
H11 270 350 1.296 0.666
H12 250 362 1.448 0.744
To draw such comparison, we implicitly accept that fewer staff and more admitted
patients are desirable. In general, there are two mutually exclusive items for
measuring the efficiency score: input and output. The smaller input and larger
output amounts are preferable. According to this definition, in this numerical
example, the number of staff and the number of patients are input and output,
respectively.
In the aforementioned discussion, we have just relatively compared the
performance of two dominating and dominated hospitals and could not measure
their efficiency scores. In addition, such analysis fails to compare two arbitrary
hospitals. To measure the efficiency score of all hospitals, the following ratio can
be utilized:
Efficiency = Output
Input
This ratio can be interpreted as a patient per staff (output per input) and it can be
found that more output amounts and/or less input amounts lead to more efficiency
score.
In Table 1–1, the fourth column indicates the ratio of patients to staff. The
normalized efficiency score are presented in the last column, that is, the maximum
one, H10, becomes unity. The main difference between these measures will be
discussed subsequently. Based on this ratio definition for measuring efficiency, as
we expect, the efficiency score of H5, 0.736, is larger than H1, 0.659 and also H8 with
0.877 is more efficient than H12 with 0.744
General Discussion 3
Note that the efficiency measured in this approach is not absolute and hence if a
hospital is removed or added, then the efficiency can be changed. It is easy to show
that the efficiency scores will not change if an inefficient hospital is removed or a
hospital with 1.947 is added. On the other hand, with removing H10 or adding
a hospital with 1.947 , the efficiency scores will change.
To generalize the aforementioned method, suppose that there are n homogeneous
organizations for evaluating in which each organization utilizes a single input to
produce a single output. We call these similar organizations as decision making
units (DMUs1) and denote by DMU 1,… , . With these assumptions the
efficiency score of the under evaluation unit, DMU ∈ ,…, , can be measured as
max ∶ 1,… ,
(1.1)
where 0 and 0 are input and output of DMU , respectively. Clearly,
efficiency score, , is a positive number which is less than or equal to 1, i.e. ∀ ∈
0, 1 and hence DMU is efficient if and only if 1; otherwise it is inefficient. In
this method, since the performance of a DMU is relatively measured, can be
called relative efficiency score of DMU and there is always at least one efficient unit.
Consider the following set
, : maximum,…,
Let DMU , ∈ . If 1 then EF is called efficient frontier since it
involves all the efficient points. We will discuss more about the efficient frontier
shortly.
Note that the above formula (1.1) is a ratio of ratios form and hence is able to present
the two properties: units invariant and constant returns to scale (CRS). A measure is
called units invariant if it is independent of the units of measurements. In other
words, the efficiency scores remain unchanged if one multiplies all inputs by a
positive constant and all outputs by a positive constant :
max ∶ 1, … , max ∶ 1, … ,
If inputs and outputs of a DMU are multiplied by a positive constant and the
efficiency scores remain unchanged, the unit operates under CRS assumption.
1 In general, a DMU converts input(s) into output(s) and whose efficiency score is desirable.
In managerial applications, DMUs may include hospitals, banks, schools and et cetera. In
engineering, DMUs may take such forms as airplanes or their components such as jet engines.
4 Chapter 1
2014 M. Toloo
Figure 1–1 Data set and efficient frontier
If we return to the numerical example we can notice that H10 is efficient and relative
to this hospital, the worst hospital (H3) attains only 0.440 of efficiency score. In a
similar manner, the efficiency score of other hospitals can be interpreted. In
addition, if the number of patients were restated in units of persons/month (instead
of 100 persons/month), then the ratio of output to input, the fourth column in Table
1–1, would multiply by 100 whereas the ratio of ratios, the last column, remains
unchanged. Furthermore, multiplying the number of staff and the number of
patients of a DMU by a positive number does not change the obtained efficiency
score.
To provide some geometric insights, we plot number of staff and number of
patients on the horizontal and vertical axes, respectively. The line that connects
each observation to the origin shows the set of points with an identical slope (i.e.,
ratio of output to input). Considering the CRS assumption, the efficiency score of
all points on the line is the slope of the line, which is a constant measure.
The efficient frontier is the highest slope ( 1.947 ) and as a result this frontier
envelops all data2. The following figure depicts the data set and the efficient frontier.
Mathematically, DMU dominates DMU when and and if we also
have or , then DMU strictly dominatesDMU . A DMU is efficient if
other DMUs cannot strictly dominate it. To improve the efficiency of an inefficient
unit such as H1, we look for a point on the efficient frontier that strictly dominates
H1. Figure 1–1 depicts the set of all points on the efficient frontier that dominate
H1, i.e. the line segment . This line is called project points because one of them
2 The name Data Envelopment Analysis (DEA) stems from this property.
H1H2
H3
H4H5
H6H7
H8
H9
H10
H11
H12
0
50
100
150
200
250
300
350
400
450
0 50 100 150 200 250 300 350
Patients
Staff
Efficient Frontier
General Discussion 5
Figure 1–2 Projection Points
must be selected as a target for improving H1. Note that it is assumed that
increasing in input and/or decreasing in output are not preferable and hence it is
not allowed. Considering the line 1.947 as the efficient frontier, the efficient
point A with coordinates .
, 222 is achieved by decreasing the input of H1
(input orientation) and similarly the efficient point B with coordinates 173,173
1.947 is gained by increasing the output of H1 (output orientation).
As it can be extracted from Figure 1–2, the set of all points that dominates H1 is the
triangle ABH1 and subsequently the efficiency score of H1 is equal to the ratio of
114.02 to 173 or equivalently the ratio of 222 to 336.83:
114.02
173
222
336.830.659
The DM believes that obtaining more details about the data set of hospitals leads
to more accurate efficiency scores. In such case, we can consider three following
scenarios:
1. Decomposing the number of staff into the number of doctors and the
number of nurses, which implies two inputs and one output case.
2. Decomposing the number of patients into the number of inpatients and
the number of outpatients that gives us one input and two outputs case.
3. Decomposing the number of staff and the number of patients,
simultaneously, (scenario 1 plus scenario 2) which leads to two inputs and
two outputs case.
To gain basic insights, we discuss these scenarios in following sections:
H1H2
H3
H4H5
H6H7
H8
H9
H10
H11
H12
0
50
100
150
200
250
300
350
400
450
0 50 100 150 200 250 300 350
Patients
Staff
A
B
6 Chapter 1
2014 M. Toloo
1. 2 Two inputs and one output
Suppose that the number of staff is decomposed into the number of doctors and
the number of nurses for all hospitals. The columns in Table 1–2 labeled Doctors
and Nurses show this decomposition, for example in H1, the number of staff, 173,
is equal to the number of doctors plus the number of nurses, 25 148. We apply
the CRS property and normalize the input values to get 1 unit for output which is
shown in the columns labeled Normalized Inputs and Unitized Output in Table 1–2.
The efficiency scores are shown in the last column and the calculation method will
be discussed successively.
With this slight manipulation, we can plot the new data set on new inputs axes.
Since the new output is unitized to 1, we can ignore it and just consider unitized
axes, Doctors/Patients ( ) and Nurses/Patients ( ), in Figure 1–3. The figure
illustrates the intersection of efficient frontier in when Patients=1. The line
connecting H8, H9 and H10 indicates the non‐dominated space to be built the
efficient frontier. Notice that the vertical and horizontal dashed lines and the
efficient frontier envelop all data set. H8, H9 and H10 are the efficient hospitals,
which illustrate that more efficient hospitals might be obtained if we use more input
and output factors.
There are many alternative methods to project an inefficient hospital on the
efficient frontier. One method is to radially decrease the input values, as much as
possible, meaning that it identifies a set of points with an identical ratio of inputs3.
Having the same ratio of inputs for a projection point implies that all inputs can be
simultaneously reduced without altering the mix (=proportions) in which they are
utilized. Therefore, to improve an inefficient hospital, H5, we connect it to the origin
by a green solid line. This line shows the set of all points with the same mix of H5’s
inputs and hence an efficient point in this line indicates a suitable projection point
for H5, as specified by in Figure 1–4.
The efficiency score of H5 can be measured by comparing the current position, H5,
with the projection point , as below
,
,
√0.0713 0.465
√0.093 0.6050.769
where , and , are distance from origin to and , respectively.
The coordinates of P can be easily calculated from intersection of .
.,
associated with the segment line OH5, and 2.834 0.667 associated with
the segment line H9H10. Since the projection point is on the line connecting H9
and H10, the set containing these efficient hospitals is called reference set for H5. In
3 In this example, the improvement of an inefficient hospital cannot be implemented by
increasing the outputs because it is assumed that they are be fixed at 1.
General Discussion 7
Table 1–2 Two inputs and one output case
Hospitals Inputs Output
Normalized
Inputs
Unitized
Output
Efficiency
Doctors Nurses Patients
H1 25 148 222 0.113 0.667 1 0.677
H2 46 234 240 0.192 0.975 1 0.441
H3 32 193 212 0.151 0.910 1 0.499
H4 39 207 297 0.131 0.697 1 0.624
H5 23 150 248 0.093 0.605 1 0.769
H6 36 287 316 0.114 0.908 1 0.542
H7 50 203 342 0.146 0.594 1 0.724
H8 16 196 323 0.050 0.607 1 1
H9 21 176 353 0.059 0.499 1 1
H10 31 159 370 0.084 0.430 1 1
H11 45 225 350 0.129 0.643 1 0.668
H12 43 207 362 0.119 0.572 1 0.752
Figure 1–3 Two inputs and one output case
Doctors
Patients
Nurses
Patients
Patients
Patients
H1
H2H3
H4
H5
H6
H7H8
H9
H10
H11
H12
0,000
0,200
0,400
0,600
0,800
1,000
1,200
0,000 0,050 0,100 0,150 0,200 0,250
Nurses/Patients
Doctors/Patients
8 Chapter 1
2014 M. Toloo
Figure 1–4 Improvement of H5
The efficiency score of H5 is equal to 0.769, meaning that reducing the coordinates,
or equivalently both inputs of this hospital by 0.769 leads to a point on the efficient
frontier. In other words, 0.769(0.093, 0.605) is equal to the coordinates of , which
is located on the efficient frontier.
In Figure 1–4, it is impossible to figure out the output orientation improvement
because the number of patients is fixed at one. One approach to show the output
orientation is to unitize one input and plot the new normalized data. Toward this
end, we consider one input such as Nurse that are one for all hospitals as well as
normalizing Doctors and Patients. Although the efficiency scores will remain
unchanged the form of efficient frontier and subsequently the way of efficiency
measurement will be changed. Table 1–3 demonstrates the alternative normalized
data set.
Figure 1–5 shows the new normalized data set where the new horizontal and
vertical unitized axes are Doctors/Nurses ( ) and Patients/Nurses ( ), respectively.
The efficient frontier in this figure is the intersection of efficient frontier in when
Nurses=1. As a result, the dashed segment line OH8 and the vertical segment line
AH5 in Figure 1–5 correspond to vertical dashed line and the segment line OH5 in
Figure 1–3, respectively.
In this figure, the vertical segment line AH5 identifies the set of all points with the
same ratio of H5’s inputs, 1.653. As we expect, the projection point of H5 is located
on the line connecting H9 and H10. The optimal achievement of H5‘s output is as
follows:
2.1502
1.6531.3
H1
H2H3
H4
H5
H6
H7H8
H9
H10
H11
H12
0,000
0,200
0,400
0,600
0,800
1,000
1,200
0,000 0,050 0,100 0,150 0,200 0,250
Nurses/Patients
Doctors/Patients
P
General Discussion 9
Table 1–3 Data with unitized Nurses
Hospitals
Normalized
Input
Unitized
Input
Normalized
Output Efficiency
H1 0.169 1 1.500 0.677
H2 0.197 1 1.026 0.441
H3 0.166 1 1.098 0.499
H4 0.188 1 1.435 0.624
H5 0.153 1 1.653 0.769
H6 0.125 1 1.101 0.542
H7 0.246 1 1.685 0.724
H8 0.082 1 1.648 1
H9 0.119 1 2.006 1
H10 0.195 1 2.327 1
H11 0.200 1 1.556 0.668
H12 0.208 1 1.749 0.752
Figure 1–5 Alternative perspective of two inputs and one output case
where 0.153,2.1502 is obtained from the intersection of 4.248 1.299,
the segment line H9H10, and 0.153. Put differently, if H5 increases its output by
1.3, then it would be efficient. It should be noticed that the efficiency score of H5 is
equal to .
0.769 which practically shows the CRS property.
As a matter of fact, vertically moving up from H5 to the efficient frontier in Figure
1–4 (Doctors/Nurses, Patients/Nurses) space is equivalent to radially moving from
H5 to the efficient frontier in Figure 1–5 (Doctors/Patients, Nurses/Patients) space
Doctors
Nurses
Nurses
Nurses
Patients
Nurses
H1
H2H3
H4
H5
H6
H7H8
H9
H10
H11
H12
0,000
0,500
1,000
1,500
2,000
2,500
0,000 0,050 0,100 0,150 0,200 0,250 0,300
Patients/N
urses
Doctors/Nurses
P
10 Chapter 1
2014 M. Toloo
and these movements are two specific projections from a general movement in
(Doctors, Nurses, Patients) space.
1. 3 One input and two outputs
Now let us consider the second scenario and suppose that the number of patients
is decomposed into the number of inpatients and the number of outpatients. The
third and fourth columns of Table 1–4 show the decomposition, for example in H1,
the number of patients, 222, is equal to the number of inpatients plus the number
of outpatients, 33 189. The unitized input and normalized outputs are shown in
the 5th–7th columns of this table. The efficiency scores are reported in the last
column and the utilized geometrical method will be discussed successively.
Figure 1–6 illustrates one input and two outputs case geometrically where the
horizontal and vertical unitized axes are inpatients/staff and outpatients/staff,
respectively. There are three efficient hospitals, i.e. H9, H10 and H9, and the lines
that connect these hospitals are efficient frontier.
To improve an inefficient hospital we radially increase its outputs (remember that
inputs are fixed). Consider an inefficient hospital, H7, and the segment line that
connects origin to H7 and crosses the efficient frontier at A. This segment line, OA,
which shows the set of all points with the same ratio of H7’s outputs, is plotted in
Figure 1–7.
Table 1–4 One input and two outputs case
Hospitals Input Outputs
Unitized
input
Normalized
output
Efficiency
Staff Inpatients Outpatients
H1 173 33 189 1 0.191 1.092 0.687
H2 280 65 175 1 0.232 0.625 0.513
H3 225 70 142 1 0.311 0.631 0.633
H4 246 63 234 1 0.256 0.951 0.634
H5 173 87 161 1 0.503 0.931 1
H6 323 91 225 1 0.282 0.697 0.606
H7 253 84 258 1 0.332 1.020 0.766
H8 212 68 294 1 0.321 1.387 0.887
H9 197 33 320 1 0.168 1.624 1
H10 190 75 295 1 0.395 1.553 1
H11 270 73 277 1 0.270 1.026 0.675
H12 250 69 293 1 0.119 0.572 0.751
S
Staff
taff
Inpatients
Staff
Outpatients
Staff
General Discussion 11
Figure 1–6 One input and two outputs case
Figure 1–7 Improvement of H7
Similarly, the efficiency score of H7 can be measured using the formula below
,
,
√0.332 1.020
√0.433 1.331 0.766
The coordinates of P can be calculated from intersection of .
. and
5.751 3.828. The reference set for the inefficient hospital H7 involves H5 and
H10.
To implement the input orientation improvement under CRS, we normalize the
input value staff and the output value inpatients to get the unity value for the
output outpatients, which is shown in Figure 1–8.
H1
H2 H3
H4H5
H6
H7
H8
H9
H10
H11
H12
0,000
0,200
0,400
0,600
0,800
1,000
1,200
1,400
1,600
1,800
0,000 0,100 0,200 0,300 0,400 0,500 0,600
Outpatients/Staff
Inpatients/Staff
H1
H2 H3
H4H5
H6
H7
H8
H9
H10
H11
H12
0,000
0,200
0,400
0,600
0,800
1,000
1,200
1,400
1,600
1,800
0,000 0,100 0,200 0,300 0,400 0,500 0,600
Outpatients/Staff
Inpatients/Staff
P
12 Chapter 1
2014 M. Toloo
Figure 1–8 Alternative perspective of one input and two outputs case
Horizontal moving from H7 to the efficient frontier in Figure 1–8 is equivalent to
radially moving from this hospital to the efficient frontier in Figure 1–7. The similar
approach can be utilized to perform all necessary computations and obtain
instructive details.
1. 4 Two inputs and two outputs
Now, suppose that for each DMU there are two inputs, the number of doctors and
nurses, and two outputs, the number of inpatients and outpatients as
demonstrated in Table 1–5. One approach to measure the efficiency scores in this
case, might unitize one input or one output. We then plot the data set by taking
three normalized items as axes. However, it is not easy to analyze a three‐
dimensional graph and more importantly this approach cannot be extended if
more than four items exist or DMUs operate under variable returns to scale (VRS)
assumption. One practical method to tackle this issue is to assign a fixed weight to
inputs and outputs in order to obtain one input as the weighted sum of inputs, and
one output as the weighted sum of outputs. Therefore, the one‐input‐one‐output
method can be applied as discussed above. Nevertheless, the main issue in this
approach is how to select weights, because various weights might lead to different
efficiency scores. To illustrate the role of selected weights, we consider the
numerical example with 12 hospitals in terms of two decomposed inputs and two
decomposed outputs as exhibited in Table 1–5.
The imposition of the fixed weights is very critical since the efficiency score is
driven from these weights. Having the fixed weights, the ratio of weighted sum of
outputs to weighted sum of inputs for all hospitals can be calculated and then the
normalized ratio would then provide an index for evaluating efficiency scores. Let
and be the weights for Doctors and Nurses, respectively. The data set in Table
H1
H2
H3
H4
H5
H6
H7
H8
H9
H10 H11H12
0,000 0,200 0,400 0,600 0,800 1,000 1,200 1,400 1,600 1,800 2,000
Inpatients/O
utpatients
Staff/Outpatients
P
General Discussion 13
Table 1–5 Two inputs and two outputs case
Hospital Inputs Outputs
Doctors Nurses Inpatients Outpatients
H1 25 148 33 189
H2 46 234 65 175
H3 32 193 70 142
H4 39 207 63 234
H5 23 150 87 161
H6 36 287 91 225
H7 50 203 84 258
H8 16 196 68 294
H9 21 176 33 320
H10 31 159 75 295
H11 45 225 73 277
H12 43 207 69 293
1–4 will be obtained if we select , 1,1 . Similarly, selecting , 1,1
leads to the data set in Table 1–2 where and are the weights for inpatients
and outpatients, respectively, also the unit weights , , , 1,1,1,1 result
in the data in Table 1–1.
Table 1–6 reports the different efficiency scores when various fixed weights are
selected. The efficiency scores in the second column are obtained by fixing
, , , 5,1,1,4 which means that a doctor is weighted five times more
than a nurse and also an outpatient is weighted four times more than an inpatient.
The efficiency scores presented in the third and fourth column are the fixed weights
2.5,0.3,1.5,4.5 and 0.3,1,0.3,1 , respectively. As can be seen, different fixed
weights lead to different efficiency scores. A main question here is which weights
should be selected? Which one is better? Obviously, the best (optimal) weights
must be selected not the better one; otherwise, it is not clear how much the
efficiency scores are due to assigning a non‐optimal weights and how much
inefficiency is associated with the data. To have the optimal weights we must have
some criteria which lead to different types of optimization problems: A multi‐
objective mathematical programming approach can be developed if maximizing
the individual efficiency score of all units is desirable. An integrated minimax
mathematical programing method can be formulated if minimizing the deviation
from efficiency score of the worst inefficient DMU is preferred. Another criterion
might be maximizing the sum of efficiency score of all DMUs. The last column in
Table 1–6, labeled CSW, indicates the efficiency score via an aggregation minimax
mathematical approach.
14 Chapter 1
2014 M. Toloo
Table 1–6 Various weights
Hospitals 5,1,1,4 2.5,0.3,1.5,4.5 0.3,1,0.3,1 CSW
H1 0.619 0.584 0.678 0.662
H2 0.353 0.331 0.416 0.440
H3 0.387 0.374 0.426 0.486
H4 0.532 0.498 0.613 0.621
H5 0.590 0.578 0.632 0.741
H6 0.454 0.452 0.449 0.509
H7 0.527 0.480 0.689 0.689
H8 0.965 1 0.830 0.895
H9 1 0.981 0.959 0.932
H10 0.855 0.797 1 1
H11 0.562 0.522 0.664 0.665
H12 0.629 0.581 0.756 0.742
sum 7.473 7.180 8.113 8.382
The values at the bottom of each column show the sum of various efficiency scores
via different weights. The aggregation minimax method gives a set of efficiency
scores with higher value of the sum of efficiency than the other methods. Most
importantly, as will be discussed in the next section, an advantage of the
optimization approaches is that it is not a need to pre‐select the fixed weights for
inputs and outputs.
The weights obtained from all the aforementioned optimization methods are
common to all DMUs and hence are called common set of weights (CSW). Beside
these approaches, there are some others that allow the weights to vary from one
DMU to another. The maximum efficiency score for each DMU obtains with such
flexible weights. More details about the approaches and the proposed models are
explained in the succeeding chapters. However, to illustrate the benefit of flexible
weights, Table 1–7 exhibits optimal weights and the efficiency scores that are
calculated by the basic DEA model, i.e. CCR4.
In this table, the last column which is labeled CCR shows the efficiency score and ∗, ∗, ∗ and ∗ are the optimal weights for the number of doctors, nurses,
inpatients and outpatients, respectively. There are four efficient DMUs, i.e. H5, H8,
H9 and H10, in this method while in the previous approaches only one efficient
DMU was determined. In some cases, some optimal weights are zero which means
the corresponding item must be ignored to gain the maximum efficiency score. For
4 Originated by Charnes, Cooper and Rhodes in 1978.
General Discussion 15
Table 1–7 The CCR efficiency scores and optimal weights
Hospitals ∗ ∗ ∗ ∗ CCR
H1 1.80×10–03 6.45×10–03 0 3.67×10–03 0.693
H2 0 4.27×10–03 5.86×10–03 8.12×10–03 0.523
H3 0 5.18×10–03 7.11×10–03 9.85×10–03 0.638
H4 9.91×10–03 2.96×10–03 5.38×10–03 1.27×10–03 0.636
H5 1.47×10–02 4.41×10–03 7.99×10–03 1.89×10–03 1
H6 2.31×10–02 5.81×10–04 7.12×10–03 0 0.648
H7 0 4.93×10–03 6.76×10–03 9.36×10–04 0.809
H8 2.87×10–02 2.76×10–03 2.76×10–03 2.76×10–03 1
H9 1.20×10–02 4.25×10–03 2.82×10–03 2.83×10–03 1
H10 2.70×10–03 5.76×10–03 2.70×10–03 2.70×10–03 1
H11 0 4.44×10–03 6.10×10–03 8.45×10–04 0.679
H12 0 4.83×10–03 0 2.60×10–03 0.763
instance, the optimal weight for the first output of H1 is zero ( ∗ 0) and hence the
efficiency score for H1 (0.693) is calculated in the ignorance of the number of
inpatients item for all hospitals.
17
CHAPTER 2
Basic DEA Models
This chapter provides some basic DEA models. Charnes et al. (1978) formulated a
linear programming (LP) model, named CCR, to evaluate the efficiency of a set of
similar DMUs with multiple inputs and multiple outputs. Then, Banker et al. (1984)
proposed a model, named BCC, which is an extension version of the CCR model
from CRS to VRS. After that, the field of DEA has been rapidly developing with
new theoretical and practical topics.
2.1 The CCR model
In general, suppose there are DMUs (DMU 1, … , ) with semipositive5
inputs, , … , , and semipositive outputs, , … , . A DMU
has at least one positive input and one positive because it is not possible to be
assumed no input and no output for a DMU. In addition, all inputs and outputs
are assumed to be nonnegative. Inputs and outputs must be selected somehow to
reflect thoroughly the production process. Let and be the input data and output
data matrixes, respectively:
, , … ,
……
⋮ ⋮ ⋮…
∈
, , … ,
… …
⋮ ⋮ ⋮…
∈
We denote the unknown input and output weights by m‐vector , … , and
s‐vector , … , , respectively. These weights, also called multipliers or
shadow prices, are chosen in a manner that are used to maximize the efficiency score
of each DMU relative to other DMUs. DEA is based on some mathematical models
to find the optimal weights for each DMU. However, the weights can be differed
5 A vector ∈ is called semipositive if and where is the origin in .
18 Chapter 2
2014 M. Toloo
from one DMU to others since for each DMU one distinct optimization problem
must be solved. One of the main contributions of DEA models is that the weights
are driven from inputs and outputs instead of being fixed them in advance. The
following ratio measures the efficiency of an unit under evaluation, DMU ∈
1,… , , which must be maximized:
∑
∑
where and are called virtual input and virtual output, respectively.
In addition, if we assume this ratio of virtual output to virtual input is less than or
equal to one for all DMUs, then the efficiency score of DMU is related to the
performance of other DMUs. In this respect, Charnes et al. (1978) proposed the
following fractional programming to measure the efficiency score of DMU :
max
∑
∑s. t.∑
∑1 1, … ,
0 1, … ,0 1, … ,
(2.1)
It is implicitly assumed that weights, the importance of inputs and outputs, are not
negative and subsequently non‐negativity constraints are imposed in the model.
Note that the fractional programming problem (2.1) is nonconvex and nonlinear.
To deal with the non‐linearity, Charnes et al. (1978) utilized Charnes and Cooper
(1962)’s transformation6 approach to formulate the following equivalent LP which
is named as CCR:
max θ ∑s. t.∑ 1
∑ ∑ 0 1, … ,
0 1, … ,0 1, … ,
(2.2)
This model involves 1 constraints and decision variables (weights).
Similar to (or even simpler than) other LPs it can be solved by the simplex method.
More importantly, mathematical duality relations in this model provide
opportunities for extending results and simplifying proofs that are not available
from other approaches. Primal‐Dual relations lead to rich economic interpretations
related to DEA models. We will use these useful relations to prove that the CCR
model is always feasible and its optimal objective value is bounded, i.e. 0 ∗
1. The primal form of CCR model (2.2) is called the multiplier form because it
6 Letting ∑ and then and .
Basic DEA Models 19
contains the multipliers (weights) and, as will be demonstrated subsequently, the
dual of the multiplier form of CCR is called the envelopment form, because its feasible
region envelops all data.
We sometimes use the following vector form of the input‐oriented CCR model:
: max. .
1 0 1, … , (2.3)
Suppose that the CCR model (2.3) is solved and ∗, ∗, ∗ is the optimal solution.
The following definition partitions all DMUs into two sets: efficient and inefficient:
Definition 2–1 (CCR‐efficiency). DMU is CCR‐efficient if and only if
(i) ∗ 1
(ii) There exists at least one strictly positive optimal solution ∗, ∗ 7
As a result, DMU is CCR‐inefficient if either ∗ 1 or ∗ 1 and for every optimal
solutions there exists at least one zero weights. It must be noticed here that that
although ∗ 1 is a necessary condition for being efficient, it is not sufficient.
The multiplier form of the CCR model (LP ) finds the optimal weights (multipliers)
for DMU , under evaluation DMU, and hence to gain the optimal weights for all
DMUs, LPs must be solved, i.e. LP ,… , LP . These models have constraints in
common, 0 1, … , , and their feasible region differ only from the
first constraint, 1. The first constraint of model (2.2), which is called
normalization condition, ensures that the weights are relative.
It is worth pointing out that the ∗ and ∗ represent the importance of input and
output , respectively. If ∗ 0 when DMU is being evaluated, then ∗ 0 for
all 1,… , which means does not effect on the efficiency score of DMU . In
other words, the efficiency score is obtained meanwhile the ith input is removed
from related calculations. The similar discussion can be done when ∗ 0. This
makes a conceptual problem in economic concept of efficiency and must be avoid.
The optimal solution of model (2.2) for the efficient DMUs is often highly
degenerate and consequently, there are alternative optimal solutions for the
weights. As a result, if there is an optimal solution of the CCR model with ∗
1and at least one zero weight, then we cannot conclude that DMU is CCR‐efficient.
If we ignore the normalization constraint in model (2.2), then the maximum value
of objective function over the cone8 , | , , , as
7 e.g. ∗ and ∗ or equivalently ∀ , ∗ 0 and ∀ , ∗ 0. 8 A set is called cone when for all ∈ and all scalar 0 we have ∈ .
20 Chapter 2
2014 M. Toloo
the feasible region will be unbounded. However, the normalization constraint can
be fixed at any other positive number:
Lemma 2–1 The CCR model (2.3) is equivalent to the following model:
max. .
(2.4)
where is a positive parameter.
The proof is straightforward and left as an exercise to the reader. It is enough to
verify that ∗, ∗, ∗ is an optimal solution of the multiplier form of CCR model
if and only if ∗, ∗, ∗ is an optimal solution of model (2.4).
The following theorem proves that there always exist specific bounded weights:
Theorem 2–1 There exists a positive parameter , which makes the following model
equivalent to the CCR model (2.3):
max. .
(2.5)
where 1,… ,1 ∈ .
Proof: Suppose ∗, ∗ be an optimal solution of the CCR model (2.3). Let
max ∗,… , ∗ , ∗, … , ∗ . If 1, then the constraints and are
redundant and, according to Lemma 2–1, models (2.3) and (2.5) are equivalent.
Otherwise, let ∗∗
, ∀ , ∗∗
, ∀ , and . Clearly, ∗, ∗ is an optimal
solution of model (2.4) where ∗ 1, ∀ and ∗ 1, ∀ . Hence, ∗, ∗ is an
optimal solution of model (2.5). The reverse of the theorem is obvious. □
Although, theoretically, for determining a suitable value for the parameter in
model (2.5) some computational effort must be done, the following theorem proves
that it is not necessary to determine a value for .
Theorem 2–2 There exists a scale of data such that the CCR model (2.3) is
equivalent to model (2.6):
Basic DEA Models 21
max. .
1
(2.6)
Proof. Suppose ∗, ∗ be an optimal solution of the CCR model (2.3). Consider
two cases:
Case A ( 1, ∀ & 1, ∀ ): In this case from the constraint 1 and
1, ∀ we have 1, ∀ . Furthermore, the constraint 0 (or
equivalently 1 and 1, ∀ give 1, ∀ . Under these assumptions,
the constraints and are redundant in model (2.6), hence models (2.3)
and (2.6) are equivalent.
Case B (∃ , 1 or ∃ , 1): Consider the following two subcases:
Subcase B1 (∃ , 0 1 or ∃ , 0 1r): In the input oriented case, let
min 0, 0 1
,otherwise
for ( 1,… , ). Clearly, 1 for all and according to case A the proof is
complete. Similarly, the output oriented case, ∃ , 1, can be proved.
Subcase B2 (∃ , 0 or ∃ , 0): In this case, under ∗ 1, let ∗ . In
terms of this scaling, the associated multiplier must be less than 1. Similarly, the
output oriented case, ∃ , 0, can be proved. □
Let us return to the traditional the multiplier form of the CCR model (2.3). If ∗
1, then the constrain 0is tight (because 0). In this case,
if there exists a positive ∗, ∗ such that ∀ , 0, then DMU is called
extreme efficient. On the other hand, if ∗ 1, then we have ∗ ∗ and, so,
the constraint 0is not tight and there exist some other tight
constraints. Let be the index set of the efficient DMUs that forces the DMU to
be inefficient; mathematically ∗ ∗ . The reference set or peer group of
DMU is a subset of . The set spanned by all efficient DMUs in the reference
set is called the efficient frontier of DMU and, as will be discussed later, the set
spanned by means of all efficient DMUs establish the efficient frontier.
In DEA point of view, DMU dominates DMU if and only if , is a
semi‐positive vector. In this case, DMU uses fewer inputs than DMU (with at least
the same outputs) and/or produces more outputs than DMU (with at most the
same inputs), which shows DMU is more efficient than DMU . If a DMU is efficient,
then it is not dominated by means of other DMUs; otherwise there exists at least
22 Chapter 2
2014 M. Toloo
one point (not necessary a DMU) that dominates it. In the following theorem we
are looking for a point that dominates an inefficient unit.
Theorem 2–3 The optimal solution of the following model is ∗
∗,
∗
∗:
max. .
∗ 1∗ (2.7)
where ∗, ∗ is the optimal solution of the multiplier form of the CCR model.
Proof. It is easy to show that ∗
∗,
∗
∗ is a feasible solution of (2.7). To obtain a
contradiction, suppose that , be the optimal solution of model (2.7) with ∗
∗ . Under these assumptions, ∗ , ∗ is a feasible solution of the multiplier
form of the CCR model that its objective value is higher than the optimal solution, ∗ ∗ , which contradicts the optimality conditions. □
The following corollaries can be obtained from Theorem 2–3:
Corollary 2–1 The optimal objective value of model (2.7) is always equal to one.
It should be mentioned here that if ∗, ∗ , , then ∗
∗,
∗
∗ and
∗ , is located on the efficient frontier which results in the following
corollary:
Corollary 2–2 If ∗, ∗ , then ∗ , is an efficient point which
dominates inefficient DMU .
It follows easily that radially decreasing the inputs of an inefficient DMU by ∗
projects , into a point (not necessarily a DMU) on the efficient frontier, i.e. ∗ , , that dominates , when ∗, ∗ . If ∗ 0 , then there is
an excess in the ith input ( ) and similarly, the shortage in the rth output ( ) can
be recognized when ∗ 0 such that ∗ , is located on the efficient
frontier. We will subsequently show that how these variables, known as slacks, can
be measured.
2.2 Numerical Example
We consider a simple example of 6 DMUs with 2 inputs and 1 output, adapted
from Cooper et al. (2007b). Table 2–1 shows the data and the optimal solution of
multiplier form of the CCR model, which is obtained by DEA‐Solver software:
Basic DEA Models 23
Table 2–1 Data set and results
DMUs ∗ ∗ ∗ ∗
A 4 3 1 0.143 0.143 0.857 0.857
B 7 3 1 0.053 0.211 0.632 0.632
C 8 1 1 0 1 1 1
D 4 2 1 0.083 0.333 1 1
E 2 4 1 0.5 0 1 1
F 10 1 1 0 1 1 1
The multiple form of CCR model for DMU is:
max θs. t.4 3 1
4 3 0 7 3 08 0 4 2 02 4 0 10 0
, , 0
(2.8)
There are seven constraints in this model: the first constraint is the normalization
condition for DMU , 1, and the other six constraints are corresponding to all
DMUs, i.e. 0, … , 0. The last six constraints, which are
labeled by A–F, are common in the CCR models for all other DMUs.
As it is shown in Table 2–1, the optimal solution is ∗, ∗, ∗, ∗
0.143,0.143,0.857,0.857 and so this unit is CCR‐inefficient with CCR‐efficiency
score 0.857.
To obtain the multiple form of CCR model for DMU , we need only change the
normalization constraint9 of model (2.8) based on the data of DMU :
max θs. t.8 1
4 3 0 7 3 08 0 4 2 02 4 0 10 0
, , 0
(2.9)
In a similar manner, the CCR model for the other DMUs can be formulated and
solved.
According to the results reported in Table 2–1, there are four DMUs with ∗ 1
and among these units only the optimal weights vector of DMU is positive. Under
9 There is only one unitized output and hence the objective function for all DMUs is identical.
24 Chapter 2
2014 M. Toloo
Table 2–2 The desired weights
DMUs ∗ ∗ ∗ ∗
A 0.143 0.143 0.857 0.857
B 0.053 0.211 0.632 0.632
C 0.083 0.333 1 1
D 0.167 0.167 1 1
E 0.167 0.167 1 1
F 0 1 1 1
the CCR–efficiency definition, DMU is efficient and the status of DMU , DMU and
DMU is not clear. The main question here is whether or not there are the alternative
positive optimal weights for these DMUs? Obviously, if there are the alternative
solutions, then different software lead to different solutions and to obtain a specific
optimal solution an appropriate approach must be adopted. Table 2–2 reports a set
of (possibly) positive optimal weights for the given data set.
As can be seen, there are the positive optimal weights for DMU and DMU ,
whereas there is no positive optimal weights for DMU and consequently DMU
and DMU are efficient but DMU is inefficient.
This numerical example shows the importance of the positive weights in DEA
models. For more details see Cooper et al. (2007a, 2009). Charnes et al. (1978)
imposed a positive lower bound on the weights and improved the CCR model as
follows:
max θ ∑s. t.∑ 1
∑ ∑ 0 1, … ,
1, … ,1, … ,
(2.10)
where parameter 0 is named non‐Archimedean infinitesimal. We call this
model as CCR to show that it includes the non‐Archimedean epsilon .
Archimedean property of real numbers implies that for any real number 0
there exists another positive real number such that 010. It is assumed
that 0 is smaller than any positive real number such that for all positive
multipliers we have 0, which means that is infinitesimal (and hence is not
a real number) without the Archimedean property. Practically, we have to consider
a positive real number for the non‐Archimedean infinitesimal. The epsilon
10 Mathematically, ∀ 0, ∃ ∈ : . This essentially means that there are no
infinitesimally small elements in the real line, no matter how small 0 gets we will always
be able to find an even smaller positive real number 2.
Basic DEA Models 25
forestalls weights from being zero, although model (2.10) might be infeasible for
an unsuitable value of epsilon. As it has been mentioned in Charnes et al. (1993):
… if one uses a small number in place of the infinitesimal epsilon, one is caught between
Scylla and Charybdis, i.e. for decent convergence to an optimum, the numerical zero
tolerance should be as large as possible, whereas the numerical value approximating the
infinitesimal should be as small as possible!
A suitable value for the epsilon must be found. Clearly, all the obtained weights in
model (2.10) are positive which makes the CCR‐efficiency definition easier: DMU
is CCR‐efficient if and only if in the CCR model ∗ 1. However, selecting a
suitable value for epsilon plays an important role in this definition.
Arnold et al. (1996) provided more detailed treatment of these non‐Archimedean
infinitesimal. Mehrabian et al. (2000) proposed the following model to obtain a
suitable value for epsilon in the CCR model:
maxs. t.∑ 1 1, … ,
∑ ∑ 0 1, … ,
0 1, … ,0 1, … ,
(2.11)
It is proved that the CCR model is feasible for all ∈ 0, ∗ . The interval 0, ∗ is
called assurance interval and ∈ 0, ∗ is called an assurance value. Amin and Toloo
(2004) designed a polynomial‐time algorithm to obtain an assurance value. As will
be seen subsequently, using the primal and dual optimality conditions provide an
epsilon free approach, which is called a two‐phase approach, to determine the
CCR‐efficient DMUs without dealing with the epsilon parameter.
2.3 The Output‐oriented CCR model
The purpose of the multiplier form of CCR model (3.2) is to find the minimum
that reduces, as much as possible, the input vector radially to . Looking for
the maximum such that increases the output vector radially to can be
considered as an alternative approach. The former and later approaches are called
input‐oriented and output‐oriented CCR model. Similar to input‐oriented model,
the output‐oriented model can be formulated as follows:
min s. t.
1 (2.12)
The output‐oriented version of the multiplier form of the CCR model is always
feasible and its optimal objective value is bounded, i.e. ∗ ∈ 1,∞ . DMU is CCR‐
26 Chapter 2
2014 M. Toloo
efficient if and only if ∗ 1 and there exists at least one positive optimal solution ∗, ∗ . If either ∗ 1 or there is no positive optimal weights with ∗ 1, then
DMU is CCR‐inefficient. Let ∗, ∗, ∗ and ∗, ∗, ∗ be the optimal solution of
models (2.3) and (2.12), respectively. The following theorems clarify the relation
between the input‐oriented and output‐oriented multiplier forms of the CCR
model:
Theorem 2–4 The following properties are satisfied:
∗1∗, ∗
∗
∗, ∗
∗
∗
Proof: See Cooper et al. (2007b).
Similar to the proof of Theorem 2–3, the following theorem can be proven:
Theorem 2–5 The optimal solution of the following model is ∗
∗,
∗
∗:
min. .
∗ 1∗ (2.13)
Corollary 2–3 The optimal objective value of model (2.13) is always equal to one.
Having positive weights, , ∗ is located on the efficient frontier which render
the following corollary:
Corollary 2–4 If ∗, ∗ , then , ∗ is an efficient point which
dominates inefficient DMU .
Note that, similar to the input‐orientation model, radially increasing the outputs of
an inefficient DMU by ∗ projects , into a point on the efficient frontier. i.e.
, ∗ , that dominates , when ∗, ∗ . If ∗ 0 , then there is
a shortage in the rth output ( ) and similarly, the excess in the ith input ( ) can be
recognized when ∗ 0 such that , ∗ is located on the efficient
frontier.
Table 2–3 summarizes the optimal solution of model (2.13) with (possibly) positive
optimal weights for the given numerical example in Table 2–1.
Comparing the optimal solutions in Tables 2–2 and 2–3 can be instructive to
understand the relationships between the input‐oriented and output‐oriented
multiplier forms of CCR model. For instance, ∗∗ .
1.167 and ∗
∗
∗
.
.0.167.
Basic DEA Models 27
Table 2–3 The weights of model (2.12)
DMUs ∗ ∗ ∗ ∗
A 0.167 0.167 1.167 1.167
B 0.084 0.334 1.582 1.582
C 0.083 0.333 1 1
D 0.167 0.167 1 1
E 0.167 0.167 1 1
F 0 1 1 1
The following model is the output‐oriented multiplier form of the CCR model for
DMU :
min φ 4ν 3s. t.
14 3 0 7 3 08 0 4 2 02 4 0 10 0
, , 0
(2.14)
The optimal solution is ∗, ∗, ∗, ∗ 0,1,1,1 and so this unit is CCR‐
inefficient, although ∗ 1. As a result, 10,1, ∗ 1 10,1,1 is not located on
the efficient frontier and there is an excess 0 such that 10 , 1,1 , which
dominates 10,1,1 , lies on the efficient frontier.
2.4 Production Possible Set
In microeconomics, a production function is a function that transforms all
combinations of inputs of an entity, firm or organization into the output. Given
that a set of all technically feasible combinations of output and inputs, only the
combinations encompassing a maximum output for a specified set of inputs would
constitute the production function. Alternatively, a production function can be
defined as the specification of the minimum input requirements needed to produce
the output under a given technology. To estimate the production function we first
define production possibility set (PPS).
The semi‐positive vector , ∈ is called an activity and the set of feasible
activities is called PPS. In other words,
PPS , | , isafeasibleactivity
, | canbeproducedfrom
Various PPS can be achieved with different definitions for feasible activities. To
define a PPS based on CRS technology and obtain an accurate mathematical
definition, we consider the following axioms:
28 Chapter 2
2014 M. Toloo
Figure 2–1 Feasibility axiom
(A1) Feasibility. All observed activities, i.e. DMU , for 1,… ,
∀ , ∈ PPS .
(A2) Free disposability. All dominated activities of a feasible activity are feasible
, ∈ PPS ⇒ ∀ , ∀ , , ∈ PPS .
(A3) CRS. If an activity , is feasible, then ∀ 0, , is a feasible activity
, ∈ PPS ⟹ ∀ 0, , ∈ PPS .
(A4) Convexity. The convex combination of each two feasible activities is a feasible
activity, i.e. , , , ∈ ⇒ ∀ ∈ 0,1 , , 1 , ∈
Note that the change of these axioms leads to the modification of the definition of
feasible activity.
We first geometrically interpret the axioms and then mathematically define the
resulting PPS set. Consider three arbitrary DMUs with single input and single
output, e.g. DMU , DMU and DMU .
Under the feasibility axiom, all DMUs belong to PPS as depicted in Figure 2–1.
Free disposability regions for the observed DMUs are indicated in Figure 2–2. In
this figure, the south‐east region of each DMU indicates the free disposability of
both inputs and outputs.
The ray from origin through each feasible activity is feasible based on CRS axiom.
These rays for observed DMUs are portrayed in Figure 2–3. More specifically, their
rays must be considered for all other feasible activities of free disposability regions.
A
B
C
Input
Output
Basic DEA Models 29
Figure 2–2 Free disposability regions for 1 input and 1 output case
Figure 2–3 CRS axiom
Finally, the minimal set that satisfies the axioms (A1)–(A4) is the PPS as shown in
Figure 2–4.
Efficient frontier is a non‐dominated subset of the PPS. DMU is efficient if it lies on
the efficient frontier and otherwise is inefficient and its inefficiency score is
measured by comparing DMU with an efficient activity that dominates this unit.
Efficient frontier for single input and single output case is a ray that passing a DMU
with maximum slope as in Figure 2–4.
A
B
C
Input
Output
A
B
C
Input
Output
30 Chapter 2
2014 M. Toloo
Figure 2–4 PPS for single input and single output case
Figure 2–5 Free disposability region for 2 inputs and 1 output case
Assume that a fixed value for the output of DMUs A, B and C. Free disposability
regions for two inputs and a constant output case are shown in Figure 2–5 where
the free disposability of both inputs of each DMU is its north‐east region.
Note that let us to plot the dominated regions in space. Considering all axioms
under the CRS axiom, the corresponding PPS can be presented in Figure 2–6.
As it was mentioned before, all points in the efficient frontier must be non‐
dominated and as a result the vertical and horizontal dashed lines passing A and
C, respectively, are not belonging to the efficient frontier.
A
C
Input
Output
BPPS
A
B
C
Input1/Output
Input2/O
utput
Basic DEA Models 31
Figure 2–6 PPS for 2 inputs and 1 output case
Figure 2–7 PPS for 2 inputs and 1 output
Figure 2–7 depicts a typical PPS for two inputs and one output as a convex cone in
. It should be mentioned here the PPS in Figure 2–6 is the intersection of the PPS
in Figure 2–7 with plane 1. Note that the PPS shown in Figure 2–6 is equivalent
to the PPS in Figure 2–7 under the CRS assumption,
Analogously, Figures 2–8 and 2–9 represent the free disposability region and the
PPS, respectively, for a single constant input and two outputs case. The set of all
south‐west regions of DMUs consists the free disposability regions.
A
B
C
Input1/Output
Input2/O
utput
PPS
Efficient
Frontier
32 Chapter 2
2014 M. Toloo
Figure 2–8 Free disposability region for 1 input and 2 outputs case
Figure 2–9 PPS for 1 input and 2 outputs case
A typical PPS for one input and two outputs case under CRS assumption is
presented in Figure 2–10. In fact, Figure 2–9 shows the intersection of the PPS in
Figure 2–10 with plane 1.
To develop an envelopment form of the CCR model we first need to
mathematically define the PPS. To do so, we take care of the following theorem.
Theorem 2–6 The PPS which satisfies the axioms (A1)–(A4) can be mathematically
defined by
P , | , , .
A
B
C
Output1/Input
Output2/Input
A
B
C
Output1/Input
Output2/Input
PPS
Efficient
Frontier
Basic DEA Models 33
Figure 2–10 PPS for 1 input and 2 outputs
Proof: It suffices to show that the axioms (A1)–(A4) hold for P :
Let for 1,… , where is the jth unit vector11 shows ∀ DMU
, ∈ P which means that the feasibility axiom is hold.
If , ∈ P , then there is such that , and
subsequently if , , then , which implies
, ∈ P . This therefore shows that the free disposability axiom is
hold.
Suppose , ∈ P and 0 is given. There is such that
, . Let lead to , and hence , ∈
P shows that the CRS axiom holds.
Suppose , , , ∈ P and 0 1 is given. Hence, there are
and such that , and ,
. If we assume 1 , then 1
and 1 and consequently 1 ,
1 ∈ P shows that the convexity axiom holds. □
Note that the border of P envelops all data and, as will be seen shortly, the
developed model based on this set is called an envelopment form of the CCR model.
2.5 Envelopment form of the CCR model
To measure the technical or radial efficiency score of under evaluation unit,
DMU , , the input vector must be radially decreased to as small as
possible. Mathematically, the following optimization problem must be solved:
11 A vector which having zero components, except for a 1 in the jth position.
34 Chapter 2
2014 M. Toloo
min θs. t.θ , ∈
(2.15)
Considering Theorem 2–6, model (2.15) is equivalent to the following model, so‐
called an envelopment form of CCR model:
min θs. t.∑ 1, … ,
∑ 1, … ,
0 1, … ,
(2.16)
This model contains the following properties:
1. , 1, is a feasible solution for this model to be used to show the
feasibility of the envelopment form of CCR model all the time.
2. Given the feasibility of 1, we obtain ∗ 1 , since the objective value
of this feasible solution is 1 and model (2.16) is a minimization problem.
3. The objective value for every optimal solution is positive, i.e. 0;
otherwise from the first type of constraints we have which
implies and from the second type of constraints we obtain
which is impossible.
4. From properties 2 and 3 we have 0 ∗ 1.
As a result, the following model is equivalent to model (2.16):
min θs. t.∑ 1, … ,
∑ 1, … ,
0 1, … ,
0
(2.17)
Since models (2.16) and (2.17) are equivalent, their duals are equivalent as well.
Hence, the following lemma holds.
Lemma 2–2 The following model is equivalent to the multiplier form of the CCR
model:
max. .
1 (2.18)
Basic DEA Models 35
In the envelopment form of the CCR slack variables play an important role in CCR‐
efficiency definition. The slack variables associated with the input constraints
∈ presents the input excesses while the slack variables associated with
the output constraints ∈ presents the output shortfalls.
We now define the CCR‐efficiency score in the envelopment form of the CCR
model as follows:
Definition 2–2 (CCR‐efficiency). DMU is CCR‐efficient if and only if ∗ 1 and
for all optimal solutions there is no input excess and output shortfall ( ∗ , ∗
).
Note that an optimal solution ∗, ∗, ∗ 1, , is not sufficient for being
efficient. It is necessary to show that there is no positive excess and shortfall for any ∗ 1. To deal with this issue, the following two‐phase approach is developed:
Phase I. Solve model (2.16) to obtain ∗.
Phase II. Solve the following model:
max ∑ ∑s. t.∑ ∗ 1, … ,
∑ 1, … ,
0 1, … ,
0 1, … ,
0 1,… ,
(2.19)
The aim of the model is to find the maximum value for the summation of input
excesses and output shortfalls among the optimal solutions of model (2.16). In fact,
the feasible region of Phase II is the set of alternative solutions for Phase I.
Clearly, if ∗ 1 and ∗ 0, then for all optimal solutions of model (2.16) there is
no input excesses and output shortfalls and consequently the unit under evaluation
is CCR‐efficient.
Definition 2–3 Model (2.19) is called max‐slack and its optimal solution is known
as max‐slack solution. In addition, a max‐slack with ∗ , ∗ is called zero‐
slack.
Hence, the following CCR‐efficiency definition can be driven:
Definition 2–4 (CCR‐efficiency). DMU is CCR‐efficient if and only if ∗ 1 with
zero‐slack.
In the next section, we will apply the CCR model to a real data set.
36 Chapter 2
2014 M. Toloo
2.6 Application (Bank industry)
This section illustrates the proposed approach through a real data set involving 14
banks in the Czech Republic, a member state of European Union12. Table 2–4
exhibits 14 banks in the Czech Republic (see Table A–1 in Appendix A for their
brief description) with two inputs, the number of employees and assets, and two
outputs, deposits and loans.
We apply the input‐oriented CCR model to measure the optimal weights and
efficiency score, as reported in Table 2–5.
The optimal solution for the first bank is ∗, ∗, ∗, ∗, ∗ 0.987,0,2.98
10 , 3.22 10 , 0 which shows the CCR‐efficiency score of Air bank is
0.987and radially decreasing the inputs of bank, 400,33600 , by 0.987 projects
the bank on 394.96,33176.64 . It will be shown that 0.987 ,
9394.96,33176,30696,11135 is not an efficient point even if it dominates
0.987 , 400,33600,30696,11135 . By reference to Table 2–5 there are five
efficient banks: CMZRB, EQB, FIO, RB, and UCB.
The optimal weights and objective values of the output‐oriented CCR model are
summarized in Table 2–6.
Table 2–4 Bank data in the Czech Republic
Banks Inputs Outputs
Employees Assets Deposits Loans
AIR 400 33600 30696 11135
CMZRB 217 111706 86967 16813
CS 10760 920403 688624 489103
CSOB 7801 937174 629622 479516
EQB 296 8985 7502 5611
ERB 72 33614 2940 1762
FIO 59 18561 17174 6465
GEMB 3346 135474 97063 101898
ING 293 128425 92579 19216
JTB 407 85087 62085 39330
KB 8758 786836 579067 451547
LBBW 365 31300 20274 2528
RB 2927 197628 144143 150138
UCB 2004 318909 195120 192046
12 However, has not joined the Eurozone yet.
Basic DEA Models 37
Table 2–5 Optimal weights and CCR‐efficiency score (input‐oriented)
Banks ∗ ∗ ∗ ∗ ∗
AIR 0.9874 0 2.98E‐05 3.22E‐05 0
CMZRB 1 2.05E‐03 4.98E‐06 1.05E‐05 4.98E‐06
CS 0.9146 1.81E‐06 1.07E‐06 9.65E‐07 5.11E‐07
CSOB 0.8886 2.80E‐05 8.34E‐07 5.93E‐07 1.07E‐06
EQB 1 6.06E‐05 1.09E‐04 8.80E‐05 6.06E‐05
ERB 0.2233 1.39E‐02 0 0 1.27E‐04
FIO 1 3.64E‐03 4.23E‐05 4.23E‐05 4.23E‐05
GEMB 0.9901 0 7.38E‐06 0 9.72E‐06
ING 0.8804 1.13E‐03 5.21E‐06 9.51E‐06 0
JTB 0.964 2.00E‐03 2.17E‐06 0 2.45E‐05
KB 0.9245 2.12E‐06 1.25E‐06 1.13E‐06 5.98E‐07
LBBW 0.7 0 3.19E‐05 3.45E‐05 0.00E+00
RB 1 3.40E‐06 5.01E‐06 3.40E‐06 3.40E‐06
UCB 1 8.70E‐05 2.59E‐06 1.84E‐06 3.34E‐06
Table 2–6 Optimal weights and objective value (output‐oriented)
Banks ∗ ∗ ∗ ∗ ∗
AIR 1.0128 0 3.02E‐05 3.26E‐05 0
CMZRB 1 2.05E‐03 4.98E‐06 1.05E‐05 4.98E‐06
CS 1.0934 1.98E‐06 1.17E‐06 1.06E‐06 5.59E‐07
CSOB 1.1254 3.15E‐05 9.39E‐07 6.67E‐07 1.20E‐06
EQB 1 6.06E‐05 1.09E‐04 8.80E‐05 6.06E‐05
ERB 4.4783 6.22E‐02 0 0 5.69E‐04
FIO 1 3.64E‐03 4.23E‐05 4.23E‐05 4.23E‐05
GEMB 1.0100 0 7.45E‐06 0 9.82E‐06
ING 1.1358 1.28E‐03 5.92E‐06 1.08E‐05 0
JTB 1.0373 2.07E‐03 2.25E‐06 0 2.54E‐05
KB 1.0817 2.29E‐06 1.35E‐06 1.22E‐06 6.47E‐07
LBBW 1.4286 0 4.56E‐05 4.93E‐05 0.00E+00
RB 1 3.40E‐06 5.01E‐06 3.40E‐06 3.40E‐06
UCB 1 8.70E‐05 2.59E‐06 1.84E‐06 3.34E‐06
The optimal solution for Air bank is ∗, ∗, ∗, ∗, ∗ 1.0128,0,3.02
10 , 3.2 10 , 0 . This bank is inefficient and its efficiency score is .
0.9874. In addition, Air bank is proportionally increased its outputs by 1.10125 to
arrive at 31055.90,11265.55 as a benchmarking point.
The PPS for this example is described as follows:
, , , :
400 217 10760 ⋯ 2927 200433600 111706 ⋯ 197628 31890930696 86967 ⋯ 144143 195120 11135 16813 ⋯ 150138 192046 0 1,2, … ,14
38 Chapter 2
2014 M. Toloo
Table 2–7 Reference sets ( ∗)
Banks CMZRB EQB FIO RB UCB
AIR 1.7874
CMZRB 1.0000
CS 8.2412 18.3840 2.1581
CSOB 14.6244 1.5083 0.8254
EQB 1.0000
ERB 0.2725
FIO 1
GEMB 0.6787
ING 0.7347 1.6701
JTB 2.1355 0.1329
KB 0.9204 13.0940 2.4093
LBBW 1.1805
RB 1
UCB 1
Table 2–7 reports the reference set ∗ of the envelopment form of the CCR model.
Note that ∗ for the inefficient DMUs are always zero which are ignored in Table
2–7.
Let us again consider AIR bank. We have ∗ 1.787 which means that
multiplying all inputs and outputs of FIO bank by 1.787, i.e.
105.4566,33175.93,30696.81,11555.54 , makes AIR bank inefficient. In addition,
reference set for CS bank is , , . The set spanned by EQB, FIQ
and RB is the efficient frontier of CS bank; mathematically
, , , :
296 59 2927 8985 18561 1976287502 17174 1441435611 6465 150138 0, 0, 0
Table 2–8 summarizes the optimal max‐slack solutions as model (2.19) of Phase II.
There are three inefficient banks with the zero‐slack solutions, i.e. CS, CSOB and
KB. Hence, radially decreasing their inputs by ∗ (or decreasing their outputs by ∗) projects these three banks into an efficient point. On the other hand, AIR, ERB,
GEMB, ING, JTB and LBBW are six inefficient banks with the nonzero max‐slack
solutions, meaning that the projection of these type of inefficient banks on the
efficient frontier max‐slack solutions must be considered. More precisely, ∗ , ∗, ∗ is an efficient projection point for such DMUs.
Basic DEA Models 39
Table 2–8 Max‐slack solutions
Banks ∗
∗
∗
∗
AIR 289.487 0 0 420.237
CMZRB 0 0 0 0
CS 0 0 0 0
CSOB 0 0 0 0
EQB 0 0 0 0
ERB 0 2448.487 1740.679 0
FIO 0 0 0 0
GEMB 1326.245 0 766.22 0
ING 0 0 0 3934.215
JTB 0 0 522.434 0
KB 0 0 0 0
LBBW 185.866 0 0 5103.968
RB 0 0 0 0
UCB 0 0 0 0
For instance, the following feasible activity is an efficient projection point for
inefficient AIR bank:
0.987 ∗, ∗ 105.473, 33176.64, 30696, 11555.24
In the next section, we will consider DEA models with the VRS assumption, which
are originated by Banker et al. (1984).
2.7 The BCC model
To formulate a DEA model with VRS property, Banker et al. (1984) proposed the
following new PPS by removing the CRS axiom:
, : , , 1 ,
Note that the set is a convex polyhedral set. Figure 2–11 depicts a typical
for a single input and single output case.
Figure 2–12 also displays a typical PPS for two inputs and one output case which
is a convex polyhedral in . In this case, since the assumption is VRS, we cannot
unitize the output and plot the PPS in space.
40 Chapter 2
2014 M. Toloo
Figure 2–11 PPS for single input and single output case (VRS)
Figure 2–12 PPS for 2 inputs and 1 output (VRS)
The aim of the following model, which is called the envelopment form of BCC, is
to find the minimum such that , ∈ :
min θs. t.∑ 1, … ,
∑ 1, … ,
∑ 1
0 1, … ,
(2.20)
The last constraint of the model, ∑ 1, is called convexity constraint. The BCC
model differs from the CCR model in the presence of the convexity constraint. The
A
C
Input
Output
BPPS
Efficient Frontier
Basic DEA Models 41
convexity constraint enables the model to evaluate the efficiency score in regions
of increasing, constant, or decreasing returns to scale. We can develop a two‐phase
approach to obtain max‐slack for the BCC model.
Suppose that ∗ and max‐slack ∗, ∗ are obtained. So, one can define the BCC‐
efficiency as follows:
Definition 2–5 (BCC‐efficiency). DMU is BCC‐efficient if and only if ∗ 1 with
zero‐slack.
From mathematical point of view, since the feasible region of model (2.20) is subset
of the feasible region of model (2.16), ⊆ , and hence the BCC‐efficiency is
always greater than or equal to the CCR‐efficiency, ∗ ∗ .
The following model as a dual of model (2.20) is called the multiplier form of BCC
model:
max θ ∑s. t.∑ 1
∑ ∑ 0 1, … ,
0 1, … , 0 1,… ,
(2.21)
where is the free variable corresponding to the convex condition in model (2.20),
i.e. ∑ 1.
Definition 2–6 (BCC‐efficiency). DMU is called BCC‐efficient if and only if ∗ 1
and there exists at least one strictly positive optimal solution ∗, ∗ ; otherwise is
BCC‐inefficient.
In contrast to envelopment forms, the feasible region of CCR model (2.2) is not
subset of the multiplier form of BCC model (2.21). Nevertheless, if , is a
feasible solution for the multiplier form of CCR model (2.2), then , , 0 is a
feasible solution for the multiplier form of BCC model (2.21) and hence ∗
∗ .
2.8 The output‐oriented BCC model
The output‐oriented BCC model can be formulated as follows:
min θ ∑s. t.∑ 1
∑ ∑ 0 1, … ,
0 1, … ,0 1,… ,
(2.22)
42 Chapter 2
2014 M. Toloo
Table 2–9 Optimal weights and BCC‐efficiency score (input‐oriented)
Banks ∗ ∗ ∗ ∗ ∗ ∗
AIR 1 1.12E‐05 2.96E‐05 2.94E‐05 1.12E‐05 –2.62E‐02
CMZRB 1 5.48E‐04 7.89E‐06 1.14E‐05 7.89E‐06 –1.23E‐01
CS 1 1.07E‐06 1.07E‐06 1.07E‐06 1.07E‐06 –2.65E‐01
CSOB 1 3.79E‐05 7.52E‐07 1.24E‐06 5.12E‐07 –2.39E‐02
EQB 1 7.37E‐05 1.09E‐04 7.37E‐05 7.37E‐05 3.32E‐02
ERB 0.819 1.39E‐02 0 0 0 8.19E‐01
FIO 1 1.59E‐03 4.88E‐05 4.88E‐05 4.88E‐05 –1.54E‐01
GEMB 0.994 0 7.38E‐06 0 9.64E‐06 1.23E‐02
ING 1 3.10E‐03 7.14E‐07 4.38E‐05 7.14E‐07 –3.07E+00
JTB 1 3.34E‐04 1.02E‐05 1.02E‐05 1.02E‐05 –3.26E‐02
KB 1 3.46E‐05 8.86E‐07 1.28E‐06 1.28E‐06 –3.14E‐01
LBBW 0.703 0 3.19E‐05 3.55E‐05 0 –1.72E‐02
RB 1 4.03E‐06 5.00E‐06 4.03E‐06 4.03E‐06 –1.87E‐01
UCB 1 7.51E‐05 2.66E‐06 2.66E‐06 2.82E‐06 –6.06E‐02
In comparison with the CCR model, the input‐oriented BCC‐efficiency score is not
necessarily the inverse of the output‐oriented BCC‐efficiency score.
The following Table 2–9 exhibits the optimal solution of the BCC model (2.22) for
the given data set in Table 2–4.
Table 2–9 shows that the BCC‐efficiency score of a bank is larger than its CCR‐
efficiency score which was reported in Table 2–5. There are 11 BCC‐efficient banks
in this data set while 9 banks are CCR‐efficient. Therefore, the CCR‐efficient banks
must be BCC‐efficient.
2.9 Without explicit inputs/outputs DEA models
DEA models have generally developed for a data set with multiple inputs and
multiple outputs, however there are some applications with multiple outputs and
without explicit inputs (WEI), which means that inputs are not directly defined.
On the other hand, some applications deal with multiple inputs and without
explicit outputs (WEO). Several researches have been conducted in the DEA
literature to consider WEI/WEO data set: Dyson and Thanassoulis (1988)
developed a regression approach for restricting weight flexibility in DEA when
there is a single input and multiple outputs or multiple inputs and single output.
Lovell and Pastor (1999) investigated DEA models with a single constant input and
multiple outputs or with multiple inputs and a single constant output and achieved
some simplified versions of DEA models with fewer constraints and variables.
Basic DEA Models 43
Some other studies have been accomplished in performance evaluation according
to a data set with multiple outputs and WEI (for more details we refer the readers
to Fernandez‐Castro and Smith, 1994; Despotis, 2005 and Liu et al., 2011).
2.9.1 WEI‐DEA models
Suppose that there are DMUs (DMU : 1, … , ) where each DMU uses a single
input to produce outputs , … , . We first consider the CRS
assumption and then extend it to the VRS condition. As will be seen shortly, the
single input (dummy) output will be removed under CRS assumption. Under these
conditions, the output‐oriented multiplier form of CCR model for measuring the
efficiency of DMU can be written as follows:
min φs. t.∑ 1∑ 0 1, … ,
00 1,… ,
(2.23)
where and are the weights associated with single input and rth output,
respectively.
Similarly, model (2.24) is the input‐oriented multiplier model.
max θ ∑s. t.
1∑ 0 1, … ,
00 1,… ,
(2.24)
By means of the normalization constraint, we simplify the above model as:
max θ ∑s. t.
∑ 1, … ,
0 1,… ,
(2.25)
If we utilize the CRS assumption and unitize the single input, then we obtain an
equivalent data set involving a single input 1 and outputs . In this
case, the following model calculate the efficiency score of DMU :
max θ ∑s. t.∑ 1 1,… ,
0 1,… ,
(2.26)
Model (2.26) that is the absence of inputs is called DEA without explicit inputs (DEA‐
WEI) model under CRS.
44 Chapter 2
2014 M. Toloo
To formulate envelopment form of the DEA‐WEI model, we first axiomatically
construct feasible activity set (FAS) as:
(A1) Feasibility. The observed units 1,… , belong to the FAS.
(A2) Convexity. If two activities ′ and ′′ belong to the FAS, then all convex
combinations of these activities belong to the FAS.
(A3) Free disposability. For any activity in the FAS, any activity is
included in FAS.
Under all the aforementioned axioms, FAS can be mathematically defined as
, … , ∑ 1, … , , ∑ 1, 0, ∀
We formulate the following envelopment form of the DEA‐WEI model for
measuring the technical efficiency score of DMU :
max φs. t.φ ∈
(2.27)
where is equivalent to the following model, which is called the envelopment form
of DEA‐WEI model:
max φs. t.∑ φ 1,… ,
∑ 1
0 1, … ,
(2.28)
Compared with the standard DEA models, the envelopment form of the DEA‐WEI
model (2.28) is not explicitly dual of its multiplier form. Nevertheless, we
manipulate the model (2.28) to obtain an equivalent model which is the dual of
model (2.26). We assume that and , ∀ to transfer model (2.28) to the
following model:
mins. t.∑ t 1, … ,
∑
0 1,… ,
(2.29)
which is equivalent to
min ∑
s. t.∑ t 1, … ,
0 1,… ,
(2.30)
Therefore, models (2.30) and (2.26) are dual to each other.
Basic DEA Models 45
If there is a single input and multiple outputs, then the following model evaluates
the BCC‐efficiency score of DMU :
max θ ∑s. t.
∑ 1, … ,
0 1,… ,
(2.31)
The input‐oriented envelopment model measure the efficiency score under the
VRS assumption.
max φs. t.∑
∑ φ 1, … ,
∑ 1
0 1,… ,
(2.32)
If we consider a data set involving a single constant input and multiple outputs,
then constraint ∑ is equal to ∑ 1. Lovell and Pastor (1999)
proved that the constant ∑ 1 is active at optimality which shows the
convexity constraint ∑ 1 is redundant. Hence, when there is a single
constant input, the results can be obtained using the output‐oriented BCC model
(2.32) and the CCR model (2.28) which are identical.
2.9.2 Industrial robot evaluation problem
During the past two decades it has been seen a wide use of robots in industry.
Robots can be programmed to keep a constant speed and a predetermined quality
when performing a task repetitively. Robots can work under hazardous conditions
such as excessive heat or noise, heavy load, toxic gases, etc. Therefore,
manufacturers prefer to use robots in many industrial applications where
repetitive, difficult or hazardous tasks need to be performed, such as assembly,
machine loading, materials handling, spray painting and welding (Karsak and
Ahiska, 2005).
In this section, we evaluate the relative efficiency of 12 industrial robots which is
adapted from Braglia and Petroni (1999) and shown in Table 2–10. There is a single
input cost ( ) and four engineering attributes including handling coefficient, load
capacity, repeatability and velocity ( , … , ).
In order to measure the CCR‐efficiency score of robots, we firstly normalize the
data, then utilize model (2.26). Table 2–11 reports the normalized output and the
CCR‐efficiency scores. As can be seen, there are three efficient robots, R5, R8, and
R12.
46 Chapter 2
2014 M. Toloo
Table 2–10Input and output data for 12 industrial robots
Robots
R1 100000 0.995 85 1.7 3
R2 75000 0.933 45 2.5 3.6
R3 56250 0.875 18 5 2.2
R4 28125 0.409 16 1.7 1.5
R5 46875 0.818 20 5 1.1
R6 78125 0.664 60 2.5 1.35
R7 87500 0.88 90 2 1.4
R8 56250 0.633 10 8 2.5
R9 56250 0.653 25 4 2.5
R10 87500 0.747 100 2 2.5
R11 68750 0.88 100 4 1.5
R12 43750 0.633 70 5 3
Table 2–11 Normalized outputs and results obtained
Robots CCR
efficiency
R1 9.950E‐06 8.500E‐04 1.700E‐05 3.000E‐05 0.6534
R2 1.244E‐05 6.000E‐04 3.333E‐05 4.800E‐05 0.8217
R3 1.556E‐05 3.200E‐04 8.889E‐05 3.911E‐05 0.9547
R4 1.454E‐05 5.689E‐04 6.044E‐05 5.333E‐05 0.9509
R5 1.745E‐05 4.267E‐04 1.067E‐04 2.347E‐05 1
R6 8.499E‐06 7.680E‐04 3.200E‐05 1.728E‐05 0.5639
R7 1.006E‐05 1.029E‐03 2.286E‐05 1.600E‐05 0.6836
R8 1.125E‐05 1.778E‐04 1.422E‐04 4.444E‐05 1
R9 1.161E‐05 4.444E‐04 7.111E‐05 4.444E‐05 0.7656
R10 8.537E‐06 1.143E‐03 2.286E‐05 2.857E‐05 0.7143
R11 1.280E‐05 1.455E‐03 5.818E‐05 2.182E‐05 0.9091
R12 1.447E‐05 1.600E‐03 1.143E‐04 6.857E‐05 1
As we earlier stated, the BCC‐efficiency score for the normalized data is not
identical to the original data set.
2.9.3 WEO‐DEA models
In this section, we follow the DEA‐WEI approach to formulate DEA models when
there are inputs , … , and a single output . In this case, the input‐
oriented and output‐oriented multiplier forms of the CCR model are respectively
as follows:
Basic DEA Models 47
maxs. t.∑ 1 1, … ,
∑ 0 1, … ,
0 1, … ,0
(2.33)
min ∑s. t.
1
∑ 0 1, … ,
0 1, … ,0
(2.34)
By means of the alteration variable the following equivalent model is
obtained:
min ∑s. t.
∑ 1, … ,
0 1, … ,
(2.35)
Considering the CRS assumption, we unitize the single output to formulate the
following equivalent model:
min ∑ s. t.∑ 1 1, … ,
0 1, … ,
(2.36)
where . Model (2.36) contains only input data without explicit outputs,
which is known as DEA‐WEO model.
Similar to DEA‐WEI approach, we can make the FAS which contains all feasible
activities by means of the following axioms:
(A1) Feasibility. The observed units 1,… , belong to the FAS.
(A2) Convexity. If two activities ′ and ′′ belong to the FAS, then all convex
combinations of these activities belong to the FAS.
(A3) Free disposability. For any activity in the FAS, any activity is included
in the FAS.
The FAS can be therefore defined as
, … , ∑ 1,… , , ∑ 1, 0, ∀
We formulate the envelopment form of DEA‐WEO model (2.37) as
48 Chapter 2
2014 M. Toloo
max θs. t.∑ 1, … ,
∑ 1
0 1,… ,
(2.37)
Furthermore, the following output‐oriented multiplier and envelopment DEA‐
WEO models evaluates the BCC‐efficiency score of DMU , respectively:
min ∑s. t.
∑ 1, … ,
0 1, … ,
(2.38)
max φs. t.∑ 1, … ,
∑ φ
∑ 1
0 1,… ,
(2.39)
On the other hand, the following models are input‐oriented multiplier and
envelopment DEA‐WEO models:
maxs. t.∑ 1
∑ 0 1, … ,
0 1, … ,0
(2.40)
mins. t.∑ 1, … ,
∑
∑ 1
0 1,… ,
(2.41)
If we consider a data set involving multiple inputs and a single constant output,
then constraint ∑ is equal to ∑ 1. Lovell and Pastor (1999)
proved that the convexity constraint ∑ 1 is redundant and concluded that
the input‐oriented BCC model (2.41) and the CCR model (2.37) and are identical
when there is a constant output.
Basic DEA Models 49
2.9.4 Financing decision problem
The financing decision problem is a controversial topic consisting of selecting the
best asset‐financing alternative offered by banks and leasing companies (see Levy
and Sarnat 1994). More specifically, consider a company or an entrepreneur
(decision maker) who wants to make a decision about long‐term asset financing. In
this situation, it is necessary for an entrepreneur to find out which alternatives are
efficient and which are not. There are various cost‐type and other measures related
to each alternative, which should be minimized. Toloo and Kresta (2014)
considered these measures as inputs and handled a pure input data set without
explicit outputs. In their practical application the authors considered an
entrepreneur to acquire a long‐term asset (specifically a personal car). The cost of
the car is 726,000 CZK13 and the entrepreneur has two possibilities of financing:
bank loans or finance leases. The main difference between these two alternatives is
in the ownership of the car. In the case of bank loan, the entrepreneur is the owner
of the car and thus can charge a depreciation, which leads to decreasing the profit.
Whereas, in the case of finance leases, factual owner of the car is the leasing
company, however the costs of depreciation are included in lease payments and
subsequently car becomes the property of the entrepreneur after the end of the
lease term.
Moreover, as it can be extracted from Table A–2 in Appendix A, there is a broad
offer for each type of financing alternatives. In this table, the different financing
products, as DMUs, are listed together with four input measures without explicit
outputs: Down payment is the amount of money the entrepreneur has to pay in
advance. For the finance lease, it is prepayment; for the bank loan it is the amount
of own money the entrepreneur uses to finance the long‐term asset. Generally, we
can say that the bigger down payment the lower principal the entrepreneur has to
repay. However, this relationship is not strictly linear – there can be some fees and
costs related to the acquisition of financing. Annuities measure involves two
components: monthly installment of the principal (repayment portion) and
interests. For each DMU the monthly payment is constant, however the proportion
of these two components changes – the interests are decreasing and instalments are
increasing over time. Other fees includes all the cost paid by the entrepreneur in
order to obtain the bank loan or the finance lease. These costs incur for all DMUs
at the beginning of the bank financing or the lease, so there is no need to compute
their net present value. Bank loan coefficient (or coefficient of overpayment)
measures the relationship of the amount of re‐payments in total (monthly
repayments and other fees) and the principal the entrepreneur borrows from the
bank or leasing company (the cost of the car minus down payment). The financing
period was not considered in the study because it is a function of the principal,
annuities and coefficient of overpayment.
13 Czech Republic Krona
50 Chapter 2
2014 M. Toloo
It is obvious that entrepreneur wants to minimize all the costs that causes that these
measures are the inputs. If one considered the repayment period, it would be also
more measures, which must be minimized, however the problem would become a
kind of multi criteria decision‐making problem with two conflicting criteria.
Toloo and Kresta (2014) considered the information about whether or not the car
was acquired as single implicit output and logically assigned a value of 1 for the
implicit output of all DMUs. The column labeled as CCR‐efficiency in Table A–2
indicates the CCR‐efficiency score of DMUs which is obtained by model (2.36).
2.10 The additive model
The optimal solution of both CCR and BCC models does not take into account the
input excesses and output shortfalls. However, there is another variant DEA
model, which is referred to additive (ADD) model; this model has been designed
based on the slacks of the input excesses and the output shortfalls. Also, this model
combines both of the input‐oriented and output‐oriented models in a single model
(For more details Ali and Seiford 1993). The basic ADD model is formulated as
follows:
∗ max∑ ∑s. t.∑ 1, … ,
∑ 1, … ,
0 1, … ,
0 1, … ,
0 1,… ,
(2.42)
where and are the slacks for the inputs and output, respectively. The optimal
objective value of model (2.42), s∗ , is the maximum slacks. An efficient DMU in the
ADD model is defined as follows:
Definition 2–7 DMU is ADD‐efficient if and only if ∗ 0 and otherwise, DMU is
called ADD‐inefficient.
There is a symmetric relationship between CCR‐efficient and ADD‐efficient
DMUs. In this regard, Ahn et al. (1988) proved that a DMU is CCR‐efficient if and
only if it is ADD‐efficient, which means that the CCR and the ADD models, for
efficient DMUs are equivalent. The ADD model fails to measure the efficiency score
of the inefficient DMUs based on slacks, however slacks based measure (SMB)
model is extended to overcome this shortage.
Table 2–12 summarizes the optimal solution of the additive model (2.42) for the
data of the banks in the Czech Republic. The optimal objective value which is
labeled as ‘ ∗’ shows that there are five ADD‐efficient banks, i.e. CMZRB, EQB,
FIO, RB, and UCB. Table 2–5 shows that these banks are CCR‐efficient as well. This
Basic DEA Models 51
Table 2–12 Results of the additive model
Banks ∗ ∗ ∗ ∗ ∗
AIR 1934.05 163.6291 0 0 1770.425
CMZRB 0 0 0 0 0
CS 1.40E+05 0 0 31053.07 108560.8
CSOB 1.74E+05 0 0 156306.9 17452.41
EQB 0 0 0 0 0
ERB 35108.88 0 10963.29 18018.1 6127.492
FIO 0 0 0 0 0
GEMB 4108.18 1339.541 0 1747.031 1021.604
ING 33698.63 0 22942.39 0 10756.24
JTB 5526.55 0 0 5526.549 0
KB 87396.24 0 0 43591.53 43804.71
LBBW 21979.68 0 0 4215.332 17764.35
RB 0 0 0 0 0
UCB 0 0 0 0 0
numerical example illustrates that the CCR and additive models are equivalent
when the DMU under evaluation is efficient.
One of the interesting properties of the additive model is translation invariant. It
means the model is invariant if the original input and/or output data are translated.
In other words, adding a constant to each input or output for all DMUs does not
effect on the optimal solution of the ADD model (2.42). This property helps us to
handle DEA with negative data.
The dual problem to the ADD model (2.42) can be expressed as follows:
min∑ ∑s. t.∑ ∑ 0 1, … ,
1 1, … ,1 1,… ,
(2.43)
The following model is equivalent to model (2.43):
min∑ ∑s. t.∑ ∑ 0 1, … ,
1, … ,1, … ,
(2.44)
where is a positive number. In fact, if ∗, ∗ is an optimal solution for model
(2.44), then ∗
,∗
is optimal for model (2.43). Conversely, if ∗, ∗ is an optimal
solution for model (2.43) , then ∗, ∗ is optimal for model (2.44). On the other
hand, we can manipulate model (2.43) to obtain the following equivalent model:
52 Chapter 2
2014 M. Toloo
∗ mins. t.∑ ∑ 0 1,… ,
1 1, … ,1 1,… ,
(2.45)
From the optimality conditions for the primal and dual problems (2.42) and (2.45)
we have ∗ s∗ .
There is another variant of the additive model under VRS assumption, which can
be formulated by adding the convexity constraint to the ADD‐model (2.42):
max∑ ∑s. t.∑ 1, … ,
∑ 1, … ,
∑ 1
0 1, … ,
0 1, … ,
0 1,… ,
(2.46)
Similarly, adding a free variable to model (2.44) results in the following
multiplier form of the ADD model under VRS:
min∑ ∑s. t.∑ ∑ 0 1, … ,
1, … ,1, … ,
(2.47)
53
CHAPTER 3
GAMS Software
It is a need to exploit suitable software for solving an optimization problem. This
necessity increases in DEA when there are different optimization models to be
solved for dealing with a problem. Some respective software has been produced to
solve the DEA models that can be divided into two main groups: commercial and
non‐commercial. The aim of non‐commercial software is to facilitate the use of
DEA models for the beginners (students) that of course contain some restrictions
such as the problem size. Some typical non‐commercial DEA software are: DEA‐
SOLVER‐LV, DEAFrontier and MaxDEA. Commercial software, which refers to
software produced for sale contains no limitation in use.
For learning DEA‐SOLVER software, we refer to the well‐known DEA textbook
written by Cooper et al. (2007b). This software has capability to solve standard
DEA models with up to 50 DMUs and unlimited number of inputs and outputs.
Student version of DEAFrontier is accompanied with Quantitative Models for
Performance Evaluation and Benchmarking written by Zhu (2009) where it is able to
solve standard DEA models with up to 100 DMUs and unlimited number of inputs
and outputs. Although there is no limitation on the number of DMUs in the basic
version of MaxDEA software, available at http://www.maxdea.cn/, this software can
only solve some basic DEA models. The core of DEA‐SOLVER and DEAFrontier
software is Excel Solver and hence indeed they are Add‐In for Microsoft® Excel.
Beside non‐commercial version for the DEA software, there are some others
involving PIM‐DEA14, Frontier Analyst and Konsi. PIM‐DEA, which is developed
by Emrouznejad, is available at http://deazone.com/en/software. Frontier Analyst is
provided by Banxia Frontier Analyst and can be downloaded from
http://www.banxia.com/frontier/. Konsi DEA software that is provided by Konsi Ltd
creates visual graphic images of classical DEA concepts in three‐dimensional
parameter space and is available at http://www.dea‐analysis.com/.
14 Performance Improvement Management.
54 Chapter 3
2014 M. Toloo
This chapter is concerned with the General Algebraic Modeling System (GAMS),
which is a high‐level modeling system for linear, nonlinear and mixed integer
optimization problems. The power of the algebraic modeling languages like GAMS
leads to the creation and modification of the equations and inequalities by hand as
well as reporting results in the convenient way.
The GAMS consists of a language compiler and a stable of integrated high‐
performance solvers. This optimization software is suitable for complicated and
large scale modeling applications such as DEA.
3.1 The GAMS software
The GAMS software is available for use on personal computers, workstations,
mainframes and supercomputers. It is written for Microsoft windows, UNIX and
Macintosh Operations systems that are available at www.gams.com.
The GAMS is a programming language and it is essential to be aware the structure
of GAMS codes, how to debug the possible errors, and finally investigate the
outputs from GAMS.
3.2 GAMS IDE
The GAMS IDE is a general text editor designed for GAMS with the ability to
launch and monitor the compilation/execution of GAMS models. Progress of a
compilation/execution can be monitored in the process window. Figure 3–1 shows
the GAMS IDE. We need to create a project at the beginning step. The project file
is a need to remember the various settings for the editor but the file does not
contain any GAMS source code. We created in this respect a new project entitled
DEA which is shown in the title bar of the main window of Figure 3–1. After
creating the project, we will see the main window; no text file is shown.
From File menu select New to create a new GAMS file so as to write a code. A Tab
is added to the current edit window, and a temporary name (Untitled_1.gms) is
assigned. Form File menu select Save as and enter CCR to change the current file
name to CCR.gms.
GAMS Software 55
Data Envelopment Analysis with Selected Models and Applications
Figure 3–1 GAMS IDE
3.3 Structure of GAMS codes
A GAMS code involves one or more statements (sentences) that define data
structures, initial values, data modifications, and symbolic relationships
(equations). The only rule concerning the ordering of statements is that an entity of
the model cannot be referenced before its definition. The GAMS compiler does not
distinguish between upper‐and lowercase letters, so you are free to use either. The
basic components of a GMAS code are listed as bellow:
Sets (Declaration, Assignment of members)
Data15 (Declaration, Assignment of values)
Variables (Declaration, Assignment of type)
Equations (Declaration, Definition)
Model and Solve statements
Put statement (optional)
Declaration means declaring the existence of a component something by giving it
a name while assignment or definition means giving the component with a specific
value or form. The typical syntax of a statement in GAMS begins with the keyword,
followed by the name, and its members.
Let us write a GAMS code for the CCR model with the 14 active banks in the Czech
Republic data set given in Chapter 2. The data set contains 14 banks with two
15 Tables, Parameters
56 Chapter 3
2014 M. Toloo
inputs, Employee and Assets, and two outputs, Deposits and Loans. In other
words,
Bank01, Bank02,… , Bank14 ,
Employee, Assets ,
, .
These index sets must be declared and initialized in a GAMS code as follows:
SETS
j “Number of DMUs” /Bank01*Bank14/
i “Number of Inputs” /Employee, Assets/
r “Number of Outputs” /Deposits, Loans/;
Explanatory text (quoted label) may place between double quotes (”). The entire
list must be placed between slashes (/). Note that /Bank01*Bank14/ in GAMS
notation is equal to /Bank01, Bank02,…,Bank14/. Each statement must be
terminated with a semicolon (;).
These statements must be entered in the GAMS IDE, as in Figure 3–2.
Input matrix, , can be declared and initialized as follows:
TABLE X(j,i) “Input matrix” Employee Assets Bank01 400 33600 Bank02 217 111706 Bank03 10760 920403 Bank04 7801 937174 Bank05 296 8985 Bank06 72 33614 Bank07 59 18561 Bank08 3346 135474 Bank09 293 128425 Bank10 407 85087 Bank11 8758 786836 Bank12 365 31300 Bank13 2927 197628 Bank14 2004 318909;
Due to the fact that the number of DMUs is generally much more than the number
of inputs we consider instead of .
The statement TABLE declares the parameter and also specifies its domain as the
set of ordered pairs in the Cartesian product of and . The values (components) of
must be entered under the appropriate heading. Note that GAMS will perform
domain check to make sure that the row and column names of the table are
members of the appropriate sets.
GAMS Software 57
Data Envelopment Analysis with Selected Models and Applications
Figure 3–2 GAMS code for the CCR model
In a similar manner, output matrix, , can be declared and initialized as follows:
TABLE Y(j,r) “Output matrix” Deposits Loans Bank01 30696 11135 Bank02 86967 16813 Bank03 688624 489103 Bank04 629622 479516 Bank05 7502 5611 Bank06 2940 1762 Bank07 17174 6465 Bank08 97063 101898 Bank09 92579 19216 Bank10 62085 39330 Bank11 579067 451547 Bank12 20274 2528 Bank13 144143 150138 Bank14 195120 192046;
To declare inputs, Xo(i), and outputs, Yo(r), of DMU under evaluation, DMU , we
use PARAMETERS statement as follows:
PARAMETERS
Xo(i) “Input Vector of DMUo”
Yo(r) “Output Vector of DMUo”;
It should be noted that, the value of these parameters must be assigned before
solving the CCR model.
58 Chapter 3
2014 M. Toloo
The statement VARIABLES declares all decision variables in an optimization
problem. Each variable is given a name, a domain if appropriate, and explanatory
text. Therefore, we need to define four non‐negative decision variables (weights)
concerning the data set. The objective function value must be also defined as a free
variable
VARIABLES
Theta “Objective value (efficiency score)”
v(i) “Input Weights”
u(r) “Output weights”;
There exist five type of variable in GAMS, which are exhibited in Table 3–1.
Table 3–1 Type of variables
Variable type Allowed Range of Variable
free(default) ∞ to ∞
positive 0 to ∞
negative ∞ to 0
binary 0 or 1
integer 0, 1, …, 100 (default)
POSITIVE VARIABLES statement declares non‐negative variables and FREE
VARIABLES statement declares free variables:
FREE VARIABLES
Theta;
POSITIVE VARIABLES
v(i)
u(r);
EQUATION statement declares all equations (constraints) of the model:
EQUATIONS
Object “Objective function”
Normal “Normalization constraint”
Common (j) “Common constraints”;
Nevertheless it seems two constraints are defined, GAMS (as an algebraic
modeling language) creates all required equations and inequalities.
To define each equation we require to pursue the following steps:
1. Define the name of the equation by EQUATION statement.
2. Define the domain (if there is).
3. Enter the symbol ʹ..ʹ
4. Enter the left‐hand‐side expression of each constraint.
GAMS Software 59
Data Envelopment Analysis with Selected Models and Applications
5. Define a suitable relational operator: =L= (means less than or equal to), =E=
(means equal to), or =G= (means greater than or equal to).
6. Enter the right‐hand‐side expression.
The CCR constraints can be defined as follows:
Object.. Theta =E= SUM(r,u(r)*Yo(r));
Normal.. SUM(i,v(i)*Xo(i)) =E= 1;
Common(j).. SUM(r,u(r)*Y(j,r))- SUM(i,v(i)*X(j,i)) =L= 0;
where SUM(r,u(r)*yo(r)) is equivalent to ∑ .
The following statement defines the CCR model as CCR_UV_I16 that contains all
the defined equations:
MODEL CCR_UV_I /Object, Normal, Common/;
If all the equations are to be included in a model, we can simply enter /all/ as shown
below:
MODEL CCR_UV_I /all/;
To call the solver we use SOLVE statement, which in our example is written as:
SOLVE CCR_UV_I USING LP maximizing z;
Keyword USING is a reserved word in GAMS and must be written. The term of LP
presents the linear programming solver. GAMS as a powerful software is able to
solve a wide range of programming models as listed below:
1. LP: linear programming
2. QCP: quadratic constraint programming
3. NLP: nonlinear programming
4. DNLP: nonlinear programming with discontinuous derivatives
5. MIP: mixed integer programming
6. RMIP: relaxed mixed integer programming
7. MIQCP: mixed integer quadratic constraint programming
8. MINLP: mixed integer nonlinear programming
9. RMIQCP: relaxed mixed integer quadratic constraint programming
10. RMINLP: relaxed mixed integer nonlinear programming
11. MCP: mixed complementarity problems
12. MPEC: mathematical programs with equilibrium constraints
13. CNS: constrained nonlinear systems
The term of MINIMIZING indicates that the problem is a minimization problem.
Similarly, the term of MAXIMIZING is used for maximization problems. Finally, z
is the name of the variable to be optimized or the value of objective function.
16 It means CCR model in UV (multipliers) space and input orientation.
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2014 M. Toloo
Note that the defined model CCR_UV_I cannot be solved, because the input and
output vector of under evaluation DMU, i.e. Xo(i) and Yo(r), are still not defined.
On the other hand, these vectors must be updated when the DMU under
evaluation is changed, i.e., we need to solve 12 different LP models. In doing so,
we first define another name such as o to j, the index set of DMUs, as follows:
ALIAS (j,o);
In fact, the name o is identical to a j’. Therefore, ∈ 1,… ,12 . For a fixed o, the
following loop is used to assign input i consumed by DMUo, i.e. X(o,i), to parameter
xo(i) for all i.
LOOP (i,xo(i)=X(o,i));
Note that the ʹ=ʹ symbol here is differ from the ʹ=E=ʹ symbol used in equation
statement. The former symbol is used only for the direct assignments. A direct
assignment gives a value to a parameter before calling the solver.
Similarly, the following loop can be used to allot all outputs of DMUo to parameter
yo:
LOOP (r,yo(r)=y(o,r));
We then put these two loops and the SOLVE statement into a new loop over the
index set o as follows:
LOOP (o,
LOOP (i,xo(i)=X(o,i));
LOOP (r,yo(r)=y(o,r));
SOLVE CCR_UV_I USING LP maximizing theta;
);
A GAMS code of the multiplier form of the CCR model is prepared as summarized
below:
SETS
j “Number of DMUs” /Bank01*Bank14/
i “Number of Inputs” /Employee, Assets/
r “Number of Outputs” /Deposits, Loans/;
TABLE X(j,i) “Input matrix” Employee Assets Bank01 400 33600 Bank02 217 111706 Bank03 10760 920403 Bank04 7801 937174 Bank05 296 8985
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Data Envelopment Analysis with Selected Models and Applications
Bank06 72 33614 Bank07 59 18561 Bank08 3346 135474 Bank09 293 128425 Bank10 407 85087 Bank11 8758 786836 Bank12 365 31300 Bank13 2927 197628 Bank14 2004 318909;
TABLE Y(j,r) “Output matrix” Deposits Loans Bank01 30696 11135 Bank02 86967 16813 Bank03 688624 489103 Bank04 629622 479516 Bank05 7502 5611 Bank06 2940 1762 Bank07 17174 6465 Bank08 97063 101898 Bank09 92579 19216 Bank10 62085 39330 Bank11 579067 451547 Bank12 20274 2528 Bank13 144143 150138 Bank14 195120 192046;
PARAMETERS
Xo (i) “Input Vector of DMUo”
Yo(r) “Output Vector of DMUo”;
VARIABLES
Theta “Objective value (efficiency score)”
v(i) “Input Weights”
u(r) “Output weights”;
FREE VARIABLES Theta; POSITIVE VARIABLES v(i) u(r);
EQUATIONS
Object “Objective function”
Normal “Normalization constraint”
Common (j) “Common constraints”;
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2014 M. Toloo
Object.. Theta =E= SUM(r,u(r)*Yo(r));
Normal.. SUM(i,v(i)*Xo(i)) =E= 1;
Common (j).. SUM(r,u(r)*Y(j,r))- SUM(i,v(i)*X(j,i)) =L= 0;
MODEL CCR_UV_I /Object, Normal, Common/;
ALIAS (j,o);
LOOP (o,
LOOP (i,xo(i)=X(o,i));
LOOP (r,yo(r)=y(o,r));
SOLVE CCR_UV_I USING LP maximizing theta;
);
3.4 Running a model
To run the CCR model, select Run command from File menu or press the F9 key.
You can also use the mouse and click on the run button17 on the main window.
Note that when moving the mouse over various buttons, a small yellow box will
appear to indicate its function.
3.5 Navigating the listing file
After starting the run, a new window, called the Process Window, will be shown
as in Figure 3–‐3. The Process Window shows the progress of the GAMS execution.
The top of the window shows the name of GAMS code (here is CCR). When the
run finishes, the title of Process Window changes to No active process. To read
solution, we double‐click on the line ‐‐‐Reading solution for model CCR_UV_I which
is shown in blue.
17
GAMS Software 63
Data Envelopment Analysis with Selected Models and Applications
Figure 3–3 Process Window
3.6 Listing Window
After running the model, a new tab for the listing file (a file with the same code
name with ‘.lst’ suffix) will be generated in the edit window. Click on CCR.lst tab
on GAMS IDE to open it. As it is represented in Figure 3–4, the listing window
shows two panes; the left pane is the index of the listing file and the right pane is
the listing file. The index pane contains the list for the whole of the solving process
as a tree.
This window can be considered as the GAMS output which helps us to check and
comprehend a model. It may contain three parts: the echo print of the program, an
explanation of any errors detected, and the maps.
64 Chapter 3
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Figure 3–4 Listing window
3.7 Compilation
The first section of output from a GAMS run is an echo to report the errors. For the
sake of future reference, GAMS puts the line numbers on the left‐hand side of the
echo. Our CCR example contains no errors, the echo print is as presented below:
1 SETS
2 j "Number of DMUs" /Bank01*Bank14/
3 i "Number of Inputs" /Employee, Assets/
4 r "Number of Outputs" /Deposits, Loans/;
5
6 TABLE X (j,i) "Input matrix"
7 Employee Assets
8 Bank01 400 33600
9 Bank02 217 111706
10 Bank03 10760 920403
11 Bank04 7801 937174
12 Bank05 296 8985
13 Bank06 72 33614
14 Bank07 59 18561
15 Bank08 3346 135474
16 Bank09 293 128425
17 Bank10 407 85087
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Data Envelopment Analysis with Selected Models and Applications
18 Bank11 8758 786836
19 Bank12 365 31300
20 Bank13 2927 197628
21 Bank14 2004 318909;
22
23 TABLE Y (j,r) "Output matrix"
24 Deposits Loans
25 Bank01 30696 11135
26 Bank02 86967 16813
27 Bank03 688624 489103
28 Bank04 629622 479516
29 Bank05 7502 5611
30 Bank06 2940 1762
31 Bank07 17174 6465
32 Bank08 97063 101898
33 Bank09 92579 19216
34 Bank10 62085 39330
35 Bank11 579067 451547
36 Bank12 20274 2528
37 Bank13 144143 150138
38 Bank14 195120 192046;
39
40 PARAMETERS
41 Xo(i) "Input Vector of DMUo"
42 Yo(r) "Output Vector of DMUo";
43
44 VARIABLES
45 Theta "Objective value (efficiency score)"
46 v (i) "Input Weights"
47 u(r) "Output weights";
48
49 FREE VARIABLES
50 Theta;
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51
52 POSITIVE VARIABLES
53 v (i)
54 u(r);
55
56 EQUATIONS
57 Object "Objective function"
58 Normal "Normalization constraint"
59 Common (j) "Common constraints";
60
61 Object.. Theta =E= SUM(r,u(r)*Yo(r));
62 Normal.. SUM (i,v(i)*Xo(i)) =E= 1;
63 Common (j).. SUM(r,u(r)*Y(j,r))- SUM(i,v(i)*X(j,i)) =L= 0
64
65 MODEL CCR_UV_I /Object, Normal, Common/;
66
67 ALIAS (j,o);
68
69
70 LOOP (o,
71 LOOP (i,xo(i)=X(o,i));
72 LOOP (r,yo(r)=y(o,r));
73 SOLVE CCR_UV_I USING LP maximizing theta;
74 );
To show how GAMS detects errors, we intentionally make a mistake in the code.
For instance, we change ‘=E=’ in line 61 to ‘=’ which is a common mistake for
beginners. In this case, the GAMS compiler inserts a coded error message inside
the echo print on the line. These messages always start with **** and contain a ʹ$ʹ
directly below the line where the compiler finds the error. The $ is followed by a
numerical error code, which is explained after the echo print.
The following shows the result in the echo:
61 Object.. Theta = SUM(r,u(r)*Yo(r));
**** $37
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Data Envelopment Analysis with Selected Models and Applications
which means something is wrong in line 61. At the bottom of the echo print, we see
the interpretation of error code 37:
37 '=l=' or '=e=' or '=g=' operator expected
Note that practically one might receive more than one error message. It is
suggested to concentrate on fixing the errors in turn. In many cases, all the errors
will be removed after debugging the first error.
3.8 Equation listing
The ability to generate a model is an additional interesting advantage of the GAMS
software. It is important to verify whether the GAMS generated is our model. The
equation listing shows the specific instance of the model that is created when the
current values of the sets and parameters are plugged into the general algebraic
form of the model. For example, the generic common constraint given in the line
63 is
Common (j).. SUM(r,u(r)*Y(j,r))- SUM(i,v(i)*X(j,i)) =L= 0;
while the equation listing of specific constraints is
---- Common =L= Common constraints
Common (Bank01).. - 400*v(Employee) - 33600*v(Assets) + 30696*u(Deposits)+ 11135*u(Loans) =L= 0 ; (LHS = 0)
Common (Bank02).. - 217*v(Employee) - 111706*v(Assets) + 86967*u(Deposits) + 16813*u(Loans) =L= 0 ; (LHS = 0)
Common (Bank03).. - 10760*v(Employee) - 920403*v(Assets) + 688624*u(Deposits)+ 489103*u(Loans) =L= 0 ; (LHS = 0)
REMAINING 11 ENTRIES SKIPPED
The default output is a maximum of three specific equations for each generic
equation. To change the default, insert an input statement prior to the solve
statement:
Option limrow = r ;
where r is the desired number. Hence, if we want to observe all created equations
the following statement must be added to the code:
Option limrow = 14.
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3.9 Column listing
Similar to the equation listing which shows the constraints, the column listing deals
with variables. Note that each column in the coefficient matrix is related to a
variable. For each variable, its lower bound (.LO), initial level value (.L), upper
bound (.UP), initial marginal value18 (.M) are presented. In addition, the column
listing lists the individual coefficients for each variable. Consider the following
column listing for the first input weight (variable):
---- v Input Weights
v(Employee)
(.LO, .L, .UP, .M = 0, 0, +INF, 0)
400 Normal
-400 Common (Bank01)
-217 Common (Bank02)
-10760 Common (Bank03)
-7801 Common (Bank04)
-296 Common (Bank05)
-72 Common (Bank06)
-59 Common (Bank07)
-3346 Common (Bank08)
-293 Common (Bank09)
-407 Common (Bank10)
-8758 Common (Bank11)
-365 Common (Bank12)
-2927 Common (Bank13)
-2004 Common (Bank14)
In this list, (.LO, .L, .UP, .M = 0, 0, +INF, 0) means
1. The lower bound of v(Employee) is equal to zero, i.e. v(Employee).LO=0
2. The initial value of v(Employee) is equal to zero, i.e. v(Employee).L=0
3. There is no upper bound for v(Employee), i.e. v(Employee).UP= ∞
4. The initial marginal value of v(Employee) is equal to zero, i.e. v(Employee).M=0
18 Dual value or its complementary pair in a dual problem.
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Data Envelopment Analysis with Selected Models and Applications
Moreover, 400 Normal means the coefficient of v(Employee) in the Normal
constraint is equal to 400. In a similar manner, the other components can be
interpreted.
3.10 Model Statistic
The model statistic is the last part of the output that is generated by GAMS before
solving the problem. It deals with the size of the model, as shown below for the
CCR model:
LOOPS o Bank01
MODEL STATISTICS
BLOCKS OF EQUATIONS 3 SINGLE EQUATIONS 16
BLOCKS OF VARIABLES 3 SINGLE VARIABLES 5
NON ZERO ELEMENTS 61
The first line shows that Bank01 is a DMU under evaluation. The BLOCK levels
refer to the number of generic equations and variables. There are 3 blocks
associated with the equations (Object, Normal and Common) and 3 blocks
associated with the variables (theta, v and u). The SINGLE levels refer to individual
rows and columns in the specific model. There are 16 single equations (one
objective function, one normalization condition and 14 common constraints) and 5
variables, i.e. theta, v(Employee), v(Assets), u(Deposits) and u(Loans). Note that
the coefficient matrix in this problem involves 16 rows and 5 column and hence
there are 16×5=80 elements in the matrix. NON ZERO ELEMENTS’ indicates the
number of non‐zero elements of the coefficient matrix. In the CCR model of the
example we have 19 zero elements: theta is not in the 14 constraints. Two input
variables are excluded from the objective function and two output variables are
excluded from the normalization condition.
3.11 Solution report
This section provides the results of the optimization involving different parts. In
the first part, it reports general information about the solving process, as shown
below for the CCR model:
S O L V E S U M M A R Y
MODEL CCR_UV_I OBJECTIVE Theta
TYPE LP DIRECTION MAXIMIZE
SOLVER CPLEX FROM LINE 73
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This list indicated that CCR_UV_I is the name of the model, the objective function
variable is Theta, the CCR model is a maximization LP problem, the solver is the
well‐known solver CPLEX19, and finally the solve statement is located in line 73.
The second part of solution report presents the solver and model status, and the
objective function value, as shown below for the first bank (Bank01):
**** SOLVER STATUS 1 Normal Completion
**** MODEL STATUS 1 Optimal
**** OBJECTIVE VALUE 0.9874
The solver status reports the way that solver is terminated. Normal completion
means that the solver terminated in a normal way. Table B–1 in Appendix B
provides all possible solver status.
The model status describes the characteristics of the accompanying solution.
Optimal means that the solution is optimal; in other words, it is feasible (within
tolerances) and it has been proven that no other feasible solution with better
objective value exists. There are three possible model status for LP termination
involving 1 OPTIMAL, 3 UNBOUNDED and 4 INFEASIBLE. Table B–2 in
Appendix B provides all possible model status for all types of optimization models.
The last line shows that the objective value for Bank01 is 0.9874 which indicates
that the bank under evaluation is inefficient.
The third part of solution report lists the lower bound, level of value, upper bound
and marginal for all equations (rows) as is shown in below for the common
equation:
---- EQU Common constraints
LOWER LEVEL UPPER MARGINAL
Bank01 -INF -0.013 . .
Bank02 -INF -0.527 . .
Bank03 -INF -5.243 . .
Bank04 -INF -7.640 . .
Bank05 -INF -0.026 . .
Bank06 -INF -0.906 . .
Bank07 -INF . . 1.787
Bank08 -INF -0.910 . .
19 To change solver form File menu select Options and the click on Solvers tab.
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Data Envelopment Analysis with Selected Models and Applications
Bank09 -INF -0.844 . .
Bank10 -INF -0.535 . .
Bank11 -INF -4.792 . .
Bank12 -INF -0.279 . .
Bank13 -INF -1.245 . .
Bank14 -INF -3.215 . .
It indicates that considering Bank01, as the bank under evaluation, there is not
lower bound on this type of constants whereas its upper bound is zero,
mathematically ∈ ∞, 0 . Moreover, the level for Bank01 is –0.013 which
means 0.013. From the normal condition we have 1 and hence ∗ 1 0.013 0.987 which already reported in the previous part. The 7th
common constraints 0 shows that Bank07 lead to inefficiency of
Bank01. The marginal value for this constraint is 1.78 and subsequently we have ∗ 1.787. In other words, the optimal solution for the dual problem can be
obtained from the GAMS’s solution report.
Similar to the previous part, the last part of solution report provides more details
about all variables (columns).
3.12 $Include option
There are two special symbols, the asterisk ʹ*ʹ and the dollar symbol ʹ$ʹ, in GAMS
which can be used in the first position on a line to indicate a non‐language input
line. An asterisk in the first column means that the line will not be processed, but
treated as a comment. A dollar symbol in the same position indicates that compiler
options are contained in the rest of the line.
$include helps us to use multiple files in a GAMS program. Suppose that we put
the input and output tables into two text files such as input.txt and output.txt. In
this case we replace lines 7–21 with the following option :
$include "input.txt";
where the external file input.txt is as follows:
Employee Assets Bank01 400 33600 Bank02 217 111706 Bank03 10760 920403 Bank04 7801 937174 Bank05 296 8985 Bank06 72 33614 Bank07 59 18561 Bank08 3346 135474 Bank09 293 128425 Bank10 407 85087
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Bank11 8758 786836 Bank12 365 31300 Bank13 2927 197628 Bank14 2004 318909;
Similarly, we can replace lines 24–38 with the following option:
$include "output.txt";
where the external file output.txt is as follows:
Deposits Loans Bank01 30696 11135 Bank02 86967 16813 Bank03 688624 489103 Bank04 629622 479516 Bank05 7502 5611 Bank06 2940 1762 Bank07 17174 6465 Bank08 97063 101898 Bank09 92579 19216 Bank10 62085 39330 Bank11 579067 451547 Bank12 20274 2528 Bank13 144143 150138 Bank14 195120 192046;
It is also supposed that dad.txt file is located in the GAMS system directory20 (the
GAMS system directory is the directory where the GAMS system files should
reside). However, we can add a path to ask GAMS for finding the file in the
required folder, as shown below:
$include "C:\My GAMS CODES\data.txt";
3.13 The Put Writing Facility
Although the GAMS software provides various lists, we must verify all parts to
find our desirable list. PUT statement is one way to get rid of this issue. The put
writing facility enables one to generate the structural documents using information
that is stored by the GAMS system. This information is available using numerous
suffixes connected with identifiers, models, and the system. The put writing facility
generates documents automatically when GAMS is executed. A document is
written to an external file sequentially, a single page in place. First, we need to
define a name for which is used inside the GMAS and then connect it to an external
file name. The following statement defines the result as an internal file name and
connects it to CCR‐efficiency.txt as an external file.
FILE result/CCR-effciency.txt/;
20 The GAMS system directory is located at gamsdir folder in documents.
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Data Envelopment Analysis with Selected Models and Applications
Suppose that we solve both the CCR and BCC models. In this case we can define
two files as follows:
FILE result/CCR-effciency.txt/, result2/BCC-efficiency.txt/;
The following statement assigns the CCR‐efficiency.txt file as the current file:
PUT result;
The following statement writes the textual item CCR efficiency score into the current
file:
PUT ‘CCR efficiency score’///;
Notice that the text is quoted. The 3 slashes following the quoted text represent 3
carriage returns, which leads to 2 free lines.
The following statement writes the efficiency score of under evaluation DMU on
the current file:
PUT theta.l/;
To write the label of the DMU under evaluation we can use the suffix .tl as shown
below:
PUT o.tl theta.l/;
To report the efficiency score in 6 digits and 4 decimal places one can use the
following statement
PUT o.tl theta.l:6:5/;
The following statements present the optimal input and output weights:
LOOP (i, PUT v.l(i):10:5);
LOOP (r, PUT u.l(r):10:5);
In conclusion, the following codes give us more suitable information about the
optimal solution of the CCR model:
File result /C:\My GAMS CODES\CCR-efficiency.txt/;
PUT result;
PUT 'CCR efficiency score'///;
PUT ' DMUs eff. v1 v2 u1 u2'/;
PUT '========================================================='/;
LOOP (o,
LOOP (i,xo(i)=X(o,i));
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2014 M. Toloo
LOOP (r,yo(r)=y(o,r));
SOLVE CCR_UV_I USING LP maximizing theta;
PUT o.tl theta.l:6:5;
LOOP (i, PUT v.l(i):10:5);
LOOP (r, PUT u.l(r):10:5);
PUT /;
);
To see the output file CCR‐effiicency.txt, double click on the following link which
will be appeared at Process Window:
--- putfile result C:\My GAMS CODES\CCR-efficiency.txt
The following GAMS model runs the CCR and BCC models for the given data set
and puts their optimal solutions into two different external files:
SETS
j "Number of DMUs" /Bank01*Bank14/
i "Number of Inputs" /Employee, Assets/
r "Number of Outputs" /Deposits, Loans/;
TABLE X (j,i) "Input matrix"
Employee Assets
Bank01 400 33600
Bank02 217 111706
Bank03 10760 920403
Bank04 7801 937174
Bank05 296 8985
Bank06 72 33614
Bank07 59 18561
Bank08 3346 135474
Bank09 293 128425
Bank10 407 85087
Bank11 8758 786836
Bank12 365 31300
Bank13 2927 197628
Bank14 2004 318909;
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Data Envelopment Analysis with Selected Models and Applications
TABLE Y (j,r) "Output matrix"
Deposits Loans
Bank01 30696 11135
Bank02 86967 16813
Bank03 688624 489103
Bank04 629622 479516
Bank05 7502 5611
Bank06 2940 1762
Bank07 17174 6465
Bank08 97063 101898
Bank09 92579 19216
Bank10 62085 39330
Bank11 579067 451547
Bank12 20274 2528
Bank13 144143 150138
Bank14 195120 192046;
PARAMETERS
Xo (i) "Input Vector of DMUo"
Yo(r) "Output Vector of DMUo";
VARIABLES
Theta "Objective value (efficiency score)"
v(i) "Input Weights"
u(r) "Output weights";
FREE VARIABLES
Theta
u0;
POSITIVE VARIABLES
v(i)
u(r);
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EQUATIONS
Object "Objective function"
Normal "Normalization constraint"
Common (j) "Common constraints"
Trick "makes uo=0";
Object.. Theta =E= SUM(r,u(r)*Yo(r))+u0;
Normal.. SUM (i,v(i)*Xo(i)) =E= 1;
Common (j).. SUM(r,u(r)*Y(j,r))+u0- SUM(i,v(i)*X(j,i)) =L= 0;
Trick.. u0 =E=0;
MODEL CCR_UV_I /Object, Normal, Common, trick/;
MODEL BCC_UV_I /Object, Normal, Common/;
ALIAS (j,o);
File result /C:\My GAMS CODES\CCR-efficiency.txt/;
File result2/C:\My GAMS CODES\BCC-efficiency.txt/;
PUT result;
PUT 'CCR efficiency score'///;
PUT ' DMUs eff. v1 v2 u1 u2'/;
PUT '========================================================='/;
PUT result2;
PUT 'BCC efficiency score'///;
PUT ' DMUs eff. v1 v2 u1 u2'/;
PUT '========================================================='/;
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Data Envelopment Analysis with Selected Models and Applications
LOOP (o,
LOOP (i,xo(i)=X(o,i));
LOOP (r,yo(r)=y(o,r));
SOLVE CCR_UV_I USING LP maximizing theta;
PUT result;
PUT o.tl theta.l:6:5;
LOOP (i, PUT v.l(i):10:5);
LOOP (r, PUT u.l(r):10:5);
PUT /;
SOLVE BCC_UV_I USING LP maximizing theta;
PUT result2;
PUT o.tl theta.l:6:5;
LOOP (i, PUT v.l(i):10:5);
LOOP (r, PUT u.l(r):10:5);
PUT /;
);
3.14 GAMS Data Exchange (GDX)
Microsoft Excel is the best software for storing, organizing and manipulating data.
GAMS data exchange (GDX) facilities import Excel files to GAMS. A GDX file is a
file that stores the values of one or more GAMS symbols such as sets, parameters,
variables and equations. Figure 3‐5 shows an Excel file, named data.xlsx,
containing two sheets: Inputs and Outputs. The input and output data set in the
given data set is located in Inputs and Outputs sheets, respectively. In this section,
we show how to import data.xlsx file to GAMS.
The following statement allows GDXXRW.EXE program to read data.xlsx file and
also put range A1:C15 of inputs sheet (rng=inputs!A1:C15) into parameter X
(par=X):
$CALL GDXXRW.EXE data.xlsx par=X rng=inputs!A1:C15
In this case, a GDX file with the same name of Excel file, i.e. data.gdx, will be
created.
To put parameter X from GDX to GAMS, we need to firstly declare it as follows:
PARAMETER
X(j,i);
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Figure 3–5 Excel file (data.xlsx)
The following statement opens data.gdx:
$GDXIN data.gdx
To load parameter X from the opened GDX file to GAMS and then close the current
GDX file, we use the following statements, respectively:
$LOAD X Y
$GDXIN
Likewise, the output data set can be imported from the Excel file to GDX and then
transfer to GAMS. Nevertheless, the following statements can be used to import
both inputs and output data set from data.xlsx to GAMS:
$CALL GDXXRW.EXE data.xlsx par=X rng=inputs!A1:C15 par=Y rng=outputs!A1:C15
PARAMETER
X(j,i)
Y(j,r);
$GDXIN data.gdx
$LOAD X Y
$GDXIN
An easy way to change the current GAMS system directory is to create new project
into the new directory (folder).
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Data Envelopment Analysis with Selected Models and Applications
Figure 3–6 A GDX file
Figure 3–6 depicts the created data.gdx file which is opened by GAMS interface. It
helps us to verify the imported data from Excel to GDX.
81
CHAPTER 4
Weights in DEA
The efficiency of a DMU with multiple inputs and multiple outputs is defined as
the ratio of the weighted sum of outputs to the weighted sum of inputs. Hence,
weights of the inputs and outputs play an important role in the performance
evaluation in terms of the multiplier DEA model. Traditional DEA models seek the
optimal weights for a DMU in the best light in comparison to all the other DMUs.
This flexibility may cause different weights for one input or output; in addition,
they are not unique for an efficient DMU. In other words, there are alternative
optimal solutions for multiplier form of the CCR and BCC models when DMU
under evaluation is efficient. In this case, if software reports some zero weights for
an efficient DMU, then there exists at least one strictly positive optimal set of
weights (by definition 2–1). A systematic approach for finding a specific solution
among alternative optimal solutions is to formulate a new model. Hence, we
formulate a minimax model that explores entire alternative optimal weights for
finding a set of (possibly) positive optimal weights.
Suppose that the multiplier form of the CCR model is solved and the optimal
solution is ∗, ∗, ∗ in place. The set , : ∗ , 1,
0, 1, … , , , represents all alternative optimal weights. The
following non‐linear programming model determines a specific optimal set of
weights which are known as maximin weights (see Wang et al., 2009):
maxmin , … , , , … ,s. t.∑ ∗
∑ 1
∑ ∑ 0 1, … ,
0 1, … ,0 1,… ,
(4.1)
where the parameter ∗ is the known efficiency score of DMU . Note that the
feasible region of model (4.1) is the set and subsequently the model finds
minimax weights among whole alternative optimal weights achieved from the
CCR model. The non‐linear maximin objective function in model (4.1) can be
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2014 M. Toloo
transformed to a linear function by including an additional decision variable, say
, which indicates the minimum value of weights, min ,… , , , … , . In
order to establish this relationship, the following extra constraints must be
imposed:
1, … , 1, … ,
Now, when is maximized, these constraints ensure that will be less than or equal
to weights. At the same time, the optimal value of will be no less than the
minimum of all weights. Therefore the optimal value of z will be as small as
possible and exactly equal to the minimum of weights. The following model
utilizes the efficiency obtained by the CCR model to find a maximin weight:
maxs. t.∑ ∗
∑ 1
∑ ∑ 0 1, … ,
0 1, … ,0 1, … ,
(4.2)
If ∗ 1 and ∗ 0, then DMUo is efficient by definition 2–1. Note that if ∗ 0
then DMUo is inefficient even if ∗ 1.
Consider the real data set of active banks in the Czech involves 14 banks with two
inputs and two outputs (see Chapter 2). We first utilize both DEA‐Solver and
GAMS software to solve the CCR model for the given data and then compare the
results obtained.
The following Table 4–1 exhibits the efficiency score and optimal weights gained
by DEA‐Solver software.
There are 9 banks with ∗ 1 which are absolutely inefficient, UCB is a single bank
with ∗ 1 and ∗, ∗, ∗, ∗ 0,0,0,0 which is efficient by definition, and the
status of the other banks is unclear.
Table 4–2 presents the efficiency score and optimal weights obtained by a written
code in GAMS software.
Comparing Table 4–1 with Table 4–2 illustrates a unique efficiency score is
obtained for each DMU by both DEA‐Solver and GAMS software, however, the
optimal weights are not necessary identical. In addition, two zero weights are
obtained for CMZRB and RB banks by GAMS while DEA‐Solver identified one
zero weight for each one. In the presence of alternative optimal weights, these
weights are determined based on the solution method or software used to solve
the problem. It is worth noting that FIO is found as an efficient bank in Table 4–2
with positive weights.
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Data Envelopment Analysis with Selected Models and Applications
Table 4–1 Optimal weights obtained by DEA‐Solver
Banks ∗ ∗ ∗ ∗ ∗
AIR 0.9874 0 2.98E‐05 3.22E‐05 0
CMZRB 1 4.61E‐03 0 6.92E‐06 2.37E‐05
CS 0.9146 1.81E‐06 1.07E‐06 9.65E‐07 5.11E‐07
CSOB 0.8886 2.80E‐05 8.34E‐07 5.93E‐07 1.07E‐06
EQB 1 0 1.11E‐04 1.07E‐04 3.50E‐05
ERB 0.2233 1.39E‐02 0 0 1.27E‐04
FIO 1 0 5.39E‐05 5.82E‐05 0
GEMB 0.9901 0 7.38E‐06 0 9.72E‐06
ING 0.8804 1.13E‐03 5.21E‐06 9.51E‐06 0
JTB 0.964 2.00E‐03 2.17E‐06 0 2.45E‐05
KB 0.9245 2.12E‐06 1.25E‐06 1.13E‐06 5.98E‐07
LBBW 0.7 0 3.19E‐05 3.45E‐05 0
RB 1 1.23E‐04 3.24E‐06 0 6.66E‐06
UCB 1 8.70E‐05 2.59E‐06 1.84E‐06 3.34E‐06
Table 4–2 Optimal weights obtained by GAMS
Banks ∗ ∗ ∗ ∗ ∗
AIR 0.9874 0 3.00E‐05 3.20E‐05 0
CMZRB 1 4.61E‐03 0 1.10E‐05 0
CS 0.9146 2.00E‐06 1.00E‐06 1.00E‐06 1.00E‐06
CSOB 0.8886 2.80E‐05 1.00E‐06 1.00E‐06 1.00E‐06
EQB 1 0 1.11E‐04 1.07E‐04 3.50E‐05
ERB 0.2233 1.39E‐02 0 0 1.27E‐04
FIO 1 9.10E‐05 5.40E‐05 4.90E‐05 2.60E‐05
GEMB 0.9901 0 7.00E‐06 0 1.00E‐05
ING 0.8804 1.13E‐03 5.00E‐06 1.00E‐05 0
JTB 0.964 2.00E‐03 2.00E‐06 0 2.50E‐05
KB 0.9245 2.00E‐06 1.00E‐06 1.00E‐06 1.00E‐06
LBBW 0.7 0 3.20E‐05 3.50E‐05 0
RB 1 0 5.00E‐06 0 7.00E‐06
UCB 1 8.70E‐05 3.00E‐06 2.00E‐06 3.00E‐06
Now, we apply model (4.2) on the given data to obtain maximin weights, which
leads to a set of possibly positive weights as listed in Table 4–3.
Table 4–3 shows that there is a strictly positive set of weights for each DMU with ∗ 1 and hence these units are efficient by definition 2–1. However, there are still
some zero weights for some inefficient DMUs. For instance, consider the first
DMU, AIR, and note that the optimal weight for the first input, the number of
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2014 M. Toloo
Table 4–3 Optimal maximin weights
Banks ∗ ∗ ∗ ∗ ∗
AIR 0.9874 0 3.00E‐05 3.20E‐05 0
CMZRB 1 2.05E‐03 5.00E‐06 1.10E‐05 5.00E‐06
CS 0.9146 2.00E‐06 1.00E‐06 1.00E‐06 1.00E‐06
CSOB 0.8886 2.80E‐05 1.00E‐06 1.00E‐06 1.00E‐06
EQB 1 6.10E‐05 1.09E‐04 8.80E‐05 6.10E‐05
ERB 0.2233 1.39E‐02 0 0 1.27E‐04
FIO 1 3.64E‐03 4.20E‐05 4.20E‐05 4.20E‐05
GEMB 0.9901 0 7.00E‐06 0 1.00E‐05
ING 0.8804 1.13E‐03 5.00E‐06 1.00E‐05 0
JTB 0.964 2.00E‐03 2.00E‐06 0 2.50E‐05
KB 0.9245 2.00E‐06 1.00E‐06 1.00E‐06 1.00E‐06
LBBW 0.7 0 3.20E‐05 3.50E‐05 0
RB 1 3.00E‐06 5.00E‐06 3.00E‐06 3.00E‐06
UCB 1 8.70E‐05 3.00E‐06 2.00E‐06 3.00E‐06
employee, and the second output, loans, is zero which means the model did not
take these measures into account for obtaining the maximum possible efficiency
score (0.9874). Obviously, if we develop an approach to consider all measures in
this evaluation, i.e. positive weights, then the efficiency score of such DMUs may
be decreased.
Since the status of DMUs with ∗ 1 might be in question we can apply model
(4.2) only for these units and utilize the following maximin model (for more
discussion about weight restrictions we refer the reader to Wang et al., 2009):
maxs. t.∑ ∑ 0
∑ ∑ 0 1, … ,
0 1, … ,0 1,… ,
(4.3)
The maximin weight method firstly utilizes the CCR model to measure the
efficiency score and then tries to find a lower bound (as large as possible) for the
weight and subsequently it may fail to find a set of positive optimal weights for
some inefficient DMUs. In fact, the efficiency score of such DMUs must be
decreased to achieve a positive set of weights. A method that provides positive
weights for all measures can be obtained by a positive lower bound on all weights,
which has been discussed in Chapter 2. Next section provides another possible
method by means of another type of weight restrictions.
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4.1 Weight restrictions
The main concern in DEA is to reduce the need for a priori knowledge.
Nevertheless, in some cases additional information concerning the importance of
factors leads to restriction imposed on the feasible region of weights. Assurance
Region (AR) method imposes some constraints on the relative importance of the
weights for special measures. AR method was firstly proposed by Thompson et al.
(1986) to help in choosing a best site for the location of a high‐energy physics
laboratory.
Consider two pairs of input measures, and . The following constraints put a
restriction on the ratio of weights for the given measures:
where and are two the lower and upper bounds that the ratio may
assume. Without loss of generality, we can assume that . Given that
we have and hence it is implicitly assumed and
. Furthermore, two restrictions and give
the similar restrictions for the ratio as . As a result, we consider the
following constraint on the ratio for 2,… , :
Imposing these restrictions on the DEA models may decrease the efficiency score
and we subsequently observe a decrease in the number of efficient DMUs. This
property increases the discriminating power in the AR models. Note that since the
efficiency score is extremely sensitive to weight restrictions, some more care needs
to be taken in choosing these bounds.
If more information about weights variations is available, then some more general
restrictions can be considered. For instance, Roll et al. (1991) suggested the
following restrictions:
where and are the lower and upper bounds for , respectively. From the
given lower and upper bounds for weights in the proposed restriction by Roll et
al. (1991) we can obtain theses bounds for the ratio of as and ,
respectively.
Allen et al. (1997) considered the following weight restrictions to generalize the AR
method which is called Assurance Region Global (ARG) method:
∑
86 Chapter 4
2014 M. Toloo
where the ratio of a weighted input to the weighted sum of all inputs is restricted.
Note that by summing up the constraint ∑
over from 1 to ,
we have ∑∑
∑1 ∑ . As a result, (1) the sum of the lower
bounds of inputs must be less than or equal to one and (2) the sum of the upper
bounds must be greater than or equal to one in the case all inputs are restricted;
otherwise the ARG model is infeasible.
It is worth noting that the similar weight restrictions can be considered on the
outputs.
Adding some weight restrictions to the multiplier form of the CCR model leads to
a model with restricted feasible region. For instance, the following model is known
as CCR‐AR:
max ∑s. t.∑ 1
∑ ∑ 0 1, … ,
0 2, … ,0 2, … ,0 2, … ,0 2, … ,
0 1, … ,0 1,… ,
(4.4)
where and are the lower and upper bounds on ratio , respectively.
The following dual program of model (4.4) is the envelopment form of the AR
model:
min θs. t.∑ ∑ ∑
∑ ∑ ∑ 2, … ,
∑ ∑ ∑
∑ ∑ ∑ 2, … ,
0 1, … ,
, 0 2, … ,
0 2, … ,
(4.5)
where , , , and are the dual variables associated with the last four
constraints of model (4.4) respectively.
A more general weight restriction method, known as cone‐ration, was proposed
by Charnes et al. (1990). Cone‐ratio method restricts the feasible region of a weight
to places in the polyhedral convex cone. For a deeper discussion of weight
restriction methods we refer readers to Cooper et al. (2007b).
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Now, we apply the assurance region method on the real data set of active banks in
the Czech Republic. Suppose that an expert in bank industry believes that:
1. The importance of employees is at least 2 times more than the weight of
assets.
2. The importance of employees is at most 5 times less than the weight of
assets.
3. The importance of Deposits is at least equal to the weight of loans.
4. The importance of Deposits is at most 2 times less than the weight of
loans.
The following four constraints can be formulated to reflect the above additional
information:
2 5; 1 2
Table 4–4 summarizes the optimal weights achieved by considering the above
weight restriction:
As can be extracted from the table, all the lower and upper bounds for the input
weights are achieved and also this is the case for 6 output weights. Moreover, all
obtained weights are strictly positive and hence all DMUs with ∗ 1 are efficient.
Comparing Table 4–3 with Table 4–4 it is seen that the AR‐efficiency scores are less
than or equal to the CCR‐efficiency score. The status of CMZRB, EQB, and UCB
banks, which are CCR‐efficient, is changed to AR‐inefficient. Definitely, the ratio
of optimal weights obtained by the CCR model for these DMUs are not in the
assurance region imposed by additional constraints. For instance, considering
Table 4–3 the ratio of optimal input weights for CMZRB is ∗
∗
.
.410
which is quite far from the interval 2,5 . There are two AR‐efficient DMUs, i.e. FIO
and RB, which are CCR‐efficient. This example illustrates that the assurance region
method increases the discriminating power and enables further discriminate
among efficient DMUs.
Figure 4–1 depicts the changes made by applying the assurance region constraints.
CMZRB and UCB, which were evaluated as the efficient banks without imposing
the assurance region are now inefficient. In fact, the efficiency of banks generally
declines in the presence of weight restrictions.
In sum, weight restriction methods restrict the feasible region and lead to more
discriminating power. Nonetheless, this method is inapplicable when there is no
information about the weights. Next section introduces some more variant models
with a higher discriminating power.
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2014 M. Toloo
Table 4–4 AR‐Efficiency and weights
Banks ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗
∗
AIR 0.9649 5.81E‐05 2.91E‐05 2 2.66E‐05 1.33E‐05 2
CMZRB 0.7813 4.43E‐05 8.87E‐06 5 8.19E‐06 4.10E‐06 2
CS 0.9140 2.12E‐06 1.06E‐06 2 9.60E‐07 5.17E‐07 1.86
CSOB 0.8481 5.12E‐06 1.02E‐06 5 9.08E‐07 5.77E‐07 1.57
EQB 0.9937 2.09E‐04 1.04E‐04 2 9.44E‐05 5.09E‐05 1.86
ERB 0.1059 1.47E‐04 2.94E‐05 5 2.61E‐05 1.66E‐05 1.57
FIO 1 2.65E‐04 5.30E‐05 5 4.90E‐05 2.45E‐05 2
GEMB 0.9677 1.41E‐05 7.03E‐06 2 4.86E‐06 4.86E‐06 1
ING 0.7269 3.85E‐05 7.70E‐06 5 7.11E‐06 3.56E‐06 2
JTB 0.8856 5.74E‐05 1.15E‐05 5 1.02E‐05 6.47E‐06 1.57
KB 0.9245 2.49E‐06 1.24E‐06 2 1.12E‐06 6.06E‐07 1.86
LBBW 0.6155 6.24E‐05 3.12E‐05 2 2.86E‐05 1.43E‐05 2
RB 1 9.83E‐06 4.91E‐06 2 4.44E‐06 2.39E‐06 1.86
UCB 0.8544 1.52E‐05 3.04E‐06 5 2.69E‐06 1.71E‐06 1.57
Figure 4–1 CCR‐efficiency and CCR‐AR‐efficiency scores
4.2 Some other approaches
There are some various versions of multiplier DEA models in the literature. To
introduce these models, we first add a variable to each inequality constraints of the
multiplier form of the CCR model:
1 1 1
0,990
0,987
0,925
0,915
0,964
1
0,889
1
0,880
0,700
0,223
1 1
0,9937
0,9677
0,9649
0,9245
0,9140
0,8856
0,8544
0,8481
0,7813
0,7269
0,6155
0,1059
CCR‐Efficiency CCR‐AR‐efficiency
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∗ min or max ∑
s. t.∑ 1
∑ ∑ 0 1, … ,
0 1, … ,
0 1, … ,0 1,… ,
(4.6)
From the second set of constraints for an unit under evaluation, ∑
∑ 0 and the normalization constraint, ∑ 1, we result in
∑ 1 . Hence, (i) 0 if and only if ∑ =1, (ii) ∈ 0,1 ,
and (iii) 1 ∗ is efficiency score of DMUo. The variable is called deviation from
efficiency and DMUo is called efficient if and only if there exists at least a strictly
positive set of optimal weights ∗, ∗ with ∗ 0.
Sexton et al. (1986) formulated the following LP with considering the weighted
sum of all deviation variables as the objective function:
min∑
s. t.∑ 1
∑ ∑ 0 1, … ,
0 1, … ,
0 1, … ,0 1,… ,
(4.7)
where parameter presents the weight (importance) of deviation variable . In
this model, 1 ∗ ∑ ∗ is called weighted minsum efficiency score for DMUo.
The evaluated efficiency score by this method is closely related to the given
parameters ( , … , . Let ∗, ∗, ∗ be the optimal solution for model (4.6). ∗, ∗ is a feasible solution for the CCR model and given optimality conditions
1 ∗ ∗. In other words, the efficiency defined under minsum criterion in
model (4.6) is more restrictive than that of the CCR model. As a result, if DMUo is
weighted minsum efficient, then it is CCR‐efficient (1 ∗ 1 ∗ ⇒ ∗ 1) but
the reverse is not always true. If we also suppose , … , 1, … ,1 , then 1 ∗
is known as minsum efficiency score of DMU .
The minsum model can be simplified as follows:
min∑ ∑ ∑
s. t.∑ 1
∑ ∑ 0 1, … ,
0 1, … ,0 1,… ,
(4.8)
The feasible region of model (4.8) and the multiplier form of the CCR model are
identical.
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2014 M. Toloo
Table 4–5 Minsum efficiency and weights
Banks ∗ ∗ ∗ ∗ ∗
AIR 0.7690 7.14E‐04 2.10E‐05 1.50E‐05 2.70E‐05
CMZRB 0.7750 1.50E‐05 9.00E‐06 8.00E‐06 4.00E‐06
CS 0.8736 2.60E‐05 1.00E‐06 1.00E‐06 1.00E‐06
CSOB 0.8422 2.00E‐06 1.00E‐06 1.00E‐06 1.00E‐06
EQB 0.6637 1.78E‐03 5.30E‐05 3.80E‐05 6.80E‐05
ERB 0.1040 5.00E‐05 3.00E‐05 2.70E‐05 1.40E‐05
FIO 1 9.10E‐05 5.40E‐05 4.90E‐05 2.60E‐05
GEMB 0.8081 1.36E‐04 4.00E‐06 3.00E‐06 5.00E‐06
ING 0.7221 1.30E‐05 8.00E‐06 7.00E‐06 4.00E‐06
JTB 0.8757 2.00E‐05 1.20E‐05 1.10E‐05 6.00E‐06
KB 0.9191 3.10E‐05 1.00E‐06 1.00E‐06 1.00E‐06
LBBW 0.4057 7.71E‐04 2.30E‐05 1.60E‐05 3.00E‐05
RB 1 1.14E‐04 3.00E‐06 2.00E‐06 4.00E‐06
UCB 0.8342 5.00E‐06 3.00E‐06 3.00E‐06 1.00E‐06
The following Table 4–5 shows the minsum efficiency and optimal weights for the
banks in the Czech Republic given in Chapter 2.
As can be seen, all weights are positive without having need of defining a positive
lower bound on the weights, however this result is not always true all the time.
Since all weights are positive, each DMU with ∗ 1, such as FIFO or RB, is
minsum‐efficient.
Stewart (1996) suggested an alternative approach for dealing with deviation
variables in DEA. The author formulated the following minimax form of deviation
variables as the objective function:
mins. t.∑ 1
∑ ∑ 0 1, … ,
0 1, … ,
0 1, … ,
0 1, … ,0 1,… ,
(4.9)
where ∗ max ∗: 1, … , . In this model, 1 ∗ is known as minimax
efficiency score of DMUo. Similar to the minsum approach, the minimax efficiency
score evaluated by model (4.9) is more restrictive than the CCR efficiency.
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Table 4–6 Minimax efficiency and weights
Banks ∗ ∗ ∗ ∗ ∗
AIR 0.8143 5.63E‐04 2.30E‐05 1.80E‐05 2.40E‐05
CMZRB 0.7220 2.08E‐04 9.00E‐06 7.00E‐06 9.00E‐06
CS 0.8830 2.10E‐05 1.00E‐06 1.00E‐06 1.00E‐06
CSOB 0.8773 2.20E‐05 1.00E‐06 1.00E‐06 1.00E‐06
EQB 0.7206 1.51E‐03 6.20E‐05 4.70E‐05 6.50E‐05
ERB 0.1163 6.90E‐04 2.80E‐05 2.20E‐05 3.00E‐05
FIO 1 1.22E‐03 5.00E‐05 3.80E‐05 5.30E‐05
GEMB 0.8384 1.12E‐04 5.00E‐06 4.00E‐06 5.00E‐06
ING 0.6736 1.80E‐04 7.00E‐06 6.00E‐06 8.00E‐06
JTB 0.9382 2.57E‐04 1.10E‐05 8.00E‐06 1.10E‐05
KB 0.9204 2.40E‐05 1.00E‐06 1.00E‐06 1.00E‐06
LBBW 0.4533 6.07E‐04 2.50E‐05 1.90E‐05 2.60E‐05
RB 1 9.10E‐05 4.00E‐06 3.00E‐06 4.00E‐06
UCB 0.9580 6.60E‐05 3.00E‐06 2.00E‐06 3.00E‐06
The following nonlinear programming model is equivalent to model (4.9) with the
same feasible region as the multiplier form of the CCR model:
minmax ∑ ∑ : 1,… ,
s. t.∑ 1
∑ ∑ 0 1, … ,
0 1, … ,0 1,… ,
(4.10)
Table 4–6 reports minimax efficiency score and optimal weights for the given real
data set.
Comparison between Table 4–5 and Table 4–6 shows that two minsum and
minimax approaches lead to the similar results, however it is not always true.
Li and Reeves (1999) considered all three mentioned objective functions and
formulated the following Multiple Objective Linear Programming DEA
(MOLPDEA) model:
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2014 M. Toloo
min or max ∑
min
min∑
s. t.∑ 1
∑ ∑ 0 1, … ,
0 1, … ,
0 1, … ,
0 1, … ,0 1,… ,
(4.11)
The first objective function (criterion) is identical to the objective function of model
(4.6), the second one is minimax of all deviation variables and finally the last
criterion is the sum of the deviation.
Typically, there is no unique optimal solution for MOLP problems that can be
obtained without incorporating preference information. The concept of an optimal
solution is often replaced by the set of non‐dominated solutions. A non‐dominated
solution contains the property that it is not possible to move away from it to any
other solution without sacrificing in at least one criterion. For more details about
multi‐objective optimization we refer readers to Steuer (1989) and Cohon (2013).
4.3 Common set of weights
Thus far, the presented DEA models must be solved times, one model for each
DMU, and hence the optimal weight can differ from one DMU to others. Whereas
CSW models integrate by solving a single optimization problem to obtain a CSW.
As a result, there is no DMU under evaluation in this type of DEA models. Instead
of having a single normalization constrain, CSW models usually involve
normalization constraints ∑ 1 1, … , . The CSW models have
several advantages over the traditional DEA models; (i) the optimal set of weights
is obtained by solving only a single integrated problem, (ii) there is no need to solve
corresponding individual LP problem for evaluating all efficiencies, (iii) these
models render more discriminating power among efficient DMUs. Cook et al.
(1990) and Roll et al. (1991) introduced the CSW approach to measure the relative
efficiency of highway maintenance patrols in Ontario.
4.3.1 Integrated minsum approach
Consider the following model as an integrated version of model (4.7):
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min∑
s. t.∑ 1 1, … ,
∑ ∑ 0 1, … ,
0 1, … ,
0 1, … ,0 1,… ,
(4.12)
The model involves normalization constraints (∑ 1, 1, … , ). The
model obtains a CSW to evaluate the efficiency score of all DMUs with solving a
single LP.
Clearly, , , , , is a feasible solution for this model. On the other
hand, due to the minimization of the objective function the optimal objective value
is always zero i.e., , , is an optimal solution. Note that unlike traditional
DEA models, model (4.12) fails to find a non‐zero weights when there is no positive
lower bounds for , . To overcome the problem we must consider a positive
lower bound for the weights. We also point out that the two‐phase approach for
finding the CCR‐efficient DMUs without dealing with the epsilon (see Chapter 2)
is inapplicable for the model.
The following integrated minsum is obtained by imposing the non‐Archimedean
epsilon on the previous model:
min∑
s. t.∑ 1 1, … ,
∑ ∑ 0 1, … ,
0 1, … ,
1, … ,1,… ,
(4.13)
In this model , is called a CSW. Here the efficiency score of DMUj is ∑ ∗
∑ ∗
∗
∑ ∗ and hence DMUk is efficient if and only if ∗ 0.
The following theorem proves that the efficiency score of DMUs remains
unchanged when the first constraints of the model are disregarded, although this
omission changes the feasible region. This type of redundant constraint is called
non‐geometrically redundant constraint. As will be proved subsequently, the
ignorance of these non‐geometrically redundant constraint helps us formulate an
epsilon‐free model.
Theorem 4–1 The normalization constraints in the integrated minsum model
(4.13) are non‐geometrically redundant constraint.
Proof. Consider the following dual problem of model (4.13):
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2014 M. Toloo
max ∑ ∑ ∑
s. t.∑ 0 1, … ,
∑ 0 1, … ,
1, 0 1, … ,
0 1, … ,
0 1, … ,
(4.14)
It is sufficient to show that in the optimality ∗ 0 for all . On the contrary,
suppose that ∗,
∗, ∗, ∗ be the optimal solution for model (4.14) where ∗
. Clearly, ∗,
∗, , ∗ is a feasible solution for this model with the better
objective value, which is a contradiction. Hence, model (4.14)is equivalent to the
following model:
max ∑ ∑ s. t.∑ 0 1, … ,
∑ 0 1, … ,
1 1, … ,
0 1, … ,
0 1, … ,
(4.15)
And the dual of model (4.15) is as follows
min∑
s. t.∑ ∑ 0 1, … ,
0 1, … ,
1, … ,1,… ,
(4.16)
It completes the proof. □
Removing the extra normalization constraints not only leads to a simpler model,
but also changes the role of epsilon in the model. The following theorem proves
that model (4.16) is feasible for all values of epsilon.
Theorem 4–2 The optimal objective value of the following model is unbounded:
∗ maxs. t.∑ ∑ 0 1, … ,
0 1, … ,0 1, … ,
(4.17)
Proof. Clearly, model (4.17) is always feasible. Considering the weak duality
property, we can show that the following dual model of (4.17) is infeasible:
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min 0s. t.∑ 0 1, … ,
∑ 0 1, … ,
∑ ∑ 1
0 1, … ,
0 1, … ,
0 1, … ,
(4.18)
From the first set of constraints, ∑ we have 0∀ , 0∀ .
Considering the second set of constraints we obtain 0∀ which is impossible
due to the constraint ∑ ∑ 1. □
As a result, in model (4.16) is an Archimedean parameter. More importantly, if
the vector ∗, ∗, ∗ be the optimal solution of model (4.16) for a fixed parameter
0, then for all 0 the vector ∗, ∗, ∗ is an optimal solution of model (4.16)
if we select as a parameter. In other words, the efficiency scores of model (4.16)
are identical for any positive value for . The most suitable value for epsilon is 1
that leads to the following integrated epsilon‐free equivalent model:
min∑
s. t.∑ ∑ 0 1, … ,
0 1, … ,
1 1, … ,1 1,… ,
(4.19)
This model excludes epsilon and in comparison with model (4.16) is simpler, and
more practical. As it was earlier mentioned DMU is minsum efficient if and only if ∗ 0. Now, we compare the integrated minsum model (4.19) with the CCR
model.
Theorem 4–3 The following model is equivalent to the CCR model:
max∑
s. t.∑
∑ ∑ 0 1, … ,
0 1, … ,0 1,… ,
(4.20)
where is a positive number.
Proof. By reference to Toloo (2009), the CCR model (4.6) is equivalent to model
(4.21):
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2014 M. Toloo
max∑
s. t.∑
∑ ∑ 0 1, … ,
0 1, … ,0 1,… ,
(4.21)
Consequently, it is sufficient to show that models (4.20) and (4.21) are equivalent.
Toward the end, consider the following dual of these models, respectively:
mins. t.∑ 1, … ,
∑ 1, … ,
0 1, … ,
0 1, … ,
0 1, … ,
free is sign
(4.22)
mins. t.∑ 1, … ,
∑ 1, … ,
0 1, … ,
0 1, … ,
0 1, … ,0
(4.23)
From the constraints of model (4.22), it is implied that 0. Therefore, this model
is equivalent to model (4.23) and the related primal models are equivalent. It
completes the proof. □
Suppose that the integrated minsum model (4.19) is solved; let the set be the set
of indices of efficient DMUs, ∗ 0 1, … , . In the following theorem,
we prove that a minsum efficient DMU with CSW is CCR‐efficient.
Theorem 4–4 DMU ∈ is CCR‐efficient.
Proof. Let ∈ and ∗, ∗, ∗ be an optimal solution of the integrated minsum
model (4.19) with ∗ 0. With these assumptions we have ∑ ∗
∑ ∗ 0 and consequently ∑ ∗ ∑ ∗ . Let ∑ ∗ and
note that from the constraints 1, 1, … , , we have 0. Under these
conditions ∗, ∗ is a feasible solution of model (4.20) and the related objective
function value is equal to and hence, according to Theorem 4–3, DMU is CCR‐
efficient. □
Let us utilize the real data set of active banks in the Czech Republic to evaluate
their performance based on minsum‐integrated approach. The optimal objective
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Table 4–7 Minsum efficiency score (CSW)
Banks ∗ Banks ∗
AIR 0.8674 GEMB 0.8782
CMZRB 0.7428 ING 0.6925
CS 0.8940 JTB 0.9145
CSOB 0.8646 KB 0.9218
EQB 0.7997 LBBW 0.5092
ERB 0.1115 RB 1
FIO 1 UCB 0.9120
value and the optimal solution of model (4.19) for the given data set is 6.2985E+5
and ∗, ∗, ∗, ∗ 18.525, 1.215,1,1 , respectively. Table 4–7 reports the
minsum efficiency score.
There are two minsum efficient DMUs, FIO and RB, and the worst inefficient bank
is ERB.
4.3.2 Integrated minimax approach
An alternative technique involving less deviation variables minimizes the
maximum value of these variables. We formulate the following integrated epsilon‐
free minimax model that minimizes the maximum value of deviation variables for
all DMUs:
mins. t.∑ ∑ 0 1,… ,
0 1, … ,
0 1, … ,
1 1, … ,1 1,… ,
(4.24)
Similar to the minsum model (4.19), this epsilon‐free model has very interesting
properties (for more details see Toloo 2014a).
We apply the integrated minimax model (4.11) to the data set of banks in the Czech
Republic. The optimal CSW is ∗, ∗, ∗, ∗ 18.525, 1.215,1,1 which is
identical to the CSW of model (4.19). This example illustrates that there are some
similarities between the minsum and minimax approaches. However, the
following section demonstrates that in some cases a DMU might be efficient in a
model and inefficient in another model.
4.3.3 Facility layout design problem
Facility Layout Design (FLD) referees to the arrangement of all equipment,
machinery, and furnishings after considering the various objectives of the facility.
98 Chapter 4
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The layout consists of production areas, support areas, and the personnel areas in
the building. Traditionally, there are four types of layout designs that
manufacturing organizations employ, namely, process layout, product layout,
fixed position layout, and Group Technology (GT) layout (Stevenson 2011). The
need for facility layout design arises both in the process of designing a new layout
and in redesigning an existing layout. The concept of FLD is usually considered as
a multi‐objective problem. For this reason, a layout generation and its evaluation
are often challenging and time consuming due to their inherent multiple objectives
in nature and their data collection process. As a result, the determination of the
best layout for a facility is regarded as a multiple objectives optimization problem.
The best layout can optimize measurements of production efficiency, such as
throughput, cycle time, or resource utilization.
To have an efficient evaluation of facility layout design, it is necessary to consider
both qualitative and quantitative measures. Ertay et al. (2006) considered two
qualitative measures, flexibility in volume and variety and quality related to the
product and production, and four quantitative measures; material handling cost,
adjacency score, shape ratio, and material handling vehicle. The authors presented
a decision‐making methodology based on DEA, which uses both quantitative and
qualitative measures for evaluating FLD. The material handling cost and adjacency
score measures that are to be minimized are viewed as inputs whereas shape ratio,
flexibility, quality and material handling vehicle measures that to be maximized
are considered as outputs.
Material handing cost ( ) is the cost of movement, protection, storage and
control of materials and products throughout manufacturing, warehousing,
distribution, consumption and disposal. In general, 20–50% of the
production cost is for the material handling and alleviating any amount of
cost in this section has a crucial role in the production time, the production
cost, and the system efficiency (see Singh and Sharma 2005).
The adjacency score ( ) is computed as the sum of all flow values (or
relationship values) between those departments (activities) that are
adjacent in the layout.
The shape ratio ( ) is the ratio between the sizes of shape in different
dimensions. For example, the aspect ratio of a rectangle is the ratio of its
longer side to its shorter side. In unequal plant layout, the maximum aspect
ratio is a constraint that should be satisfied. When it exist more than two
dimensions, such as hyper rectangles, the aspect ratio can still be defined as
the ratio of the longest side to the shortest side.
Flexibility ( ). In designing the plant layout taking into account the
changes over short and medium terms in the production process and
manufacturing volumes. The concept of plant layout is not static but it is
dynamic because of continuous manufacturing and technological
Weights in DEA 99
Data Envelopment Analysis with Selected Models and Applications
Table 4–8 Inputs and outputs of 19 FLDs
FLDs Inputs Outputs
1 20309.56 6405 0.4697 0.0113 0.041 30.89
2 20411.22 5393 0.438 0.0337 0.0484 31.34
3 20280.28 5294 0.4392 0.0308 0.0653 30.26
4 20053.2 4450 0.3776 0.0245 0.0638 28.03
5 19998.75 4370 0.3526 0.0856 0.0484 25.43
6 20193.68 4393 0.3674 0.0717 0.0361 29.11
7 19779.73 2862 0.2854 0.0245 0.0846 25.29
8 19831 5473 0.4398 0.0113 0.0125 24.8
9 19608.43 5161 0.2868 0.0674 0.0724 24.45
10 20038.1 6078 0.6624 0.0856 0.0653 26.45
11 20330.68 4516 0.3437 0.0856 0.0638 29.46
12 20155.09 3702 0.3526 0.0856 0.0846 28.07
13 19641.86 5726 0.269 0.0337 0.0361 24.58
14 20575.67 4639 0.3441 0.0856 0.0638 32.2
15 20687.5 5646 0.4326 0.0337 0.0452 33.21
16 20779.75 5507 0.3312 0.0856 0.0653 33.6
17 19853.38 3912 0.2847 0.0245 0.0638 31.29
18 19853.38 5974 0.4398 0.0337 0.0179 25.12
19 20355 17402 0.4421 0.0856 0.0217 30.02
improvements taking place necessitating quick and immediate changes in
production processes and designs. A flexible layout may be necessary
because of technological changes in the products as well as simple change
in processes, machines, and materials.
Quality ( ). In manufacturing, the quality of product/service is defined as
fitness for pre‐determined purposes.
Material handling vehicle ( ) encompasses a diverse range of tools,
vehicles, storage units, appliances and accessories for transporting, storing,
controlling, enumerating and protecting products at any stage of
manufacturing, distribution consumption or disposal. Material handling
equipment is the mechanical equipment, which is generally divided into
four main categories: storage and handling equipment, engineered systems,
industrial trucks, and bulk material handling.
Table 4–8 exhibits inputs and outputs of 19 FLDs which is adapted from Ertay et
al. (2006).
Table 4–9 reports the efficiency scores of 19 FLDs: the second column indicates the
CCR‐efficiency score of the CCR model, the third and fourth columns show
100 Chapter 4
2014 M. Toloo
Table 4–9 Different efficiencies for 19 FLDs
FLDs CCR Minsum Minimax
1 0.9846 0.9473 0.5965
2 0.9884 0.9846 0.7283
3 0.9974 0.9878 0.7057
4 0.9493 0.9426 0.6487
5 1 0.8603 0.9445
6 0.9733 0.9323 0.9153
7 1 0.9217 0.6242
8 0.8568 0.8134 0.5379
9 0.8892 0.8138 0.7963
10 1 1 1
11 0.9983 0.9505 0.9777
12 1 0.9854 1
13 0.7759 0.7454 0.5951
14 1 1 1
15 1 1 0.7360
16 1 0.9853 0.9746
17 1 1 0.6803
18 0.8517 0.8118 0.6487
19 1 0.6374 0.6846
minsum‐efficiency and minimax efficiency of models (4.19) and (4.24),
respectively.
The optimal CSW for the minsum and minimax models is ∗, ∗, ∗, ∗ , ∗ , ∗
1,1, 2.32 10 , 2.24 10 , 2.79 10 , 5.81 10 and ∗, ∗, ∗, ∗ , ∗ , ∗
1.08, 1, 9.12 10 , 1.43 10 , 1, 3.55 10 , respectively.
As can be extracted from Table 4–9, there are 9 CCR‐efficient, 4 minsum‐efficient
and 3 minimax‐efficient FLDs. All minsum‐efficient and minimax‐efficient FLDs
are also CCR‐efficient. FLD10 and FLD14 are both minsum and minimax efficient
FLD15 and FLD17 are minsum‐efficient and minimax‐inefficient, FLD12 is minimax‐
efficient and minsum‐inefficient. In addition, there is a considerable difference
between the minsum‐efficiency and minimax‐efficiency score of FLD1. The last row
of table indicates the biggest difference between the CCR‐efficiency and minsum
and minimax efficiency scores.
101
CHAPTER 5
Best Efficient Unit
One of the purposes of DEA in practice is to provide the prioritization among
DMUs. However, DEA models partition all the DMUs into two sets: efficient and
inefficient, where an efficient and inefficient DMU respectively have a score of 1 and
less than 1. Hence, these models fail to provide more information about the efficient
DMUs. On the other hand, standard DEA models must be solved times, once for
each unit, to get the maximum ratio of weighted sum of outputs to weighted sum
of inputs. The flexibility of weights in these methods leads to the maximum
efficiency score for each unit. One method to improve the discriminating power of
DEA models is to reduce the flexibility of weights. Although a discriminating
power can be improved by the weight restriction methods, a priori knowledge on
the weights is needed which is often unavailable. The CSW approach is an
alternative method that leads to a DEA model with more discriminating power.
However, they may suffer from discrimination among efficient DMUs. Adler et al.
(2002) comprehensively reviewed all ranking methods in DEA context including
linear discriminant analysis (Torgersen et al., 1996), discriminant analysis of ratios
(Sinuanay‐Stern et al., 1994), super efficiency ranking methods (Andersen and
Petersen 1993), benchmark ranking methods (Seuyoshi 1999), cross efficiency
ranking methods (Dyson et al., 2001).
In some cases, the main concern is to find a single efficient DMU, called the most
efficient DMU, which can be determined by ranking approaches. However, it is not
necessary to rank all efficient DMUs for identifying the best efficient DMU.
Noteworthy, ranking methods usually deal with a variable set of optimal weights
for each DMU, whereas the best efficient DMU may be determined in an identical
situation by the CSW approaches. Consequently, instead of solving at least one
optimization problem for each DMU, an integrated model can be applied to find
the best efficient DMU. Recently, the problem of finding the most (best) efficient
DMU in DEA has called the attention to some researchers: Shang and Sueyoshi
(1995) suggested a DEA/AHP approach to find the most efficient flexible
manufacturing system (FMS). The proposed framework first involved the
integrated use of analytic hierarchy process (AHP), simulation and an accounting
102 Chapter 5
2014 M. Toloo
procedure to determine the necessary outputs and inputs of FMS alternatives, and
then, the application of DEA with restricted weights and cross‐efficiency analysis
to select the most efficient FMS. Baker and Talluri (1997) dealt with finding the
most efficient robot in advanced manufacturing technology (AMT). Cook et al.
(1996) proposed a structure for decision problems where some factors are
measurable only on an ordinal scale, which enables the treatment of qualitative
factors in an ordinal sense within the standard DEA model. Karsak and Ahishka
(2005) followed the approach of Cook et al. (1996) in order to incorporate ordinal
outputs into the AMT evaluation process. The authors proposed a MCDM method
to handle industrial robot selection problem. Ertay et al. (2006) developed a robust
layout framework based on the DEA/AHP methodology to find the most efficient
FLD. Farzipoor Saen (2007), Toloo and Nalchigar (2011a), and Toloo (2014b)
addressed the supplier selection problem in the presence of both cardinal and
ordinal data. Toloo et al. (2009) and Toloo and Nalchigar (2011b) formulated an
integrated model for finding the most efficient discovered rule in data mining and
designed an algorithm to rank all efficient discovered rules. Amin and
Emrouznejad (2010) suggested an approach to find the most relevant information
among a return set of ranked lists of documents retrieved by distinct search
engines. Farzipoor Saen (2011) formulated a new DEA model to tackle the media
selection problem. Toloo (2013) proposed an approach to find the most efficient
professional tennis player. Toloo and Kresta (2014) dealt with the problem of
finding the most efficient long‐term asset financing provided by Czech banks and
leasing companies. Toloo and Etray (2014) suggested a minimax approach to find
the most cost efficient automotive vendor with price uncertainty.
5.1 Multi‐input and multi‐output case
The minimax efficiency formulation can reduce the number of efficient DMUs,
although it may have deficiency in determining the best one. To overcome this
weakness, Ertay et al. (2006) formulated the following minimax model:
mins. t.∑ 1
∑ ∑ 0 1, … ,
0 1, … ,
0 1, … ,
1, … ,1,… ,
(5.1)
where ∈ 0,1 is a constant, called discriminating parameter, that is determined by
trial‐and‐error method to obtain a single relatively efficient alternative. In this
model, ∗ is a deviation from efficiency of DMU when DMU is being evaluated.
The efficiency score of DMU is 1 ∗ and this unit is efficient if and only if ∗ 0.
The second term of the objective function plays the role of a penalty function. In
Best Efficient Unit 103
Data Envelopment Analysis with Selected Models and Applications
Table 5–1 Different efficiency scores
FLDs CCR Minimax
( 0)
Minimax
( 0.3)
1 0.985 0.952 0.932
2 0.988 0.959 0.952
3 0.997 0.933 0.926
4 0.949 0.872 0.872
5 1 0.794 0.794
6 0.973 0.897 0.897
7 1 0.794 0.793
8 0.857 0.787 0.776
9 0.889 0.775 0.775
10 1 0.847 0.811
11 0.998 0.9 0.9
12 1 0.868 0.868
13 0.776 0.776 0.776
14 1 0.97 0.97
15 1 1 0.994
16 1 1 1
17 1 0.973 0.924
18 0.852 0.796 0.785
19 1 0.806 0.806
this approach, firstly model (5.1) must be solved for 0. If a single efficient DMU
is determined, then the model succeeds in finding most efficient unit. Otherwise, a
suitable value for (by trial‐and‐error approach) should be assigned to obtain a
single efficient DMU.
Table 5–1 reports several efficiency scores obtained from the CCR and Ertay et al.
(2006)’s models. The minimax approach successfully decreases the number of
efficient FLDs from 9 to 2 as well as determining FLD15 and FLD16 as the most
efficient candidates. However, we require a single efficient DMU to find most
efficient DMU. The last column shows the efficiency score obtained from the model
of Ertay et al. (2006) for 0.3. The efficiency scores with 0.3 are less than or
equal to the efficiency scores computed by other methods. The results show that
the FLD16 is the single efficient unit.
Ertay et al. (2006) ignored the importance of epsilon’s role in discriminating
between efficient DMUs and practically considered 0. Table 5–2 shows the
optimal weights for the minimax model (5.1) for the data set with 0.
As can be seen, the second input and output are omitted in the evaluation process.
In other words, model (5.1) ignores two out of six measures (i.e., more than 30% of
measures) to identify the best efficient candidates. To tackle this issue and have
maximum discriminating power, we consider the maximum non‐Archimedean
epsilon for the data set, ∗ 2.64852 10 , in the method of Ertay et al. (2006).
Consequently, the following results from Table 5–3 can be listed:
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2014 M. Toloo
Table 5–2 The optimal weights for the minimax model
FLDs ∗ ∗ ∗ ∗ ∗ ∗
1 0.000049 0 0.074192 0 0.024538 0.029672
2 0.000049 0 0.073822 0 0.024415 0.029524
3 0.000049 0 0.074299 0 0.024573 0.029715
4 0.000050 0 0.075140 0 0.024851 0.030051
5 0.000050 0 0.075345 0 0.024919 0.030133
6 0.000050 0 0.074618 0 0.024678 0.029842
7 0.000051 0 0.076179 0 0.025195 0.030467
8 0.000050 0 0.075982 0 0.025130 0.030388
9 0.000051 0 0.076845 0 0.025415 0.030733
10 0.000050 0 0.075197 0 0.024870 0.030074
11 0.000049 0 0.074115 0 0.024512 0.029641
12 0.000050 0 0.074760 0 0.024726 0.029899
13 0.000051 0 0.076714 0 0.025372 0.030681
14 0.000049 0 0.073232 0 0.024220 0.029288
15 0.000048 0 0.072836 0 0.024089 0.029130
16 0.000048 0 0.072513 0 0.023982 0.029001
17 0.000050 0 0.075897 0 0.025101 0.030354
18 0.000050 0 0.075897 0 0.025101 0.030354
19 0.000040 0 0.077202 0 0 0.025723
1. The CCR model with maximum epsilon has more discriminating power
and determines 7 efficient DMUs. Hence, FLD5 and FLD19 are not the most
efficient candidates.
2. The minimax model (5.1) for 0 with 2.64852 10 identifies
FLD10 and FLD14 as the most efficient candidates. Note that neither FLD15
nor FLD16 are efficient with a set of strictly positive weights while they
were identified as the efficient units in Ertay et al. (2006).
3. The most efficient FLD with 0.4 is FLD14.
In conclusion, considering a set of positive weights as an essential issue in DEA
models may lead to different results in fining the most efficient DMU.
Amin and Toloo (2007) provided the following numerical example, see Table 5‐4,
to criticize the trial‐and‐error method of Ertay et al. (2006).
There are 4 DMUs with a single input and 4 outputs in the data set. The maximum
epsilon for the data is ∗ 0.0714. The following model represents model (5.1) for
DMU1.
Best Efficient Unit 105
Data Envelopment Analysis with Selected Models and Applications
Table 5–3 Different efficiency scored with maximum epsilon
FLDs CCR Minimax
( 0)
Minimax
( 0.4)
1 0.9655 0.7405 0.7405
2 0.9884 0.8365 0.8365
3 0.9950 0.8170 0.8170
4 0.9493 0.7655 0.7655
5 0.9892 0.9049 0.8684
6 0.9733 0.9288 0.9288
7 1 0.7319 0.7319
8 0.8568 0.6684 0.6684
9 0.8532 0.7989 0.7989
10 1 1 0.8535
11 0.9853 0.9622 0.9468
12 1 0.9657 0.9440
13 0.7644 0.6724 0.6724
14 1 1 1
15 1 0.8468 0.8468
16 1 0.9883 0.9883
17 1 0.8184 0.8184
18 0.8449 0.7332 0.7332
19 0.6714 0.6714 0.6714
Table 5–4 A counter example adapted from Amin and Toloo (2007)
DMUs
DMU1 1 2 3 4 5
DMU2 1 3 2 4 5
DMU3 1 4 2 3 5
DMU4 1 5 2 3 4
mins. t.2 3 4 5 13 2 4 5 14 2 3 5 15 2 3 4 1
0 1, … ,4
0 1, … ,4
0.0714 1,… ,
(5.2)
Note that the input weight is equal to 1 from the normalization constraint of
model (5.1) and hence is substituted in model (5.2) The optimal solution of model
(5.2) for all ∈ 0,1 is ∗ ∗ ∗ ∗ 1 which shows DMU1 is efficient by
trial‐and‐error method of Ertay et al. (2006). It is of interest to verify that model
(5.2)is the same for all the DMUs and the method proposed by Ertay et al. (2006)
106 Chapter 5
2014 M. Toloo
has deficiency in detecting the most efficient DMU. More precisely, it may not
converge to a single efficient DMU.
Amin and Toloo (2007) utilized the CSW approach to evaluate the performance
attributes and tried to eliminate the requirement of using parameter in the objective
function which is used in Ertay et al. (2006). Indeed, unlike model (5.1), which
modifies the objective function of minimax model, Amin and Toloo (2007)
attempted to restrict the feasible region of the integrated minimax model.
Furthermore, Toloo and Nalchigar (2009) utilized the approach of Amin and Toloo
(2007) to develop a model for finding the most efficient DMU under VRS
assumption. However, Amin (2009) showed these models may fail to find the most
efficient DMU and formulated the following mixed integer non‐linear programing
(MINLP) DEA model:
mins. t.∑ 1 1, … ,
∑ ∑ 0 1, … ,
0 1, … ,
∑ 1
0 1, … ,
∈ 0,1 1, … ,
1 1, … ,
0 1, … ,
1, … ,1, … ,
(5.3)
Here ∗ is a deviation from efficiency of DMU and the efficiency score of DMU is ∑ ∗
∑ ∗
∗
∑ ∗ and subsequently the unit is efficient if and only if ∗ 0. The
constraints of model are formulated such that the model can find a single efficient
DMU. In each feasible solution of the model there is an auxiliary binary variable
with a zero value, say 0, and the others are equal to 1, 1. Under this
assumption, the non‐linear constraint 0 for leads to 0 and
also for gives 0. Hence, there exists a unique deviation variable with
a value of zero and model (5.3)correctly finds the most efficient DMU with a
minimax criterion.
Amin (2009) did not mention how to determine an assurance value for the non‐
Archimedean for model (5.3) Hence, Foroughi (2011) showed model (5.3) might
be infeasible. Toloo (2014c) first proved that model (5.3) is always feasible for a
suitable value of epsilon that can be obtained using model (5.4), then applied the
well‐known separable programming indirect technique (see Taha, 2011) to solve
MINLP model (5.3).
Best Efficient Unit 107
Data Envelopment Analysis with Selected Models and Applications
ε∗ maxs. t.∑ 1 1, … ,
∑ ∑ 0 1, … ,
∑ 1
0 1, … ,
0 1, … , 0 1, … ,
∈ 0,1 1, … ,
1 1, … ,
0 1,… ,
(5.4)
Furthermore, the following theorems prove the validity and some properties of the
MINLP model.
Theorem 5–1 Model (5.4) is always feasible.
Proof. Consider the following epsilon form of the envelopment CCR model:
min ∑ ∑s. t.∑ 1, … ,
∑ 1, … ,
0 1, … ,
0 1, … ,
0 1,… ,
(5.5)
Cooper et al. (2007b) proved that there exists at least one extreme efficient unit.
Without loss of generality, suppose that DMU is such extreme efficient unit. Then ∗, ∗, ∗,
∗1, , , is a feasible solution of model (5.5) and its
objective value is equal to 1. Consequently, this vector is an optimal max‐slack
solution of model (5.5).
From the strong complementary slackness conditions, there is an optimal solution ∗, ∗, ∗ of the multiplier CCR model such that
∗ ∗ ∗ 0, ∗ ∗ ∗
where ∗ ∑ ∗ ∑ ∗ .These conditions imply that ∗ 0, ∗ 0.
Therefore, the vector ∗, ∗, ∗, , is a feasible solution of the MINLP model of
Amin (2009) where:
1 ∗⁄ ,
1,,
1,0,
Note that due to the constraints of the model ∗ ∑ ∗ ∑ ∗ 1
∑ ∗ 1
. As a result, we have ∀ , 1, which completes the proof.
Theorem 5–20 ∗ ∞
108 Chapter 5
2014 M. Toloo
Proof. Using Theorem 5–1, a similar procedure proves ∗ 0. On the other hand,
from the first and the second types of the constraints of model (5.4), we achieve
∑ 1 ⇒ ∃ . . ∀
∑ ∑ ∑ 1 ⇒ ∃ . . ∀
In addition, the last two sets of constraints result in
∗ min , … , , , … , min , ∞
which completes the proof.
Corollary 5–1 Model (5.3) is always feasible for the non‐Archimedean value
achieved from model (5.4).
Proof. According to Theorem 5–1, model (5.4) is always feasible. Let
, , , , , be a feasible solution of this model, then obviously
, , , , is a feasible solution of model (5.3) if 0 is considered as an
assurance value.
To illustrate the validity and discriminating power of MINLP model (5.3) we utilize
the data set in Table 5–4. The maximum value for non‐Archimedean epsilon for the
data is ∗ 0.071363. The following optimal solution of model (5.3) clarifies that
DMU is the most efficient DMU:
0,if 20,if 2
,0,if 21,if 2
Practically, dealing with non‐linear constraints and solving the non‐linear
programming problems might be difficult. Hence, we should show that the MINLP
models (5.3) and (5.4) are solvable. Unfortunately, there is no single algorithm for
dealing with the general nonlinear models, because of the unstable behavioral of
the nonlinear functions. Nevertheless, (nonlinear) DEA models usually have
special structure what with the nature of DEA method. The Karush‐Kuhn‐Tucker
(KKT) conditions are the most general approach to handle nonlinear programs (for
more details see Bazaraa et al., 2013). The objective functions of MINLP models are
affine but their feasible regions are non‐convex sets and consequently, the KKT
conditions are impractical. The constraints 0 1, … , make these
models nonlinear and therefore we utilize the separable programming indirect
technique to deal with these models. In doing so, let , then log
log log where the variable is non‐negative and the logarithmic
function is undefined for non‐positive values. Hence we assume that
where is an arbitrary positive number. For simplicity, let 1 that is
substituted in model (5.3)to obtain the following equivalent model:
Best Efficient Unit 109
Data Envelopment Analysis with Selected Models and Applications
mins. t.∑ 1 1, … ,
∑ ∑ 1 1, … ,
0 1, … ,
∑ 1
0 1, … ,
log log log 0 1, … ,
∈ 0,1 1, … ,
1 1, … ,
1 1, … ,
1, … ,1, … ,
(5.6)
Note that since the left hand side of the 5th and 6th set of constraints, i.e.
and log log , are separable functions, model (5.6) can be solved
using a separable programming approach (for more details see Taha, 2011).
Now, we apply the MINLP model proposed by Amin (2009) to the FLDs data set.
The optimal solution of the epsilon model (5.4) is ∗ 0.0000269. Considering the
maximum value for epsilon, the following results can be obtained by using model
(5.3):
∗ 0.3226, ∗ 1, 140, 14
, ∗ 0, 140, 14
Note that Amin (2009)’s method seeks the most efficient DMU without having need
of applying a trial‐and‐error approach. Table 5–5 presents the efficiency score
obtained from model (5.3). The table shows that Amin (2009)’s model with the
maximal value for epsilon selects FLD14 as the most efficient DMU. This result is
the same as the model of Eraty et al. (2006) with the maximum value for the epsilon.
Now, we consider the real data set containing 14 banks in the Czech Republic.
Firstly, we use model (5.4) to calculate the maximum epsilon, ∗ 7.366 10 ,
which is a very small number. Hence, we normalize the data to have a more
suitable result. Then, we utilize the method of Amin (2009) to find the most efficient
bank. Table 5–6 exhibits the normalized data and the efficiency score of DMUs with ∗ 0.3816.
The results show that FIO is the best bank in the Czech Republic. Although the
outputs of CS bank are more than the others, this bank is not selected as efficient
since it consumes more inputs.
Foroughi (2011) proposed the following alternative MIP model (5.7) for
determining a single efficient DMU.
110 Chapter 5
2014 M. Toloo
Table 5–5 Efficiency score obtained by the model of Amin (2009)
FLDs Efficiency FLDs Efficiency
1 0.60847 11 0.97681
2 0.73726 12 0.99979
3 0.71503 13 0.59968
4 0.65872 14 1
5 0.94248 15 0.74510
6 0.91725 16 0.97394
7 0.63407 17 0.69105
8 0.54882 18 0.65513
9 0.79538 19 0.67740
10 0.99994
Table 5–6 The normalized data and efficiency score
Banks Efficiency
AIR 0.0372 0.0359 0.0446 0.0228 0.6991
CMZRB 0.0202 0.1192 0.1263 0.0344 0.7410
CS 1 0.9821 1 1 0.7633
CSOB 0.7250 1 0.9143 0.9804 0.7978
EQB 0.0275 0.0096 0.0109 0.0115 0.5162
ERB 0.0067 0.0359 0.0043 0.0036 0.1195
FIO 0.0055 0.0198 0.0249 0.0132 1
GEMB 0.3110 0.1446 0.1410 0.2083 0.6357
ING 0.0272 0.1370 0.1344 0.0393 0.6858
JTB 0.0378 0.0908 0.0902 0.0804 0.9092
KB 0.8139 0.8396 0.8409 0.9232 0.8022
LBBW 0.0339 0.0334 0.0294 0.0052 0.3888
RB 0.2720 0.2109 0.2093 0.3070 0.8328
UCB 0.1862 0.3403 0.2833 0.3926 0.9042
maxs. t.∑ 1 1, … ,
∑ ∑ 0 1, … ,
∑ ∑ 1 1, … ,
∑ 1
∈ 0,1 1, … ,
1, … ,1, … ,
s free
(5.7)
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In this model, DMU is efficient unit if and only if ∗ 1. Foroughi (2011) claimed
that model (5.7) can be applied to rank the extreme efficient DMUs. To do this,
suppose that this model is solved and DMU ( ∗ 1) is identified as the first one
in the ranking from the top. Then a new constraint, 0, must be added to this
model to determine the second DMU, and this is repeated until all extreme efficient
DMUs are ranked.
Furthermore, Foroughi (2011) also claimed that we can ignore the role of epsilon
in the model:
Note that, in DEA, is usually used to discriminate between efficient
and weak efficient DMUs so it is assumed to be positive. However, as
it will be seen, the single efficient DMU obtained from the proposed
model is extreme efficient and so, even if we select 0, it will be
efficient which is also an advantage of the model.
We illustrate that if we select zero value for the non‐Archimedean epsilon in the
model of Foroughi (2011), then the model can be inapplicable to determine the best
efficient unit and consequently this approach fails to rank all extreme efficient units
as well. Consider the numerical example in Table 5–4. The optimal solution of
model (5.7), with 0, indicates that DMU is the best efficient unit with omitting
the role of three outputs, , , and ( ∗ ∗ ∗ 0):
∗ 0.3333, ∗ 1, ∗ 0.3333, 30, 3
, ∗ 1, 20, 2
However, it should be noted that if and are excluded from computations, then
DMU and DMU are identical. The following is an alternative optimal solution:
∗ 0.3333, ∗ 1, ∗ 0.3333, 20, 2
, ∗ 1, 10, 1
In conclusion, DMU can be selected as the best unit in addition to DMU .
Therefore, the numerical example clarifies that the Foroughi’s model (2011) is
unable to find the best single efficient unit when zero belongs to the assurance
interval for the non‐Archimedean epsilon. Indeed, in this case one of the alternative
solutions will be selected randomly (depending on the solution method or the
software used to solve the problem).
We illustrate that the in some cases method of Foroughi (2011) is unable to find a
single efficient DMU. Now, we show it may also fail to rank efficient units. Suppose
that we want to rank the units that are listed in Table 5–4. When 0, we apply
Foroughi’s approach (Foroughi, 2011). Solving model (5.7) results in ∗ 1 and
subsequently DMU is at the top of the ranking. Next, we obtain ∗ 1 after solving
the model with new constraint 1, which implies that DMU is at the 2nd of the
ranking. In a similar manner, DMU and DMU are the next ranked units,
successively.
112 Chapter 5
2014 M. Toloo
Table 5–7 The optimal solution of Foroughi’s model with 0
Added
Constraint ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
‐‐‐‐ 0.333 0 0 0.333 0 1 0 1 0 0
0 0.333 0 0.333 0 0 1 1 0 0 0
00 0 0 0 0 0 0 0 0 1 0
As a result, it seems that this approach is able to rank these units. The following
Table 5–7 shows that the optimal solution of Foroughi’s model (Foroughi, 2011)
after adding these two constraints ( 0, 0):
It can be easily seen that in model (5.7), alternative optimal solutions exist and
consequently the best DMU cannot be determined, correctly. In addition, in this
stage this model fails to rank the DMUs, as well. On the other hand, as shown in
the last row of Table 5–7, the optimal weights of all inputs and outputs are equal
to zero; hence DMU is at the 3rd position of the ranking without considering any
inputs and outputs, which is irrational. Indeed, the ranking approach of Foroughi
(2011) suffers from afore‐mentioned drawback as well.
One way to deal with this drawback is to consider an appropriate lower bound on
the weights. In doing so, we first formulate the following MIP model to determine
the maximum non–Archimedean epsilon for model (5.7):
ε∗ maxs. t.∑ 1 1, … ,
∑ ∑ 0 1, … ,
∑ ∑ 1 1, … ,
∑ 1
0 1, … ,0 1, … ,
∈ 0,1 1,… ,
s free
(5.8)
This model has some interesting properties.
Theorem 5–3 Model (5.8) is always feasible.
Proof. Similar to the proof of Theorem 5–1, let ∗, ∗, ∗ be an optimal solution of
the multiplier CCR model that is obtained from the complementary slackness
conditions and ∗ be an assurance value. Clearly, ∗
,∗
,∗
, , is a feasible
solution to model (5.8) where
Best Efficient Unit 113
Data Envelopment Analysis with Selected Models and Applications
min ∗ : : 1, … , ,
min∗ ∗
: 1, … , ,
1, 0,
which completes the proof. □
Similar to Theorem 5–2, Theorem 5–7 shows that the optimal solution value of
model (5.8) is positive and finite. Hence, in the model of Foroughi (2011) the
assurance interval for the non‐Archimedean epsilon is 0, ∗ where ∗ is the
optimal objective value of the MIP model (5.8) and the best efficient unit can be
determined for any epsilon value which belongs to the assurance interval.
The optimal objective value of model (5.8) for the data set in Table 5–4 is ∗
0.0769. Table 5–8 reports the optimal solution of model (5.7) when the epsilon
value is 0.0769:
Table 5–8 The optimal solution of Foroughi’s model with ∗ 0.0769
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
0.1767 0.0522 0.1245 0.0522 0.0522 1 1 0
It is easy to show that there is no alternative optimal solution here and DMU is the
most efficient unit. Similar to the approach of Foroughi (2011), to rank all DMUs
we add a new related constraint ( 1) to the model, then the optimal solution
declares that DMU is at the 2nd position in the ranking order. In a similar manner,
DMU and DMU will be ranked at the 3rd and 4th positions, respectively. It is
noteworthy that considering the maximum non‐Archimedean epsilon value, the
weights of inputs and outputs always remain strictly positive.
Wang and Jiang (2012) showed that the model of Foroughi (2011) involves many
redundant constraints. The authors formulated some novel MIP models for
identifying the most efficient DMU under different returns to scales, which contain
only essential constraints and decision variables and are much simpler and more
succinct compared to Foroughi’s.
5.2 DEA‐WEI approach
In this section, we provide some approaches for finding the most efficient DMU
with pure output measures or a single input measure under CRS assumption.
Practically, there are some problems where either all measures are output type or
there is a single constant input measure, for example, in technology selection
(Karsak and Ahiska, 2005; Amin et al., 2006), discovered association rules from data
mining (Toloo et al., 2009), and professional tennis players (Ramón et al., 2012;
Toloo, 2013).
114 Chapter 5
2014 M. Toloo
5.2.1 Penalty function approach
Karsak and Ahiska (2005) introduced a practical common weight multi‐criteria
decision making (MCDM) method with an improved discriminating power for
technology selection. In contrast with conventional DEA models, the authors
formulated the following integrated DEA model for dealing with single (but not
necessary constant) input and multiple outputs , … , data set with CSW:
mins. t.∑
1 1, … ,
0 1, … ,
0 1, … ,
1, … ,
(5.9)
Let ∗ , ∗, ∗ be the optimal solution of model (5.9). In this model, the
efficiency score of DMU with the common set of optimal weights ∗ is equal to 1∗and consequently this unit is efficient if and only if ∗ 0 (or equivalently ∗ ). Now, let be indexes set of efficient DMUs, i.e. : ∗ 0 ,
which is named the efficient set. If the efficient set is singleton and ∈ , then
DMU is the unique efficient DMU with the common set of optimal weights ∗and
can be considered as the most efficient unit. Otherwise, this model fails to find the
most efficient unit. To tackle this problem Karsak and Ahiska (2005) formulated
the following model:
min ∑ ∈
s. t.∑
1 1, … ,
0 1, … ,
0 1, … ,
1, … ,
(5.10)
where ∈ 0,1 is a discriminating parameter. Note that model (5.10) for 0
measures the minimax efficiency score of DMUs.
The second term of objective function, ∑ ∈ , enhances the discriminating power
of the model and can be interpreted as a penalty function. This function is
considered to make ∈ variables as large as possible. The discriminating
parameter plays an important role for obtaining the best efficient unit. Karsak
and Ahiska (2008) designed a bi‐section algorithm to obtain an appropriate value
of over an interval , at iteration where the search interval is 0,1 at the
initial iteration. At each iteration the value to be assigned to parameter , say
for iteration , is set to be the midpoint of the interval , , and based on the
degree of discrimination. In this regard, two cases arise: (i) all DMUs are inefficient
in the current solution which means that must be varies within interval , ,
(ii) more than a single efficient DMU exist in the current solution which updates
Best Efficient Unit 115
Data Envelopment Analysis with Selected Models and Applications
Table 5–9 Various efficiency scores (Karsak and Ahiska, 2005)
Robots CCR‐
efficiency Model (5.10) with 0 0.1 0.2
R1 0.653 0.653 0.653 0.634
R2 0.821 0.753 0.753 0.694
R3 0.954 0.883 0.883 0.777
R4 0.95 0.862 0.862 0.784
R5 1 1 1 0.886
R6 0.563 0.563 0.563 0.55
R7 0.683 0.683 0.683 0.677
R8 1 0.631 0.631 0.55
R9 0.765 0.687 0.687 0.623
R10 0.714 0.617 0.617 0.632
R11 0.909 0.89 0.89 0.892
R12 1 1 1 1
the new interval , for the next iteration. This procedure is repeated until
either one of the two stopping conditions for the algorithm is satisfied. One
stopping condition is defined as stop when the value assigned to results in a single
efficient DMU. The other stopping condition is defined as stop when the length of the
current interval is less than or equal to the accuracy of the given tolerance. For a deeper
discussion and more details about steps on the bi‐section algorithm we refer
readers to Karsak and Ahiska (2008). Karsak and Ahiska (2005, 2008) applied their
approaches to deal with industrial robot selection problem, which was introduced,
in Chapter 2. The authors considered 0.00001and determined R12 as the most
efficient robot, as is shown in Table 5–9.
The second column indicates the efficiency scores of DMUs and there are three
CCR‐efficient robots, i.e. R5, R8 and R12. The minimax efficiency score under 0
in model (5.10)improves the discriminating power when the number of efficient
DMUs is reduced into two. However, the model fails to identify the most efficient
DMU when 1. Finally, in order to determine the best robot, Karsak and Ahiska
(2005) considered the min 0.2 objective function and determined
a single efficient robot, R12.
5.2.2 Minimax approach
Karsak and Ahiska (2005, 2008) presented some numerical examples to illustrate
that their approach always finds a suitable value for with a single efficient DMU,
nevertheless their approach was criticized by Amin et al. (2006) and Toloo (2013).
The existence of the most efficient DMUs in the model of Karsak and Ahiska (2005)
means that there would exist a suitable ∈ 0,1 such that | | 1. Amin et al.
(2006) formulated an MIP model in which it assures that | | 1 without
considering the parameter . In doing so, the authors instead of utilizing a penalty
116 Chapter 5
2014 M. Toloo
function proposed by Karsak and Ahiska (2005), tried to restrict the feasible region
as much as possible. In fact, Amin et al. (2006) defined as deviation from
the efficiency (instead of in the model of Karsak and Ahiska, 2005) where 0
1 and suggested to impose the constraint ∑ 1 where is a binary
variable:
mins. t.∑
1 1, … ,
0 1, … ,
0 1 1, … ,
∑ 1
∈ 0,1 1, … ,
1, … ,
(5.11)
Here 1 ∗ ∗ ∑ ∗
is efficiency score of DMU . Due to the constaint
∑ 1 there exists only a single binary variable with a vlaue of zero, say
d∗ 0. In this case, the efficiency score of DMU is 1 ∗ whereas the efficiency
score of DMU , ∀ is ∗. As a result, model (5.11) allows the efficiency score of
DMU to be larger than one, whereas the efficiencies of the other DMUs are all less
than or equal to one.
Amin et al. (2006) did not formulated a model for finding the maximum epsilon
value for their model, however, to do this, we suggest the following model:
maxs. t.∑
1 1, … ,
0 1 1, … ,
∑ 1
0 1,… ,
∈ 0,1 1, … ,
(5.12)
Now, we apply the model of Amin et al. (2006) for the data set of 12 industrial
robots. The maximum value of epsilon is ∗ 6.46 10 . Similar to Karsak and
Ahiska (2005, 2008), Table 5–10 shows that R12 is the most efficient robot.
Although the efficiency score of R11 is equal to one, this unit is not identified as the
most efficient DMU because the efficiency score of R12 is greater than one.
Toloo (2013) formulated a new DEA‐WEI model to find the most efficient DMU
where the efficiency score of a DMU is equal to one and the efficiency score for
other DMUs is less than one. He firstly criticized the model of Karsak and Ahiska
(2005) and then utilized a real data set containing 40 professional tennis players to
illustrate a drawback of the approach of Karsak and Ahiska (2005, 2008). More
Best Efficient Unit 117
Data Envelopment Analysis with Selected Models and Applications
Table 5–10 Efficiency scores of Amin et al. (2006)
Robots Efficiency Robots Efficiency
R1 0.5861 R7 0.6963
R2 0.4484 R8 0.2428
R3 0.2996 R9 0.3694
R4 0.4506 R10 0.7773
R5 0.3711 R11 1
R6 0.5337 R12 1.1616
precisely, Karsak and Ahiska (2008) designed a bi‐section algorithm to compute
the values of discriminating parameter in a systematic manner. Indeed, the bi‐
section algorithm avoids the burden of using a trial‐and‐error procedure and
calculates the value of in an iterative manner with the benefit of bi‐section search.
It is also claimed that the aim of the bi‐section algorithm is to find the value that
decreases the number of efficient DMUs as much as possible. Karsak and Ahiska
(2008) presented an illustrative numerical example to validate the proposed the bi‐
section algorithm. Mathematically, the penalty function ∑ ∈ results in
positive ∈ variable (as much as possible), which can increase the discriminating
power of the model. However, there is no guarantee that only one efficient DMU
will be obtained as the most efficient unit. In the ensuing section, we present a
numerical example of professional tennis players that is adapted from Toloo (2013).
This example enable us to illustrate a drawback of the approach of Karsak and
Ahiska (2005, 2008).
5.2.3 Professional tennis players
We utilize a real data set involving of 40 professional tennis players that is taken
from Ramón et al. (2012). This data set is chosen because there is a severe
competition between these professional tennis players, which makes it hard to find
the best one. The data set, which is exhibited in Table (1), is originally located in
the official website of the association of tennis professionals (ATP) at
http://www.atpworldtour.com. The ATP provides a ranking of players that reflects
their performance in the tournaments during a season. To be specific, the players
are given a different amount of points, which depends mainly on the rounds they
obtained in the tournaments of a season, and the ranking is eventually determined
according to the total points. Thus, the ATP ranking is concerned with the
competitive performance of the players. Contrarily, Ramón et al. (2012) derived a
ranking of 53 players that reflects the efficiency performance of their game. The
ATP statistics reports data regarding the different aspects of the game of the players
such as the percentage of the 1st serve points won or the percentage of return games
won. These aspects can be used to derive a ranking of players regarding their
performance. Ramón et al. (2012) used the CSW approach in order to define an
overall index of performance of the players.
118 Chapter 5
2014 M. Toloo
Table 5–11 The data set of 40 professional tennis players
No. Player
1 Federer 62 79 57 90 69 31 51 41 24
2 Nadal 68 71 57 84 65 33 57 47 34
3 Djokovic 63 79 54 85 66 33 54 42 31
4 Murray 58 76 54 85 65 35 56 46 33
5 del Potro 62 74 53 84 65 31 53 42 27
6 Davydenko 67 71 55 83 64 34 54 41 31
7 Roddick 70 79 57 91 64 26 49 37 19
8 Soderling 60 78 54 86 65 31 51 44 25
9 Verdasco 69 72 54 85 66 31 53 45 28
10 Tsonga 63 80 54 89 67 28 47 38 19
11 Gonzalez 63 77 53 88 71 29 49 38 22
12 Stepanek 61 73 50 80 62 31 53 40 25
13 Monfils 62 76 50 84 63 30 50 42 25
14 Cillic 56 75 54 84 65 33 51 38 27
15 Simon 55 74 54 82 67 30 52 43 25
16 Robredo 63 72 54 81 61 31 51 44 25
17 Ferrer 61 69 52 77 60 32 55 43 32
18 Haas 60 77 53 85 66 28 48 39 21
19 Youzhny 62 71 52 80 60 31 51 40 26
20 Berdych 59 74 53 81 61 30 50 37 22
21 Wawrinka 58 73 52 81 66 32 50 38 25
22 Hewitt 53 76 53 81 62 31 53 39 28
23 Ferrero 67 68 54 78 60 29 53 43 26
24 Ljubicic 59 78 51 85 67 27 48 35 16
25 Querrey 60 79 52 86 60 27 48 39 19
26 Almagro 59 75 52 82 60 28 49 40 21
27 Kohlschreiber 66 70 56 82 66 30 50 41 24
28 Melzer 60 74 51 81 59 30 47 40 21
29 Troicki 59 71 45 73 63 30 50 44 25
30 Montantes 58 71 50 76 60 30 51 45 24
31 Chardy 59 75 52 82 62 26 47 36 19
32 Mathieu 57 71 50 78 61 26 49 42 20
33 Isner 67 75 56 89 70 22 42 32 11
34 Andreev 62 71 52 80 61 29 48 37 20
35 Karlovic 67 85 54 92 69 21 42 32 10
36 Tipsarevic 56 74 52 80 61 28 50 43 24
37 Beck 59 72 52 81 66 28 50 33 19
38 Garcia‐López 61 69 49 75 59 30 53 40 27
39 Blake 57 74 52 82 63 27 48 38 19
40 Bennetau 66 70 48 77 59 29 50 40 22
We consider the data for 40 professional tennis players, which are shown in Table
5–11. For each player, the following measures are considered that are concerned
with different aspects of both service and return:
Best Efficient Unit 119
Data Envelopment Analysis with Selected Models and Applications
Table 5–12 Efficiency status for various
Efficiency status of players
0 Djokovic and Nadal are efficient
0.5 All players are inefficient
0.25 Djokovic and Nadal are efficient
0.375 Djokovic and Karlovic are efficient
0. 4375 Karlovic is efficient!
= percentage of 1st serve.
= percentage of 1st serve points won.
= percentage of 2nd serve points won.
= percentage of service games won.
= percentage of break points saved.
= percentage of points won returning 1st serve.
= percentage of points won returning 2nd serve.
= percentage of break points converted.
= percentage of return games won.
The problem of professional tennis players is solved using the proposed bi‐section
algorithm of Karsak and Ahiska (2008); the results are summarized in Table 5–12.
As can be seen, Nadal (Player 2) and Djokovic (Player 3) are minimax efficient (
0). As Karsak and Ahiska claimed, to improve the discriminating power of their
model, we apply the bi‐section algorithm. The initial search interval is 0,1 , hence
we assume 0.5 in the first iteration of the algorithm. In this case, all players
have an efficiency score less than 1. Next assume 0.25 by setting the second
search interval to be 0,0.5 . Therefore, Djokovic and Nadal are efficient. When we
assume 0.375 an unexpected result is obtained: Djokovic and Karlovic are
efficient while Karlovic is not even a minimax efficient. The final result is
completely incorrect when we assume 0.4375 (based on bi‐section algorithm):
Karlovic is the most efficient professional tennis player. In other words, an
inefficient player is determined as the most efficient player. Obviously, it is illogical
to select an inefficient unit as the most efficient unit.
5.2.4 New minimax method
The professional tennis players example illustrated the main drawback of Karsak
and Ahiska (2005)’s approach. However, we tackle this problem with resorting to
auxiliary binary variables. Notwithstanding a penalty function ∑ ∈ on the
objective function of the integrated minimax DEA model of Karsak and Ahiska
(2005), we appropriately restrict the feasible region of the model to obtain a single
efficient unit. The author formulated the following integrated MIP model for
selecting the most efficient unit without explicit inputs:
120 Chapter 5
2014 M. Toloo
mins. t.∑ 1 1, … ,
0 1, … ,
∑ 1
1, … ,
∈ 0,1 1, … ,
0 1,… ,
1, … ,
(5.13)
where M is a large enough positive number.
In this model, the third set of constraints lead to only one auxiliary binary variable
with a value of zero, 0. Consequently, the constraint for index
implies 0. In this case, constraint is redundant. On the other hand,
we have 1, ∀ and hence, for a large enough value of , the constraints
lead to 0, ∀ also constraints , ∀ are redundant.
Therefore, 0 if and only if 0 and more importantly 0 if and only if
1.
In integrated DEA models the non‐Archimedean epsilon plays an important role
and must be determined correctly. Otherwise, the related model might be
infeasible. In order to find an appropriate value for epsilon in model (5.13), the
following MIP model was suggested:
maxs. t.∑ 1 1, … ,
∑ 1
1, … ,
0 1, … ,
∈ 0,1 1,… ,
0 1,… ,
(5.14)
The following theorems validate the suggested approach (for the proofs we refer
the reader to Toloo, 2013).
Theorem 5–4 Model (5.14) is always feasible.
Theorem 5–5 Model (5.13) is feasible when the optimal objective value of model
(5.14) is selected as the non‐Archimedean epsilon.
Theorem 5–6 Model (5.13) determines a single efficient unit.
Up to this point, we theoretically showed some interesting features of the Toloo
(2013)’s approach. Now, we apply this approach for the real data set without
explicit inputs in Table 5–11. The optimal solution of model (5.14) is ∗ 0.00194.
Considering this value as an assurance value for the non‐Archimedean epsilon,
model (5.13) is solved and the following results are achieved
Best Efficient Unit 121
Data Envelopment Analysis with Selected Models and Applications
∗ 0, 21, 2
, ∗ 0, 20, 2
In the other words, the second player (Rafael Nadal) is the most efficient
professional tennis player. It is important to verify that no alternative solution
exists for model (5.13). Alternative solutions play an important role in DEA models
(for instance see Toloo, 2012). If we add a new constraint 1 and resolve this
model, then the optimal objective value will increase and there is no alternative
optimal solution. Thereby, Rafael Nadal is the most efficient professional tennis
player.
5.3 DEA‐WEO approach
In this section, we suggest an approach for finding the most efficient DMU when
there is a pure input data set. The idea stems from the use of the minimax method
for WEI condition in order to deal with a pure output data set. We firstly formulate
an integrated DEA‐WEO model and then suggest imposing some suitable
constraints on the model to obtain a single efficient DMU.
We propose the following integrated nonlinear minimax model for measuring
CSW‐efficiency under without explicit outputs condition:
minmax : 1, … ,s. t.∑ 1 1, … ,
0 1,… ,
1, … ,
(5.15)
The minimax objective can be transformed by including an additional decision
variable , which represents the maximum of deviations, max :
1, … , . In order to establish this relationship, the extra constraints
0 1, … , must be imposed to model (5.15). When is minimized, these
constraints ensure that will be greater than or equal to , ∀ . At the same
time, the optimal value of will be no greater than the maximum of all
because has been minimized. Therefore, the optimal value of will be
both as small as possible and exactly equal to the maximum over :
mins. t.∑ 1 1, … ,
0 1, … ,
0 1,… ,
1, … ,
(5.16)
In this model, ∗ is the CSW‐efficiency score of DMU and hence DMU is CSW‐
efficient if and only if ∗ 0; otherwise, is CSW‐inefficient. The following
122 Chapter 5
2014 M. Toloo
theorems express the relation between CSW‐efficiency and CCR‐efficiency scores
(for the proofs we refer readers to Toloo and Kresta, 2014):
Theorem 5–7 The CSW‐efficiency score of a DMU with pure input data is greater
than or equal to its CCR‐efficiency score.
Theorem 5–8 The CSW‐efficient DMU with pure input data is CCR‐efficient.
These theorems ensure that the discrimination power of integrated model (5.16) is
more than the CCR model when there is explicit outputs. It is therefore of interest
to develop model (5.16) to deal with finding the most efficient DMU without
explicit outputs. In doing so, the following integrated MIP model can be utilized:
mins. t.∑ 1 1, … ,
0 1, … ,
∑ 1
1, … ,
1, … ,
∈ 0,1 1,… ,
1, … ,
(5.17)
where and are large enough positive numbers. Note that since the output‐
oriented CCR model is utilized, unlike Toloo (2013), can be greater than one. To
meet this condition, the constraint (for a large enough ) is imposed to
the model. We suggest the following integrated MIP model to obtain the maximum
value of epsilon for model (5.17):
maxs. t.∑ 1 1, … ,
0 1, … ,
∑ 1
1, … ,
1, … ,
0 1, … ,∈ 0,1 1,… ,
(5.18)
The following theorem help us to verify the validity of the DEA‐WEO method (for
the proofs we refer readers to Toloo and Kresta, 2014).
Theorem 5–9 Model (5.17) determines a single CCR‐efficient unit.
Theorem 5–10 Model (5.18) is always feasible
Theorem 5–11 Model (5.17) is feasible when the optimal objective value of model
(5.18) is selected as the non‐Archimedean epsilon.
Best Efficient Unit 123
Data Envelopment Analysis with Selected Models and Applications
Table 5–13 Efficiency score of asset financing alternatives
DMUs Efficiency DMUs Efficiency DMUs Efficiency DMUs Efficiency
1 0.608273 36 0.055018 71 0.059766 106 0.072422
2 0.211282 37 0.047188 72 0.060234 107 0.073153
3 0.127845 38 0.041309 73 0.060569 108 0.073687
4 0.091659 39 0.036732 74 0.043908 109 0.049848
5 0.071434 40 0.033069 75 0.047781 110 0.055298
6 0.058524 41 0.03007 76 0.049225 111 0.057386
7 0.049549 42 0.027569 77 0.049978 112 0.058486
8 0.042972 43 0.025453 78 0.050439 113 0.059165
9 0.037938 44 0.023638 79 0.050751 114 0.059623
10 0.033958 45 0.022065 80 0.050974 115 0.059956
11 0.030733 46 0.065963 81 0.038976 116 0.043584
12 0.028068 47 0.06958 82 0.041764 117 0.047398
13 0.025822 48 0.071418 83 0.042783 118 0.048819
14 0.023913 49 0.072036 84 0.043311 119 0.049559
15 0.022267 50 0.072532 85 0.043632 120 0.050013
16 0.655738 51 0.073298 86 0.043846 121 0.050317
17 0.21645 52 0.07385 87 0.044001 122 0.050538
18 0.129618 53 0.181587 88 0.035039 123 0.038719
19 0.092533 54 0.348432 89 0.037095 124 0.041471
20 0.071932 55 0.501756 90 0.037833 125 0.042475
21 0.058834 56 0.642674 91 0.038212 126 0.042994
22 0.049774 57 0.772201 92 0.038442 127 0.043311
23 0.043126 58 0.890472 93 0.038597 128 0.043524
24 0.038045 59 1 94 0.038707 129 0.043676
25 0.034038 60 0.058792 95 0.17618 130 0.034832
26 0.030794 61 0.067114 96 0.329056 131 0.036862
27 0.028114 62 0.070432 97 0.462535 132 0.037593
28 0.025858 63 0.072213 98 0.579374 133 0.037967
29 0.023942 64 0.073319 99 0.682594 134 0.038194
30 0.022289 65 0.074074 100 0.773994 135 0.038346
31 0.318878 66 0.074616 101 0.855432 136 0.038456
32 0.162681 67 0.050271 102 0.058214 137 0.545554
33 0.109409 68 0.055822 103 0.066361 138 0.582072
34 0.082318 69 0.057947 104 0.069604 139 0.647249
35 0.065963 70 0.05907 105 0.071342
In order to illustrate the potential uses of the suggested approach, we utilize a real
data set involving 139 different alternatives for long term asset financing provided
by Czech banks and leasing companies which was presented in Chapter 2.
The optimal solution of model (5.17) with the maximum value of epsilon ∗
0.000087 is summarized in Table 5–13.
124 Chapter 5
2014 M. Toloo
Hence, DMU59 is selected as the best efficient DMU. The 59th financing product
denotes bank loan from Komerční banka (group Société Générale) with zero down
payment, minimal annuities and no other fees, thus all the costs (the first three
inputs) are minimized for the entrepreneur.
To verify whether there is an alternative solution or not, we impose a new
constraint 1 on model (5.17) and resolve the model. In this case, the optimal
objective value increases and it proves that there is no alternative solution and
DMU59 is the single efficient unit.
Furthermore, some interesting results can be extracted from Table 5–12: it is better
to utilize bank loan since the six most efficient DMUs are 59, 58, 101, 100, 57 and
99, respectively. All of them are bank loan offers provided by Komerční Banka. In
addition, the worst six DMUs are 45, 15, 30, 44, 14, and 29, respectively. These
DMUs denote two product types from the same group (KBC group): finance lease
from ČSOB leasing and bank loan from the ČSOB bank. Hence, it can be concluded
that for car financing it is more efficient to utilize bank loan provided by Komerční
banka than the financing offers of the other companies. Products from KBC group
(both the bank loan and finance lease) are inefficient for the entrepreneur.
5.4 Epsilon‐free approaches
Wang and Jiang (2012) introduced the following MIP model for selecting the best
efficient DMU under CRS assumption:
∗ min∑ ∑ ∑ ∑
s. t.∑ ∑ 1, … ,
∑ 1
∈ 0,1 1, … ,
1, … ,
1,… ,
(5.19)
where 1, … , are the binary variables. In this model, DMU is determined
as the best efficient unit if and only if ∗ 1. Note that to have positive weights,
Wang and Jiang (2012) borrowed max and max for
the lower bounds from slack‐adjusted DEA models suggested by Sueyoshi (1999).
Wang and Jiang (2012) proved that (5.19) is feasible and can always produce an
optimal solution as well as presenting some important advantages over the
previous models: fewer variables and constraints, more reliable, and more
practical. Furthermore, it is also asserted that the main aim of their model is to find
a CSW that maximizes the efficiency scores of all DMUs. Toloo (2015) manipulated
the model of Wang and Jiang (2012) to formulate a new minimax model.
Best Efficient Unit 125
Data Envelopment Analysis with Selected Models and Applications
We begin our discussion with proving that in model (5.19), the efficiency score for
just one unit is larger than one and for the others is less than or equal to one.
Toward this end, we suppose that model (5.19) is solved and the optimal solution, ∗, ∗, ∗ , is in place. In addition, we assume
1
max
1
max
∑ ∗ ∑ ∗ ∗ 0
Theorem 5–12 In model (5.19), .
Proof. There is no loss of generality if we assume that at optimality ∗ 1, ∀
∗ 0. Now, suppose ∗ is fixed and consider the following corresponding LP
∗ min∑ ∑ ∑ ∑
s. t.∑ ∑ 1
∑ ∑ 0 1, … ,
1, … ,
1, … ,
(5.20)
Clearly, ∗, ∗ is the optimal solution of model (5.20) with 21∗ 22
∗ . Now,
consider the dual of model (5.20) as follows:
∗ max ∑ ∑s. t.∑ ∑ 1, … ,
∑ ∑ 1, … ,
0 1, … ,
0 1,… ,0 1, … ,
(5.21)
The following complementary slackness conditions hold for any optimal solution ∗, ∗ of model (5.21) and ∗, ∗, ∗ of model (5.21):
∗ 1 ∑ ∑ 0∗ ∑ ∑ 0 1, … ,∗ 0 1, … ,∗ 0 1, … ,
Suppose, contrary to our claim, that . The complementary slackness
conditions provided that ∀ ∗ 0, but from the dual feasibility conditions, i.e. the
first set of constraints in model (5.21) we obtain ∀ ∑ which is a
contradiction. □
Theorem 5–13 In model (5.19), ∗ 1, ∀ ∗ 1.
126 Chapter 5
2014 M. Toloo
Proof. To obtain a contradiction, assume ∀ ∗ 1. Obviously, with this
assumption, model (5.20) is equivalent to the following LP model:
24∗ min∑ ∑ ∑ ∑
s. t.∑ ∑ 0 1, … ,
1, … ,
1, … ,
(5.22)
It should be noticed here that 24∗
22∗
21∗ . Now, considering the special
structure of the model (5.22), we select a tight constraint, ∑ ∑
0, such that increasing its right hand side will decrease the optimal objective
function value ( ∗ ). Note that we have ; otherwise ∗ 1. Consider the
following LP model that can be gained by increasing the right hand side of the tight
constraint ∑ ∑ ` 0 from zero to one:
z25∗ min∑ ∑ ∑ ∑
s. t.∑ ∑ 1
∑ ∑ 0 1, … ,
1, … ,
1, … ,
(5.23)
Let ∗, ∗ be the optimal solution of this model and ∗ 1, ∀ ∗ 0. Clearly, ∗, ∗, ∗ is a feasible solution of model (5.19) with an objective function value less
than 1∗ which is a contradiction. □
So far, we showed that the selected efficient DMU outperforms other DMUs. In the
remainder of this section, we manipulate the model of Wang and Jiang (2012) to
obtain an equivalent model.
Theorem 5–14 Model (5.19) is equivalent to the following model:
min∑
s. t.∑ ∑ 1, … ,
∑ 1
∈ 0,1 1, … ,
0 1, … ,
1, … ,
1, … ,
(5.24)
Proof. From the first set of constraints in model (5.24), for 1,… , we have
∑ ∑ ; summing up these constraints over from 1 to
, results in ∑ ∑ ∑ ∑ ∑ ∑ . On the other
hand, considering the second set of constraints in this model, ∑ 1, and
substituting it for the obtained relation ∑ ∑ ∑
∑ ∑ 1 will be in place. Ignoring the constant 1 enables us to
Best Efficient Unit 127
Data Envelopment Analysis with Selected Models and Applications
consider the objective function of model (5.24) as ∑ ∑
∑ ∑ which is equal to the objective function of model (5.19). The
reverse is also true which completes the proof. □
We also prove that model (5.24) is equivalent to the following model:
min∑
s. t.∑ ∑ 0 1, … ,
∑ 1
∈ 0,1 1, … ,
1 1, … ,
1, … ,
1, … ,
(5.25)
Theorem 5–15 Models (5.24) and (5.25) are equivalent.
Proof. In model (5.24), let 1 and 1 for 1,… , . Evidently,
∑ ∑ 1 and ∑ ∑ ∑ ∑ ∑ .
Given max∑ is equal to min∑ ∑ ∑ ∑ the proof
completes. □
On the other hand, ∑ is a constant and hence model (5.25) is equivalent to the
following model:
min∑
s. t.∑ ∑ 0 1, … ,
∑ 1
∈ 0,1 1, … ,
1 1, … ,
1, … ,
1, … ,
(5.26)
Corollary 5–2 The model of Wang and Jiang (2012) is equivalent to model (2.28).
Note that according to Theorem 5–13 ( ∗ 1, ∀ ∗ 1) and Corollary 5–2 we
conclude that ∗ ∗ 0 and ∀ , ∗ ∗ 0. We can then identify whether
a DMU is the most efficient DMU as follows:
Definition 5–1 In model (2.28), DMU is the most efficient unit if and only if ∗
∗ min ∗ ∗ 1, … , .
Toloo (2015) formulated the following minimax model as an alternative MIP for
identifying the most efficient unit under CRS assumption:
128 Chapter 5
2014 M. Toloo
mins. t.∑ ∑ 0 1,… ,
0 1, … ,
∑ 1
∈ 0,1 1, … ,
1 1, … ,
1, … ,
1, … ,
(5.27)
Let us firstly show some essential properties of model (5.27) and then compare this
model with some afore‐mentioned models.
Theorem 5–16 Model (5.27) is always feasible.
Proof. Let , , be a feasible solution to model (5.19). Note that Wang and
Jiang (2012) proved that such solution exists. Let 1 and 1
for 1,… , and max 1,… , . It is easy to verify
that , , , , is a feasible solution to model (5.27). □
Theorem 5–17 The optimal objective value of model (5.27) is bounded.
Proof. Let , , , , be a feasible solution to model (5.29). From the
constrains of this model we have 1 which means the
objective function value of any feasible solution is bounded. This fact that model
(5.27) is a minimization problem completes the proof. □
Similar to the proof of Theorem 5–13, we can demonstrate that in the new proposed
minimax model (5.27), only the efficiency score of most efficient DMU is more than
unity and the efficiencies of the other DMUs are all less than or equal to one.
We utilize the epsilon‐free common set of weight minimax approach to formulate
the following model:
mins. t.∑ ∑ 0 1,… ,
0 1, … ,
∑ 1
1, … ,
1, … ,
∈ 0,1 1, … ,
1 1, … ,1 1, … ,
(5.28)
where M and N are large positive numbers.
In this model, the auxiliary binary variables, 1,… , , are
introduced to provide a group of mutually exclusive alternatives, so the
Best Efficient Unit 129
Data Envelopment Analysis with Selected Models and Applications
Table 5–14 Efficiency scores by epsilon‐free approaches.
Banks Wang and Jiang (2012) Toloo (2014a) Toloo (2015)
AIR 0.7330 0.8674 0.8913
CMZRB 0.6835 0.7428 0.8418
CS 0.8661 0.894 0.9174
CSOB 0.8979 0.8646 0.9127
EQB 0.6239 0.7997 0.7234
ERB 0.1252 0.1115 0.1255
FIO 1 1 1.115
GEMB 0.7861 0.8782 0.8254
ING 0.6388 0.6925 0.7816
JTB 0.9792 0.9145 1
KB 0.9181 0.9218 0.9498
LBBW 0.3676 0.5092 0.5259
RB 1 0.9999 1
UCB 1.0355 0.9120 0.9802
deviation from efficiency variable, , for just one DMU can take a
value of zero.
The following theorem, which can be proved similar to Theorem 4 in Toloo
(2013), ensures that the formulated model (5.28) is valid.
Theorem 5–18 In model (5.28) there exists just a single efficient DMU.
Proof. Let ∗ , ∗, ∗, ∗, ∗ be the optimal solution of model (5.28). There is
no loss of generality if we assume ∗ 0, ∗ 1, implies ∗ 0, ∗
0. In this case, by means of the second set of constraints we have ∗ ∗
0. This shows that DMUk is an efficient unit by using the optimal CSW, ∗, ∗ .
Now, suppose that two efficient units are determined using the optimal CSW,
say DMU and DMU . With this assumption, we have ∗ ∗ 0 and
subsequently ∗ ∗ 0 which leads to a contradiction with the feasibility of
the given optimal solution. Thus, DMU and DMU could not be efficient units at
the same time. □
In order to compare these methods, we consider the data set containing 14 active
banks in the Czech Republic. Table 5–14 summarizes efficiency scores obtained
by epsilon‐free approaches.
The bold values in the table highlight the efficiency score for the most efficient
banks. The results reveal that UCB is selected as the most efficient bank based on
the method of Wang and Jiang (2012). Nevertheless, it is different from the
selection made by the methods of Toloo (2014a, 2015). It should be noted here
that it is understandable that different models may result in different selection.
In addition, the methods of Toloo (2015) and Wang and Jiang (2012) allows the
efficiency score of one unit to be larger than one and the efficiency score of other
DMU are less than or equal to one. In these approaches, the most efficient DMU
130 Chapter 5
2014 M. Toloo
is a unit when its efficiency score is more than one while in the method of Toloo
(2014a), the efficiency score of all DMUs are less than or equal to one. In Toloo
(2014a)’s method, there is only a single DMU with efficiency score of one which
is selected as the most efficient DMU.
5.5 Most cost efficient DMU
Cost efficiency (CE) is a useful tool to evaluate the ability of a DMU to produce the
current outputs at minimal cost, given its input prices. In other words, CE can be
expressed as a measure of the potential cost reduction reachable given the outputs
and the current input prices at each DMU. The concept of CE with price uncertainty
two approaches are developed: the most favorable price (optimistic perspective)
and the least favorable price (pessimistic perspective). In the following sub‐section,
we extend an approach for finding the most cost efficient DMU with input price
uncertainly.
5.5.1 CE with input price uncertainly
Thompson et al. (1996), Schaffnit et al. (1997) and Taylor et al. (1997) utilized
assurance region (AR) method in the multiplier form of DEA models to develop
new DEA/AR models for measuring CE under input price uncertainly. The aim of
these studies was to propose an optimistic DEA model for measuring the CE,
whereas Camanho and Dyson (2005) enhanced these methods to account for
different scenarios relating to input price information.
The following model measures the optimistic CE with input price uncertainty:
max ∑s. t.∑ 1
∑ ∑ 0 1, … ,
, , 1, … ,
1, … ,
(5.29)
where and are minimum and maximum bounds estimated for the price
of input of DMU . This model is formulated by imposing 2 input weight
restrictions on the standard DEA model. The bounds of the CE measure are
obtained from assessments in the light of the most favorable price scenario and the
least favorable price scenario.
Note that changing the objective function of model (5.29) from maximization to a
minimization leads to approximately zero CE for all DMUs assessed. Hence it is not
easy to extend a model for measuring the pessimistic CE with input price
uncertainty. However, Camanho and Dyson (2005) tackled this issue and extended
the following model:
Best Efficient Unit 131
Data Envelopment Analysis with Selected Models and Applications
min ∑s. t.∑ 1
∑ ∑ 0
∑ ∑ 0 1, … ,
, , 1, … ,
1, … ,
(5.30)
where the index represents the peer DMU underlying the optimistic CE
assessment of DMU (see model (5.29)). Finding the set of peer DMUs in this model
can be considered as an arguable topic.
Toloo and Ertay (2014) considered two optimistic and pessimistic scenarios for
finding optimistic/pessimistic cost efficient using a basic LP model and a MIP
model for determining the most cost efficient unit among these cost efficient
candidates. Toloo and Ertay (2014) formulated the following integrated optimistic
CE model to measure the cost efficiency score with CSW under an optimistic
perspective:
min ∑
s. t.∑ 1 1, … ,
∑ ∑ 0 1, … ,
01
01
1, … ,0 1, … ,
(5.31)
Definition 5–2 DMU is optimistic cost efficient with CSW if and only if ∗ 0.
The following LP model can be utilized for attaining a suitable value of epsilon in
model (5.31):
ε∗ maxs. t.∑ 1 1, … ,
∑ ∑ 0 1, … ,
01
0 1
0 1,… ,
(5.32)
Toloo and Ertay (2014) proved that the optimal objective value of model (5.32) is
bounded and hence model (5.31) is feasible for ∈ 0, ∗ . Nevertheless, to have the
maximum discrimination among cost efficient DMUs, let ∗.
Let be the index set of optimistic cost efficient DMU, i.e. : ∗ 0 . If
is singleton and ∈ , then model (5.33) can determine DMU as the most
132 Chapter 5
2014 M. Toloo
cost efficient unit under an optimistic perspective. One method to have a single
cost efficient DMU is to add some suitable variables and constraints to the model.
In doing so, Toloo and Ertay (2014) formulated the following MIP model for
obtaining the most optimistic cost efficient DMU:
min ∑
s. t.∑ 1 1, … ,
∑ ∑ 0 1, … ,
01
01
∑ 1
1, … ,
∈ 0,1 1, … ,
1, … ,0 1, … ,
(5.33)
where is a large enough positive number.
In contrast with the approach proposed by Camanho and Dyson (2005), we can
transform the objective function of model (5.33) from a minimization to a
maximization to formulate the pessimistic integrated model:
max ∑
s. t.∑ 1 1, … ,
∑ ∑ 0 1, … ,
01
01
1, … ,0 1, … ,
(5.34)
Definition 5–3 DMU is pessimistic cost efficient with CSW if and only if ∗ 0.
Model (5.32) provides a suitable value for the epsilon in model (5.34) because the
feasible region of models (5.33) and (5.34) is identical. Let be the index set of
pessimistic cost efficient DMU. If is singleton and ∈ , then model (5.34)
enables us to select DMU as the most cost efficient unit under an pessimistic
perspective.
Analogously, if one changes the objective function of model (5.33) to maximization,
then the following model finds the most pessimistic cost efficient DMU:
Best Efficient Unit 133
Data Envelopment Analysis with Selected Models and Applications
max ∑
s. t.∑ 1 1, … ,
∑ ∑ 0 1, … ,
01
01
∑ 1
1, … ,
∈ 0,1 1, … ,
1, … ,0 1, … ,
(5.35)
Note that since the feasible region of models (5.33) and (5.35) is identical, then the
following model obtains the maximum value for their epsilon:
maxs. t.∑ 1 1, … ,
∑ ∑ 0 1, … ,
01
01
∑ 1
1, … ,
0 1, … ,
∈ 0,1 1,… ,
0 1, … ,
(5.36)
Next section provides a case study of an automotive company located in Turkey to
verify the effectiveness and the validity of the approach.
5.5.2 Turkish automotive company
This section considers a Turkish automotive company that manufactures both
passenger car and light commercial vehicle as an application of the CE‐DEA
models. This company is a global player that has been manufacturing five brands
in its factory, which has a privileged position in the highest rank by achieving Silver
production level within 170 group factories of World Class Manufacturing (WCM).
As the pioneer of the Turkish automotive industry, company exported its products
to 80 countries by 2011 and its received share from the total automotive industry
export was at the level of 22.8% with 180698 units. Turkey’s leading automotive
company continued to create long‐term value for its investors in 2010. Hence, the
company succeeded in generating strong financial results and maintaining high
levels of customer and distributor satisfaction. Company manufactured 312000
vehicles in 2010, or 28.5% of the automotive sector total to become Turkey’s largest
automobile and light commercial vehicle manufacturer. More than 85% of
134 Chapter 5
2014 M. Toloo
company sales in 2010 were models manufactured at its plant. Among global
brands that manufacture in Turkey, the company is the top performer in terms of
the ratio of local production to total sales.
As it is shown in Table C–1 in Appendix C, this real data set contains seven inputs
and five outputs:
Input 1 ( ): number of branch offices
Input 2 ( ): total number of exhibiting vehicles (including branch office)
Input 3 ( ): total number of test vehicles
Input 4 ( ): total number of selling advisers
Input 5 ( ): total number of employee of vendors
Input 6 ( ): number of the exhibitions and presentation activities realized by
vendors
Input 7 ( ): number of advertisement broadcasting by multi‐media
Output 1 ( ): number of the sold vehicles
Output 2 ( ): average satisfaction value for all vendors determined by the whole
customers
Output 3 ( ): service endorsement of vendors after selling (in Turkish Lira)
Output 4 ( ): credit endorsement loaned by means of financial founding for
customers (in TL)
Output 5 ( ): vendor’s selling endorsement on the spare parts (in TL)
The last two rows of Table C–1 in Appendix C report the minimum and maximum
input prices. The objective of the formulated CE‐DEA models is to determine the
most cost efficient vendor in both the most and the least favorable price. The
maximum epsilon for these models is ∗ 0.001114. Regarding this suitable value,
the integrated model (5.31) implies that 40,49,56 which shows that
Vendor40, Vendor49, and Vendor56 are optimistic cost efficient vendors which are
the most cost efficient candidates under an optimistic perspective. Finally,
Vendor40 is determined as the most cost efficient vendor by model (5.33).
To find the most cost efficient unit under a pessimistic point of view, we apply
model (5.34), which leads to 56 . In this case, is singleton and the
model can individually find Vendor56 as the most cost efficient with a pessimistic
point of view. Therefore, it is unnecessary to utilize MIP model (5.35) with this data
set.
135
CHAPTER 6
Data Selection in DEA
DEA is a data‐driven based mathematical approach for measuring the efficiency of
DMUs. Hence, data play an important and critical role in DEA and selecting input
and output measures is an essential issue in this method. The measures should
reflect an analystʹs or a managerʹs interest in the factors that will enter into the
relative efficiency evaluations of the DMUs. There are two debatable problems in
selecting measures in DEA; one is related to status of a measure (input or output)
and the next one is related to obtaining a measures. In general, it is assumed that
the input versus output status of the chosen performance measures is known.
Nevertheless, in some situations, certain performance measures can play either
input or output roles, which are called flexible measures. On the other hand, if the
number of measures is high in comparison with the number of DMUs, then most
of DMUs are evaluated efficient. One method to deal with this situation is to use
some suitable measures, referred to as selective measures. The main idea to
accommodate flexible and selective measures is to utilize auxiliary binary variable
6.1 Flexible measures
Beasley (1990, 1995) firstly faced with a data selection issue in DEA. Indeed, he
found that research income measure in the evaluation of research productivity by
universities can be considered either as input or output. Some other such measures
are: uptime measure in evaluating robotics installations (Cook et al., 1992), outages
measure in the evaluation of power plants (Cook et al., 1998), deposits measure in
the evaluation of bank efficiency Cook and Zhu (2005), medical interns have a
similar interpretation in the evaluation of hospital efficiency,. In many situations,
the input versus output status of certain measures can be deemed as flexible. There
are two alternative approaches to deal with flexible measures, which are based on
the multiplier and envelopment models.
136 Chapter 6
2014 M. Toloo
6.1.1 Multiplier approach
Cook and Zhu (2007) modified the standard CRS‐DEA model to formulate both
individual DMU and aggregate models to derive the most appropriate designations
for flexible measures. An individual DMU model classifies measures for each DMU
and must be solved times, meanwhile an aggregate model classifies measures for
all the DMUs simultaneously and hence must be solved once. Suppose there are
flexible measures, , … , , that may be input or output status. For each
measure , a binary variable ∈ 0,1 is introduced where 0 presents an input
status, and 1 presents an output presents. Let be the weight for each
measure . Cook and Zhu (2007) established the following mathematical
programming classifier model:
max
∑ ∑
∑ ∑ 1s. t.∑ ∑
∑ ∑ 11 1, … ,
0 1, … , 0 1, … ,0 1, … ,
∈ 0,1 1, … ,
(6.1)
Note that under these assumptions 2 various cases for flexible measures (two
cases for each flexible measure) and hence model (6.1) choses the best combination
of these cases for DMU . However, as will be seen subsequently, this combination
is not necessarily unique.
Model (6.1) is equivalent to the following programming problem by the same
transformation of Charnes and Cooper (1962):
max∑ ∑s. t.∑ ∑ 1 1
∑ ∑ ∑ ∑ 1 0 1, …
0 1, … , 0 1, … ,0 1, … ,
∈ 0,1 1, … ,
(6.2)
Although model (6.2) is non‐linear due to term , the following method can be
applied to eliminate the product of a binary and a continuous variable:
Let be a binary variable, and be a continuous variable for which 0
holds. Now a continuous variable, , is introduced to replace the product .
The following constraints must be added to force to take the value of :
, , 1 , 0
Data Selection in DEA 137
Data Envelopment Analysis with Selected Models and Applications
The validity of these constraints can be checked by examining all following
possible situations:
If 0,then from the constraint we have 0. In this case, other
constraints and 1 are redundant.
If 1, then form the constraint 1 , or equivalently , and
also we have ; subsequently the constraint is redundant.
Cook and Zhu (2007) utilized this method and let for 1,… , to
formulate the following mixed integer classifier model:
∗ max∑ ∑s. t.∑ ∑ ∑ 1
∑ 2∑ ∑ ∑ 0 1, … ,
0 1, … ,1 1, … ,
0 1, … , 0 1, … ,0 1, … ,
∈ 0,1 1, … ,
(6.3)
where M is a large positive number.
Let ∗, ∗, ∗, ∗, ∗ be the optimal solution of model (6.3), : ∗ 0 and
: ∗ 1 . Furthermore, suppose that the model is solved for 1,… , .
One possible approach for deciding the overall input versus output status of
flexible measure would then be to compare | | with | |: If | |
| |, then flexible measure must be selected as input and if | | | |,
then is must be selected as output. Ties for determining the status of flexible
measure , i.e. | | | |, may be broken by applying an aggregated model.
Cook and Zhu (2007) formulated the following aggregated model to evaluate the
aggregate efficiency, which optimizes the aggregate or the average ratio of outputs
to inputs:
max
∑ ∑ ∑ ∑
∑ ∑ ∑ 1 ∑
s. t.∑ ∑
∑ ∑ 11 1, … ,
0 1, … , 0 1, … ,0 1, … ,
∈ 0,1 1, … ,
(6.4)
Similarly, the following equivalent MIP model can be obtained:
138 Chapter 6
2014 M. Toloo
max∑ ∑ s. t.∑ ∑ ∑ 1
∑ 2∑ ∑ ∑ 0 1, … ,
0 1, … ,1 1, … ,
0 1, … , 0 1, … ,0 1, … ,
∈ 0,1 1, … ,
(6.5)
where ∑ ∀ , ∑ ∀ , ∑ ∀ .
Now, we formulate some models exclude parameter . According to Theorem 2–
1, we can conclude the following Lemmas:
Lemma 6–1 There exists at least one positive , which makes the following models
equivalent to models (6.3) and (6.5), respectively:
max∑ ∑s. t.∑ ∑ ∑
∑ 2∑ ∑ ∑ 0 1, … ,
0 1, … ,1 1, … ,
0 1 1, … , 0 1 1, … ,0 1 1, … ,∈ 0,1 1, … ,
(6.6)
max∑ ∑ s. t.∑ ∑ ∑
∑ 2∑ ∑ ∑ 0 1, … ,
0 1, … ,1 1, … ,
0 1 1, … , 0 1 1, … ,0 1 1, … ,∈ 0,1 1, … ,
(6.7)
As we earlier stated, in general, there are 2 various cases for flexible measures.
Let ∗ 1,2, … , 2 be the CCR‐efficiency for each case. The following theorem
proves that the optimal objective value of model (6.3) equals to the maximum value
of objective function for all the possible combinations.
Theorem 6–1 ∗ max ∗: 1, … , 2 .
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Data Envelopment Analysis with Selected Models and Applications
Proof: Let ∗, ∗, ∗, ∗, ∗ be the optimal solution of model (6.3) and : ∗
0 . Consider the following model:
∗ max∑ ∑ ∉
s. t.∑ ∑ ∈ 1
∑ ∑ ∉ ∑ ∑ ∈ 0 1, … ,
0 1, … , 0 1, … ,
0 ∈ 0 ∉
(6.8)
Note that model (6.8) is the CCR model when we consider , ∀ ∈ as input and
also , ∀ ∉ as output. Let ∗, ∗, ∗, ∗ be the optimal solution of model (6.8).
Clearly, we have ∗ ∗ ∀ , ∗ ∗ ∀ , ∗ ∗ ∀ ∈ and ∗ ∗ ∀ ∉
which imply ∗ ∗. Hence, we have proved that ∗ ∈ ∗, ∗, … , ∗ and what is
left is to show that ∗ max ∗, ∗, … , ∗ . Let ∗ max ∗, ∗, … , ∗ and contrary
to our claim, ∗ ∗ . Without loss of generality, we assume that the following CCR
model obtains ∗ :
∗ max∑ ∑ ∈
s. t.∑ ∑ ∈ 1
∑ ∑ ∈ ∑ ∑ ∈ 0 1, … ,
0 1, … , 0 1, … ,
0 ∈ 0 ∈
(6.9)
Let ∗, ∗, ∗, ∗ be the optimal solution of model (6.9). Clearly, ∗, ∗, ∗, ∗, ∗
is a feasible solution of model (6.3) where
∗ ∗ ∀ , ∗ ∗ ∀ ,
∗ ∗, ∈
0, otherwise, ∗
∗, ∈
0, otherwise, ∗
1, ∈0, otherwise
The objective value of this feasible solution, ∗, is larger than the optimal objective
function, ∗, which is a contradiction. □
In the following section, we apply these developments to assist in determining the
input output status of flexible variables in reality.
6.1.2 University evaluation
This section deals with evaluating of 50 universities, shown in Table 6‐1, which was
firstly used by Beasley (1990). This real application involves General Expenditure
(GE) and Equipment Expenditure (EE) as two inputs, and Under Graduate
140 Chapter 6
2014 M. Toloo
Table 6–1 University data set (Beasley, 1990)
DMU GE ( ) EE ( ) UGS ( ) PGT ( ) PGR ( ) RI ( )
University 1 528 64 145 0 26 254
University 2 2605 301 381 16 54 1485
University 3 304 23 44 3 3 45
University 4 1620 485 287 0 48 940
University 5 490 90 91 8 22 106
University 6 2675 767 352 4 166 2967
University 7 422 0 70 12 19 298
University 8 986 126 203 0 32 776
University 9 523 32 60 0 17 39
University 10 585 87 80 17 27 353
University 11 931 161 191 0 20 293
University 12 1060 91 139 0 37 781
University 13 500 109 104 0 19 215
University 14 714 77 132 0 24 269
University 15 923 121 135 10 31 392
University 16 1267 128 169 0 31 546
University 17 891 116 125 0 24 925
University 18 1395 571 176 14 27 764
University 19 990 83 28 36 57 615
University 20 3512 267 511 23 153 3182
University 21 1451 226 198 0 53 791
University 22 1018 81 161 5 29 741
University 23 1115 450 148 4 32 347
University 24 2055 112 207 1 47 2945
University 25 440 74 115 0 9 453
University 26 3897 841 353 28 65 2331
University 27 836 81 129 0 37 695
University 28 1007 50 174 7 23 98
University 29 1188 170 253 0 38 879
University 30 4630 628 544 0 217 4838
University 31 977 77 94 26 26 490
University 32 829 61 128 17 25 291
University 33 898 39 190 1 18 327
University 34 901 131 168 9 50 956
University 35 924 119 119 37 48 512
University 36 1251 62 193 13 43 563
University 37 1011 235 217 0 36 714
University 38 732 94 151 3 23 297
University 39 444 46 49 2 19 277
University 40 308 28 57 0 7 154
University 41 483 40 117 0 23 531
University 42 515 68 79 7 23 305
University 43 593 82 101 1 9 85
University 44 570 26 71 20 11 130
University 45 1317 123 293 1 39 1043
University 46 2013 149 403 2 51 1523
University 47 992 89 161 1 30 743
University 48 1038 82 151 13 47 513
University 49 206 1 16 0 6 72
University 50 1193 95 240 0 32 485
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Table 6–2 Results from the multiplier model
DMU ∗ ∗ DMU ∗ ∗
University 1 1 1 University 26 0 0.565
University 2 0 0.640 University 27 1 0.855
University 3 0 0.837 University 28 0 1
University 4 1 0.686 University 29 1 0.825
University 5 0 1 University 30 0 0.930
University 6 0 1 University 31 1 0.776
University 7 1 1 University 32 0 0.896
University 8 1 0.812 University 33 1 1
University 9 0 1 University 34 0 1
University 10 1 0.907 University 35 1 1
University 11 0 0.890 University 36 0 0.837
University 12 1 0.709 University 37 1 0.831
University 13 0 0.803 University 38 0 0.833
University 14 0 0.768 University 39 0 0.791
University 15 0 0.704 University 40 1 0.741
University 16 0 0.543 University 41 1 1
University 17 1 0.819 University 42 0 0.847
University 18 1 0.628 University 43 0 0.921
University 19 1 1 University 44 0 1
University 20 0 0.898 University 45 0 0.889
University 21 0 0.700 University 46 0 0.851
University 22 1 0.717 University 47 1 0.688
University 23 0 0.617 University 48 0 0.939
University 24 0 1 University 49 0 1
University 25 1 1 University 50 0 0.842
Students (UGS), PostGraduate Teaching (PGT), and PG Research (PGR) as three
outputs. Also the flexible measure here is the Research Income (RI).
Table 6–2 reports the results of the classifier model proposed by Cook and Zhu
(2007), where the columns labeled ‘ ∗’ shows the optimal and the columns
labeled ‘ ∗’, the optimal value to model of Cook and Zhu (2007). The authors
concluded that the flexible measure RI must be considered as input: In this case, 20
out of the 50 universities treat the research income measure as an output, i.e., the majority
of 30 treat it as an input. See Cook and Zhu (2007).
Practically, to validate the results of Cook and Zhu (2007) we first evaluate the
efficiency of DMUs when RI is considered as an input and then re‐evaluate it when
RI is considered as an output. Since there is only one flexible measure, there are
only two possible cases for the status of RI. Table 6–3 summarizes the efficiency
scores obtained from these two individual substitutions.
The second and third columns of Table 6–3 reports the efficiency score when RI is
treated as input and output, respectively. As can be seen, the maximum value of
these two columns leads to the efficiency score in Table 6–2. The last column is
142 Chapter 6
2014 M. Toloo
Table 6–3 The comparison of max and min of efficiency scores
DMU Efficiency,
RI as Input
Efficiency,
RI as output ∗
University 1 1 1 0 or 1
University 2 0.615 0.640 0
University 3 0.837 0.663 0
University 4 0.645 0.686 1
University 5 1 0.893 0
University 6 1 1 0 or 1
University 7 1 1 0 or 1
University 8 0.750 0.812 1
University 9 1 0.658 0
University 10 0.892 0.907 1
University 11 0.890 0.747 0
University 12 0.691 0.709 1
University 13 0.803 0.772 0
University 14 0.768 0.702 0
University 15 0.704 0.688 0
University 16 0.543 0.520 0
University 17 0.536 0.819 1
University 18 0.593 0.628 1
University 19 1 1 0 or 1
University 20 0.858 0.898 0
University 21 0.700 0.669 0
University 22 0.664 0.717 1
University 23 0.617 0.560 0
University 24 0.484 1 0
University 25 0.952 1 1
University 26 0.425 0.565 0
University 27 0.853 0.855 1
University 28 1 0.809 0
University 29 0.775 0.825 1
University 30 0.831 0.930 0
University 31 0.728 0.776 1
University 32 0.896 0.841 0
University 33 1 1 0 or 1
University 34 1 1 0 or 1
University 35 1 1 0 or 1
University 36 0.837 0.735 0
University 37 0.782 0.831 1
University 38 0.833 0.806 0
University 39 0.791 0.789 0
University 40 0.740 0.741 1
University 41 1 1 0 or 1
University 42 0.847 0.835 0
University 43 0.921 0.643 0
University 44 1 1 0 or 1
University 45 0.883 0.889 0
University 46 0.848 0.851 0
University 47 0.655 0.688 1
University 48 0.939 0.883 0
University 49 1 0.637 0
University 50 0.842 0.835 0
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obtained by comparing the efficiency scores in the second and third columns. The
efficiency score of 9 universities, i.e. universities 1, 6, 7, 19, 33, 34, 35, 41, and 44, is
identical when RI is considered as either an input or an output. In other words,
∩ 1, 6, 7, 19, 33, 34, 35, 41, 44 . The status of these measures are
randomly determined in Cook and Zhu (2007), depending on the solution method
or the software used to solve the problem.
This analysis illustrates that in some cases the classifier model of Cook and Zhu
(2007) might has an alternative solution. Logic dictates that these DMUs must not
be taken into account for classifying inputs and outputs. Considering this fact, 21
out of the 41 remaining universities consider RI as an output and the rest, i.e. 20
universities, consider RI as an input. As a result, RI must be considered as an input,
however Cook and Zhu (2007) ignored these alternative solutions and
accommodated it as an output which clearly is an incorrect result. This critical
difference illustrates the necessity of considering the alternative solutions in
classifying inputs and outputs. Furthermore, Cook and Zhu (2007) applied their
aggregate model (6.5), the optimal ∗ 1, with the optimal objective function
value being 0.69329. The authors tried to provide some explanations for the
different results between their two‐multiplier models, however there is no any
difference at all. In the next section a new envelopment approach is introduced to
overcome this problem
6.1.3 Envelopment approach
We show that Cook and Zhu (2007) did not consider alternative solutions for their
models and assume that ∩ . However, in some situations we may
have ∩ , which can effect on the final decision about the status of
the flexible measure. In other words, alternative optimal solutions of these models
require to be considered to deal with the flexible measures, otherwise incorrect
results might occur. Practically, the efficiency scores of a DMU could be equal
when the flexible measure is considered either as input or output. We refer to these
measures as share case. We illustrate that share cases must not be taken into account
for classifying inputs and outputs.
Cook and Zhu (2007) developed their approach based on multiplier form of the
CCR model. However, we utilize the envelopment form of the CCR model to
develop an alternative classifier approach that enables us to accommodate flexible
measure. In the envelopment form of the CCR model, if flexible measure is an
input measure, then the constraint ∑ must be imposed, otherwise
the constraint ∑ . The condition that either ∑ or
∑ must hold cannot be formulated in a linear programming model,
because in a linear program all constraints must hold. The following auxiliary
binary variable is introduced to transform the either‐or constraints to a
simultaneous constraint:
144 Chapter 6
2014 M. Toloo
0, isaninput1, isanoutput
In the presence of a sufficient large amount for, , the either‐or constraints are
converted to the following simultaneously constraints:
∑
∑ 1
The conversion guarantees that only one of the two constraints can be active at any
one time. If 0, then the constraint ∑ is active and the constraint
∑ is redundant due to the constraint ∑ ; otherwise
the first constraint is redundant and the last one, i.e. ∑ , is active. In
other words, flexible measure will end up with either an input, 0, or an
output, 1.
Hence, we formulate the following classifier MIP problem to determine the status
of flexible measures:
∗ mins. t.∑ 1, … ,
∑ 1, … ,
∑ 1, … ,
∑ 1 1, … ,
∈ 0,1 1, … ,0 1, … ,
(6.10)
It is easy to verify that model (6.10) has the properties of units invariance and CRS
We also suggest the following envelopment aggregate model to evaluate the
aggregate efficiency, which optimizes the aggregate or the average ratio of outputs
to inputs:
mins. t.∑ 1, … ,
∑ 1, … ,
∑ 1, … ,
∑ 1 1, … ,
∈ 0,1 1, … ,0 1, … ,
(6.11)
where ∑ ∀ , ∑ ∀ , ∑ ∀ .
Although the multiplier form of CCR model is the dual of the envelopment form
of the CCR model with equal optimal objective value, model (6.10) is not the dual
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of model (6.3) and consequently might have different optimal objective values.
Model (6.3), as a maximization problem, shows the existence or non‐existence of a
flexible measure multiplier (variable); whereas in model (6.10) either‐or constraint
of a flexible measure constraint is considered as a minimization problem. In the
following, we prove that there might be a gap between the optimal objective value
of multiplier and envelopment approaches.
Theorem 6–2 ∗ min ∗: 1,… , 2 .
Proof: Let ∗, ∗ be the optimal solution of model (6.10) and : ∗ 0 . Under
these assumptions, model (6.10) is equivalent to the envelopment form of CCR
model where for ∀ ∈ is input and also for ∀∉ is output. As a result, ∗
∗, … , ∗ . Similar to the proof of Theorem 6–1, let ∗ min ∗, … , ∗ and on the
contrary suppose ∗ ∗. Without loss of generality we assume that the following
envelopment form of the CCR model obtains ∗:
∗ mins. t.∑ 1, … ,
∑ ∈
∑ 1, … ,
∑ ∈
0 1, … ,
(6.12)
Let ∗ be an optimal solution of model (6.12) and also ∗ 0 if ∈ ; otherwise ∗ 1. A trivial verification shows that ∗, ∗ is a feasible solution of model (6.10)
with an objective value that is smaller than the optimal objective function, which
is impossible. □
Consider the university application, which was introduced in the previous section.
The optimal solution of classifier model (6.3) leads to the maximum value of
efficiency score, however the minimum value of efficiency score will be obtained
from model (6.10). As a result, the classifier models (6.3) and (6.10) classify inputs
and outputs for the best and worst situations, respectively.
Corollary 6–1 ∗ ∗ ∗ for 1,… , 2 .
Practically, it is not easy to pre‐determine which model has alternative optimal
solutions via the classifier model. However, the following corollary provides an
easy condition to find the alternative solutions.
Corollary 6–2 IF ∗ 1, then ∗ 1 for 1,… , 2 .
Corollary 6–2 is a key tool to tackle alternative solutions, especially when 1
which is the case in the afore‐mentioned university application. Because when
there is a single flexible measure, the optimal value of single binary variable for
146 Chapter 6
2014 M. Toloo
model (6.3), i.e. ∗, can be obtained from the optimal value of single binary variable
for model (6.10). In fact, under these assumptions ∗ 1 ∗.
In addition, Corollary 6–1 shows that the envelopment approach has more
discriminating power than the multiplier approach. However, considering the
following cases, we can handle the flexible measure by model (6.10):
If ∗ 1, then ∗ 1 and hence the flexible measure is a share case and
must not be taken into account for classifying inputs and outputs.
If ∗ 1 and ∗ 0, then ∗ 1 and consequently flexible measure has
to be designated as an output. In this case minimum efficiency score
(worst situation) with more discriminating power will obtain if we
consider flexible measure as an input.
If ∗ 1 and ∗ 1, then ∗ 0. The flexible measure has to be selected
as an input if the best situation is desirable. Clearly, the worst situation
with maximum discriminating power will obtain if we consider flexible
measure as an outputs.
If | | | |, then the flexible measure has to be selected as an
input.
If | | | |, then the flexible measure has to be selected as an
output.
If | | | |, then an aggregate approach should be utilized.
6.2 Selective measures
There is a statistical and empirical rule, known as the rule of thumb, which if the
number of performance measures is high in comparison with the number of units,
then a large percentage of the units will be determined as efficient. It also implies
that the selection of performance measures is very crucial for finding an acceptable
number of efficient DMUs. In this section, we suggest two alternative approaches
to choose some inputs and outputs in a way that produces acceptable results.
6.2.1 The rule of thumb in DEA
If a performance measure is added or deleted from consideration, it will influence
the relative efficiencies. Empirically, when the number of performance measures is
high in comparison with the number of DMUs, then most DMUs are evaluated
efficient so that such results may be questionable. A rough rule of thumb expresses
the relation between the number of DMUs and the number of performance
measures (see Cooper et al., 2007b) for more details):
max 3 ,
Table D–1 in Appendix D practically represents the number of DMUs and the
number of performance measures applied in some studies. Statistically, we find
out that in nearly all of these studies the number of inputs and outputs do not
Data Selection in DEA 147
Data Envelopment Analysis with Selected Models and Applications
exceed 6. A simple calculation shows that if 6 and 6, then 3
and under these assumptions, 3 max 3 , . As a result,
3 can be considered as the rule of thumb in DEA, which is simpler.
There are some cases that the number of performance measures and DMUs does
not satisfy the rule of thumb. To rectify this issue, in many scenarios we select some
performance measures in a manner which complies with the rule of thumb and
imposes progressive effect on the efficiency scores. Selecting measures among a set
of suggested factors is an important issue for the decision maker. For example,
consider the problem of evaluating 50 branches of a bank; performance measures
suggested by the manager may practically run into more than 25 inputs and 30
outputs. The total number of measures, i.e. 55, and DMUs do not satisfy the rule of
thumb and we subsequently encounter many efficient units. To make the problem
easier, suppose that the manager pre‐selected three inputs, e.g. employees,
expenses and space, and three outputs, e.g. loans, profits and deposits. With this
assumptions, if the manager wants to select 2 out of 22 remaining inputs and 1 out
of 27 remaining outputs and also consider all possible combinations of
performance measures, then an optimization problem must be solved at most
196350 50222
271
times, which is illogical.
Consider n DMUs, each using m inputs to produce s outputs, where
max 3 , . In this situation, one approach for improving discrimination
power among DMUs is to adopt an assurance region method or the cone ratio
model (see Cooper et al., 2007b). However, a decision maker requires to have more
information about the value of these inputs and outputs and subsequently the final
result is closely related to these information. In the lack of such information, two
main methods can be applied to hold the rule of thumb: increasing the number of
units ( ) or decreasing the number of performance measures ( ). In the first
method, we add some more DMUs for the evaluation, while in the second method
we ignore some inputs and outputs. In practice, it is hardly possible to increase the
number of DMUs and consequently we resort to some models, which optimally
decrease the number of performance measures. We refer these models as selecting
models.
To analyze the number of possible combinations of selective measures we firstly
suppose 3 max 3 , . In this case, /3 performance
measures should be ignored in a way that the efficiency scores of all DMUs
decrease as much as possible. Note that decreasing the efficiency score of all DMUs
leads to maximum discrimination among efficient DMUs. If one considers all
possible input and output combinations, then the multiplier form of the CCR
model or the BCC model must be solved ∑ /3/
times. In a
similar manner, the number of times that these models must be solved is
∑ / when max 3 , . Obviously, this method is
148 Chapter 6
2014 M. Toloo
inappropriate even for a very small set of performance measures. In this study, a
new approach is proposed to tackle this issue.
In some cases, the decision maker believes that some performance measures are
more important than others and we fix these measures (as fixed measures), and
select performance measures from the remaining inputs and outputs (selective
measures). In the case of no preferences between different inputs (or outputs), the
formulated model covers all performance measures.
In the following sub‐sections, we extend two multiplier and envelopment forms of
selecting model. However, in some situations, some specific measures must appear
in evaluation model so that the experts and managers believe that these measures
reflect thoroughly the production process. We refer these measures as fixed
measures and we also consider them for developing our approach. In particular, we
develop two individual and aggregate models for each forms of selecting models.
In individual models, the performance of each DMU is considered, while the
aggregate models deal with overall performance of the collection of DMUs.
6.2.2 Multiplier form of selecting model
Let and denote subsets of outputs corresponding to fixed‐output and
selective‐output measures, respectively. Similarly, assume that and are the
parallel subsets of inputs. We extend the multiplier form of the CCR model to
formulate the individual and aggregate selecting models to derive the most
appropriate designations for selective measures with keeping the rule of thumb.
We consider the epsilon form of the CCR model to keep the weights positive.
What’s more, for each selective input and output measure we introduce the binary
variables and , respectively. In a nutshell, by adopting the fractional CCR
model, we propose the following mixed integer non‐linear fractional problem to
obtain the efficiency of DMU :
max∑
∑
s. t.∑
∑1 1, … ,
∑ ∈ ∑ ∈ min , 2√ | | | |
∈
∈
, ∈ 0,1 ∈ , ∈, ∈ , ∈
(6.13)
If we apply Charnes and Cooper (1962)’s method, the following MINLP model is
obtained:
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max ∑s. t.∑ 1
∑ ∑ 0 1, … ,
∑ ∈ ∑ ∈ min , 2√ | | | |
∈
∈
, ∈ 0,1 ∈ , ∈, ∈ , ∈
(6.14)
where 0is a small number and is a large positive number. The following
theorem validates model (6.14).
Theorem 6–3 The selecting model (6.14) will meet the rule of thumb.
Proof. Let ∗, ∗,∗,
∗ be the optimal solution of selecting model (6.14).
Consider a selective input measure, denoted by p. If ∗0, then from the
constraint we have ∗ 0, meaning that the selective measure is not
selected. Otherwise, i.e. ∗1, then the selective input measure p is designated
since the constraint leads to ∗ 0. Similarly, we obtain ∗ 0 when ∗
0, i.e., the selective output q is not selected. We also have ∗ 0 when ∗
1. As a result, the total number of considered inputs and output, including fixed
and selective measures, in selecting model (6.14) is equal to | | ∑∗
∈
| | ∑∗
∈ .
Now, consider the constraint ∑ ∈ ∑ ∈ min /3 , 2√ | |
| | . If /3 min /3 , 2√ , then we have 3 | | ∑∗
∈ | |
∑∗
∈ ; otherwise, i.e. 2√ min /3 , 2√ , the constraint ∑ ∈
∑ ∈ 2√ | | | | leads to 2√ | | ∑∗
∈ | |
∑∗
∈ and consequently an easy calculation shows that | |
∑∗
∈ | | ∑∗
∈ . As a result, from the optimal solution of the
selecting model (6.14) we obtain max 3 | | ∑∗
∈ | |
∑∗
∈ , | | ∑∗
∈ | | ∑∗
∈ which completes the proof. □
We now eliminate the min function form the right hand side of third set of
constraints of model (6.14). Not that if 36, then we have /3 2√ and
subsequently /3 min /3 , 2√ . In this case the constraint ∑ ∈
∑ ∈ min /3 , 2√ | | | | can be replaced with the constraint
∑ ∑ | | | | and otherwise we have 2√ min /
3 , 2√ and the mentioned constraint should be replaced with the constraint
∑ ∑ 2√ | | | | .
150 Chapter 6
2014 M. Toloo
Now suppose that a manger is interested in selecting exactly p out of m2 selective
inputs and q out of s2 selective outputs, where min /3 , 2√ | |
| | . This conditions will be easily established in model (6.14), when the constraint
∑ ∈ ∑ ∈ min /3 , 2√ | | | | is replaced with two
constraints ∑ ∗∈ and ∑ ∗
∈ . Note that to select input and output
measures, we first solve the selecting model (6.14) and obtain a set of optimal ∗ ∗
for each DMU. If ∗1
∗ 1 for the majority of the units, then
we select as an input (output) measure. Now, the multiplier form of the CCR
model can be used to evaluate DMUs with the fixed and selective measures.
If there is no priority in choosing selective measures , we propose
the following selecting model:
max ∑s. t.∑ 1
∑ ∑ 0 1, … ,
∑ ∈ ∑ ∈ min , 2√
1, … ,
1, … ,
∑ 1
, ∈ 0,1 1, … , ; 1, … ,
(6.15)
The constraint ∑ 1 is added to have at least one output measure. Note that
the condition ∑ 1 can be extracted from the first constraint ∑
1 . Similar to the proof of Theorem 6–3, one can prove that model (6.15) fulfills the
rule of thumb.
The following MINLP model measures the maximum value of epsilon for model
(6.15), which can be linearized via the afore‐mentioned method in previous section.
maxs. t.∑ 1
∑ ∑ 0 1, … ,
∑ ∈ ∑ ∈ min , 2√
1, … ,
1, … ,
∑ 1
, ∈ 0,1 1, … , ; 1, … ,
(6.16)
Similarly, we can obtain the maximum value of epsilon for the afore‐mentioned
models.
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6.2.3 Envelopment form of selecting model
This sub‐section extends the envelopment form of the selecting model. Unlike
traditional DEA models, the envelopment and multiplier forms of the selecting
models are not mutually dual.
We formulate the following envelopment form of the selecting model:
max ∑ ∈ ∑ ∈ ∑ ∈ ∑ ∈
s. t.∑ ∈
∑ ∈
∑ 1 ∈
∑ 1 ∈
∑ ∈
∑ ∈
, ∈ 0,1 ∈ , ∈
, 0 ∀ , ∀
0 ∀
(6.17)
where M is a very large positive number and also p and q are two positive
parameters such that min , 2√ | | | | . Note that model (6.17)
is nonlinear due to the two nonlinear terms and , which will be
eliminated from the model.
In order to show the contribution of the proposed model (6.17), we state and prove
a theorem, which ensures an acceptable number of efficient DMUs by keeping the
rule of thumb:
Theorem 6–4 The presented model (6.17) will meet the rule of thumb.
Proof. Let ∈ and ∈ . If ∗ 1, then the constraint ∑
1 changes to ∑ which means that the input measure
is selected. Otherwise, the constraint ∑ 1 converts
to ∑ which, for , implies that ∑ is
redundant and subsequently the input measure is not selected. Note that in the
objective function, the term considers slack variables of the selected inputs.
Considering the number of the fixed‐input measures, | |, and the constraint
∑ ∈ the total number of input measures is | | . Similarly, if 1,
then the constraint ∑ is active; otherwise it is redundant. The
constraint ∑ ∈ implies that the total number of the fixed‐output and
selective‐output measures is . As a result, the number of measures in model
(6.17) is min /3 , 2√ . This completes the proof. □
152 Chapter 6
2014 M. Toloo
Now, we eliminate the nonlinear terms from the formulated mixed integer
nonlinear programming model (6.17). In doing so, we replace the nonlinear term
by a new variable on which a number of constraints is imposed. First let
for ∈ . We need to add some suitable constraints when 1 and
otherwise 0. This can be done by imposing the following restrictions on the
model:
0
1
where is a very large positive number. If 0, then the constraint
1 is redundant and on the other hand from the constraint 0
we have 0. Otherwise (i.e. 1) 0 is redundant and
the constraint 1 converts to and hence
. In a similar manner, we assume for ∈ and impose the following
restrictions on the model:
0
1
As a result, the following mixed integer linear programming is formulated:
max ∑ ∈ ∑ ∈ ∑ ∈ ∑ ∈
s. t.∑ ∈
∑ ∈
∑ 1 ∈
∑ 1 ∈
∑ ∈
∑ ∈
0 ∈
1 ∈
0 ∈
1 ∈
, ∈ 0,1 ∈ , ∈
, 0 ∈ , ∈
, 0 ∀ , ∀
0 ∀
(6.18)
If the manager has no priority over choosing the selective measures, in model (6.18)
we can set to achieve the following envelopment form for the selecting
model:
Data Selection in DEA 153
Data Envelopment Analysis with Selected Models and Applications
max ∑ ∑
s. t.∑ 1, … ,
∑ 1, … ,
∑ 1 1, … ,
∑ 1 1, … ,
∑
∑
0 1, … ,
1 1, … ,
0 1, … ,
1 1, … ,
, ∈ 0,1 1, … , ; 1, … ,
, 0 1, … , ; 1, … ,
, 0 1, … , ; 1, … ,
0 1,… ,
(6.19)
where 1, 1, min , 2√ | | | | . Notice that from 1
( 1) we have at least one input (output).
It is worth noting that unlike the multiplier and envelopment forms of CCR model,
which are LP, the proposed forms of the selecting models are MIP. As a result, the
optimal objective value of the multiplier form of selecting model (6.14) is not equal
to the optimal objective value of the envelopment form model (6.18).
Indeed, models (5.14) and (5.18) are not mutually dual and, as will be illustrated
by a real data set of banks, the multiplier form of the selecting model selects the
selective measures based on the maximum performance whereas the maximum
discrimination among DMUs is considered in the envelopment form. The
following theorem expresses the relation between these models:
Theorem 6–5 0 ∗ ∗ 1.
Proof. Let ∗, ∗,∗,
∗and ∗, ∗,
∗,
∗,
∗,
∗,
∗,
∗ be the optimal
solution of models (6.14) and (6.18), respectively. Consider model (6.17) and let ∗
1 and ∗
1 . The optimal solution of the following
envelopment form of CCR model is ∗, ∗,∗,
∗:
154 Chapter 6
2014 M. Toloo
max ∑ ∈ ∪ ∑ ∈ ∪
s. t.∑ ∈ ∪
∑ ∈ ∪
0 ∈ ∪
0 ∈ ∪
0 1, … ,
(6.20)
Since the optimal objective value of CCR model is positive and less than or equal
to 1 we have 0 ∗ 1. Consider the following dual model of (6.20):
max ∑ ∈ ∪
s. t.∑ ∈ ∪ 1
∑ ∈ ∪ ∑ ∈ ∪ 0 1, … ,
∈ ∪∈ ∪
(6.21)
Let ∗, ∗ be the optimal solution of model (6.21). Under the optimality
conditions (primal‐dual relationships) ∗ ∗ . Let
1, ∈0, ∈ /
,1, ∈ 0, ∈ /
A simple computation shows that ∗, ∗,∗,
∗is a feasible solution of model
(6.14) with objective value ∗ . Hence, we have ∗ ∗ ∗ and what is left is to
show that ∗ 1. To do this, consider the given optimal solution of model (6.14), ∗, ∗,
∗,
∗, and fix
∗and
∗ If we redefine
∗1 and
∗1 , then ∗, ∗ , ∗ is an optimal solution of the multiplier form of CCR
model (6.21) and hence 0 ∗ 1 which completes the proof. □
In the proposed individual approaches, the manager solves the proposed
multiplier or envelopment forms of the selecting models for each DMU and obtains
a set of optimal ∗ and
∗ (or
∗ and
∗
) specific to that DMU. One criterion for
selecting measures is to consider the majority choice among the DMUs. However,
in the aggregate approach, the manager makes the decision based on a set of the
optimal ∗
and
∗ (or
∗
and
∗
) for all DMUs. Such a model would be helpful
if ties are encountered using the proposed selecting models on an individual DMU
basis. In the following sub‐section, we adopt this aggregate approach for both
multiplier and envelopment forms.
6.2.4 Aggregate approach
The overall performance of the collection of DMUs can be considered as an
alternative approach for selecting performance measures. To follow this viewpoint,
we examine the efficiency of the aggregate outputs to aggregate inputs with the
Data Selection in DEA 155
Data Envelopment Analysis with Selected Models and Applications
outlook on the selective measures. Specifically, we suggest the following MIP to
obtain the aggregate efficiency in the multiplier form:
max∑s. t.∑ 1
∑ ∑ 0∀
∑ ∈ ∑ ∈ min , 2√
∀
∀
∑ 1
, ∈ 0,1 ∀ , ∀
(6.22)
It should be mentioned here that the aggregate approach optimizes the aggregate
efficiency of the collection of DMUs and it is recommended when ties are
encountered using an individual DMU model.
Similarly, we formulate the following aggregate MIP envelopment model to deal
with the aggregate efficiency:
max ∑ ∑
s. t.∑ ∀
∑ ∀
∑ 1 ∀
∑ 1 ∀
∑
∑
0 ∀
1 ∀
0 ∀
1 ∀
, ∈ 0,1 ∀ , ∀
, 0∀ , ∀
, 0 ∀ , ∀
0 ∀
(6.23)
All models in this section are presented under CRS assumption. Straightforwardly,
their VRS version of the model can be formulated by adding a free variable with
free in sign, i.e. ∈ ∞, ∞ , to the multiplier forms or adding the convexity
constraint ∑ 1 to the envelopment forms (see Toloo et al., 2015 and Toloo
and Tichý, 2015).
In the next section, we utilize some real data sets of banks in order to show the
applicability of the proposed multiplier and envelopment selecting approaches.
156 Chapter 6
2014 M. Toloo
6.2.5 Banking industry applications
In this section, we first consider a real data set of the largest private bank in Iran
and then deal with active banks in the Czech Republic. The former bank has
approximately 3150 branches in different cities in Iran with 127 branches in one of
the Northern provinces, Gilan. The data set, which is presented in Table D–3
(Appendix D), involves 20 branches of the capital of Gilan.
To choose inputs and outputs measures, we studied different papers on bank
industry. Mostafa (2009) in a study on bank industry evaluated the efficiency of
top Arab banks using two different approaches, namely DEA and neural networks.
Although Emrouznejad and Anouze (2009) in a note on Mostafa (2009) indicated
that the efficiency scores that are obtained in this paper are incorrect and the results
need to be revised, we just use the information provided by Mostafa (2009) that
summarizes the previous research conducted to evaluate bank efficiency using
DEA. The inputs and outputs that were investigated in the previous studies are
showed in Table D–2 (Appendix D). We can come to some helpful results from this
table. For example, in more than 40% of the studies expenses is used as an input
measure. Therefore, it is reasonable to enter this factor in the inputs set. Moreover,
based on the bank policies, the bank manager provided some inputs and outputs.
Some of these measures are the same as the ones in Table D–2, but the rest of them
are different. For instance, One‐Time Password (OTP) is an output measure that the
manager wishes to take into account though it is not in Table D–2.
Finally, considering both the manager’s interests and the resulted information
from Table D–2, the inputs and outputs measures are suggested as follows:
Inputs: Employees, Expenses, Space, Assets, Number of accounts, Costs.
Outputs: Loans, Number of transactions, Deposits, Check card, Credit card, OTP.
Now, we want to obtain the efficiency of these 20 branches with 6 inputs and 6
outputs hence the number of DMUs and measures does not satisfy the rule of
thumb in DEA, i.e. 20 36 max 3 , . The last column of Table
D–3 indicates efficiency scores after applying the standard CCR model to the data.
Results show that 80% of all the branches are reported efficient and, hence these
efficiency scores have insufficient discrimination power. Moreover, the data in the
Table D–2 shows that in more than 84% of the studies Employees has been used as
input and Loans with 46% as output. Practically, Deposits is an important factor in
the bank industry and has a special influence on shares and earnings. Regarding
these considerations, prior to designating inputs and outputs via selecting models
we set Employees, Deposits and Loans as fixed measures.
Imposing the above restrictions, we apply the aggregate multiplier model (6.22) to
the data set in Table D–3. Results indicate that Assets and Costs, are recognized as
the selective input measures, and No. of transactions as the selective output measure,
Data Selection in DEA 157
Data Envelopment Analysis with Selected Models and Applications
Table 6–4 Results of applying the aggregate multiple model
DMUs Inputs
Outputs
Efficiency No. of
Employees Assets Costs
No. of
transactions Deposits Loans
1 11 1753 10020
5214 72149 57537 0.86
2 17 2604 11440 5343 89781 51114 0.70
3 7 1155 8,427 5145 42654 52485 1.00
4 12 1899 11816 3249 97812 67298 0.85
5 14 2215 12426 6706 77031 43487 0.64
6 14 2357 9907 6259 75923 41442 0.75
7 9 1370 10365 3652 47763 43262 0.65
8 5 829 5283 3913 45732 14237 1.00
9 6 985 11061 3566 55222 41062 0.88
10 6 1023 5856 4559 53323 37418 1.00
11 8 1311 8745 4441 69734 57883 0.99
12 9 1536 7326 5031 49153 47139 0.98
13 8 1367 8326 5053 92365 55543 1.00
14 7 1193 6525 4762 64235 22347 0.89
15 9 1359 8158 6876 89104 45717 1.00
16 7 1111 11135 4307 42012 73925 1.00
17 7 1182 6920 5331 69360 27246 0.99
18 7 1069 5864 4004 51438 26531 0.81
19 6 992 5039 2342 39948 20223 0.66
20 7 1180 8378 4238 154284 43928 1.00
with the aggregate efficiency score being 0.78. The last column of Table 6–4
demonstrates the efficiencies, which are obtained by applying the conventional
CCR model to the data of this table. Comparing the last columns of Table D–3 and
Table 6–4 indicates that the number of efficient DMUs is reduced after using the
selecting model and then applying the CCR model to the DMUs with the selective
measures. More precisely, efficient DMUs 1, 4, 5, 6, 11, 12, 14, 17 and 18 in Table
D–3 are now inefficient with the efficiency scores, 0.86, 0.85, 0.64, 0.75, 0.99, 0.98,
0.89, 0.99, and 0.66. In summary, with the selective measures 7 out of 20 DMUs are
reported efficient while there are 16 efficient DMUs with considering all measures.
This means that while the conventional CCR model with all measures reported
80% of DMUs efficient, by using the selecting model the percentage of efficient
DMUs is reduced to 35.
It is worth to point out that to obtain the same results without utilizing the
proposed multiplier or envelopment forms of the selecting models, 400 20
optimization problems must be solved. In other words, our approach choices
the selective measures with fairly fewer computations.
158 Chapter 6
2014 M. Toloo
Now, we apply the individual multiplier form of the selecting model to evaluate
14 active banks in the Czech Republic with the VRS assumption. The Czech
Republic has relatively stable economy with close tights to Germany and its
banking sector is highly internationalized (majority of banks are owned by
financial institutions from Eurozone) with just a few large banks, although it can
currently be regarded as highly competitive, especially due to aggressive policy of
small banks in the retail segment.
Even if data of some banks were omitted, Table D–4 clearly shows that just three
or four large banks make the market. Even though, the market can currently be
regarded as competitive and innovative due to the presence of small banks. Note
finally that the Czech banking sector can be regarded as very stable and no serious
impact of recent subprime crises was recorded, mainly due to public rescue of large
banks in late 90’s and their subsequent acquisition of European financial
institutions.
In order to select the most important bank features, we follow Mostafa (2009) (see
table D–2 in Appendix D). In addition, considering the manager’s interests,
availability of information and the most utilized measures in Mostafa (2009), the
inputs and outputs measures are suggested as follows:
Inputs: Employees, Number of branches, Assets, Equity, Expenses.
Outputs: Deposits, Loans, Non‐interest income, Interest income.
Table D–4 exhibits the data set and the BCC‐efficiency score. The last column in the
table shows that 93% of banks (i.e. 13 of 14) are BCC‐efficient. Such result is
questionable and is due to an inadequate number of performance measures. To get
an acceptable result, we have to decrease the number of performance measure such
that excess 3 . Toward this end, we first apply the multiplier form of
selecting model (6.14) (with considering a free variable ) to the data set in Table
D–4. Reference to the optimal solution shows that Expenses is a selected input and
also Deposits and Loans are selected outputs. Table 6–5 summarizes all the fixed,
selected data and also efficiency scores which are obtained by using the BCC
model. As can be extracted from this table, there are 5 efficient banks out of 14
banks, which show that the proposed multiplier model succeeds in decreasing the
percentage of efficient DMUs from 93% to 36% with the maximum number of
efficient DMU. Nevertheless, as will be illustrated, the proposed envelopment form
can decrease the number of efficient DMUs even more than multiplier form, but
with the minimum number of efficient DMU.
Notwithstanding the BCC‐efficiency scores in Table D–3, these scores in Table 6–5
illustrate that the proposed multiplier form of the selecting model succeeds in
holding the rule of thumb and achieving the satisfactory efficiency scores.
Data Selection in DEA 159
Data Envelopment Analysis with Selected Models and Applications
Table 6–5 Selective measures obtained by multiplier model
Banks Inputs Outputs BCC‐
Efficiency Employees Expenses Deposits Loans
AIR 400 745
30696 11135 0.5607
CMZRB 217 566 86967 16813 1
CS 10760 18259 688624 489103 1
CSOB 7801 16087 629622 479516 1
EQB 296 601 7502 5611 0.4551
ERB 72 173 2940 1762 1
FIO 59 347 17174 6465 1
GEMB 3346 5276 97063 101898 0.5875
ING 293 1034 92579 19216 1
JTB 407 1333 62085 39330 1
KB 8 758 13511 579 067 451 547 1
LBBW 365 1138 20274 2528 0.25
RB 2927 57112 144143 150138 0.5346
UCB 2004 13804 195120 192046 1
Table 6–6 Selective measures by envelopment model
Banks
Inputs Outputs BCC‐
Efficiency Employees No. of
branches Non‐interest
income Loans
AIR 400 18
14 11135 0.3686
CMZRB 217 5 634 16813 1
CS 10760 658 15412 489103 1
CSOB 7801 322 8,747 479516 1
EQB 296 13 19 5611 0.3419
ERB 72 1 15 1762 1
FIO 59 36 211 6465 1
GEMB 3346 260 3943 101898 0.7704
ING 293 10 468 19216 0.773
JTB 407 3 487 39330 1
KB 8 758 399 8834 451 547 0.8591
LBBW 365 18 128 2528 0.256
RB 2927 125 2829 150138 0.7506
UCB 2004 98 2740 192046 1
To decrease the number of efficient DMUs and discriminate between them we
apply the proposed envelopment form of selecting model (6.17) to the data set in
Table D–3 with 1, 2. The optimal solution implies that Number of Branches,
Non‐interest income and Loans are the selective measures. The last column of Table
6–6 demonstrates the efficiencies, which are calculated using the BCC model. It can
160 Chapter 6
2014 M. Toloo
be extracted that there are 7 efficient banks, and as a result, the formulated selecting
model successfully decreased the percentage of efficient DMUs from 93% to 50%.
161
CHAPTER 7
Conclusion
Nowadays, it is necessary to evaluate the efficiency (doing things right),
effectiveness (doing the right things) and economic efficiency (doing things at a
lowest price) in an organization. However, it is a challenge to implement these
evaluations when there are multiple inputs and multiple outputs. Data
Envelopment Analysis (DEA) is a powerful, non parametric and quantitative
method for evaluating the relative efficiency of a set of Decision Making Units
(DMUs), such as universities, car makers, hospitals, banks and so on.
The book Data Envelopment Analysis with Selected Models and Applications provides
some basic concepts of DEA methodology. We mainly focus on some selected DEA
models and their extensions. In this respect, we first present the concept of relative
efficiency and mathematically construe both envelopment and multiplier forms of
the DEA models. To solve DEA models the General Algebraic Modeling System
(GAMS) software, which is a high‐level modeling system for linear, nonlinear and
mixed integer optimization problems, is introduced. We present the weights in
DEA and their importance followed by considering common set of weights (CSW)
approach as a main means finding the most efficient DMU. Finally, we introduce
the role of flexible and selective measures in DEA. Throughout the book we
illustrate the models using different applications such as banking industry in the
Czech Republic and Iran, Technology Selection (TS) and Facility Layout Design
(FLD).
The book is intended mainly for Ph.D. students and academic staff, researchers and
public institutions, which are interested in studying the field of performance
evaluation.
This publication has been elaborated in the framework of the project Support
research and development in the Moravian‐Silesian Region 2013 DT 1 –
International research teams (02613/2013/RRC). Financed from the budget of the
Moravian‐Silesian Region. It has been also supported by the Czech Science
162 Chapter 7
2014 M. Toloo
Foundation (GACR project 14‐31593S) and through European Social Fund (OPVK
project CZ.1.07/2.3.00/20.0296).
163
Appendix
Appendix A
Table A–1 Banks in the Czech Rep
ublic
Bank
Full name
Description
AIR
Air Ban
k
small and very new m
arket participant owned by Czech conglomerate, which is,
however, registered in the Netherlands
CMZRB
Českomoravská záruční a rozvojová ban
ka
a specialized banking institution m
anaged by the government representatives in
order to support selected m
arket segments
CS
Česká spořitelna
large, traditional ban
k, owned by Erste Group, Austria
CSOB
Československá obchodní banka
large, traditional ban
k, owned by KBC Ban
k, B
elgium
EQB
Equa Ban
k
though existing for more than 20 years, form
er IC Banka, currently owned by Equa
Group, M
alta
ERB
Evropsko‐ruská ban
ka
small and relatively new
, owned by a natural person
FB
Fio banka
small, though exiting m
ore than 20 years, owned by Fio holding, C
zech Republic
GEMB
GE M
oney Bank
mid‐size, owned by GE Holding, U
SA
ING
ING Ban
k N.V.
midsize branch of foreign bank
JTB
J&T bank
small to m
id‐size, owned indirectly by two natural persons
KB
Komerční banka
large, traditional ban
k owned by Societe General, France
LBBW
LBBW Bank CZ
small, owned by Lan
desban
k Baden‐W
urttemberg, Germany
RB
Raiffeisen Ban
k
mid‐size, owned by Austrian bank of the same nam
e
UCB
UniCredit Ban
k Czech Rep
ublic
fourth largest ban
k after several m
ergers (H
VB Ban
k, Ban
k A
ustria Creditan
stalt
and traditional Czech ban
k Živnostenská banka), currently owned by Austrian ban
k
of the respective name, w
hich is owned by Italian UniCredit
164 Appendix
2014 M. Toloo
Table A–2 The Czech banks’ and leasing companies’ asset financing alternatives with their
efficiency score
DMU
Down
payment
Annuities
Other fees
Bank loan
coefficient
CCR‐
efficiency
DMU
Down
payment
Annuities
Other fees
Bank loan
coefficient
CCR‐
efficiency
1 0 17696 1210 1.318 0.9165 71 181500 10911 0 1.202 1
2 36300 16917 1210 1.326 0.9069 72 181500 9415 0 1.245 0.999998
3 72600 16137 1210 1.335 0.8973 73 181500 8352 0 1.288 1
4 108900 15344 1210 1.345 0.8884 74 217800 44091 0 1.041 0.999985
5 145200 14565 1210 1.356 0.8786 75 217800 22869 0 1.080 1
6 181500 13774 1210 1.368 0.8704 76 217800 15809 0 1.120 0.999969
7 217800 13067 1210 1.391 0.8578 77 217800 12288 0 1.161 1
8 254100 12284 1210 1.408 0.8501 78 217800 10184 0 1.202 0.999954
9 290400 11502 1210 1.429 0.8412 79 217800 8787 0 1.245 1
10 326700 10719 1210 1.453 0.8313 80 217800 7795 0 1.288 1
11 363000 9954 1210 1.484 0.8230 81 254100 40941 0 1.041 1
12 399300 9185 1210 1.522 0.8185 82 254100 21236 0 1.080 0.999978
13 435600 8526 1210 1.590 0.8048 83 254100 14680 0 1.120 0.999951
14 471900 7763 1210 1.655 0.8556 84 254100 11411 0 1.161 0.999961
15 508200 7012 1210 1.744 0.9374 85 254100 9456 0 1.202 1
16 0 16327 1210 1.351 0.9104 86 254100 8160 0 1.245 0.9999
17 36300 15615 1210 1.360 0.9006 87 254100 7239 0 1.289 0.999955
18 72600 14902 1210 1.370 0.8907 88 290400 37792 0 1.041 1
19 108900 14160 1210 1.379 0.8821 89 290400 19602 0 1.080 1
20 145200 13448 1210 1.391 0.8718 90 290400 13550 0 1.120 1
21 181500 12737 1210 1.406 0.8632 91 290400 10533 0 1.161 1
22 217800 12026 1210 1.422 0.8553 92 290400 8729 0 1.202 1
23 254100 11337 1210 1.444 0.8450 93 290400 7532 0 1.245 1
24 290400 10644 1210 1.469 0.8348 94 290400 6682 0 1.289 1
25 326700 9930 1210 1.495 0.8254 95 0 65268 0 1.079 0.9702
26 363000 9215 1210 1.526 0.8160 96 0 34947 0 1.155 0.9443
27 399300 8516 1210 1.568 0.8138 97 0 24865 0 1.233 0.9217
28 435600 7900 1210 1.636 0.8423 98 0 19841 0 1.312 0.9019
29 471900 7197 1210 1.704 0.9165 99 0 16842 0 1.392 0.8845
30 508200 6521 1210 1.802 1 100 0 14854 0 1.473 0.8690
31 0 36060 0 1.192 0.9152 101 0 13444 0 1.556 0.8553
32 36300 34381 0 1.196 0.9103 102 145200 52335 0 1.081 0.9636
33 72600 32502 0 1.194 0.9107 103 145200 28082 0 1.160 0.9329
34 108900 30791 0 1.198 0.9064 104 145200 20013 0 1.240 0.9068
35 145200 29125 0 1.204 0.9005 105 145200 15990 0 1.321 0.8845
36 181500 27510 0 1.213 0.8939 106 145200 13585 0 1.403 0.8653
37 217800 25888 0 1.223 0.8874 107 145200 11989 0 1.486 0.8487
38 254100 24270 0 1.234 0.8800 108 145200 10855 0 1.570 0.8553
39 290400 22655 0 1.248 0.8713 109 181500 49186 0 1.084 0.9613
40 326700 21039 0 1.265 0.8630 110 181500 26449 0 1.166 0.9286
41 363000 19420 0 1.284 0.8564 111 181500 18884 0 1.249 0.9009
42 399300 17802 0 1.308 0.8485 112 181500 15112 0 1.332 0.8771
43 435600 16189 0 1.338 0.8370 113 181500 12857 0 1.417 0.8566
44 471900 14572 0 1.376 0.8282 114 181500 11361 0 1.502 0.8387
45 508200 12960 0 1.428 0.8127 115 181500 10298 0 1.589 0.8513
Appendix 165
Data Envelopment Analysis with Selected Models and Applications
DMU
Down
payment
Annuities
Other fees
Bank loan
coefficient
CCR‐
efficiency
DMU
Down
payment
Annuities
Other fees
Bank loan
coefficient
CCR‐
efficiency
46 145200 29125 0 1.204 0.9005 116 217800 46037 0 1.087 0.9588
47 145200 20070 0 1.244 0.9043 117 217800 24815 0 1.172 0.9244
48 145200 15816 0 1.307 0.8937 118 217800 17755 0 1.258 0.8954
49 145200 14437 0 1.342 0.8877 119 217800 14234 0 1.344 0.8707
50 145200 13336 0 1.378 0.8804 120 217800 12130 0 1.432 0.8495
51 145200 11688 0 1.449 0.8690 121 217800 10733 0 1.521 0.8313
52 145200 10516 0 1.521 0.8774 122 217800 9741 0 1.610 0.8513
53 0 63322 0 1.047 1 123 254100 42887 0 1.091 0.9560
54 0 33001 0 1.091 1 124 254100 23182 0 1.179 0.9195
55 0 22919 0 1.136 1 125 254100 16626 0 1.268 0.8891
56 0 17895 0 1.183 1 126 254100 13357 0 1.359 0.8634
57 0 14896 0 1.231 1 127 254100 11402 0 1.450 0.8416
58 0 12908 0 1.280 1 128 254100 10106 0 1.542 0.8229
59 0 11498 0 1.330 1 129 254100 9185 0 1.635 0.8512
60 145200 50389 0 1.041 1 130 290400 39738 0 1.095 0.9527
61 145200 26136 0 1.080 1 131 290400 21548 0 1.187 0.9140
62 145200 18067 0 1.120 1 132 290400 15496 0 1.281 0.8819
63 145200 14044 0 1.161 1 133 290400 12479 0 1.375 0.8557
64 145200 11639 0 1.202 1 134 290400 10675 0 1.470 0.8338
65 145200 10043 0 1.245 1 135 290400 9478 0 1.567 0.8152
66 145200 8909 0 1.288 1 136 290400 8628 0 1.664 0.8512
67 181500 47240 0 1.041 0.999993 137 0 16080 5000 1.203 1
68 181500 24503 0 1.080 0.99998 138 0 14749 5000 1.226 1
69 181500 16938 0 1.120 0.999985 139 0 12760 5000 1.272 1
70 181500 13166 0 1.161 0.999999
Appendix 167
Data Envelopment Analysis with Selected Models and Applications
Appendix B
Table B–1 All possible solver status
No. Name Description
1 NORMAL COMPLETION
The solver terminated in a normal way: i.e., it was not
interrupted by a limit (resource, iterations, nodes, ...) or by
internal difficulties.
2 ITERATION INTERRUPT
The solver was interrupted because it used too many
iterations. Use option iterlim to increase the iteration limit if
everything seems normal.
3 RESOURCE INTERRUPT
The solver was interrupted because it used too much time.
Use option reslim to increase the time limit if everything
seems normal.
4 TERMINATED BY SOLVER
The solver encountered difficulty and was unable to
continue. More detail will appear following the message.
5 EVALUATION ERROR LIMIT
Too many evaluations of nonlinear terms at undefined
values. You should use variable bounds to prevent forbidden
operations, such as division by zero. The rows in which the
errors occur are listed just before the solution.
6 CAPABILITY PROBLEMS
The solver does not have the capability required by the
model, for example, some solvers do not support certain
types of discrete variables or support a more limited set of
functions than other solvers.
7 LICENSING PROBLEMS
The solver cannot find the appropriate license key needed to
use a specific subsolver.
8 USER INTERRUPT
The user has sent a message to interrupt the solver via the
interrupt button in the IDE or sending a Control+C from a
command line.
9 ERROR SETUP FAILURE
The solver encountered a fatal failure during problem set‐up
time.
10 ERROR SOLVER FAILURE
The solver encountered a fatal error.
11
ERROR INTERNAL SOLVER FAILURE
The solver encountered an internal fatal error.
12 SOLVE PROCESSING SKIPPED
The entire solve step has been skipped. This happens if
execution errors were encountered and the GAMS parameter
ExeErr has been set to a nonzero value, or the property
MaxExecError has a nonzero value.
13 ERROR SYSTEM FAILURE
This indicates a completely unknown or unexpected error
condition.
168 Appendix
2014 M. Toloo
Table B–2 All possible model status
No. Name Description
1 OPTIMAL The solution is optimal, that is, it is feasible (within
tolerances) and it has been proven that no other feasible
solution with better objective value exists.
2 LOCALLY OPTIMAL
This message means that a local optimum has been
found, that is, a solution that is feasible (within
tolerances) and it has been proven that there exists a
neighborhood of this solution in which no other feasible
solution with better objective value exists.
3 UNBOUNDED
The solution is unbounded. This message is reliable if
the problem is linear, but occasionally it appears for
difficult nonlinear problems that are not truly
unbounded, but that lack some strategically placed
bounds to limit the variables to sensible values.
4 INFEASIBLE The problem has been proven to be infeasible. If this was
not intended, something is probably misspecified in the
logic or the data.
5 LOCALLY INFEASIBLE
No feasible point could be found for the problem from
the given starting point. It does not necessarily mean
that no feasible point exists.
6 INTERMEDIATE INFEASIBLE
The current solution is not feasible, but that the solver
stopped, either because of a limit (e.g., iteration or
resource), or because of some sort of difficulty. Check
the solver status for more information.
7 FEASIBLE SOLUTION
A feasible solution has been found to a problem without
discrete variables.
8 INTEGER SOLUTION
A feasible solution has been found to a problem with
discrete variables. There is more detail following about
whether this solution satisfies the termination criteria
(set by options optcr and optca).
9 INTERMEDIATE NON-INTEGER
This is an incomplete solution to a problem with discrete
variables. A feasible solution has not yet been found.
10 INTEGER INFEASIBLE
It has been proven that there is no feasible solution to a
problem with discrete variables.
11 LIC PROBLEM - NO SOLUTION
The solver cannot find the appropriate license key
needed to use a specific subsolver.
12 ERROR UNKNOWN After a solver error, the model status is unknown.
13 ERROR NO SOLUTI0N
An error occurred and no solution has been returned. No
solution will be returned to GAMS because of errors in
the solution process.
14 NO SOLUTION RETURNED
A solution is not expected for this solve. For example,
the convert solver only reformats the model but does not
give a solution.
15 SOLVED UNIQUE
This is used when a CNS model is solved and the solver
somehow can be certain that there is only one solution.
The simplest examples are a linear model with a non‐
singular Jacobian, a triangular model with constant non‐
Appendix 169
Data Envelopment Analysis with Selected Models and Applications
No. Name Description
zero elements on the diagonal, and a triangular model
where the functions are monotone in the variable on the
diagonal.
16 SOLVED Locally feasible in a CNS models ‐ this is used when the
model is feasible and the Jacobian is non‐singular and
we do not know anything about uniqueness.
17 SOLVED SINGULAR
Singular in a CNS models – this is used when the model
is feasible (all constraints are satisfied), but the Jacobian
/ Basis is singular. In this case there could be other
solutions as well, in the linear case a linear ray and in the
nonlinear case some curve.
18 UNBOUNDED – NO SOLUTION
The model is unbounded and no solution can be
provided.
19 INFEASIBLE – NO SOLUTION
The model is infeasible and no solution can be provided.
170 Appendix
2014 M. Toloo
Appendix 171
Data Envelopment Analysis with Selected Models and Applications
Appendix C Table C–1 The data set for 73 vendors
Vendors
inputs
outputs
1
2
3
4
5
6
7
1
2
3 (T
L)
4 (T
L)
5 (T
L)
Vendor1
1 9
6 4
30
24
52
640
97.04
1,910,370
280,500
1,003,590
Vendor2
1 7
7 4
20
28
24
277
90.57
1,059,404
872,136
1,918,123
Vendor3
1 13
6 3
21
30
32
455
89.23
1,008,395
1,203,813
659,653
Vendor4
1 14
16
12
53
36
102
1294
89.49
3,133,756
1,838,444
13,603,758
Vendor5
1 7
7 9
30
30
48
617
90
223,217
1,026,029
318,097
Vendor6
1 13
15
8 56
32
121
1237
88.29
2,835,982
531,900
1,952,176
Vendor7
2 21
15
19
48
44
155
1832
95.69
2,662,772
8,583,540
2,035,702
Vendor8
1 10
12
6 28
27
62
711
95.06
1,466,014
3,144,548
8,675,483
Vendor9
1 8
8 10
49
36
95
1308
94.35
3,370,414
3,695,887
14,623,374
Vendor10
1 8
13
6 41
40
146
1500
96.84
2,769,254
8,092,843
1,744,480
Vendor11
3 13
13
9 62
38
95
1167
91.1
2,903,007
1,251,241
2,554,249
Vendor12
1 14
4 4
24
25
57
623
93.4
1,054,126
3,016,386
641,379
Vendor13
3 13
3 10
58
34
52
674
85.9
4,099,307
1,972,924
7,118,558
Vendor14
1 3
6 5
13
24
38
361
88.1
569,169
1,405,713
347,915
Vendor15
2 11
4 4
24
24
36
412
88
1,009,351
1,144,857
858,399
Vendor16
1 8
5 7
22
30
66
839
90.82
1,661,328
3,018,817
1,085,838
Vendor17
1 5
1 4
21
24
32
140
91.67
800,394
448,270
575,468
Vendor18
4 23
11
19
82
72
152
1978
91.67
4,831,579
8,602,663
16,859,827
Vendor19
1 15
8 11
56
80
262
4270
90
4,148,022
10,805,356
2,488,767
Vendor20
1 31
6 14
63
56
157
2260
90.52
4,316,224
3,581,663
16,333,754
Vendor21
2 16
2 7
60
49
164
2577
92.14
4,939,996
8,084,737
3,314,862
Vendor22
2 7
14
6 32
26
51
652
86.48
1,201,713
793,400
842,618
Vendor23
1 12
6 13
52
50
135
2052
94.32
3,632,682
2,175,415
1,549,335
172 Appendix
2014 M. Toloo
Vendors
inputs
outputs
1
2
3
4
5
6
7
1
2
3 (TL)
4 (TL)
5 (TL)
Vendor24
1 13
9 5
37
34
79
1013
94.41
2,932,165
2,693,782
2,343,993
Vendor25
3 7
4 7
58
36
94
1352
92.06
8,308,527
3,752,045
4,318,782
Vendor26
1 18
2 7
30
13
28
41
65
18,798
579,456
147,215
Vendor27
2 14
5 4
42
24
48
481
90.94
2,281,765
2,954,104
1,746,409
Vendor28
2 17
8 12
39
27
69
967
94.9
3,116,901
3,656,699
3,427,893
Vendor29
2 12
7 10
39
32
53
646
93.28
1,761,756
2,028,868
875,419
Vendor30
3 23
25
13
107
58
176
2437
91.54
7,921,469
3,145,388
10,576,052
Vendor31
2 10
7 4
27
25
27
321
92.61
1,077,919
2,242,762
1,361,882
Vendor32
3 3
3 7
30
20
42
448
86.88
406,804
2,537,174
788,317
Vendor33
1 6
5 4
22
10
28
130
90.97
736,004
804,568
443,961
Vendor34
6 28
8 9
49
24
85
1171
86.76
2,635,455
2,943,323
2,283,596
Vendor35
1 11
8 16
59
35
96
1141
91.95
3,210,906
5,135,574
19,608,883
Vendor36
2 19
6 9
55
28
74
937
90.59
3,762,919
1,853,635
6,506,520
Vendor37
3 13
2 5
41
30
63
745
89.1
1,511,385
5,470,865
1,163,909
Vendor38
1 13
3 4
18
12
21
37
88
290,768
553,214
63,096
Vendor39
1 9
7 9
28
10
50
198
92
37,106
670,374
252,852
Vendor40
1 7
4 3
23
24
32
385
93.84
1,310,642
1,697,151
1,183,943
Vendor41
1 17
7 7
54
33
68
1130
84.6
3,542,367
2,309,716
2,587,842
Vendor42
2 8
5 5
35
27
46
854
96.35
1,641,654
4,130,970
933,502
Vendor43
3 14
3 10
57
40
107
1321
89.39
3,136,621
4,161,952
5,582,650
Vendor44
2 9
2 3
14
24
54
519
86.48
890,330
2,658,355
555,670
Vendor45
1 4
4 4
30
16
52
331
93.75
1,220,491
1,049,447
2,533,058
Vendor46
1 9
4 7
51
14
45
338
86.33
1,523,372
1,586,915
1,805,498
Vendor47
1 15
6 7
40
32
85
1256
89.92
3,183,149
2,656,590
9,671,409
Vendor48
1 17
5 12
63
47
96
1774
92.42
3,010,272
4,652,952
13,493,640
Vendor49
1 16
6 14
50
62
168
3370
90.1
4,172,978
9,059,555
22,050,080
Vendor50
1 14
6 14
53
58
145
2611
91.83
2,757,365
4,277,783
13,246,972
Appendix 173
Data Envelopment Analysis with Selected Models and Applications
Vendors
inputs
outputs
1
2
3
4
5
6
7
1
2
3 (TL)
4 (TL)
5 (TL)
Vendor51
2 19
5 5
35
44
167
1112
93.21
1,888,575
1,713,480
2,131,156
Vendor52
1 22
6 20
61
68
184
2307
93.1
5,128,151
7,607,844
21,397,558
Vendor53
2 21
5 16
57
41
145
1287
91.23
5,758,179
1,055,287
34,259,621
Vendor54
1 11
6 6
32
48
148
1310
93.86
2,660,087
2,911,758
2,338,698
Vendor55
1 15
5 20
68
58
189
3689
90.24
4,044,013
3,650,729
29,173,672
Vendor56
1 15
6 13
47
61
340
5923
91.41
5,777,861
1,753,252
4,033,602
Vendor57
2 13
6 9
60
25
64
831
91.76
1,651,355
1,937,189
1,055,978
Vendor58
1 5
13
3 30
21
49
664
90.71
1,078,255
2,679,027
749,988
Vendor59
4 12
5 6
42
38
86
1092
95.74
2,555,576
4,128,704
1,612,117
Vendor60
1 10
4 3
38
24
42
460
95.21
2,650,715
1,043,431
1,378,041
Vendor61
1 16
4 21
75
47
211
2193
88.93
5,576,965
13,186,358
7,256,506
Vendor62
1 12
6 8
37
24
59
868
94.72
2,161,652
2,738,578
918,075
Vendor63
1 9
3 10
45
19
48
621
89.53
2,071,493
2,335,022
9,708,144
Vendor64
1 5
5 5
16
11
36
288
89.41
1,056,523
424,215
4,447,047
Vendor65
2 14
5 6
44
20
78
793
96.69
2,086,320
2,088,380
1,450,859
Vendor66
3 10
8 6
47
24
115
859
93.42
2,511,645
4,371,967
1,585,457
Vendor67
2 12
2 3
25
15
46
350
91
1,165,949
1,717,854
1,679,116
Vendor68
1 15
6 9
57
55
115
1245
91.32
3,371,469
3,107,083
3,908,841
Vendor69
1 12
5 7
22
28
76
551
89.94
1,395,345
1,995,610
1,134,891
Vendor70
1 12
5 8
42
50
88
590
92.23
2,458,194
973,404
4,115,389
Vendor71
1 9
5 4
23
24
55
498
89.41
1,321,073
1,727,175
872,499
Vendor72
1 7
4 5
22
30
72
630
96.25
1,212,790
5,149,724
891,921
Vendor73
2 12
4 8
60
34
78
689
87.34
1,852,594
1,316,134
1,394,026
min price
(TL)
1000000
300000
300000
100000
90000
250000
300000
max price
(TL)
28000000
7700000
5700000
1100000
1100000
550000
650000
Appendix 175
Data Envelopment Analysis with Selected Models and Applications
Appendix D
Table D–1 Some selected papers containing practical applications
No. Title Author(s) Year
Number of.
DMUs In Out
1
Diversification‐consistent
data envelopment analysis
based on directional‐
distance measures
Branda 2015 48 10 1
2
A Bounded Data
Envelopment Analysis
Model in a Fuzzy
Environment with an
Application to Safety in the
Semiconductor Industry
Hatami‐
Marbini et al. 2015b 8 2 2
3
Finding an initial basic
feasible solution for DEA
models with an application
on bank industry
Toloo et al. 2015 50 3 3
4 Interval data without sign
restrictions in DEA
Hatami‐
Marbini et al. 2015a 20 3 2
5
A network DEA assessment
of team efficiency in the
NBA
Moreno and
Lozano 2014 30 2 1
6
Incorporating health
outcomes in Pennsylvania
hospital efficiency: an
additive super‐efficiency
DEA approach
Du et al. 2014 119 4 3
7
On relations between DEA‐
risk models and stochastic
dominance efficiency tests
Branda and
Kopa 2014 48 4 1
8
An Epsilon‐free Approach
for Finding the Most
Efficient Unit in DEA
Toloo 2014a 12 2 2
9
The most cost efficient
automotive vendor with
price uncertainty: A new
DEA approach
Toloo and
Ertay 2014 73 7 5
10
Finding the best asset
financing alternative: A
DEA‐WEO Approach
Toloo and
Kresta 2014 139 4 1
11
Diversification‐consistent
data envelopment analysis
with general
deviation measures
Branda 2013 25 4 1
176 Appendix
2014 M. Toloo
No. Title Author(s) Year
Number of.
DMUs In Out
12
Chance‐constrained DEA
models with random fuzzy
inputs and outputs
Tavana et al. 2013 30 2 2
13
The most efficient unit
without explicit inputs: An
extended
MILP‐DEA model
Toloo 2013 40 1 6
14
Assessment of the site effect
vulnerability within urban
regions by data
envelopment analysis: A
case study in Iran
Saein and
Saen 2012 20 3 1
15
Positive and normative use
of fuzzy DEA‐BCC models:
A critical view on NATO
enlargement
Hatami‐
Marbini
et al.
2013 18 6 5
16
A common set of weight
approach using an ideal
decision making unit in data
envelopment analysis
Saati et al. 2012 286 2 4
17
Alternative solutions for
classifying inputs and
outputs in
data envelopment analysis
Toloo 2012 50 2 3
18 Super‐efficiency infeasibility
and zero data in DEA Lee and Zhu 2012 15 8 1
19
The examination of pseudo‐
allocative and pseudo‐
overall efficiencies in DEA
using shadow prices
Paradi and
Tam 2012 100 5 6
20
Reducing differences
between profiles of weights:
A peer‐restricted cross‐
efficiency evaluation
Ramón et al. 2011 14 3 2
21
A cross‐dependence based
ranking system for efficient
and
inefficient units in DEA
Chen and
Deng 2011 20 2 2
22
Performance and congestion
analysis of the Portuguese
hospital services
Simões and
Marques 2011 68 3 3
23
Employing post‐DEA Cross‐
evaluation and Cluster
Analysis in a sample of
Greek NHS Hospitals
Flokou et al. 2011 27 3 4
Appendix 177
Data Envelopment Analysis with Selected Models and Applications
No. Title Author(s) Year
Number of.
DMUs In Out
24
A neutral DEA model for
cross‐efficiency evaluation
and its extension
Wang and
Chin 2010a 14 3 2
25
Some alternative models for
DEA cross‐efficiency
evaluation
Wang and
Chin 2010b 7 3 3
26
Malmquist productivity
index based on common‐
weights DEA:
The case of Taiwan forests
after reorganization
Kao 2010 17 4 3
27
A semi‐oriented radial
measure for measuring the
efficiency of
decision making units with
negative data, using DEA
Emrouznejad
et al. 2010 13 2 3
28
A distance friction
minimization approach in
data envelopment analysis:
A comparative study on
airport efficiency
Suzuki et al. 2010 30 4 2
29
Efficiency evaluations with
context‐dependent and
measure‐specific data
envelopment: An
application in a World Bank
supported project
Ulucan and
Bariş Atici
2010 81 2 6
30
Measuring the performance
of financial holding
companies
Chao et al. 2010 14 3 3
31
DEA model with shared
resources and efficiency
decomposition
Chen et al. 2010 27 3 3
32 DEA‐based production
planning Du et al. 2010 20 2 5
33
DEA game cross‐efficiency
approach to Olympic
rankings
Wu et al. 2009 62 2 3
34
A new method for ranking
discovered rules from data
mining by DEA
Toloo et al. 2009 46 1 4
35
A new integrated DEA
model for finding most
BCC‐efficient DMU
Toloo and
Nalchigar 2009 19 2 4
36
Slacks‐based efficiency
measures for predicting
bank performance
Liu 2009 24 3 3
178 Appendix
2014 M. Toloo
No. Title Author(s) Year
Number of.
DMUs In Out
37
Efficiency measurement and
ranking of the tutorial
system using IDEA
Lin 2009 20 1 2
38
The link between
operational efficiency and
environmental impacts A
joint application of Life
Cycle Assessment and Data
Envelopment Analysis
Lozano et al. 2009 62 14 1
39
Modeling the efficiency of
top Arab banks: A DEA–
neural network approach
Mostafa 2009 85 4 4
40
Methodological comparison
between DEA and DEA–DA
from the perspective of
bankruptcy assessment
Sueyoshi and
Goto 2009 23 7 2
41
Static versus dynamic DEA
in electricity regulation: the
case of US transmission
system operators
Geymueller 2009 50 4 1
42
Alternative secondary goals
in DEA cross‐efficiency
evaluation
Liang et al. 2008 18 2 3
43
Finding the most efficient
DMUs in DEA: An
improved integrated model
Amin and
Toloo 2007 19 2 4
44 Rank order data in DEA: A
general framework Cook and Zhu 2006 33 3 5
45
Technical efficiency of
peripheral health units in
Pujehun district of Sierra
Leone: a DEA application
Renner et al. 2005 37 2 6
46 Ranking efficient units in
DEA Chen 2004 20 3 1
47
Are all Scales Optimal in
DEA? Theory and Empirical
Evidence
Førsund and
Hjalmarsson 2004 163 4 4
Appendix 179
Data Envelopment Analysis with Selected Models and Applications
Table D–2 Inputs and outputs in bank studies adapted from Mostafa (2009)
Inputs Frequency % Outputs Frequency %
Employees 22 84.62 Loans 12 46.15
Expenses 11 42.31 Number of
transactions 8 30.77
Space 5 19.23 Deposits 7 26.92
Terminals 5 19.23 Non‐ interest income 7 26.92
Capital 5 19.23 Interest income 5 19.23
Deposits 4 15.38 customer response 3 11.54
Assets 4 15.38 Revenues 3 11.54
Number of branches 4 15.38 Profit 3 11.54
Rent 3 11.54 Investment in
securities 3 11.54
Number of accounts 3 11.54 Advances 2 7.69
Credit application 2 7.69 Current accounts 2 7.69
ATMs 2 7.69 Error corrections 1 3.85
Location 2 7.69 Liability sales 1 3.85
Costs 2 7.69 Maintenance 1 3.85
Marketing 2 7.69 Marketed balances 1 3.85
Selected financial
ratios 1 3.85
Selected financial
ratios 1 3.85
Supplier 1 3.85 Insurance commission 1 3.85
Acquired equipment 1 3.85 satisfaction 1 3.85
Funds from
customers 1 3.85 ROA 1 3.85
Loanable funds 1 3.85 ROE 1 3.85
Counter transactions 1 3.85
Net worth 1 3.85
Borrowings 1 3.85
Loans 1 3.85
Size 1 3.85
Sale FTE 1 3.85
City employment rate 1 3.85
180 Appendix
2014 M. Toloo
Table D–3 Iranian bank data set and CCR‐efficiency score
DMUs
Inputs Outputs
Questionable Efficiency
Employees
No. of accounts
Assets
Space
Costs
Expenses
No. of
transactions
Deposits
Loans
Check Card
Credit Card
OTP
1 11 1250 1753 97 10020 3137 5214 72149 57537 5105 4839 25 1
2 17 5019 2604 150 11440 4406 5343 89781 51114 8646 8364 24 0.96
3 7 3217 1155 61 8,427 2180 5145 42654 52485 2797 2697 5 1
4 12 1061 1899 105 11816 6477 3249 97812 67298 3373 3096 68 1
5 14 5219 2215 123 12426 3325 6706 77031 43487 8993 8787 58 1
6 14 1389 2357 123 9907 3757 6259 75923 41442 7604 7371 40 1
7 9 7166 1370 79 10365 2714 3652 47763 43262 3608 3497 9 0.68
8 5 1475 829 44 5283 2887 3913 45732 14237 3795 3500 32 1
9 6 1800 985 52 11061 2852 3566 55222 41062 3299 3182 15 0.93
10 6 1689 1023 52 5856 2606 4559 53323 37418 1858 1746 8 1
11 8 1780 1311 70 8745 4442 4441 69734 57883 3030 2882 23 1
12 9 2669 1536 79 7326 1989 5031 49153 47139 4811 4578 31 1
13 8 7175 1367 70 8326 3727 5053 92365 55543 6840 6588 45 1
14 7 2120 1193 61 6525 3473 4762 64235 22347 5382 5188 22 1
15 9 30618 1359 79 8158 3824 6876 89104 45717 7628 7292 105 1
16 7 1464 1111 61 11135 1524 4307 42012 73925 3187 2984 22 1
17 7 8924 1182 68 6920 3573 5331 69360 27246 3743 3524 24 1
18 7 2388 1069 61 5864 2523 4004 51438 26531 4360 4140 17 0.99
19 6 4714 992 52 5039 2398 2342 39948 20223 2688 2574 36 1
20 7 1866 1180 62 8378 3165 4238 154284 43928 4182 4008 18 1
Appendix 181
Data Envelopment Analysis with Selected Models and Applications
Table D–4 Active banks in the Czech Republic and BCC‐efficiency score
Banks
Inputs Outputs
Questionable
BCC‐efficiency
No. of
Employees
No. of
branches
Assets
Equity
Expenses
Deposits
Loans
Non‐ interest
income
Interest income
AIR 400 18 33600 2596 745 30696 11135 14 554 1
CMZRB 217 5 111706 4958 566 86967 16813 634 1700 1
CS 10760 658 920403 93190 18259 688624 489103 15412 37717 1
CSOB 7801 322 937174 73930 16087 629622 479516 8,747 32697 1
EQB 296 13 8985 1296 601 7502 5611 19 215 1
ERB 72 1 33614 464 173 2940 1762 15 131 1
FIO 59 36 18561 726 347 17174 6465 211 536 1
GEMB 3346 260 135474 34486 5276 97063 101898 3943 11026 1
ING 293 10 128425 913 1034 92579 19216 468 5139 1
JTB 407 3 85087 7233 1333 62085 39330 487 3686 1
KB 8 758 399 786836 100577 13511 579 067 451 547 8834 35 972 1
LBBW 365 18 31300 2774 1138 20274 2528 128 1046 0.8625
RB 2927 125 197628 18151 57112 144143 150138 2829 8563 1
UCB 2004 98 318909 38937 13804 195120 192046 2740 8891 1
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191
List of Tables
Table 1–1 Hospital case with single input and single output ................................... 2 Table 1–2 Two inputs and one output case ................................................................. 7 Table 1–3 Data with unitized Nurses ........................................................................... 9 Table 1–4 One input and two outputs case ............................................................... 10 Table 1–5 Two inputs and two outputs case ............................................................. 13 Table 1–6 Various weights ........................................................................................... 14 Table 1–7 The CCR efficiency scores and optimal weights ..................................... 15 Table 2–1 Data set and results ..................................................................................... 23 Table 2–2 The desired weights .................................................................................... 24 Table 2–3 The weights of model (2.12) ....................................................................... 27 Table 2–4 Bank data in the Czech Republic ............................................................... 36 Table 2–5 Optimal weights and CCR‐efficiency score (input‐oriented) ................ 37 Table 2–6 Optimal weights and objective value (output‐oriented) ........................ 37 Table 2–7 Reference sets ( ∗) ..................................................................................... 38 Table 2–8 Max‐slack solutions ..................................................................................... 39 Table 2–9 Optimal weights and BCC‐efficiency score (input‐oriented) ................ 42 Table 2–10Input and output data for 12 industrial robots ...................................... 46 Table 2–11 Normalized outputs and results obtained ............................................. 46 Table 2–12 Results of the additive model .................................................................. 51 Table 3–1 Type of variables ......................................................................................... 58 Table 4–1 Optimal weights obtained by DEA‐Solver ............................................... 83 Table 4–2 Optimal weights obtained by GAMS ........................................................ 83 Table 4–3 Optimal maximin weights .......................................................................... 84 Table 4–4 AR‐Efficiency and weights ......................................................................... 88 Table 4–5 Minsum efficiency and weights ................................................................. 90 Table 4–6 Minimax efficiency and weights ............................................................... 91 Table 4–7 Minsum efficiency score (CSW) ................................................................. 97 Table 4–8 Inputs and outputs of 19 FLDs .................................................................. 99 Table 4–9 Different efficiencies for 19 FLDs ............................................................ 100 Table 5–1 Different efficiency scores ........................................................................ 103 Table 5–2 The optimal weights for the minimax model ........................................ 104 Table 5–3 Different efficiency scored with maximum epsilon .............................. 105 Table 5–4 A counter example adapted from Amin and Toloo (2007) .................. 105 Table 5–5 Efficiency score obtained by the model of Amin (2009) ....................... 110
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2014 M. Toloo
Table 5–6 The normalized data and efficiency score ............................................. 110 Table 5–7 The optimal solution of Foroughi’s model with 0 ......................... 112 Table 5–8 The optimal solution of Foroughi’s model with ∗ 0.0769 .............. 113 Table 5–9 Various efficiency scores (Karsak and Ahiska, 2005) ............................ 115 Table 5–10 Efficiency scores of Amin et al. (2006) .................................................. 117 Table 5–11 The data set of 40 professional tennis players ..................................... 118 Table 5–12 Efficiency status for various ................................................................ 119 Table 5–13 Efficiency score of asset financing alternatives.................................... 123 Table 5–14 Efficiency scores by epsilon‐free approaches. ...................................... 129 Table 6–1 University data set (Beasley, 1990) .......................................................... 140 Table 6–2 Results from the multiplier model .......................................................... 141 Table 6–3 The comparison of max and min of efficiency scores ........................... 142 Table 6–4 Results of applying the aggregate multiple model ............................... 157 Table 6–5 Selective measures obtained by multiplier model ................................ 159 Table 6–6 Selective measures by envelopment model ........................................... 159
193
List of Figures
Figure 1–1 Data set and efficient frontier ..................................................................... 4 Figure 1–2 Projection Points ........................................................................................... 5 Figure 1–3 Two inputs and one output case ................................................................ 7 Figure 1–4 Improvement of H5 ...................................................................................... 8 Figure 1–5 Alternative perspective of two inputs and one output case ................... 9 Figure 1–6 One input and two outputs case .............................................................. 11 Figure 1–7 Improvement of H7 ................................................................................... 11 Figure 1–8 Alternative perspective of one input and two outputs case ................. 12 Figure 2–1 Feasibility axiom ........................................................................................ 28 Figure 2–2 Free disposability regions for 1 input and 1 output case ...................... 29 Figure 2–3 CRS axiom ................................................................................................... 29 Figure 2–4 PPS for single input and single output case ........................................... 30 Figure 2–5 Free disposability region for 2 inputs and 1 output case ...................... 30 Figure 2–6 PPS for 2 inputs and 1 output case .......................................................... 31 Figure 2–7 PPS for 2 inputs and 1 output ................................................................... 31 Figure 2–8 Free disposability region for 1 input and 2 outputs case ...................... 32 Figure 2–9 PPS for 1 input and 2 outputs case .......................................................... 32 Figure 2–10 PPS for 1 input and 2 outputs ................................................................. 33 Figure 2–11 PPS for single input and single output case (VRS) .............................. 40 Figure 2–12 PPS for 2 inputs and 1 output (VRS) ...................................................... 40 Figure 3–1 GAMS IDE .................................................................................................. 55 Figure 3–2 GAMS code for the CCR model ............................................................... 57 Figure 3–3 Process Window ......................................................................................... 63 Figure 3–4 Listing window .......................................................................................... 64 Figure 3–5 Excel file (data.xlsx) ................................................................................... 78 Figure 3–6 A GDX file ................................................................................................... 79 Figure 4–1 CCR‐efficiency and CCR‐AR‐efficiency scores ...................................... 88
195
Index
ADD‐efficient, 50
additive (ADD) model, 50
adjacency score, 98
advanced manufacturing
technology, 102
advanced manufacturing
technology (AMT), 102
aggregate efficiency, 137, 144, 155,
157
aggregate model, 136, 148
annuities, 49, 124
asset‐financing alternatives, 49
association of tennis professionals,
117
association of tennis professionals
(ATP), 117
Assurance Region (AR) method, 85
Assurance Region Global (ARG)
method, 85
auxiliary binary variable, 119
Bank loan coefficient, 49
banking industry, 82, 87, 96, 153,
155, 156
best efficient, 101
bi‐section algorithm, 114
CCR‐efficiency, 19
CEPLEX solver, 70
classifier model, 136, 137, 141, 143,
145
Common Set of Weights (CSW), 92
cone, 19
constant returns to scale (CRS), 3
convexity axiom, 28
convexity constraint, 40, 45, 48, 52,
155
cost efficiency (CE), 130
CRS axiom, 28
CSW‐efficiency, 121
DEAFrontier, 53
DEA‐SOLVER, 53
decision making units (DMUs), 3
deviation from efficiency variable,
89
discovered association rules, 113
discriminating parameter, 102, 114,
117
discriminating power, 85, 87, 92,
101, 103, 104, 108, 114, 115, 117,
119, 146
dominate, 1
down payment, 49
Echo print in GAMS, 64
efficient, 3
efficient frontier, 3, 21
envelopment form, 19
epsilon‐free approach, 95, 97
external file in GAMS, 71
extreme efficient, 21
Facility Layout Design (FLD), 97
feasibility axiom, 28
feasible activity, 27
196 Index
2014 M. Toloo
feasible activity set (FAS), 44
financing decision problem, 49
fixed measure, 148, 156
fixed weights, 12
flexibility, 98
flexible measure, 135–138, 141, 143–
146
fractional programming, 18
free disposability axiom, 28
GAMS data exchange (GDX), 77
GAMS IDE, 54
GAMS statments
ALIAS, 60
EQUATION, 58
FILE, 72
LOOP, 60
MODEL, 59
PARAMETERS, 57
PUT, 72
SETS, 56
SOLVE, 59
TABLE, 56
VARIABLES, 58
General Algebraic Modeling System
(GAMS), 54
Group Technology (GT), 98
Handling coefficient, 45
Individual DMU model, 136, 154,
155
individual multiplier form, 158
industrial robot evaluation problem,
45
inefficient, 3
infinitesimal epsilon, 24
input, 2
input data matrix, 17
input excess, 35
input orientation, 5, 11, 59
input price uncertainly, 130
input‐oriented model, 19, 25
integrated minimax approach, 97
integrated minsum approach, 92
Konsi, 53
Limrow in GAMS, 67
linear programming, 18
listing window in GAMS, 63
load capacity, 45
Material handing cost, 98
material handling vehicle, 99
MaxDEA, 53
maximin objective function, 81
maximin weights, 81, 83, 84
max‐slack, 35, 38, 41, 107
max‐slack model, 35
max‐slack solution, 35, 107
minimax efficiency score, 90, 91,
114, 115
minsum efficiency score, 89, 97
model statistic in GAMS, 69
model status in GAMS, 70, 168
most efficient, 101, 103, 104, 106,
108, 109, 113–116, 119, 121, 122,
124, 127–129, 176, 178
multiplier form, 18–22, 25, 27, 34, 41,
43, 44, 52, 60, 81, 86, 88, 89, 91,
130, 143, 144, 147, 148, 150, 153–
155, 158
multipliers, 17, 19, 24, 59
Non‐Archimedean epsilon, 24, 93,
103, 108, 111–113, 120, 122
non‐geometrically redundant
constraint, 93
normalization condition, 19, 23, 69
One‐Time Password (OTP), 156
optimistic CE model, 130, 131
option $include, 71
output, 2
output orientation, 5, 8
Index 197
Data Envelopment Analysis with Selected Models and Applications
output per input, 2
output shortfall, 35
output‐oriented model, 25
Peer group, 21
pessimistic CE model, 130
PIM‐DEA, 53
Primal‐Dual relations, 18
production function, 27
production possibility set (PPS), 27
professional tennis players, 113,
116–119
pure input data, 49, 121, 122
Quality, 45, 98, 99
Radial efficiency, 33
ratio of ratios form, 3
reference set, 6, 11, 21, 38
relational operator in GAMS, 59
relative efficiency, 3, 45, 92, 135
repeatability, 45
running a model in GAMS, 62
Selecting model, 147–154, 156–159
selective measure, 135, 147–150, 152,
153, 155, 157, 159
semipositive, 17
separable programming, 106, 108,
109
shadow prices, 17, 176
shape ratio, 98
share case, 143, 146
single constant input, 31, 42, 45, 113
single constant output, 42, 48
solution summary in GAMS, 69
solver status in GAMS, 167
supplier selection problem, 102
Technical efficiency, 33, 44
the rule of thumb, 146–151, 156, 158
translation invariant, 51
Turkish automotive company, 133
Unit under evaluation, 18, 35, 89
unit vector, 33
unitized axes, 6
unitized measure, 6
units invariance, 144
units invariant, 3
Variable returns to scale (VRS), 12
velocity, 45
virtual input, 18
virtual output, 18
Weight restrictions, 84–87, 130
weighted minsum efficiency, 89
without explicit inputs, 42, 43, 119,
120, 176
without explicit inputs (WEI), 42
without explicit outputs, 42, 47, 49,
121, 122
without explicit outputs (WEO), 42
world class manufacturing (WCM),
133
Zero‐slack, 35
199
Data Envelopment Analysis
with Selected Models and
Applications
Mehdi Toloo
Summary
DEA is a well‐known powerful quantitative, analytical tool for measuring and
evaluating performance. Such evaluations help managers optimize the use of
resources in their organisations as a main challenge in reality. This book covers
some basic DEA models, in particular the multiplier models to focus on the
importance of weights (multipliers) in DEA. It also aiming at dealing with some
innovative uses of binary variables in developing new DEA model formulations.
Some case studies from different contexts are provided to illustrate the validity of
the proposed approaches.
This book is organised into six chapters. In the first chapter the concept of relative
efficiency is illustrated. The second chapter reviews some basic DEA models and
their properties. Chapter 3 provides a platform for solving DEA models using
GAMS software. Next chapter presents the importance of weights in DEA. Chapter
5 develops various models to address the problem of selecting most efficient DMU.
Finally, Chapter 6 accommodates flexible and selective measures in DEA.
This publication has been elaborated in the framework of the project Support
research and development in the Moravian‐Silesian Region 2013 DT 1 – International
research teams (02613/2013/RRC). Financed from the budget of the Moravian‐
Silesian Region. It has been also supported by the Czech Science Foundation
(GACR project 14‐31593S) and through European Social Fund (OPVK project
CZ.1.07/2.3.00/20.0296).
200 Summary
2014 M. Toloo
About Author
Mehdi Toloo, B.Sc. (Pure Mathematics), M.Sc. (Applied
Mathematics), Ph.D. (Operations Research), Associate
professor at Department of Business Administration,
Technical University of Ostrava, Czech Republic. Areas of
interest include Data Envelopment Analysis, Multi‐
objective programming and Network Flows. He has
written fourteen books. His research has been published in
European Journal of Operational Research, Computers &
Operations Research, International Journal of Production
Research, Computers & Industrial Engineering, Applied
Mathematics and Computers, Applied Mathematical
Modelling, Expert Systems with Applications, Annals of Operations Research,
Journal of the Operational Research Society, International Journal of Advanced
Manufacturing Technology, Computers and Mathematics with Applications,
Measurement, Computational Economics, Computational and Applied
Mathematics, IMA Journal of Management Mathematics, International Journal of
Management Science, and The Journal of Supercomuting.
Series on Advanced Economic Issues Faculty of Economics, VŠB-TU Ostrava www.ekf.vsb.cz/saei [email protected]
EDITORS' OFFICE PUBLISHER VŠB-Technical University Ostrava, VŠB-TU Ostrava, IČ 61989100 Faculty of Economics, Sokolská 33 Faculty of Economics, Sokolská 33 701 21 Ostrava, Czech Republic 701 21 Ostrava, Czech Republic Assistant Editor: Irena HOLBOVÁ
SERIES EDITOR Tomáš TICHÝ VŠB-TU Ostrava, CZ CO-EDITORS
Martin MACHÁČEK Vojtěch SPÁČIL Jan SUCHÁČEK VŠB-TU Ostrava, CZ VŠB-TU Ostrava, CZ VŠB-TU Ostrava, CZ
EDITORIAL BOARD Zdeněk ZMEŠKAL VŠB-TU Ostrava, CZ Head of Editorial Board
Bahram ADRANGI John ANCHOR Milan BUČEK University of Portland, USA Huddersfield University, UK University of Economics, SK
Dana DLUHOŠOVÁ Grant FORSYTH Jan FRAIT VŠB-TU Ostrava, CZ Avista, USA Czech National Bank, CZ
Petr JAKUBÍK Yelena KALYUZHNOVA Jaroslav RAMÍK
EIOPA, D Henley University of Reading, UK Silesian University Opava, CZ
Jaap SPRONK Jan VECER Ruediger WINK Erasmus University Rotterdam, NL Columbia University, USA HTWK Leipzig, D
Series on Advanced Economic Issues is published by Faculty of Economics, VŠB-Technical University of Ostrava. It covers a broad set of topics in business/economics disciplines, but mainly current issues in economics, finance, management, business economy, and informatics. Original results of research focusing on any of the topics mentioned above are welcome.
The series addresses researchers, students, and practitioners interested in advanced treatment of all economic disciplines. Manuscripts can be submitted to [email protected]. We kindly ask potential authors to follow the instructions about the structure of the book before they proceed to submission procedure. The text can be written either in Czech or English. The text’s length shouldn’t be less 100 pages, when the template is followed. Before publishing, each manuscript must be reviewed at least by two independent reviewers. The reviewing procedure is strictly double–blind. For further information authors may visit www.ekf.vsb.cz/saei.
DATA ENVELOPMENT
ANALYSIS WITH SELECTED MODELS AND APPLICATIONS
Mehdi Toloo
Vydala VŠB‐TU Ostrava
1st Edition 2014
Tisk Grafico, s.r.o.
Náklad 300 ks
Počet stran 224
Volume 1 Prokop, L., Medvec, Z., Zmeškal, Z. (2009). Problematika oceňování nedodané energie v průmyslu, SAEI, vol. 1. Ostrava: VSB-TU Ostrava. Volume 2 Hrubý, P. a kol. (2010). Víceúrovňové modelování podnikových procesů (systém REA), SAEI, vol. 2. Ostrava: VSB-TU Ostrava. Volume 3 Kutscherauer, A. a kol. (2010). Regionální disparity. Disparity v regionálním rozvoji země, jejich pojetí, identifikace a hodnocení, SAEI, vol. 3. Ostrava: VSB-TU Ostrava. Volume 4 Hančlová, J. a kol. (2010). Makroekonometrické modelování české ekonomiky a vybraných ekonomik EU, SAEI, vol. 4. Ostrava: VSB-TU Ostrava. Volume 6 Tichý, T. (2010). Simulace Monte Carlo ve financích: Aplikace při ocenění jednoduchých opcí, SAEI, vol. 6. Ostrava: VSB-TU Ostrava. Volume 7 Doleželová, H., Halásek, D. (2011). Služby v obecném hospodářském zájmu v EU. Komparace České republiky a Německa, SAEI, vol. 7. Ostrava: VSB-TU Ostrava. Volume 8 Machová, Z., Kaštan, M., Janíčková, L., Kotlán, I. (2011). Tax Burden in OECD Countries: WTI Application, SAEI, vol. 8. Ostrava: VSB-TU Ostrava. Volume 9 Tichý, T. (2011). Lévy Processes in Finance: Selected applications with theoretical background, SAEI, vol. 9. Ostrava: VSB-TU Ostrava. Volume 10 Macurová, P. a kol. (2011). Řízení rizik v logistice, SAEI, vol. 10. Ostrava: VSB-TU Ostrava. Volume 11 Dvoroková, K. a kol. (2012). Ekonometrické modelování konvergence ekonomické a cenové úrovně. Analýza průřezových a panelových dat, SAEI, vol. 11. Ostrava: VSB-TU Ostrava. Volume 12 Melecký, M., Melecký, A. (2012). Aplikované modelování pro potřeby hospodářské politiky, SAEI, vol. 12. Ostrava: VSB-TU Ostrava. Volume 13 Halásková, M. (2012). Veřejná správa a veřejné služby v zemích Evropské unie, SAEI, vol. 13. Ostrava: VSB-TU Ostrava. Volume 14 Šebestíková, M. a kol. (2012). Daňová a sociální optimalizace ve vztahu k nezaměstnanosti v České republice, SAEI, vol. 14. Ostrava: VSB-TU Ostrava.
Volume 15 Pytlíková, M. a kol. (2012). Gender wage gap and discrimination in the Czech Republic, SAEI, vol. 15. Ostrava: VSB-TU Ostrava. Volume 16 Hučka, M. a kol. (2012). Správa společností v zemích střední a východní Evropy, SAEI, vol. 16. Ostrava: VSB-TU Ostrava. Volume 17 Vrabková, I. (2012). Perspektivy řízení kvality ve veřejné správě, SAEI, vol. 17. Ostrava: VSB-TU Ostrava.Volume 18 Marček, D. (2013). Pravděpodobnostné modelovanie a soft computing v ekonomike, SAEI, vol. 18. Ostrava: VSB-TU Ostrava. Volume 19 Čulík, M. (2013). Aplikace reálných opcí v investičním rozhodování firmy, SAEI, vol. 19. Ostrava: VSB-TU Ostrava. Volume 20 Šnapka, P. a kol. (2013). Rozhodování a rozhodovací procesy v organizaci (vybrané problémy), SAEI, vol. 20. Ostrava: VSB-TU Ostrava. Volume 21 Šimíčková, M., Drastichová, M. (2013). Ekonomie udržitelnosti – alternativní přístupy a perspektivy, SAEI, vol. 21. Ostrava: VSB-TU Ostrava. Volume 22 Kutscherauer, A., Šotkovský, I., Adamovský, J., Ivan, I. (2013). Socioekonomická geografie a regionální rozvoj. Regionální analýzy v přístupech socioekonomické geografie k regionálnímu rozvoji, SAEI, vol. 22. Ostrava: VSB-TU Ostrava. Volume 23 Kutscherauer, A., Václavková, R., Malinovský, J. a kolektiv. (2013). Komplementární přístupy k podpoře regionálního a municipálního rozvoje, SAEI, vol. 23. Ostrava: VSB-TU Ostrava. Volume 24 Komárková, Z., Frait, J., Komárek, L. (2013). Macroprudential Policy in a Small Economy, SAEI, vol. 24. Ostrava: VSB-TU Ostrava.
Volume 25 Komárek, L. a kol. (2013). Money, Pricing and Bubbles, SAEI, vol. 25. Ostrava: VSB-TU Ostrava. Volume 26 Tichý, T. (2013). Backtesting of VaR based models: Methodological review and selected applications, SAEI, vol. 26. Ostrava: VSB-TU Ostrava. Volume 27 Fojtíková, L. a kol. (2014). Postavení Evropské unie v podmínkách globalizované světové ekonomiky, SAEI, vol. 27. Ostrava: VSB-TU Ostrava.
Volume 28 Dluhošová, D. et al. (2014). Financial Management and Decision-making of a Company. Analysis, Investing, Valuation, Sensitivity, Risk, Flexibility, SAEI, vol. 28. Ostrava: VSB-TU Ostrava. Volume 29 Zmeškal, Z. a kol. (2014). Aplikace vícekriteriálních dekompozičních metod rozhodování v oblasti podnikové ekonomiky a managementu, SAEI, vol. 29. Ostrava: VSB-TU Ostrava. Volume 30 Toloo, M. (2014). Introduction to Data Envelopment Analysis with basic models and applications, SAEI, vol. 30. Ostrava: VSB-TU Ostrava.