Series on Advanced Economic Issues Faculty of Economics ...homel.vsb.cz/~tol0013/index_htm_files/SAEI_VOL30-DATA ENVELOP… · problem, technology selection problem, facility layout

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

[email protected]

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.






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


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.


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


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


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


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.


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:


∑ ∑ 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:


∑ ∑ 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


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


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.

60 Chapter 3

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,



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

GAMS Software 61


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”;

62 Chapter 3

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,




);

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


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

GAMS Software 65


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

GAMS Software 67


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.

68 Chapter 3

2014 M. Toloo

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)





-72 Common (Bank06)

-59 Common (Bank07)








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.

GAMS Software 69


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

70 Chapter 3

2014 M. Toloo

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.

GAMS Software 71


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

72 Chapter 3

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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.

GAMS Software 73


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,


74 Chapter 3

2014 M. Toloo



PUT o.tl theta.l:6:5;



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;

GAMS Software 75


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);

76 Chapter 3

2014 M. Toloo

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 '========================================================='/;

PUT result2;

PUT 'BCC efficiency score'///;


PUT '========================================================='/;

GAMS Software 77


LOOP (o,




PUT result;




PUT /;

SOLVE BCC_UV_I USING LP maximizing theta;

PUT result2;




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);

78 Chapter 3

2014 M. Toloo

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).

GAMS Software 79


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

82 Chapter 4

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.

Weights in DEA 83


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

84 Chapter 4

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.

Weights in DEA 85


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).

Weights in DEA 87


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.

88 Chapter 4

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

Weights in DEA 89


∗ 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.

90 Chapter 4

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.

Weights in DEA 91


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:

92 Chapter 4

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):

Weights in DEA 93


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):

94 Chapter 4

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:

Weights in DEA 95


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):

96 Chapter 4

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

Weights in DEA 97


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

2014 M. Toloo

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


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


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:

104 Chapter 5

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.



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).



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:



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)



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.


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



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



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



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:



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 5–5 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



∗ 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.



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.



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



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.


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


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



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




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:



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


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 .



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



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 9 0 1 University 34 0 1

















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 8 0.750 0.812 1


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 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 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 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 42 0.847 0.835 0

University 43 0.921 0.643 0


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 50 0.842 0.835 0



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



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



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:



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.



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:



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



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,



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.



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.


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


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


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


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


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


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



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


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


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


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


with general

deviation measures

Branda 2013 25 4 1

176 Appendix

2014 M. Toloo


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


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



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


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


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


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

183

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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

192 List of Tables

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


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.






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.

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