Decision Context Based Evaluation of Multiattribute ... · Multiattribute decision making (MADM) methods generally involve evaluating a set of decision alternatives by considering

Decision Context Based Evaluation of

Multiattribute Decision Making Methods

by

Subrata Chakraborty

B. Sc. (Electronics), M. C. A., University of Pune, India

A Dissertation Submitted to Monash University in

Fulfilment of the Requirements for the Degree of

Doctor of Philosophy

Faculty of Information Technology

Monash University, Australia

November 2009

Copyright NoticesNotice 1

Under the Copyright Act 1968, this thesis must be used only under the normalconditions of scholarly fair dealing. In particular no results or conclusions shouldbe extracted from it, nor should it be copied or closely paraphrased in whole or inpart without the written consent of the author. Proper written acknowledgementshould be made for any assistance obtained from this thesis.

Notice 2

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for third-party content included in this thesis and have not knowingly added

copyright content to my work without the owner’s permission.

i

Declaration

I, Subrata Chakraborty, hereby declare that this thesis contains no material

which has been accepted for the award of any other degree or diploma in any

university or other institution. To the best of my knowledge and belief, this thesis

contains no material previously published or written by other authors, except where

due reference is made in the text of the thesis.

SUBRATA CHAKRABORTY

Date:

Faculty of Information Technology

Monash University, Australia

ii

Acknowledgements

I would like to express my heartiest gratitude to my research supervisor,

Professor Chung-Hsing Yeh. This work could never have been successfully done

without his knowledgeable guidance, support and encouragement. I will always

treasure the inspiration and experience I have gained by working with him.

I am thankful to the Faculty of Information Technology and the Monash

University for providing necessary financial support during my studies. I also thank

the staff at the Clayton School of Information Technology for their great support

during this period.

I am grateful to my friends and fellow PhD students for their support and for

making the study period fun and enjoyable. The time we spent together was

encouraging and helped me in various stages of my research.

Finally, I would like to thank my parents and my family for their continuous

support, encouragement and sacrifices in every stage of my life. I must also thank my

wife for her patience, understanding and support during this research.

iii

Abstract

Multiattribute decision making (MADM) methods generally involve

evaluating a set of decision alternatives by considering a set of evaluation criteria or

attributes in order to achieve a decision outcome such as ranking and selection.

The diversity among the decision problems in terms of their problem

structures, characteristics, decision information and specific requirements has led to

the development of numerous MADM methods. With the availability of many

MADM methods, selecting the most suitable one for a given problem is a

challenging task for the decision maker. The decision maker may not have required

experience and expertise to understand the suitability of a method for a given

problem. In order to help the decision maker select a suitable method, several

guidelines have been developed along with empirical and simulation studies during

the past few decades.

Although existing studies provide valuable insights for selecting a suitable

MADM method for specific decision problems, they are inadequate and unable to

resolve several open research issues in MADM research, including: (a) unavailability

of general guidelines for specific decision settings, (b) lack of method comparison

experiments in detailed levels, (c) inability to find the most preferred method for

specific decision contexts, (d) lack of objective measures to compare a set of suitable

methods for a given problem, (e) inability to consider all the stakeholders in method

evaluation, and (f) inadequacy of comparison studies for group decision methods.

iv

In this study, various decision contexts are identified to understand the

decision settings and the decision maker’s evaluation and selection requirements. Six

new methodologies are developed to resolve decision context specific issues in the

area of MADM method evaluation, comparison and selection.

A new simulation model is developed to provide decision setting specific

method evaluation and selection guidelines. Experiments are conducted to illustrate

applications of the new simulation model. This work highlights the need for detailed

level method comparisons considering internal processes of MADM methods,

including: normalisation procedures, aggregation techniques and consensus

techniques.

A new rank similarity based approach along with an objective measure is

developed to compare a set of suitable methods for a given decision problem in order

to find the most preferred one. The approach measures the similarity between

ranking outcomes produced by the methods being evaluated.

An alternatives-oriented approach is developed to provide due considerations

to the decision alternatives in the method evaluation process, when they are key

stakeholders. This approach provides a new dimension to method evaluation and

selection.

A comparison between the TOPSIS and the modified TOPSIS methods is

conducted to justify the applications of these methods in MADM problem solving.

Simulation experiments and mathematical proofs are provided to help the decision

maker choose between them rationally.

v

A new group consensus technique is developed to provide a much needed

rational alternative to the existing techniques and to justify their usage. A novel

consensus technique selection approach is developed to compare and evaluate group

consensus techniques in an objective manner in order to find the one that most

satisfies the group of decision makers as a whole.

A new group decision method is developed based on comparative searching

into the complete solution space that consists of all the possible decision outcomes.

The method finds the solution preferred most by the whole group of decision makers.

The research study contributes to the MADM research by introducing the

concept of decision context based evaluation of MADM methods and developing

new approaches, models and techniques to address context specific requirements in

varying decision settings. This study also highlights the need for new perspectives

towards the method evaluation processes. The research outcomes of this study have a

great potential for practical problem solving. Various experimental results can be

used as insightful guidelines for selecting the most suitable method for a given

problem. With their simplicity and flexibility in concept and computation, the new

approaches developed can be easily adopted to address new requirements in MADM

method evaluation.

vi

Table of Contents

Declaration ................................................................................................................... i

Acknowledgements ..................................................................................................... ii

Abstract ...................................................................................................................... iii

List of Publications .................................................................................................. xiii

List of Tables ........................................................................................................... xiv

List of Figures .......................................................................................................... xvi

Chapter 1 Introduction .............................................................................................. 1

1.1 Preamble ....................................................................................................... 1

1.2 Multiattribute Decision Making Challenges ................................................. 2

1.3 Research Objectives ..................................................................................... 4

1.4 Research Outline ........................................................................................... 5

Chapter 2 A Review of Multiattribute Decision Making Methods and Method

Comparisons ............................................................................................ 9

2.1 Introduction .................................................................................................. 9

2.2 Classification of Multiattribute Decision Making Methods ....................... 10

2.2.1 Classification Based on the Data Type .......................................... 10

2.2.2 Classification Based on the Information Type and Features ......... 12

vii

2.2.3 Classification Based on the Number of Decision Makers ............. 14

2.3 Multiattribute Value Theory Based Methods ............................................. 15

2.3.1 Simple Additive Weighting ............................................................ 16

2.3.2 Technique for Order Preference by Similarity to Ideal Solution ... 16

2.3.3 Weighted Product ........................................................................... 17

2.4 Method Comparison Studies ...................................................................... 17

2.5 Concluding Remarks .................................................................................. 21

Chapter 3 Methodology Formulation and Development for Method Evaluation

and Selection .......................................................................................... 22

3.1 Introduction ................................................................................................ 22

3.2 The Multiattribute Decision Making Problem and Notation ...................... 23

3.3 Decision Context and Method Evaluation Challenges ............................... 25

3.3.1 Decision Context A ........................................................................ 26

3.3.1.1 Specifications for Decision Context A .............................. 26

3.3.1.2 Current Challenges for Decision Context A ...................... 27

3.3.2 Decision Context B ........................................................................ 27

3.3.2.1 Specifications for Decision Context B .............................. 27

3.3.2.2 Current Challenges for Decision Context B ...................... 28

3.3.3 Decision Context C ........................................................................ 28

3.3.3.1 Specifications for Decision Context C .............................. 28

3.3.3.2 Current Challenges for Decision Context C ...................... 28

3.3.4 Decision Context D ........................................................................ 29

3.3.4.1 Specifications for Decision Context D .............................. 29

3.3.4.2 Current Challenges for Decision Context D ...................... 30

viii

3.3.5 Decision Context E ........................................................................ 30

3.3.5.1 Specifications for Decision Context E ............................... 30

3.3.5.2 Current Challenges for Decision Context E ...................... 30

3.3.6 Decision Context F ......................................................................... 31

3.3.6.1 Specifications for Decision Context F ............................... 31

3.3.6.2 Current Challenges for Decision Context F ....................... 32

3.4 Overview of the Methodology Developments ............................................ 33


Chapter 4 Developments I: A Simulation Model for Method Evaluation and

Selection ................................................................................................. 38

4.1 Introduction ................................................................................................ 38

4.2 The Simulation Model ................................................................................ 39

4.3 Performance Measures ............................................................................... 41

4.3.1 The Ranking Consistency Index .................................................... 41

4.3.2 The Weight Sensitivity Index ......................................................... 44


Chapter 5 Applications of Developments I: Simulation Based Selection of a

Normalisation Procedure ...................................................................... 50

5.1 Introduction ................................................................................................ 50

5.2 Normalisation Procedures Evaluated .......................................................... 51

5.2.1 Vector Normalisation ..................................................................... 52

5.2.2 Linear Scale Transformation (Max-Min) ....................................... 52

5.2.3 Linear Scale Transformation (Max) ............................................... 53

ix

5.2.4 Linear Scale Transformation (Sum) ............................................... 54

5.3 Multiattribute Decision Making Methods Evaluated ................................. 55

5.3.1 The SAW Method .......................................................................... 55

5.3.2 The TOPSIS Method ...................................................................... 57

5.4 Experiments and Results for SAW ............................................................. 59

5.4.1 Simulation Experiments for SAW ................................................. 60

5.4.2 Experimental Results for SAW ...................................................... 62

5.4.2.1 Results for Change in Alternative Numbers ...................... 62

5.4.2.2 Results for Change in Attribute Numbers ......................... 64

5.4.2.3 Results for Change in Data Range ..................................... 65

5.5 Experiments and Results for TOPSIS ........................................................ 67

5.5.1 Simulation Experiments for TOPSIS ............................................. 67

5.5.2 Experimental Results for TOPSIS ................................................. 69

5.5.2.1 Results for Change in Alternative Numbers ...................... 69

5.5.2.2 Results for Change in Attribute Numbers ......................... 71

5.5.2.3 Results for Change in Data Range ..................................... 73


Chapter 6 Developments II: Rank Similarity Based Method Evaluation and

Selection ................................................................................................. 77

6.1 Introduction ................................................................................................ 77

6.2 Methodology Development ........................................................................ 78

6.2.1 Rank Similarity and Method Evaluation ........................................ 78

6.2.2 The Rank Correlation Coefficient .................................................. 79

6.2.3 Rank Similarity Index ................................................................... 79

x

6.3 Numerical Example .................................................................................... 81

6.3.1 Methods Used in the Example ....................................................... 81

6.3.2 The Example .................................................................................. 83


Chapter 7 Developments III: An Alternatives-Oriented Method Evaluation and

Selection ................................................................................................. 87

7.1 Introduction ................................................................................................ 87

7.2 The Alternatives-Oriented Approach and the Preference Level ................. 89

7.3 Numerical Example .................................................................................... 92

7.4 Application in Decision Support Systems .................................................. 95


Chapter 8 Developments IV: Comparisons between TOPSIS and Modified

TOPSIS Methods ................................................................................. 100

8.1 Introduction .............................................................................................. 100

8.2 TOPSIS and Modified TOPSIS ................................................................ 101

8.2.1 The TOPSIS Method .................................................................... 101

8.2.2 The Modified TOPSIS Method .................................................... 101

8.3 Method Comparisons ................................................................................ 103

8.3.1 Comparison with Equal Weight Settings ..................................... 103

8.3.2 Comparison with Non-Equal Weight Settings ............................. 106

8.3.2.1 Simulation Results ........................................................... 106

8.3.2.2 Mathematical Analysis .................................................... 107

8.4 Concluding Remarks ................................................................................ 110

xi

Chapter 9 Developments V: Evaluation of Consensus Techniques in

Multiattribute Group Decision Making ............................................ 112

9.1 Introduction .............................................................................................. 112

9.2 Group Consensus Techniques .................................................................. 113

9.2.1 Consensus during the Initial Stage ............................................... 113

9.2.2 Consensus during the Intermediate Stage .................................... 115

9.2.3 Consensus during the Final Stage ................................................ 115

9.3 New Consensus Technique Based on TOPSIS ......................................... 116

9.4 Consensus Technique Evaluation ............................................................. 119

9.5 Numerical Example .................................................................................. 120

9.6 A Simulation and Ties in Ranking Outcome ............................................ 123

9.7 Concluding Remarks ................................................................................ 124

Chapter 10 Developments VI: Comparison Based Group Ranking Outcome for

Multiattribute Group Decisions ......................................................... 125

10.1 Introduction ............................................................................................ 125

10.2 Methodology Development .................................................................... 126

10.2.1 Finding the Most Preferred Group Ranking Outcome ............... 126

10.2.2 The Outcome Similarity Index ................................................... 127

10.3 Numerical Example ................................................................................ 128

10.4 Concluding Remarks .............................................................................. 132

xii

Chapter 11 Conclusions ......................................................................................... 133

11.1 Research Developments Summary ......................................................... 133

11.1.1 Developments I: A simulation Model and Applications ............ 133

11.1.2 Developments II: Rank Similarity Based Approach .................. 134

11.1.3 Developments III: Alternatives-Oriented Approach .................. 135

11.1.4 Developments IV: TOPSIS and Modified TOPSIS Comparison135

11.1.5 Developments V: Group Consensus Technique......................... 136

11.1.6 Developments VI: Comparison Based Group Decision Method 137

11.2 Application of the Developments ........................................................... 137

11.3 Research Contributions ........................................................................... 139

11.4 Future Research ..................................................................................... 142

References ............................................................................................................... 144

Appendix A: Notation ............................................................................................ 159

Appendix B: Glossary of Terms ........................................................................... 164

Appendix C: Simulation Results ........................................................................... 168

C.1 Results for SAW ...................................................................................... 168

C.1.1 Results for Change in Alternative Numbers ................................ 168

C.1.2 Results for Change in Attribute Numbers ................................... 173

C.1.3 Results for Change in Data Range ............................................... 178

C.2 Results for TOPSIS .................................................................................. 181

C.2.1 Results for Change in Alternative Numbers ................................ 181

C.2.2 Results for Change in Attribute Numbers ................................... 186

C.2.3 Results for Change in Data Range ............................................... 191

xiii

List of Publications

Chakraborty S and Yeh C-H (2007a). A Simulation Based Comparative Study of

Normalization Procedures in Multiattribute Decision Making. In:

Proceedings of the WSEAS International Conference on Artificial

Intelligence, Knowledge Engineering and Data Bases (AIKED'07): 102-109.

Chakraborty S and Yeh C-H (2007b). Consistency Comparison of Normalization

Procedures in Multiattribute Decision Making. WSEAS Transactions on

Systems and Control 2 (2): 193-200.

Chakraborty S and Yeh C-H (2007c) Comparing Normalization Procedures in

Multiattribute Decision Making under Various Problem Settings. In:

Proceedings of the Fifth International Conference on Information

Technology in Asia (CITA’07): 36-42.

Chakraborty S and Yeh C-H (2009). A Simulation Comparison of Normalization

Procedures for TOPSIS. In: Proceedings of the International Conference on

Computers and Industrial Engineering (CIE39): 1815-1820.

xiv

List of Tables

Table 3-1 Decision contexts addressed in various chapters ....................................... 37

Table 4-1 RCI and WSI summary .............................................................................. 48

Table 5-1 Four commonly used normalisation procedures ........................................ 54

Table 5-2 Four MADM methods for the experiment with SAW ............................... 60

Table 5-3 Four MADM methods for the experiment with TOPSIS .......................... 67

Table 5-4 Simulation results in terms of performance ............................................... 75

Table 6-1 Nine MADM methods used in the example .............................................. 82

Table 6-2 Decision matrix used in the example ......................................................... 83

Table 6-3 Resultant rank matrix ................................................................................. 84

Table 6-4 Rank correlation coefficient between MADM methods ............................ 84

Table 6-5 Rank similarity index for suitable MADM methods ................................. 85

Table 7-1 Ranking outcomes obtained ....................................................................... 93

Table 7-2 Resultant rank matrix ................................................................................. 93

Table 7-3 The method preference degree matrix ....................................................... 94

List of Tables

xv

Table 7-4 The scaled method preference degree matrix ............................................ 94

Table 7-5 The preference level for MADM method .................................................. 94

Table 7-6 Comparison between existing DSS and alternatives-oriented DSS .......... 98

Table 9-1 Rank matrix generated by combining individual ranking outcomes ....... 121

Table 9-2 The rank score matrix .............................................................................. 122

Table 9-3 The overall rank score and group ranking outcomes ............................... 122

Table 9-4 Rank similarity index for group outcomes .............................................. 123

Table 10-1 Individual ranking outcomes for each decision maker .......................... 129

Table10-2 Solution space with all the possible ranking outcomes .......................... 130

Table10-3 OSI for each possible outcome ............................................................... 131

xvi

List of Figures

Figure 1-1 Stages of solving a multiattribute decision making problem ..................... 2

Figure 1-2 The research framework ............................................................................. 6

Figure 2-1 MADM classification based on the data type .......................................... 11

Figure 2-2 MADM classification based on the information type and features .......... 13

Figure 2-3 MADM classification based on the number of decision makers .............. 14

Figure 3-1 Overview of the methodology developments ........................................... 34

Figure 5-1 With 10 attributes, the effects on the ranking consistency for changes in

the number of alternatives .......................................................................................... 63

Figure 5-2 With 6 alternatives, the effects on the ranking consistency for changes in

the number of attributes ............................................................................................. 64



Figure 5-4 With 4 attributes and 4 alternatives, the effects on the ranking consistency

for changes in the data range ...................................................................................... 66

Figure 5-5 With 12 attributes and 12 alternatives, the effects on the ranking

consistency for changes in the data range .................................................................. 66

List of Figures

xvii


the number of alternatives .......................................................................................... 70






for changes in the data range ...................................................................................... 73


consistency for changes in the data range .................................................................. 74

Figure 7-1 Existing DSS for MADM problems ......................................................... 96

Figure 7-2 Alternatives-oriented DSS for multiattribute decision problems ............. 97

Figure 8-1 Distance in one dimensional space ......................................................... 107

Figure 8-2 Distance in two dimensional space ........................................................ 108

Figure 9-1 The group decision process in the evaluation and selection phases ....... 114

Figure 11-1 A computer based decision support system for method selection........ 138

1

Chapter 1

Introduction

1.1 Preamble

Decision making is an important aspect of our daily life. Simple decisions

like selecting a restaurant for dinner or more complex decisions like selecting a

strategy for a country or organisation require a certain decision process, often known

as decision analysis (Keeney and Raiffa, 1976; Hwang and Yoon, 1981; Deng,

1998). Making correct decisions is crucial as this may have significant impact on the

future direction of a person, an organisation, a country or the world.

Multiattribute decision making (MADM) is a special area of decision

analysis. General MADM problems involve the evaluation, selection and ranking of

a set of course of actions often referred to as decision alternatives with respect to a

set of evaluation criteria or attributes. Most of the real world decision problems are

multiattribute in nature. The suitability and applicability of MADM methods to solve

real world decision problems have attracted researchers and decision makers from

diverse areas including management, economics, engineering, computing,

mathematics, business, psychology, social science and medical science. The vast

diversity in decision problems and problem areas have led to the development of

numerous MADM methods (Kenney and Raiffa, 1976; Hwang and Yoon, 1981;

Zeleny, 1982; Hwang and Lin, 1987; Yoon and Hwang 1995).

Chapter 1 Introduction

2

1.2 Multiattribute Decision Making Challenges

Figure 1-1 shows the two major stages of solving an MADM problem: (a)

problem structuring, and (b) problem solving. The structuring of the problem

includes identifying the set of alternatives, identifying the set of attributes, and

deciding the preferences. Stage 1 is highly dependent on the decision maker.

Figure 1-1 Stages of solving a multiattribute decision making problem

Stage 1: Structuring the decision problem

Select the set of alternatives to be

evaluated

Decide the set of attributes to be

considered

Specify the preferences and

specific requirements

Use an appropriate MADM method to solve the decision problem

Stage 2: Solving the decision

Decision problem

Decision outcome


3

The decision maker is usually familiar with the problem area of interest and is

able to identify a set of decision alternatives. The decisions about a set of attributes

are sometimes challenging due to inter-attribute relations like attribute hierarchy

(Saaty, 1977). The decision maker often requires specifying preferences such as

relative importance of the attributes. The complexity of Stage 1 may increase with

diversity in data type (deterministic, probabilistic, and fuzzy) (Hwang and Yoon,

1981) in the decision problem. Although the problem structuring is a very

challenging stage, it is assumed that the decision maker have enough skills, expertise

and knowledge in the problem area to structure the decision problem.

After the decision problem is specified from Stage 1, the decision maker

simply needs to solve it with an appropriate MADM method as shown in Stage 2.

This stage requires addressing many challenging tasks in multicriteria analysis (Yeh,

2002; Chakraborty and Yeh, 2007a) due to the following reasons:

(a) Often the decision maker does not have required expertise and knowledge

about the MADM methods and is unable to make a rational selection of an

appropriate method.

(b) For a given problem there may be multiple suitable MADM methods

available. Objective evaluation is required to select the most preferred one

among the set of suitable methods under various decision contexts and

settings.


4

(c) Each MADM problem contains unique features in terms of decision

settings. Hence, a generalised method selection guideline applicable to the

entire decision problems is not available.

(d) Very few studies on method evaluation and comparison have been

conducted and the results are inadequate to cover a vast majority of MADM

problems.

(e) Many researchers have applied MADM methods to solve different MADM

problems without proper justification and validation, thus leading to

questionable further usage.

(f) Method selection in a group environment has not been addressed

adequately.

The selection of an MADM method may has a great impact on the outcome

of an MADM problem. The challenges identified and the lack of sufficient research

in the area of method evaluation and selection where multiple suitable MADM

methods available for a given problem highlight the need for further investigation

and study in this area.

1.3 Research Objectives

In order to address the challenges of MADM method evaluation and selection

and to provide the decision makers with rational and efficient method selection

techniques and approaches, the primary objectives of this study include:


5

(a) Review existing method evaluation and comparison studies to identify

important gaps worth investigating.

(b) Develop the general method selection guidelines by considering various

practical MADM decision settings.

(c) Identify various decision contexts considered by the decision makers while

evaluating MADM methods and developing context specific method

selection approaches.

(d) Develop objective measures to validate comparison results.

(e) Develop new techniques and methods for group decision settings.

(f) Develop new measures to compare group decision making methods.

1.4 Research Outline

Figure 1-2 shows the research framework of this thesis. It outlines the

improvements and developments achieved in this study. The developments are

grouped into three major categories: (a) simulation based study, (b) decision

problems with single decision maker, and (c) decision problems with multiple

decision makers. The simulation based study provides a general method evaluation

guideline. The studies for single and group decision problems develop new

approaches for evaluating and comparing MADM methods used for solving these

problems.


6

Figure 1-2 The research framework

Simulation based study

Single decision problem

Group decision problem

Simulation model for method evaluation (Chapter 4)

Normalisation procedure selection

(Chapter 5)

Rank similarity based method

evaluation (Chapter 6)

Alternatives-oriented method

evaluation (Chapter 7)

TOPSIS and modified TOPSIS

comparison (Chapter 8)

Group consensus technique evaluation (Chapter 9)

Group outcome based on ranking

comparison (Chapter 10)

A review of multiattribute decision making methods and method comparisons (Chapter 2)

Methodology formulation and development for method evaluation (Chapter 3)

Introduction (Chapter1)

Conclusions (Chapter 11)


7

In Chapter 2, a brief review on available MADM methods is presented. A

review of method evaluation and comparison studies is also presented to identify

research areas that require improvements and new developments.

In Chapter 3, the general MADM problem is formulated along with the

notation to be used in the thesis. A set of decision contexts are then identified along

with their challenging issues to pave the way for the development of new context

specific method evaluation and selection approaches and techniques.

Chapter 4 develops a new simulation model for method evaluation and

selection. The model is capable of comparing MADM methods based on the ranking

outcomes they produce. The model is also able to find out the sensitivity of any

method towards changes in information. It can provide general guidelines for method

selection under various decision settings using a large number of simulated decision

problems.

In Chapter 5, the use of a particular normalisation procedure with the simple

additive weighting (SAW) and the technique for order preference by similarity to

ideal solution (TOPSIS) methods are justified by using the simulation model

developed in Chapter 4.

Chapter 6 develops a new approach to method selection based on ranking

outcome. The approach is capable of comparing a set of suitable methods for a given

problem to find the most suitable one in an objective manner. A new measure is

developed for the purpose of objective comparison. A simple example is provided to

illustrate the new approach.


8

In Chapter 7, a novel method selection approach is developed to select the

most preferred MADM method from the perspective of the decision alternatives. The

approach provides a new way of recognizing the importance of all the stakeholders

of a decision problem. A new objective measure is developed for the method

evaluation and selection. An example and the potential application of this new

approach in decision support systems are also presented.

Chapter 8 compares the TOPSIS and the modified TOPSIS methods using

simulations and mathematical proofs to justify their applicability.

In Chapter 9, a new group consensus technique is developed along with a

comparison approach to justify the use of existing consensus techniques. An

objective measure based on the developments in Chapter 6 is also developed to

validate the comparison results. The method comparison is conducted based on the

ranking outcome produced. An example is provided for a better understanding of the

new approach.

Chapter 10 develops a new method for solving group decision problems by

using the comparison based searching in the whole solution space. The new method

is unique for its capability of considering all possible outcomes while finding the

group outcome for a given group decision problem. A worked example is provided to

illustrate the new method.

Chapter 11 summarises the developments achieved in this study along with

their potential applications. The contributions of this research are also highlighted

before suggesting future research direction.

9

Chapter 2

A Review of Multiattribute Decision Making

Methods and Method Comparisons

2.1 Introduction

Multiattribute decision making (MADM) methods have gained wide

popularity for solving practical decision problems involving a set of decision

alternatives and evaluation criteria. Various MADM methods have been developed in

the past few decades to solve different types of MADM problems. With the

availability of several MADM methods for a given problem, method comparison and

selection has become a significant research issue (Zanakis et al., 1998; Chakraborty

and Yeh, 2007a, 2007b, 2009).

Existing comparison studies have shown major interests in justifying the

suitability of certain MADM methods for a given decision problem. The existing

simulation based and the empirical studies are inadequate to handle the method

evaluation and selection in a comprehensive manner.

In this chapter a review of the commonly used MADM methods and their

classifications are first presented. A review of method comparison studies is then

presented to identify the limitations of existing studies which lead to the

methodology developments in this study.

Chapter 2 A Review of Multiattribute Decision Making Methods and Method Comparisons

10

2.2 Classification of Multiattribute Decision Making Methods

MADM methods are diverse in structure, methodology and applications.

Among various classifications available for MADM methods, the widely known ones

include classifications based on (a) the data type in the decision problem, (b) the

decision information type and features and (c) the number of decision makers

involved (Hwang and Yoon, 1981; Triantaphyllou, 2000).

2.2.1 Classification Based on the Data Type

Figure 2-1 shows the MADM method classification based on the data type.

MADM problems may consist of data with probability. MADM methods are

developed to deal with this stochastic data. Stochastic dominances for discrete cases

are defined and used in developing outranking and rough approximation models

(Hadar and Russel, 1969; Zaras and Martel, 1994; Zaras, 2001). Probability based

confidence index models are developed to obtain relative preference between

alternatives (Martel and D’Avignon, 1982; Martel et al., 1986). Stochastic utility

additive methods use ordinal regression (Siskos, 1980 and 1983; Jacquetlagreze E

and Siskos J, 1982). Interactive methods for stochastic MADM problems are also

developed (Nowak, 2006). Stochastic methods are used to solve decision problems in

various areas including risk analysis, portfolio analysis and financial planning,

strategic planning (Muhlemann et al., 1978; De et al., 1982; Vinso, 1982; Eom et al.,

1987-88; Lai and Hwang, 1993; Steuer and Na, 2003; Hanandeh and El-Zein, 2009)


11

Figure 2-1 MADM classification based on the data type

The fuzzy MADM problems contain information in linguistic terms. The

evolutionary concept of fuzzy set theory (Zadeh, 1965) is applied to formulate fuzzy

MADM problems (Bellman and Zadeh, 1970). Fuzzy MADM problems let the

decision maker express the preferences in linguistic terms rather than a crisp scale.

Over the past few decades, numerous fuzzy MADM methods have been developed to

solve various practical MADM problems. Among others, these developments include

α-cut (Baas and Kwakernaak, 1977; Kwakernaak, 1979; Cheng and McInnis, 1980;

Dubois and Prade, 1982), fuzzy arithmetic (Bonissone, 1980 and 1982), eigenvector

method (Saaty, 1977), possibility measure (Dubois et al., 1988), outranking methods

(Siskos et al., 1984; Brans et al., 1984), fuzzy TOPSIS (Rebai, 1993; Chen and Wei

1997; Chu, 2002b), fuzzy utility methods (Seo and Sakawa, 1984 and 1985). Detail

classifications of Fuzzy MADM methods with various decision issues and

applications can be found in several studies (e.g. Chen and Hwang, 1992; Deng,

1998).

Multiattribute decision Making

problem

Deterministic data

Fuzzy data

Stochastic data

SAW, TOPSIS, WP, ELECTREE, AHP

Fuzzy TOPSIS, Fuzzy Utility, Outranking

Stochastic UTA, Stochastic Dominance

Data type Method class


12

Deterministic MADM problems contain data in numeric form and the values

are given precisely. MADM methods developed for this type of decision problems

have gained wide acceptance due to their simplicity and computational efficiency.

Some widely used methods in this class include SAW (Churchman and Ackoff,

1954; MacCrimmon, 1968; Hwang and Yoon, 1981), TOPSIS (Hwang and Yoon

1981; Yoon and Hwang 1995), ELECTREE (Benayoun et al., 1966; Roy, 1968,

1971, 1973 and 1991; Nijkamp, 1974), WP (Bridgman, 1922; Starr, 1972; Yoon,

1989) and AHP (Saaty, 1980 and 1994).

2.2.2 Classification Based on the Information Type and Features

Figure 2-2 shows the MADM method classification by considering

availability and features of preference information. Non-compensatory methods are

based on the notion that a superiority of one attribute cannot be offset by inferiority

in some other attributes (Yoon and Hwang, 1995). Decision problems where no

preference of the decision maker is given can be solved by finding the non-

dominated alternatives using pairwise comparisons in Dominance method (Yu, 1973

and 1975; Hadar and Russel, 1974; Bergstresser et al., 1976; Wehrung et al., 1978).

With pessimistic and optimistic points of view, non-compensatory problems

can be solved using the Maximin (MacCrimmon, 1968; Bellman and Zadeh, 1970;

Foerster, 1979) and Maximax (Dawes, 1964; MacCrimmon, 1968; Foerster, 1979)

methods respectively. These methods evaluate an alternative based on the weakest

and strongest attributes respectively and require the attributes to be on a common

scale (Hwang and Yoon, 1981).


13

Figure 2-2 MADM classification based on the information type and features

Source: Adapted from Hwang and Yoon (1981)

The decision maker may provide the preferences in various ways including

(a) standard level, (b) ordinal preference and (c) cardinal preference (Hwang and

Yoon, 1981). Decision problems where the alternatives must satisfy a minimum

preference level for all attributes or the alternatives are to be evaluated based on its

greatest value of an attribute, can be solved using the Conjunctive method and

Disjunctive method (Simon, 1955; Dawes, 1964; Ando, 1979).

Decision problems where ordinal preference values given by the decision

maker represent the relative importance of the attributes can be solved using the

Lexicographic method (Luce, 1956; Encarnacion, 1964; Bettman, 1971 and 1974;

Fishburn, 1974) and the Elimination by Aspect method (Tversky, 1972a and 1972b;

Bettman, 1974).


Maximin

Dominance

Method class

Maximax

Conjunctive Method, Disjunctive Method

Lexicographic Method, Elimination by Aspect

Pessimistic

Optimistic

Standard

Ordinal

Cardinal

Information on Attribute

Information on

Environment

No Information

Multiattribute decision Making problem

Information type Information Feature


14

MADM problems with given cardinal preference information from the

decision maker can be solved with several widely used methods including (a)

additive utility based methods like SAW (Churchman and Ackoff, 1954;

MacCrimmon, 1968; Klee, 1971) and AHP (Charnes et al., 1973; Saaty, 1977), (b)

multiplicative utility based methods like WP (Bridgman,1922; Starr, 1972, Yoon,

1989), (c) concordance measure based methods like ELECTRE (Roy, 1971; Nijkamp

and Vandelft, 1977; Voogd, 1983) and (d) closeness to ideal solution based methods

like TOPSIS (Hwang and Yoon, 1981; Zeleny, 1982; Yoon and Hwang, 1995).

2.2.3 Classification Based on the Number of Decision Makers

Figure 2-3 shows a method classification based on the number of decision

makers involved in the problem solving process.

Figure 2-3 MADM classification based on the number of decision makers

With the single decision maker problem, the decision maker needs to

formulate the decision problem, decide the attribute preferences and choose an

Multiattribute decision Making

problem

A Group of Decision Makers

One Single decision Maker

Group TOPSIS, Social Choice Functions,

Borda Score technique


Decision maker Method class


15

appropriate method to solve the MADM problem. Widely used MADM methods for

single decision maker problems include SAW (Churchman and Ackoff, 1954;

MacCrimmon, 1968; Klee, 1971), AHP (Charnes et al., 1973; Saaty, 1977), WP

(Bridgman,1922; Starr, 1972, Yoon, 1989), ELECTRE (Roy, 1971; Nijkamp and

Vandelft, 1977; Voogd, 1983) and TOPSIS (Hwang and Yoon, 1981; Zeleny, 1982;

Yoon and Hwang, 1995).

MADM problems with more than one decision maker are known as

multiattribute group decision making (MAGDM) problems. Group decision

problems are similar to MADM problems with the added complexity that all the

decision makers need to achieve an agreed outcome which satisfies them most as a

whole. The decision makers need to agree on various decision aspects including the

alternatives, attributes, attribute weights and the method to be applied to solve the

problem. Methods to solve the MAGDM problems include extensions to MADM

methods like Group TOPSIS (Hwang and Lin, 1987; Chen, 2000; Chu 2002a; Shih et

al., 2007), various social choice functions and consensus techniques (Hwang and Lin,

1987) and scoring techniques like Borda score (DeBorda, 1781; DeGrazia, 1953;

Black, 1958; Arrow, 1963; Fishburn, 1973).

2.3 Multiattribute Value Theory Based Methods

In this study, three widely used methods based on multiattribute value theory

(MAVT) (Keeney and Raiffa, 1976) based methods are adopted in the explanation

and examples of the new developments, including (a) the simple additive weighting


16

(SAW) method, (b) the technique for order preference by similarity to ideal solution

(TOPSIS) method, and (c) the weighted product (WP) method.

2.3.1 Simple Additive Weighting

The simple additive weighting (SAW) method (Churchman and Ackoff,

1954; MacCrimmon, 1968; Klee, 1971) is probably the most widely used and well

known MADM method (Hwang and Yoon, 1981; Yeh, 2003). The basic principle

behind this method is to obtain an overall preference score for each alternative which

is used as the basis for evaluation and ranking. The overall preference score is

calculated as weighted sum of the individual performance ratings for each alternative

with respect to each attribute.

SAW requires the attributes to be comparable and in numerical form with the

given attribute weights (relative importance) from the decision maker. SAW applies

a normalisation procedure to convert performance ratings with different

measurement units into a comparable unit. The advantage of this method lies in its

simplicity, ease of use and sound mathematical grounds.

2.3.2 Technique for Order Preference by Similarity to Ideal Solution

The technique for order preference by similarity to ideal solution (TOPSIS)

method (Hwang and Yoon, 1981) is based on the notion that the preferred alternative

should have the shortest distance from the positive ideal solution and the longest

distance from the negative ideal solution. The TOPSIS method calculates the relative


17

closeness for each of the alternatives (comparable to overall preference score in

SAW) which is used to obtain ranking outcome.

TOPSIS requires the attributes to be numerical and comparable. Similar to

the SAW method, TOPSIS uses a normalisation procedure to generate a common

measurement unit for performance ratings. TOPSIS is a simple, easy and efficient

method applicable to practical MADM problem solving.

2.3.3 Weighted Product

The weighted product (WP) method (Bridgman, 1922; Starr, 1972, Yoon,

1989) is based on multiplicative utility. Instead of an addition operator in SAW, WP

uses a multiplication operator to obtain the overall utility score for each alternative

by combining the performance ratings and attribute weights. The overall score is

used to rank the alternatives.

Other than simplicity and ease of use, the WP method does not require a

normalisation procedure and is able to handle different measurement units for

performance ratings implicitly. WP method imposes heavy penalty on low

performing alternatives and is particularly useful where the decision maker wants to

screen out poor performing alternatives.

2.4 Method Comparison Studies

Multiattribute decision making (MADM) research has shown that no single

method is best for all problem settings. Different ranking outcomes may be obtained


18

when different methods are applied to solve the same decision problem (Zanakis et

al., 1998; Chakraborty and Yeh, 2007a, 2007b). Selecting an MADM method to

achieve the most preferred outcome for a given decision problem thus becomes an

important issue. The significance of this method evaluation and selection issue has

led to many studies on how to select the most preferred method for a given decision

setting. The method evaluation and selection studies conducted so far can be

classified as “the decision-maker-oriented” and “the method-oriented” approaches.

In the decision-maker-oriented approach, the decision maker usually applies

an MADM method on the basis of previous experiences or recommendations by

experts. This approach relies on the decision maker for method selection which may

introduce judgemental error in the decision outcome. A study examining the

behavioural impact on the method selection has showed that the method selection

process is largely influenced by the decision maker’s familiarity with certain method

(Buchanan, 1994). The decision-maker-oriented method selection approach provides

support and enhances the knowledge of the decision maker about different decision

settings and method suitability. A set of tentative guidelines for method selection has

been proposed for the decision maker to solve an MADM problem (Guitouni and

Martel, 1998). Studies between MAVT-based MADM methods and outranking

methods such as ELECTRE (Roy, 1968, 1991) have shown the differences in

structures and problem formulation (Simpson, 1996). An attempt to develop a unique

way of MADM method evaluation and selection has produced a set of meta-criteria

which should be satisfied by the methods (Cho, 2003). The decision-maker-oriented

approach relies on the decision maker’s understanding and subjective judgement for

method selection. Often the decision maker lacks enough knowledge and skills to


19

make justified choice of the most preferred method for the problem under

consideration. The high variability in subjective method selection by the decision

maker highlights the need for an objective way of method selection.

The issue of objective evaluation and selection of MADM methods have been

addressed in quite a few studies using the method-oriented approach. The method-

oriented approach compares suitable methods on the basis of an objective

performance measure. With known decision outcomes available, MADM methods

are compared for predictive accuracy (Olson, 2001). This line of study has produced

interesting results and is applicable for problems with historical data available, such

as weather forecasting and market trend prediction. Simulation based comparison

studies of several MADM methods have provided valuable insights for selecting a

method for a given problem (Zanakis et al., 1998; Deng and Yeh, 2006). The results

of these simulation based studies have shown the effect of the alternative numbers,

attribute numbers and distribution of information, which can be used as guidelines

for selecting a method for a problem with a given problem size and distribution.

Sensitivity analysis has been used by several studies to examine the degree of

sensitivity of various MADM methods in terms of attribute weights (Weber and

Borcherding, 1993; Triantaphyllou and Sanchez, 1997; Yeh, 2002). This approach is

very useful when the attributes weights are uncertain or the sensitivity of attributes

weights is a major concern of the decision maker.

In another line of research development, the concept of expected value loss

has been introduced as a performance measure for method selection in an objective

manner (Yeh, 2003). The expected value loss measures the deviations in decision


20

outcomes under various weight settings. The method with a minimum value loss is

the most preferred one. Other studies have introduced ranking consistency as a

performance measure to select the most preferred method for various problem

settings involving a wide range of alternative numbers, attribute numbers and

assessment data (Chakraborty and Yeh, 2007a, 2007b).

Although several method evaluation and selection studies have been

conducted over the past few decades, many research issues are still open for further

investigation, including the following:

(a) No general guideline available for MADM problems under specific

decision settings.

(b) No significant study has been conducted to evaluate a set of suitable

methods for a given decision problem to find the most preferred one.

(b) Methods are not evaluated for their internal problem solving processes.

(c) Evaluation and comparison studies are not performed considering specific

selection preferences (decision context) of the decision maker.

(d) The perspective of alternatives is not considered in existing method

comparison and selection studies.

(e) The consensus techniques in multiattribute group decision making problems

are not investigated for their suitability.


21

(f) Existing MADM methods for group decision problems often uses a limited

solution space to find the group ranking outcome. This solution space

limitation may be a major challenge in obtaining the most preferred

outcome for the group of decision makers.

2.5 Concluding Remarks

The brief review of MADM methods and their classifications presented in

this chapter has shown the wide diversity in the MADM research developments. The

review of method evaluation and comparison studies and subsequent identification of

unresolved issues provides the motivation and platform for new developments in the

area of MADM method evaluation and selection. The challenges identified in this

chapter will be further discussed in detail in terms of decision contexts to be

addressed in Chapter 3.

22

Chapter 3

Methodology Formulation and Development for

Method Evaluation and Selection

3.1 Introduction

Multiattribute decision making (MADM) methods are widely used to solve

real life decision problems. MADM problems are diverse in terms of decision

settings and decision information available. With the availability of several MADM

methods that produce different outcomes for a given problem, selecting the most

preferred one for the given problem is a challenging task for the decision maker.

Previous comparative studies on MADM method evaluation provide some general

and problem specific guidelines for method selection (Zanakis et al., 1998; Olson,

2001). Although these studies provide significant insights on method suitability and

selection, further study is required to justify the selection of a particular method for a

given decision problem under a specific decision context, for which a number of

suitable MADM methods are often available.

In this chapter, the general MADM problem is first presented along with the

formulation of the research problem. Notation for the MADM problem formulation

is then introduced which is to be used throughout the thesis. Next, various decision

contexts for the MADM problem along with their associated method evaluation and

selection challenges and context specific requirements are discussed. Finally, an

overview of the developments of context specific approaches and models for MADM

Chapter 3 Methodology Formulation and Development for Method Evaluation and Selection

23

method evaluation and selection is outlined to pave the way for their presentation in

Chapters 4-10.

3.2 The Multiattribute Decision Making Problem and Notation

The general multiattribute decision making (MADM) problem Φ involves the

following:

(a) A set of Q decision makers Dq; q = 1, 2, ...,Q.

(b) A set of I alternatives Ai; i = 1, 2, ..., I. (c) A set of J attributes Cj; j = 1, 2, ..., J.

(d) A set of J attribute weights Wj; j =1, 2, ..., J.

(e) (I*J) Performance ratings xij; i = 1, 2, ..., I; j = 1, 2, ..., J.

The general MADM problem Φ may have only one decision maker Dq (Q =

1) or a group of decision makers Dq (Q > 1). The involvement of more than one

decision maker increases the challenges in solving the MADM problem Φ.

The set of decision alternatives Ai (i = 1, 2, ..., I) includes various decision options

the decision maker is considering for the given decision problem which are to be

evaluated and ranked. For example, a buyer may have several options while buying a

car.

The set of attributes Cj (j = 1, 2, ..., J) are the selection criteria the decision

maker considers while evaluating the decision alternatives Ai (i = 1, 2, ..., I). For

example, the car buyer may evaluate the car options based on price, comfort, mileage

and performance.


24

The set of attribute weights Wj (j =1, 2, ..., J) represents the relative

importance of the attributes Cj (j = 1, 2, ..., J) to the decision maker. For example,

the car buyer may consider that price is more important than comfort and

performance, hence will have a higher attribute weight. The attribute weights are

presented as a vector W as shown in Equation (3-2).

The performance rating xij (i = 1, 2, ..., I; j = 1, 2, ..., J) represents the

assessment scores provided by the decision maker Dq (q = 1, 2, ...,Q) for each

alternative Ai (i = 1, 2, ..., I) with respect to each attribute Cj (j = 1, 2, ..., J). All the

performance ratings xij (i = 1, 2, ..., I; j = 1, 2, ..., J) for all the alternatives Ai (i = 1,

2, ..., I) in relation to all the attributes Cj (j = 1, 2, ..., J) can be represented as a

decision matrix X as shown in Equation (3-1), where rows and columns represent

alternatives and attributes respectively (Hwang and Yoon, 1981; Belton and Stewart,

2002; Yeh, 2003).

J. ..., 2, 1, j I;..., 2, 1, i ;

xxx

xxx

xxx

X

IJII

J

J

...

............

...

...

21

22221

11211

(3-1)

J. ..., 2, 1, jWW j ; (3-2)

With (a) the set of decision makers Dq (q = 1, 2, ...,Q), (b) the set of

alternatives Ai (i = 1, 2, ..., I) and (c) the set of attributes Cj (j = 1, 2, ..., J) being

defined, the general MADM problem Φ can be represented as a combination of the

decision matrix X and the weight vector W by using Equations (3-1) and (3-2) as


25

, WXΦ (3-3)

To solve the MADM problem Φ, a number of suitable MADM methods Mk (k

= 1, 2, ..., K) are available. The MADM methods Mk (k = 1, 2, ..., K) require (a) a

normalisation procedure Ne (e = 1, 2, ..., E) and (b) an aggregation technique. The

normalisation procedures are used to transform the performance rating xij (i = 1, 2,

..., I; j = 1, 2, ..., J) to a comparable unit as they may have diverse measurement

units. The aggregation technique is applied to combine normalised performance

ratings with the attribute weights Wj (j =1, 2, ..., J) to obtain an overall value Vi (i =

1, 2, ..., I) for each alternative Ai (i = 1, 2, ..., I). The overall value Vi (i = 1, 2, ..., I)

is used to evaluate and rank the decision alternatives Ai (i = 1, 2, ..., I).

3.3 Decision Context and Method Evaluation Challenges

A number of MADM methods Mk (k = 1, 2, ..., K) are often available to solve

the general MADM problem Φ under a given decision context. For any given

decision problem Φ, there may be multiple suitable methods that are acceptable to

the decision maker. The research challenge is how to evaluate and select the most

preferred MADM method among a set of suitable methods Mk (k = 1, 2, ..., K) under

various decision contexts. The most preferred method refers to the method that

satisfies a certain decision context most. The satisfaction level for the decision

context needs to be measured using objective measures.

The term “decision context” used in this thesis includes the decision settings

and other method evaluation preferences and considerations. The term “decision

settings” can be defined in terms of (a) problem type (such as problems with single


26

decision maker or a group of decision makers), (b) problem size (in terms of the

number of alternatives and attributes) and (c) information variations in terms of data

range and data type (qualitative or quantitative data).

Among different MADM methods available, this study uses multiattribute

value theory (MAVT) based methods (Kenney and Raiffa, 1976) in various

evaluation settings and examples, due to their ability to produce a complete ranking

of all the alternatives for a given problem. Six decision contexts are identified and

categorised below, along with their evaluation and selection challenges and

requirements.

3.3.1 Decision Context A

3.3.1.1 Specifications for Decision Context A

(a) The MADM problem involves one single decision maker only.

(b) The decision maker requires a general guideline for method selection under

different decision settings based on the size of the problem and variation in

decision information.

(c) The decision maker is not very confident on the assessment of attribute

weights and is concerned about its impact on the decision outcome.

(d) The decision maker wants to know if the use of a specific normalisation

procedure with an MADM method is justified under various decision

settings.


27

3.3.1.2 Current Challenges for Decision Context A

(a) A few simulation studies that have been conducted so far considered only a

relatively small set of MADM problems and a limited number of decision

settings (Zanakis et al., 1998; Olson, 2001). To address the general MADM

problem, a new simulation model is required to experiment with a large

number of decision problems with various decision settings.

(b) Existing studies use one particular method as the basis for comparison

which creates doubts on the impartiality of the comparison results. New

performance measures need to be developed for comparing methods

objectively, based on relative comparison between them.

(c) New performance measures need to be developed to measure the sensitivity

of each method with changes in certain decision information, such as

attribute weights.

3.3.2 Decision Context B

3.3.2.1 Specifications for Decision Context B


(b) The decision maker can use a set of suitable MADM methods for a given

problem. All these methods produce acceptable outcomes, but the decision

maker must choose one method as the most preferred one among them.


28

3.3.2.2 Current Challenges for Decision Context B

(a) Previous studies consider that only one method is suitable for a given

problem and all other methods are not acceptable (Simpson, 1996;

Triantaphyllou and Sanchez, 1997; Guitouni and Martel, 1998; Yeh 2002;

2003; Cho, 2003). A new method selection approach is required which can

compare a set of suitable methods to find the most preferred one.

(b)Objective performance measures need to be developed to find the most

preferred method from a set of suitable methods.

3.3.3 Decision Context C

3.3.3.1 Specifications for Decision Context C


(b) The decision maker does not have any specific method preferences or the

decision maker is not a key stakeholder.

(c).The alternatives in the decision problem are the key stakeholders and

should have greater inputs in the method selection process.

3.3.3.2 Current Challenges for Decision Context C

(a) Previous method selection studies have been conducted from two

perspectives: “method-oriented” and “decision-maker-oriented”. The

method-oriented studies compare MADM methods based on their


29

performances with certain decision settings (Weber and Borcherding, 1993;

Zanakis et al., 1998; Olson, 2001; Yeh, 2002 and 2003; Deng and Yeh,

2006; Chakraborty and Yeh 2007a and 2007b). The decision-maker-oriented

studies consider the preferences of the decision maker in method selection

(Simpson, 1996; Guitouni and Martel, 1998; Cho, 2003). No study has

considered the preferences of the decision alternatives in method selection.

In certain decision problems, the alternatives are the key stakeholders

(Jessop, 2009). Thus, there is a need for developing a new method selection

approach which provides due considerations to the preferences of the

decision alternatives in the method evaluation and selection process.

(b) New performance measures need to be developed to evaluate the MADM

methods objectively in terms of the alternatives’ preferences.

3.3.4 Decision Context D

3.3.4.1 Specifications for Decision Context D

(a) The decision problem involves one single decision maker only.

(b) The decision maker is unable to select between the TOPSIS (Hwang and

Yoon, 1981) and the modified TOPSIS (Deng et al., 2000; Yeh, 2002)

method for a given problem. These two methods are similar in structure

with the only difference in handling of the attribute weight. The decision

maker is concerned about the justification of using either method.


30

3.3.4.2 Current Challenges for Decision Context D

(a) No comparison study has been conducted to compare TOPSIS with

modified TOPSIS. Thus, there is a need to conduct a comprehensive

comparison study to justify the use of these two methods under specific

decision settings.

3.3.5 Decision Context E

3.3.5.1 Specifications for Decision Context E

(a) The decision problem involves a group of decision makers.

(b) The decision makers have their own decision problems reflecting their

preferences and wish to observe the ranking outcomes produced by the

method of their choice.

(c) The group outcome needs to be achieved by consensus among the group

based on the individual ranking outcomes.

(d) The consensus techniques to be used require objective evaluation and

justification.

3.3.5.2 Current Challenges for Decision Context E

(a) Currently the Borda score technique (DeBorda, 1781; DeGrazia, 1953;

Black, 1958; Arrow, 1963; Fishburn, 1973 and 1977; Gardenfors, 1973;

Fine and Fine, 1974a and 1974b; Young, 1974 and 1975; Pattanaik, 1978) is


31

the only available group consensus technique which is able to provide a

group outcome, using the individual ranking outcomes provided by each of

the decision makers (Hwang and Yoon, 1981). The Borda score technique

uses the average rank score of the individual ranking outcomes to produce

the group outcome. The average may not always be the most preferred

outcome to the group of decision makers as a whole.

(b) New consensus techniques need to be developed by considering other

aggregation procedures.

(c) New approaches need to be developed to compare group consensus

techniques and select the most preferred one for a given group decision

problem.

3.3.6 Decision Context F

3.3.6.1 Specifications for Decision Context F

(a) The decision problem involves a group of decision makers.

(b) The decision makers have their own ranking outcomes for the problem.

(c) The decision makers want to find the group outcome from the set of all

possible outcomes.

(d) The set of all possible outcomes is not limited by the number of available

methods or decision makers. All the possible ranking combinations with the

alternatives should be considered.


32

(e) All the ranking outcomes in the outcome set are considered valid and

acceptable to the group of decision makers and they want to find the most

preferred one among them.

3.3.6.2 Current challenges for Decision Context F

(a) Currently available methods for group decision problems apply various

aggregation procedures to achieve group solution from available individual

preferences and ranking outcomes. These methods are limited by the

number of individual ranking outcomes and aggregation procedures

(Eckenrode, 1965; Fishburn, 1966; Souder, 1972, 1973a and 1973b;

Minnehan, 1973; Keeney and Kirkwood, 1975; Dyer and Miles, 1976;

Bernardo, 1977; Cook and Seiford, 1978 and 1982; Hwang and Yoon,1981;

Hwang and Lin, 1987; Parkan and Wu, 1998; Chen, 2000; Chu 2002a and

2002b; Cook, 2006; Fu and Yang, 2007; Shih et al., 2007).

(b) There is a need for developing a new method capable of finding the most

preferred group outcome from whole solution space consisting of all the

possible decision outcomes for the given group decision problem.

(c) Objective performance measures need to be developed which can measure

group preferences for all possible outcomes. The performance measure

should be able to measure the satisfaction level of the group of decision

makers as a whole.


33

3.4 Overview of the Methodology Developments

The context specific challenges and requirements discussed in previous

sections are addressed by developing a number of new approaches, methods and

performance measures as shown in Figure 3-1. The new methodology developments

address research issues in three distinct areas of MADM research, including: (a)

simulation based generalised guidelines development for method evaluation and

selection, (b) evaluation of single decision maker methods, and (c) evaluation of

group decision methods.

Chapters 4 and 5 address the challenges and requirements for Decision

Context A. A new simulation model is developed for MADM method comparison in

Chapter 4. The simulation model is capable of comparing MADM methods that can

produce a complete ranking for all the decision alternatives. The simulation model

can compare the performances of different MADM methods under various decision

settings. The decision settings can be easily varied by changing the problem size (in

terms of the number of alternatives and attributes), the information range and the

attribute weights. Two new performance measures (ranking consistency index (RCI)

and weight sensitivity index (WSI)) are also developed. The RCI measures the level

of consistency of a particular method relative to other methods while producing a

ranking outcome for certain decision settings. The WSI indicates the level of

sensitivity of a method towards any change in attribute weights under various

decision settings. The simulation model is then used in Chapter 5 to justify the

suitability of certain normalisation procedures for SAW and TOPSIS methods under

various decision settings.


34

Figure 3-1 Overview of the methodology developments

Is group consensus justified?

Are all outcomes considered?

TOPSIS based consensus and consensus technique selection

(Chapter 9)

Similarity based group ranking with all possible outcomes

(Chapter 10)

Previous research studies

Yes

Yes

No

No

How many decision makers?

Generalised method selection?

Are uses of normalisation justified?

Are there multiple acceptable methods?

Are the alternatives preferences considered?

Is the use of modified TOPSIS justified?

No

Context dependent evaluation of MADM methods

Multiattribute decision making (MADM) problem

Multiattribute group decision making (MAGDM) problem

1 >1

Yes No

Simulation model for method selection

(Chapter 4)

Simulation based selection of normalisation procedure

(Chapter 5)

Context specific method development and selection

Rank similarity based selection of methods

(Chapter 6)

Alternatives-oriented selection of methods

(Chapter 7)

Comparing TOPSIS and modified TOPSIS methods

(Chapter 8)

Yes

No

No

Yes

Yes

No

Yes


35

Chapter 6 develops a novel method selection approach for addressing the

challenges and requirements for Decision Context B. The new approach considers

that all the methods being compared are valid and acceptable to the decision maker.

The most preferred method is selected by comparing the ranking outcomes produced

by different methods in terms of their relative similarity. A new performance

measure called ranking similarity index (RSI) is developed to measure the amount of

similarity that a ranking outcome (produced by a certain MADM method) has with

all the other ranking outcomes produced by other MADM methods.

Chapter 7 addresses the challenges and requirements for Decision Context C.

A new alternatives-oriented method selection approach is developed by considering

the preferences of the decision alternatives for selecting the most preferred method.

The approach calculates the overall method preference of all the decision alternatives

for each method being compared and uses it for selecting the most preferred method.

Chapter 8 addresses the challenges and requirements associated with Decision

Context D. A comprehensive comparison is conducted between the TOPSIS and the

modified TOPSIS methods by using simulation experiments. Mathematical

explanations are also presented to justify the use of these methods for making logical

and rational method selection decisions.

Chapter 9 addresses the challenges and requirements associated with

Decision Context E. A new group consensus technique is developed by applying the

theoretical grounds of the TOPSIS method (Hwang and Yoon, 1981). This new

technique is a well justified alternative method to the conventional Borda score

technique. A new consensus technique selection approach is also developed to


36

compare and select consensus techniques for a given group decision problem. The

approach introduces a new performance index called the group similarity index

(GSI), which calculates the degree of similarity for a group outcome (produced by a

consensus technique) with all the ranking outcomes provided by each individual

decision maker in the group.

Chapter 10 develops a new multiattribute group decision making method to

meet the challenges and requirements for Decision Context F. The new method finds

the most preferred group ranking outcome from all the possible ranking outcomes for

any given group decision problem. The method is based on the concept that the

ranking outcome which is most similar to all the ranking outcomes provided by all

the decision makers is the most preferred one by the group as a whole. The level of

similarity for each outcome in the set of all possible ranking outcomes is measured

by using a new performance measure called the outcome similarity index (OSI).


This chapter has outlined the methodology developments to be presented in

the subsequent chapters. These developments address various significant unresolved

issues in the area of MADM method evaluation and selection under various decision

contexts discussed in Section 3.3. Table 3-1 shows the chapters of the thesis and the

decision context they address.


37

Table 3-1 Decision contexts addressed in various chapters

Decision context Relevant chapter

Context A Chapters 4 and 5

Context B Chapter 6

Context C Chapter 7

Context D Chapter 8

Context E Chapter 9

Context F Chapter 10

38

Chapter 4

Developments I:

A Simulation Model for Method Evaluation and

Selection

4.1 Introduction

Multiattribute decision making (MADM) problems Φ are diverse in terms of

(a) the number of alternatives Ai (i = 1, 2, ..., I) to be evaluated and ranked, (b) the

number of attributes Cj (j = 1, 2, ..., J) to be considered, (c) the relative importance

Wj (j = 1, 2, ..., J) of the attributes and (d) the data type and measurement unit for the

performance ratings xij (i = 1, 2, ..., I; j = 1, 2, ..., J) given for each alternative Ai (i =

1, 2, ..., I) against each attribute Cj (j = 1, 2, ..., J). For a decision setting, there may

be multiple suitable MADM methods Mk (k = 1, 2, ..., K). Selecting the most suitable

one among them is a challenging task. Various MADM method taxonomies and

guidelines have been developed by several researchers to help the decision maker

select suitable methods for specific problem types (Hobbs, 1980; Hwang and Yoon,

1981; Ozernoy, 1987 and 1992). Various empirical studies consisting of real life

problem scenarios may help select suitable methods for a given problem (Currim and

Sarin, 1984; Gemunden and Hauschildt, 1985; Belton, 1986; Hobbs, 1986; Hobbs et

al., 1992; Stewart, 1992). Simulation experiments may provide the empirical studies

with the experimental supports by reducing their limitations on the sample

availability, assumptions and lack of expert users (Zanakis et al., 1998).

Chapter 4 A Simulation Model for Method Evaluation and Selection

39

Zanakis et al. (1998) have conducted an extensive simulation study to

compare several MADM methods in terms of rank and weight variations along with

the rank reversal scenarios (Saaty, 1987). Their study has highlighted the importance

of the simulation study for method selection and provided some interesting

comparison results which can be used for method selection purposes. Although the

study has shown a new dimension to method selection, the use of the simple additive

weighting (SAW) method as the basis for comparison, may limit the applicability of

the comparison results. The study has also used a limited set of decision settings in

terms of the number of attributes and alternatives.

This chapter develops a new simulation model which addresses the Decision

Context A outlined in Chapter 3 and generates method selection guidelines that have

general application. The simulation model is capable of comparing any number of

MADM methods under different decision settings. Two new performance measures

are also developed to justify the comparison results.

4.2 The Simulation Model

The simulation model is developed to address how the key decision

information settings may influence the decision outcomes when different MADM

methods are used. The key decision information settings include (a) the number of

attributes Cj (j = 1, 2, ..., J) considered, (b) the number of alternatives Ai (i = 1, 2, ...,

I) to be evaluated and ranked, (c) the diversity in the performance rating xij (i = 1, 2,

..., I; j = 1, 2, ..., J), and (d) the weights Wj (j = 1, 2, ..., J) of each attribute.


40

Step 1: Identify a set of MADM methods to be compared.

The set of suitable MADM methods Mk (k = 1, 2, ..., K) which are to be

evaluated and compared under various decision settings are selected. The

methods considered should be able to produce a complete ranking of the

alternatives Ai (i = 1, 2, ..., I).

Step 2: Determine the initial and the target decision settings.

For each of the decision information settings (for which Methods Mk (k = 1, 2,

..., K) are to be evaluated), the initial setting (the starting value for the

experiment) and target setting (the value when the experiment terminates) are

determined.

Step 3: Generate a set of decision problems.

A large number of decision problems are generated for each of the four

decision information settings. Three sets of decision problems Ω are generated

to validate the correctness of the experiment results.

Step 4: Solve the decision problems.

In each simulation run, the decision problems in each problem set Ω are solved

by Method Mk (k = 1, 2, ..., K). The ranking outcomes obtained are used for

measuring the performances of the methods.

Step 5: Evaluate the performance.

Use an appropriate performance measure to evaluate the performance of

Method Mk (k = 1, 2, ..., K). Objective measures should be applied as they

provide a rational base for comparisons.


41

Step 6: Vary the decision information settings.

Each of the decision information settings is changed one at a time by a

predefined amount (which produces significant variation in the results).

Step 7: Repeat the process

Steps 3 to 6 are repeated until the target settings for the decision information is

reached. The whole simulation is also repeated to identify and eliminate

possible irregularities in the evaluation and comparison results.

4.3 Performance Measures

Two new performance measures are developed to compare MADM methods

in terms of various decision settings, including the number of alternatives, the

number of attributes and the data range of performance ratings.

4.3.1 The Ranking Consistency Index (RCI)

The ranking of the alternatives is the final decision outcome that concerns the

decision maker. When the consistency of the rankings produced is a major concern, it

is important for the decision maker to select a method that produces the most

consistent ranking outcome among all the methods being tested with a given decision

settings. For a given MADM problem, a method is considered to be consistent with

another method if it produces the same ranking outcome. The ranking consistency

index (RCI) indicates the degree of consistency a method has with respect to all other

methods under certain decision settings when a large number of decision problems


42

are considered. A larger RCI value indicates that the corresponding method is more

consistent in terms of the ranking outcome it produces.

In order to calculate the RCI, a consistency weight (CW) for each ranking

outcome is defined. If a ranking outcome produced by a method is the same as that of

all other methods, then it has a consistency weight of 1. With Method Mk (k = 1, 2,

..., K), a set of consistency weight can be obtained as

K. ..., 2, 1, k 1;-K ..., 1, 0, n ;KnCWn )1/( (4-1)

where n represents the number of other methods that produce the same rank as

Method Mk.

The RCI can be obtained for Method Mk (k = 1, 2, ..., K) as

K. ..., 2, 1, k ;TCWTRCI n

K

nknk

/)*(1

0

(4-2)

where Tkn = total number of times Method Mk (k = 1, 2, ..., K) produces the same

ranking outcome with n number of other methods.

T = total number of decision problems used in the simulation run.

For example, consider the following simple experiment setting

There are four (K = 4) Methods Mk (k = 1, 2, ..., K) to be compared.

The total number of decision problems in the simulation run T = 1,000.

Applying Equation (4-1) we can obtain the consistency weight for the Methods

Mk (k = 1, 2, ..., K) as


43

CW0 = 0, when Method Mk produces an outcome different from other methods

CW1 = 1/3, when Method Mk produces an outcome similar to one of the other

methods

CW2 = 2/3, when Method Mk produces an outcome similar to two of the other

methods

CW3 = 1, when Method Mk produces an outcome similar to all the other

methods

In the 1,000 decision problems, the number of times Method Mk (k = 1)

produces a ranking outcome similar to the other methods for a problem is recorded as

T10 = 200, the number of times Method M1 produces an outcome different from

all the other methods.

T11 = 400, the number of times Method M1 produces an outcome similar to one

of the other methods.

T12 = 300, the number of times Method M1 produces an outcome similar to two


T13 = 100, the number of times Method M1 produces an outcome similar to all


For Method M1 we can calculate the RCI by applying Equation (4-2) with the

recorded information as


44

RCI1 = (200*0 + 400* 1/3 + 300* 2/3 +100*1)/1,000 = 0.43

Similarly, RCI for other three methods can be calculated. The method with

the highest RCI value is the most consistent one among the methods evaluated, in

terms of the ranking outcomes they produce for the given decision settings.

The ranking consistency index is particularly useful for comparing MADM

methods for decision settings with a varying number of alternatives and attributes, as

well as with various ranges of performance ratings.

4.3.2 The Weight Sensitivity Index (WSI)

The weights associated with the attributes in an MADM problem may have

significant impact on the decision outcomes. The weight sensitivity index (WSI)

indicates, to what extent, an MADM method is sensitive to the changes in attribute

weights under certain decision settings. The weight sensitivity index is measured as

the average amount of change in attribute weight required to get a change in the

ranking outcome, for a large sample set of MADM problems. The weight sensitivity

index for a method can be obtained by the following steps.

Step 1: Select a sample set of MADM problems.

A small sample set of MADM problems (usually 100) are selected randomly

for a given decision settings. The sample set can be defined as

L. ..., 2, 1, l ;Φl (4-3)


45

where Φl (l = 1, 2, ..., L) represent decision problems consisting of a decision

matrix Xl (l = 1, 2, ..., L) and weights Wlj (l = 1, 2, ..., L; j = 1, 2, ..., J), as

shown in Equation (3-3).

Step 2: Solve the decision problems.

The decision problem Φl (l = 1, 2, ..., L) is solved with Method Mk (k = 1, 2, ...,

K). The weights Wlj (l = 1, 2, ..., L; j = 1, 2, ..., J) for attributes Cj (j = 1, 2, ...,

J) are equal. The ranking outcome produced by each method for each decision

problem in the decision problem set Ω is used as the base ranking outcomes.

Step 3: Change each attribute weight.

For each decision problem Φl (l = 1, 2, ..., L) in the decision problem set Ω, the

attribute weight Wlj (l = 1, 2, ..., L; j = 1, 2, ..., J) for each attribute is gradually

changed one at a time, until the ranking outcome produced is different from the

base ranking outcome. The change in weight Wlj (l = 1, 2, ..., L; j = 1, 2, ..., J)

for Method Mk (k = 1, 2, ..., K) can be expressed as

L. ..., 2, 1, l K; ..., 2, 1, k J; ..., 2, 1, j WWW kljljklj ; (4-4)

where Wklj is the weight required for attribute Cj (j = 1, 2, ..., J) to get a ranking

outcome different from the base outcome for Method Mk (k = 1, 2, ..., K) for

the decision problem Φl (l = 1, 2, ..., L).

Step 4: Calculate the average weight change.

The average weight change for Method Mk (k = 1, 2, ..., K) for all the attribute

Cj (j = 1, 2, ..., J) in all the decision problems Φl (l = 1, 2, ..., L) in decision

problem set S can be obtained as


46

K. ..., 2, 1, k LJWWL

l

J

jkljk

;/))/)(((1 1

)( (4-5)

Step 5: Obtain the weight sensitivity index (WSI)

The Method Mk (k = 1, 2, ..., K) with a higher average weight change ΔWk (k =

1, 2, ..., K) is less sensitive to changes in attribute weights. The weight

sensitivity index can be obtained as

K. ..., 2, 1, k WWSI kk );1( (4-6)

A larger weight sensitivity index (WSIk) indicates that the corresponding

Method (Mk) is more sensitive to attribute weight changes. The weight sensitivity

index helps the decision maker select the most preferred method for various decision

settings. In decision settings where the decision maker is not confident about the

choice of attribute weights, a method with a lower WSI should be selected as it will

have a lower impact on the ranking outcome.

For example, consider the following decision problem setting

There are four (K = 4) Methods Mk (k = 1, 2, ..., K) to be compared.

The total number of decision problems Φl (l = 1, 2, ..., L) in the problem set for

a given decision setting is 100 (L = 100).

Each of the decision problems has four (J = 4) attributes Cj (j = 1, 2, ..., J).

Following Step 2, each decision problem is solved by each of the four

Methods (M1, M2, M3 and M4) where the attribute weights Wlj (l = 1, 2, ..., L; j = 1, 2,


47

..., J) are considered equal (0.25). The ranking outcome of each decision problem Φl

(l = 1, 2, ..., L) is considered as the base outcome for the corresponding decision

problem.

Following Step 3, for Method M1 and decision problem Φ1 we gradually

change the weight W1 till the outcome changes from the base outcome. Equation (4-

4) is then applied to obtain the weight change required. The weight change required

for Method M1 in decision problem Φ1 and weight W1 is 1.0111 W .

Similarly, for other three Methods (M2, M3 and M4) the weight change for

each of the attributes (W1, W2, W3 and W4) in each of the 100 decision problem can

be calculated.

Following Step 4, the average weight change required for each of the

Methods (M1, M2, M3 and M4) under given decision settings can be obtained by

Equation (4-5) as 15.01 W , 1.02 W , 3.03 W and 25.04 W respectively.

Applying Equation (4-6) in Step 5 the weight sensitivity index (WSI) for each

of the Methods (M1, M2, M3 and M4) can be obtained as 85.01 WSI , 90.02 WSI ,

70.03 WSI and 75.04 WSI respectively.

The results indicate that Method M2 is the most sensitive one and Method M3

is the least sensitive one in terms of variation in attribute weights. These results can

help the decision maker select an MADM method depending on the level of

confidence in attribute weights.


48


A new generalised simulation model together with two new performance

measures has been developed in this chapter to compare MADM methods for a large

number of decision problems.

Table 4-1 RCI and WSI summary

Ranking Consistency Index Weight Sensitivity Index

Definition RCI is the measurement of the

level of consistency an MADM

method shows with other MADM

methods under consideration in

terms of the outcomes they

produce.

WSI is the measurement of the

level of sensitivity an MADM

method shows in response to

variations in attribute weights.

Unit of

measurement

0 to 1 scale is used where a higher

value indicates a higher level of

consistency.

0 to 1 scale is used where a

higher value indicates a higher

level of sensitivity.

Application Suitable for simulation

experiments with large sample

data.

To be used when outcome

consistency is a concern for the

decision maker.

Suitable for simulation

experiments with large sample

data.

To be used when weight

sensitivity is a concern for the

decision maker.


49

The performance measures provide an efficient and objective approach to

method comparison and selection under highly diverse decision settings. Table 4-1

summarizes the two performance measures, RCI and WSI. The simulation model

developed in this chapter is applied for method comparison in Chapter 5 to

demonstrate its practical applicability.

50

Chapter 5

Applications of Developments I:

Simulation Based Selection of a Normalisation

Procedure

5.1 Introduction

In multiattribute decision making (MADM) problems, each alternative is

given a performance rating for each attribute, which represents the characteristics of

the alternative. It is common that performance ratings for different attributes are

measured in different units. To transform performance ratings into a compatible

measurement unit, normalisation procedures are used. MADM methods often use

one normalisation procedure to achieve compatibility between different measurement

units. For example, SAW uses linear scale transformation (max method) (Fishburn,

1967; Hwang and Yoon, 1981; Yeh, 2003), TOPSIS uses vector normalisation

procedure (Zeleny, 1982; Yoon and Hwang, 1995), ELECTRE uses vector

normalisation (Roy, 1991; Yoon and Hwang, 1995; Figueira et al., 2005) and AHP

uses linear scale transformation (sum method) (Saaty, 1977, 1980 and 1994).

Enormous efforts have been made to comparative studies of MADM methods, but no

significant study is conducted on the suitability of normalisation procedures used in

those MADM methods. This leaves the effectiveness of various MADM methods in

doubt and certainly raises the necessity to examine the effects of various

Chapter 5 Simulation Based Selection of a Normalisation Procedure

51

normalisation procedures on decision outcome when used with given MADM

methods.

The main purpose of this chapter is to justify and evaluate the use of a

specific normalisation procedure by two most widely used MADM methods (SAW

and TOPSIS) under various decision settings. Four widely applied normalisation

procedures are presented and then compared by simulation experiments using the

model developed in Chapter 4 to find out the most suitable ones for SAW and

TOPSIS. This chapter addresses the Decision Context A by providing generalised

guidelines for selecting the appropriate method and normalisation procedure under

various decision settings.

5.2 Normalisation Procedures Evaluated

The decision matrix X for a given MADM problem consists of performance

rating xij (i = 1, 2, ..., I; j = 1, 2, ..., J) which represents the preference for each

alternative Ai (i = 1, 2, ..., I) with respect to each attribute Cj (j = 1, 2, ..., J). The

performance ratings xij (i = 1, 2, ..., I; j = 1, 2, ..., J) may have different measurement

units and generally a normalisation procedure is applied to convert them into a single

comparable measurement unit. The four widely applied normalisation procedures in

MADM methods are briefly described below, including: (a) vector normalisation, (b)

linear scale transformation (max-min), (c) linear scale transformation (max), and (d)

linear scale transformation (sum).


52

5.2.1 Vector Normalisation

In this procedure, each performance rating xij (i = 1, 2, ..., I; j = 1, 2, ..., J) in

the decision matrix X is divided by its norm. The normalised value yij (i = 1, 2, ..., I; j

= 1, 2, ..., J) is obtained by

J. ..., 2, 1, j I; ..., 2, 1, i ;

x

xy

I

i

ij

ijij

1

2

(5-1)

This procedure has the advantage of converting all attributes into

dimensionless measurement unit, thus making inter-attribute comparison easier. But

it has the drawback of having non-equal scale length leading to difficulties in

straightforward comparison (Yoon and Hwang, 1995; Olson, 2001).

5.2.2 Linear Scale Transformation (Max-Min)

This procedure considers both the maximum and minimum of the

performance ratings xij (i = 1, 2, ..., I; j = 1, 2, ..., J) of attributes Cj (j = 1, 2, ..., J)

during calculation. For benefit and cost attributes, the normalised performance rating

yij (i = 1, 2, ..., I; j = 1, 2, ..., J) is obtained by Equations (5-2) and (5-3) respectively.

J. ..., 2, 1, j I; ..., 2, 1, i ;xx

xxy

jj

jijij

minmax

min

(5-2)

J. ..., 2, 1, j I; ..., 2, 1, i ;xx

xxy

jj

ijjij

minmax

max

(5-3)


53

where max

jx is the maximum performance rating among alternatives for attribute Cj

(j = 1, 2, …, J) and minjx is the minimum performance rating among alternatives for

attribute Cj (j = 1, 2, …, J).

This procedure has the advantage that the scale measurement is precisely

between 0 and 1 for each attribute. The drawback is that the scale transformation is

not proportional to outcome (Olson, 2001).

5.2.3 Linear Scale Transformation (Max)

This procedure divides the performance ratings xij (i = 1, 2, ..., I; j = 1, 2, ...,

J) for alternatives Ai (i = 1, 2, ..., I) with respect to each attribute Cj (j = 1, 2, …, J)

by the maximum performance rating for that attribute. For benefit and cost attributes,

the normalised performance rating yij (i = 1, 2, ..., I; j = 1, 2, ..., J) is obtained by

Equations (5-4) and (5-5) respectively.

J. ..., 2, 1, j I; ..., 2, 1, i ;x

xy

j

ijij max (5-4)

J ..., 2, 1, j I; ..., 2, 1, i ;x

xy

j

ijij max1 (5-5)

wheremax

jxis the maximum performance rating among alternatives for attribute Cj (j

= 1, 2, …, J).

The advantage of this procedure is that outcomes are transformed in a linear

way (Yoon and Hwang, 1995; Olson, 2001).


54

5.2.4 Linear Scale Transformation (Sum)

This procedure divides the performance ratings xij (i = 1, 2, ..., I; j = 1, 2, ...,

J) of each attribute Cj (j = 1, 2, …, J) by the sum of performance ratings for that

attribute as

J. ..., 2, 1, j I; ..., 2, 1, i ;

x

xy n

jj

ijij

1

(5-6)

where jx is performance rating for each alternative for attribute Cj (j = 1, 2, …, J)

(Yoon and Hwang, 1995).

Table 5-1 summarizes the four normalisation procedures described in Section 5.2.

Table 5-1 Four commonly used normalisation procedures

Notation Features Advantages / Disadvantages

Vector

Normalisation

I

i

ij

ijij

x

xy

1

2

Performance

ratings are

divided by its

norm.

Converts all measurement units

for attributes into a comparable

dimensionless unit.

Use of non-equal scale length

leads to difficulties in

straightforward comparison.

Linear Scale

(Max-Min) minmax

min

jj

jijij

xx

xxy

Performance

ratings are

divided by

the range.



dimensionless unit.

Considers the two extreme


55

performance rating values in

calculation.

Transformation is linear.

Scale transformation is not

proportional to outcome.

Linear Scale

(Max) max

j

ijij

x

xy

Performance

ratings are

divided by

the maximum

one.



dimensionless unit.


Considers only the maximum

value.

Linear Scale

(Sum)

n

jj

ijij

x

xy

1

Performance

ratings are

divided by

their sum.



dimensionless unit.


5.3 Multiattribute Decision Making Methods Evaluated

In these experiments, the SAW and TOPSIS methods are evaluated to find

the most suitable normalisation procedure under various decision settings.

5.3.1 The SAW Method

The simple additive weight (SAW) method, also known as the weighted sum

method, is probably the best known and most widely used MADM method (Hwang


56

and Yoon, 1981). The basic logic of the SAW method is to obtain a weighted sum of

the performance ratings of each decision alternative over all the attributes. The

overall weighted preference value is used as the basis for comparison between the

alternatives. This method involves the following two steps:

Step 1: Obtain the normalised decision matrix.

The performance ratings xij (i = 1, 2, ..., I; j = 1, 2, ..., J) in the decision matrix

X shown in Equation (3-1) are normalised by applying Equation (5-4). The

normalised performance ratings yij (i = 1, 2, ..., I; j = 1, 2, ..., J) can be given as

a matrix shown in Equation (5-7).

IJII

J

J

yyy

yyy

yyy

Y

...

............

...

...

21

22221

11211

(5-7)

Step 2: Obtain the overall preference value.

The overall preference value for alternative Ai (i = 1, 2, ..., I) can be obtained

by combining the attribute weights Wj (j = 1, 2, ..., J) from Equation (3-2) with

the Equation (5-7) as

I. ..., 2, 1, = i yWVJ

jijji ;

1

. (5-8)

Where Vi (i = 1, 2, ..., I) is the overall preference value of decision alternative

Ai (i = 1, 2, ..., I); Wj (j = 1, 2, ..., J) is the weight for attribute Cj (j = 1, 2, ..., J)

and yij (i = 1, 2, ..., I; j = 1, 2, ..., J) are normalised performance ratings


57

(Hwang and Yoon, 1981; Zeleny, 1982). An alternative with a greater overall

value (i = 1, 2, ..., I) will receive a higher ranking.

5.3.2 The TOPSIS Method


has been used extensively to solve various practical MADM problems, due to its

simplicity, computational efficiency and the ability to measure the performances of

the decision alternatives in simple mathematical form (Yeh and Chang, 2009). In

TOPSIS, an index known as similarity to positive-ideal solution is defined by

combining the closeness to the positive-ideal solution and remoteness to the

negative-ideal solution. This index is used to rank the alternatives (Hwang and Yoon,

1981; Zeleny, 1982). We will refer to the index as the overall preference value in

order to maintain uniformity with other methods used. The TOPSIS method involves

the following steps.

Step 1: Calculate the normalised performance ratings.

The performance ratings xij (i = 1, 2, ..., I; j = 1, 2, ..., J) in the decision matrix

X shown in Equation (3-1) are normalised by applying Equation (5-1). The

normalised performance ratings yij (i = 1, 2, ..., I; j = 1, 2, ..., J) can be given as

a matrix similar to Equation (5-7).

Step 2: Calculate weighted normalised performance rating.

The weight Wj (j = 1, 2, ..., J) from Equation (3-2) is combined with

normalised decision matrix Y from Equation (5-7) to get the weighted

iV


58

normalised performance rating vij (i = 1, 2, ..., I; j = 1, 2, ..., J) shown in

Equation (5-9). The weighted normalised decision matrix is shown in Equation

(5-10).

J. ,… 2, 1, =j I; ,… 2, 1, = i ; yWv ijjij * . (5-9)

IJII

J

J

vvv

vvv

vvv

V

...

............

...

...

21

22221

11211

(5-10)

Step 3: Identify the positive-ideal and negative-ideal solutions.

The set of positive-ideal solution A* and negative-ideal solution A- are

identified from Equation (5-10) in terms of weighted normalised performance

ratings.

*

J*2

*1 v ..., ,v ,vA * (5-11)

J21 v ..., ,v,vA (5-12)

where

Step 4: Calculate separation measure.

The separation measures for each decision alternative Ai (i = 1, 2, ..., I) is

calculated using n-dimensional Euclidean distance. The separation (distance)

attributecost a is if

attributebenifit a is if

j,vmin

j,vmaxv

ij

ij*j

attributecost a is if

attributebenifit a is if

j,vmax

j,vminv

ij

ij

j


59

of each alternative from the positive-ideal solution A* and negative-ideal

solution A- can be obtained by Equation (5-13) and (5-14) respectively.

I. ..., 2, 1, i ; vvSJ

jjiji

2** )( (5-13)

I. ..., 2, 1, i ; vvSJ

jjiji

2)( (5-14)

Step 5: Obtain the overall preference value

The overall preference value Vi (i = 1, 2, ..., I) for each alternative Ai (i = 1, 2,

..., I) can be calculated as

I. ..., 2, 1, i ; SS

SV

ii

ii

* (5-15)

A higher value of Vi (i = 1, 2, ..., I) indicates a higher ranking of alternative Ai

(i = 1, 2, ..., I).

5.4 Experiments and Results for SAW Simulation studies are conducted for the SAW method to find out the most

suitable normalisation procedure for this method under various decision settings. The

simulation model developed in Chapter 4 is used for the experiments. The

performance measure ranking consistency index (RCI) developed in Chapter 4 is

applied to compare the performance of the four normalisation procedures presented

in the last section.


60

5.4.1 Simulation Experiments for SAW

This simulation experiment is conducted to evaluate four normalisation

procedures which can be used with SAW. It is then used to identify the most suitable

one for SAW under various decision settings. The experiment is conducted using the

following seven steps.

Step 1: Identify the set of MADM Methods Mk (k = 1, 2, ..., K) to be compared.

The combination of each normalisation procedure and SAW aggregation

technique is considered as an MADM method. Table 5-2 shows the methods to

be compared for SAW.

Table 5-2 Four MADM methods for the experiment with SAW

MADM

method Normalisation procedure Aggregation technique

M1(S) N1: Vector normalisation SAW

M2(S) N2: Linear scale transformation (max-min) SAW

M3(S) N3: Linear scale transformation (max) SAW (Conventional)

M4(S) N4: Linear scale transformation, (sum) SAW


The experiments test three decision information settings including the number

of alternatives, the number of attributes and the data range for the decision

problem. The initial and target settings for each are selected as


61

(a) The number of alternatives with initial setting as 4 alternatives and target

setting as 20 alternatives.

(b) The number of attributes with initial setting as 4 attributes and target setting

as 20 attributes.

(c) The data range for performance ratings for each attribute with equally

divided range between 1 and 10,000.

The reason to choose 4 as the lower limit and 20 as the upper limit for the

number of alternatives and attributes is that it is a range wide enough to

produce significant results. The upper and lower limits (4 and 20) for the

number of alternatives and the number of criteria chosen in this study are not to

be considered as the only choice. Experiments were conducted with different

sets of lower and upper limits and it was found that the limit value between 4

and 20 provides significant results required for this study. The data range is

chosen as 1 to 10,000 as it can generate sufficient variations for problems with

a different number of alternatives and attributes. Different data ranges were

tested and it was found that the data range of 1 to 10,000 provides enough

samples to achieve significant conclusive outcomes.

Step 3: Generate a large number of decision problems for the current settings.

For each decision setting, 10,000 decision matrices are generated randomly in

each simulation run. Although the sample problem set with 10,000 matrices is

large enough to produce significant comparative results, the validity of results

is tested by generating three different sample sets with 10,000 matrices each

for each decision information setting.


62

Step 4: Solve each decision problem with Method Mk (k = 1, 2, ..., K).

Each of the 10,000 decision matrices generated in Step 3 is solved using each

of the MADM methods in Table 5-2.

Step 5: Use measures to evaluate the performances of Method Mk (k = 1, 2, ..., K).

The performances of the methods for a given decision settings are evaluated

using the ranking consistency index (RCI) obtained by applying Equations (4-

1) and (4-2).

Step 6: Vary particular decision information setting in a given amount.

The three decision information settings presented in Step 2 are varied one at a

time. The number of alternatives and the number of attributes are increased by

2 each time. The data range is narrowed by increasing the lower limit by 10%

to determine the new setting.

Step 7: Repeat Step 3 to Step 6 until the target information setting is reached.

5.4.2 Experimental Results for SAW

5.4.2.1 Results for Change in Alternative Numbers

These experiments are conducted to investigate the impact of the number of

alternatives on the ranking consistency. The number of attributes is set between 2 to

20 along with the data range 1 to 10,000 (evenly distributed to the attributes). The

number of alternatives is then changed from 2 to 20. With each setting of the

alternative the ranking consistency is measured for the four SAW methods given in

Table 5-2.


63

Figure 5-1 shows the results obtained by changing the number of alternatives

with the attribute number set to 10 (for complete results refer to Appendix C). The

results clearly show that the ranking consistency for all the methods reduces

significantly with the increase of the number of alternatives in the decision problems.

The Method M1(S) is surely the best performer is all cases where as the Method

M2(S) is the worst one. Methods M3(S) and M4(S) performs close to Method M1(S).

Method M3(S) is better than Method M4(S) with small number of alternative but

M4(S) is relatively better than M3(S) for problems with large number of alternatives.

The experiment results suggest that instead of the conventional linear scale

transformation- max (N3) normalisation procedure, the vector normalisation (N1)

procedure should be used with SAW when the number of alternatives in an MADM

problem is a concern of the decision maker.


the number of alternatives

0

0.1

0.2

0.3

0.4

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M4 (S)


64

5.4.2.2 Results for Change in Attribute Numbers

These experiments are conducted to find out how the number of attributes

involved in a decision problem can affect the ranking outcome. Four SAW methods

given in Table 5-1 are evaluated to find their consistency in ranking for decision

problems with a different number of attributes. The data range is set between 1 and

10,000. The number of alternatives involved is set between 2 and 20 for each

experiment. The number of attributes is increased from 2 to 20 for each setting to

measure change in ranking consistency index. Figures 5-2 and 5-3 show the results

from two settings. The ranking consistency for all the methods decreases gradually

with an increase in the number of attributes. Method M1(S) is most consistent over all

ranges of attributes and M2(S) is the least consistent at all times. With a small

number of alternatives, M3(S) is more consistent than M4(S). But with a larger

number, M4(S) is more consistent.


the number of attributes

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

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Number of Attributes

M1 (S)

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M3 (S)

M4 (S)


65



5.4.2.3 Results for Change in Data Range

These experiments address the issue of data variation in a decision problem.

The size of the decision problem for each experiment is set by selecting problems

with a specific number of attributes and alternatives from 4 to 20. For each decision

setting, the data range for each attribute is narrowed by 10%.

Figures 5-4 and 5-5 show the results from two decision settings (refer to

Appendix C for the complete results). The results show that Method M2(S) is not

affected by the change in data range and remains the least consistent one for all

decision settings. For all decision settings, ranking consistency for Methods M1(S),

M3(S) and M4(S) increases with narrower data ranges. Although Method M1(S) is the

best performer,

0

0.05

0.1

0.15

0.2

0.25

0.3

4 6 8 10 12 14 16 18 20

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M2 (S)

M3 (S)

M4 (S)


66


for changes in the data range


consistency for changes in the data range

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

100 90 80 70 60 50 40 30 20

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M1 (S)

M2 (S)

M3 (S)

M4 (S)

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0.1

0.2

0.3

0.4

0.5

0.6

0.7

100 90 80 70 60 50 40 30 20

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Data Range (%)

M1 (S)

M2 (S)

M3 (S)

M4 (S)


67

M4(S) and M3(S) show close performances. The performance of Method M4(S) is

almost same as M1(S) for decision problems with large problem sizes and very

narrow data ranges. Method M3(S) shows a decent performance for large problems

(in terms of attributes and alternatives).

5.5 Experiments and Results for TOPSIS

5.5.1 Simulation Experiments for TOPSIS

This simulation experiment is conducted to evaluate four normalisation

procedures which can be used with TOPSIS. It is then used to identify the most

suitable one for TOPSIS under various decision settings. The experiment is

conducted using the following seven steps.

Step 1: Identify the set of MADM Methods Mk (k = 1, 2, ..., K) to be compared.

The combination of each normalisation procedure and TOPSIS aggregation

technique is regarded as an MADM method. Table 5-3 shows the methods to

be compared for TOPSIS.

Table 5-3 Four MADM methods for the experiment with TOPSIS

MADM


M1(T) N1: Vector normalisation TOPSIS (Conventional)

M1(T) N2: Linear scale transformation (max-min) TOPSIS

M3(T) N3: Linear scale transformation (max) TOPSIS

M4(T) N4: Linear scale transformation (sum) TOPSIS


68


The experiments test three decision information settings including the number

of alternatives, the number of attributes and the data range for the decision

problem. The initial and target settings for each is selected as

(a) The number of alternatives with initial setting as 4 alternatives and target

setting as 20 alternatives.

(b) The number of attributes with initial setting as 4 attributes and target setting

as 20 attributes.

(c) The data range for performance ratings for each attribute with an equally

divided range between 1 and 10,000.

The reason to choose 4 as the lower limit and 20 as the upper limit for the

number of alternatives and attributes is that it is a range wide enough to

produce significant results. The upper and lower limits (4 and 20) for the

number of alternatives and the number of criteria chosen in this study are not to

be considered as the only choice. Experiments were conducted with different

sets of lower and upper limits and it was found that the limit value between 4

and 20 provides significant results required for this study. The data range is

chosen as 1 to 10,000 as it can generate sufficient variations for problems with

a different number of alternatives and attributes. Different data ranges were

tested and it was found that the data range of 1 to 10,000 provides enough

samples to achieve significant conclusive outcomes.

Step 3: Generate a large number of decision problems for the current settings.


69

For each decision setting, 10,000 unique decision matrices are generated

randomly in each simulation run. Although the sample problem set with 10,000

matrices is large enough to produce significant comparative results, the validity

of results are tested by generating three different sample sets with 10,000

matrices each for each decision information setting.

Step 4: Solve each decision problem with Method Mk (k = 1, 2, ..., K).

Each of the 10,000 decision matrices generated in Step 3 are solved using each

of the MADM methods in Table 5-3.

Step 5: Use measures to evaluate the performances of Method Mk (k = 1, 2, ..., K).

The performances of the methods for a given decision settings are evaluated

using the ranking consistency index (RCI) obtained by applying Equations (4-

1) and (4-2).

Step 6: Vary particular decision information setting in a given amount.

The three decision information settings presented in Step 2 are varied one at a

time. The number of alternatives and attributes are increased by 2 each time.

The data range is narrowed by increasing the lower limit by 10% to determine

the new setting.

Step 7: Repeat Step 3 to Step 6 until the target information setting is reached.

5.5.2 Experiment Results for TOPSIS

5.5.2.1 Results for Change in Alternative Numbers


70

The experiments are conducted similarly to the experiments for the SAW

method. With the set data range and attribute numbers, the number of alternatives is

increased to check the impact on ranking consistency.

The results in Figure 5-6 (the complete result is given in Appendix C) shows

that Method M1(T) is the best performer and M2(T) is the worst. Method M3(T)

performs better than M4(T) for problems with a smaller number of alternatives but

worse in case of larger ones. Method M1(T) performs similarly to M4(T) for decision

problems with the number of alternatives over 14. For all the four methods, the

ranking consistency drops dramatically with a larger number of alternatives. This

results shows that the conventional TOPSIS method M1(T) is currently using the

most consistent normalisation procedure, the vector normalisation (N1).



0

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0.3

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M2 (T)

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71

5.5.2.2 Results for Change in Attribute Numbers

The number of alternatives and the data range are set for each experiment and

the number of attributes is changed from 2 to 20 in steps of 2 in order to find the

impact on the ranking consistency. Figures 5-7 and 5-8 show the results of two

different decision settings (refer to Appendix C for the complete results). The

ranking consistency for all the methods is not affected much with the change in the

number of attributes where the problem involves a smaller number of alternatives.

In decision settings with a larger number of alternatives, all for methods show

a decrease in ranking consistency when the number of attributes is increased. For

decision settings with a smaller number of alternatives, Method M1(T) performs best

with M3(T) slightly better than M4(T). In decision settings with a larger number of

alternatives, M1(T) is matched with the performance by M4(T) when the number of

attributes is increased. However, in such settings, the ranking consistency of Method

M3(T) decreases significantly to match the poor performance of Method M2(T).


72





0

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M2 (T)

M3 (T)

M4 (T)

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M2 (T)

M3 (T)

M4 (T)

M1 (T)

M2 (T)

M3 (T)

M4 (T)


73

5.5.2.3 Results for Change in Data Range

With the number of alternatives and attributes set for each of these

experiments, the data range is narrowed by 10% steps to assess the impact on the

ranking consistency. Figures 5-9 and 5-10 presents the results of changes in data

range with two different decision settings in terms of the number of attributes and the

number of alternatives (results for the complete range is available in Appendix C).

For both the smaller and larger settings, Method M2(T) is unaffected by any change

in data range and is the worst performer in terms of ranking consistency.


for changes in the data range

0

0.1

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M1 (T)

M2 (T)

M3 (T)

M4 (T)


74



For decision settings with a smaller number of alternatives and attributes,

methods M1(T), M3(T) and M4(T) performs similarly with M1(T) being slightly

better. There is an increase in ranking consistency with a narrower data range for all

these three methods.

For decision settings with a larger number of alternatives and attributes,

M1(T) and M4(T) performs very closely and shows a sharp rise in performance with

narrower data ranges. Performance for Method M3(T) also increases but does not

perform as well as M1(T).

0

0.05

0.1

0.15

0.2

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0.3

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0.4

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M1 (T)

M2 (T)

M3 (T)

M4 (T)


75


Table 5-4 provides a quick reference to the results (trends) for different

variations of the SAW and TOPSIS methods under different decision settings. The

experiments in this chapter have presented useful results that can be used as general

guidelines for selecting the most suitable normalisation procedure for SAW and

TOPSIS under various decision settings. The results have shown that the

conventional methods are not necessarily the best performing ones in all decision

settings. The experiments prove that, using different normalisation procedures to

solve a given problem may lead to different ranking outcomes, thus highlighting the

need for a new way of method evaluation and comparison.

Table 5-4 Simulation results in terms of performance

Decision Settings SAW TOPSIS

N1 N2 N3 N4 N1 N2 N3 N4

Var

iati

on in

nu

mb

er o

f A

ttri

bu

tes

and

Alt

ern

ativ

es

Attributes: L Alternatives: L

H (Best)

M (Worst)

H (Near to N1)

H (Near to N3)

H (Best)

M (Worst)

H (Near to N4)

H

Attributes: L Alternatives: M

M (Best)

L (Worst)

M

M

M (Best)

L (Worst)

L

M

Attributes: L Alternatives: H

L (Best)

L (Worst)

L (Near to N2)

L (Near to N1)

L (Best)

L (Worst)

L (Near to N2)

L (Near to N1)

Attributes: M Alternatives: L

H (Best)

M (Worst)

H (Near to N1)

H (Near to N3)

H (Best)

M (Worst)

H

H (Near to N3)

Attributes: M Alternatives: M

M (Best)

L (Worst)

M

M

L (Best)

L (Worst)

L

L

Attributes: M Alternatives: H

L (Best)

L (Worst)

L (Near to N2)

L (Near to N1)

L (Best)

L (Worst)

L (Near to N2)

L (Near to N1)

Attributes: H Alternatives: L

H (Best)

M (Worst)

H (Near to N1)

H (Near to N3)

H (Best)

M (Worst)

H

H (Near to N3)

Attributes: H Alternatives: M

M (Best)

L (Worst)

M

M

L (Best)

L (Worst)

L (Near to N2)

L (Near to N1)

Attributes: H Alternatives: H

L (Best)

L (Worst)

L (Near to N2)

L (Near to N1)

L (Best)

L (Worst)

L (Near to N2)

L (Near to N1)


76

Var

iati

on in

nu

mb

er o

f A

ttri

bu

tes

and

Alt

ern

ativ

es a

nd

Dat

a R

ange

Attributes & alternatives: L Data Range: L

H (Best)

M (Worst)

H (Near to N4)

H (Near to N1)

H (Best)

M (Worst)

H

H (Near to N1)

Attributes & alternatives: L Data Range: M

H (Best)

M (Worst)

H (Near to N4)

H (Near to N1)

H (Best)

M (Worst)

H

H

Attributes & alternatives: L Data Range: H

H (Best)

M (Worst)

H (Near to N4)

H (Near to N1)

H (Best)

M (Worst)

H (Near to N4)

H

Attributes & alternatives: M Data Range: L

H (Best)

L (Worst)

H

H (Near to N1)

H (Best)

L (Worst)

H

H (Near to N1)

Attributes & alternatives: M Data Range: M

H (Best)

L (Worst)

H

H (Near to N1)

H (Best)

L (Worst)

M

H

Attributes & alternatives: M Data Range: H

M (Best)

L (Worst)

M

M (Near to N3)

M (Best)

L (Worst)

L

L

Attributes & alternatives: H Data Range: L

H (Best)

L (Worst)

M

H (Near to N1)

H (Best)

L (Worst)

M

H (Near to N1)

Attributes & alternatives: H Data Range: M

M (Best)

L (Worst)

L

M (Near to N1)

M (Best)

L (Worst)

L

M (Near to N1)

Attributes & alternatives: H Data Range: H

L (Best)

L (Worst)

L

L (Near to N1)

L (Best)

L (Worst)

L (Near to M2)

L (Near to N1)

H = High, M = Moderate, L = Low

77

Chapter 6

Developments II:

Rank Similarity Based Method Evaluation and

Selection

6.1 Introduction

Multiattribute decision making (MADM) problems are diverse greatly in

terms of the decision information, the decision context and the applications. With the

availability of multiple suitable MADM methods Mk (k = 1, 2, ..., K) for a given

MADM problem Φ, selecting the most suitable one is an extremely challenging task

(Yeh, 2003; Chakraborty and Yeh, 2007a). Several comparative and simulation

based studies suggest the suitability of certain methods under given decision settings

(Simpson, 1996; Zanakis et al., 1998; Olson, 2001; Chakraborty and Yeh, 2009).

Under certain decision contexts, the decision maker may use the results of these

studies for method evaluation and selection. In a decision context where a suitable

and acceptable set of MADM methods Mk (k = 1, 2, ..., K) is available for a given

problem Φ, the decision maker needs to select the most preferred one among them.

The Decision Context B identified in Chapter 3 is the decision context to be

addressed in this chapter. Previous studies cannot guarantee that the MADM method

selected is the most preferred one for the given problem when Decision Context B is

considered.

Chapter 6 Rank Similarity Based Method Evaluation and Selection

78

In this chapter, a novel method selection approach is developed to select the

most preferred method from a set of suitable and acceptable MADM methods Mk (k

= 1, 2, ..., K) for a given problem Φ. The approach considers the similarities between

the ranking outcomes produced by a given set of suitable MADM methods Mk (k = 1,

2, ..., K).

6.2 Methodology Development

6.2.1 Rank Similarity and Method Evaluation

The new method selection approach is developed for dealing with the

decision context where each of the outcomes produced by a set of suitable MADM

methods )...,,2,1( K kM k are considered valid and acceptable to the decision

maker. The most preferred method is to be chosen from the set of suitable MADM

methods )...,,2,1( K kM k depending on the most preferred ranking outcome. The

solution space is considered to be limited, as it consists of the ranking outcomes

produced by a specific set of suitable MADM methods )...,,2,1( K kM k for the

given problem Φ. Hence, the most preferred outcome must be among the ranking

outcomes produced.

For a given MADM problem Φ, the solution space consists of different

ranking outcomes )...,,2,1( K kOk produced by each suitable Method

)...,,2,1( K kM k , which are all valid and acceptable to the decision maker. The

most preferred MADM method is the one that produces the most preferred outcome.

The most preferred outcome is the one which is closest to all other outcomes. The


79

closeness between the ranking outcomes can be measured in terms of the similarity

between them. In this new approach, the similarity between two ranking outcomes is

measured by using the rank correlation coefficient (Spearman, 1904). The method

which produces the outcome most similar to all other outcomes is the most preferred

method for the given problem.

6.2.2 The Rank Correlation Coefficient

The rank correlation coefficient is widely used as a measurement of

association between different ranks (Kendall, 1955; Raju and Pillai, 1999). It has

been successfully applied in various studies to test the sensitivity and significance of

certain information in different MADM problem settings (Zanakis et.al, 1998;

Triantaphyllou and Sanchez, 1997; Yurdakul and Yusuf, 2009). The rank correlation

coefficient between two ranks can be defined as

I. ..., 2, 1, i II

dI

ii

;

6

13

1

2

(6-1)

where di is the difference between the ranks for the alternative Ai (i = 1, 2, ..., I).

6.2.3 Rank Similarity Index

The rank similarity index (RSI) is developed as a measure of decision

outcome similarity for an MADM method with all the other suitable MADM

methods in the set of acceptable methods. This measure indicates the relative

closeness of a method with other methods in terms of ranking outcome similarity.


80

The RSI is the average of the rank correlation coefficients between a ranking

outcome and all the other ranking outcomes. The method with the largest RSI

indicates that the ranking outcome it produces is most similar or closest to all other

outcomes, hence the most preferred one. The rank similarity index can be obtained

using the following five steps.

Step 1: Generate the rank matrix (Rk)

This step involves solving the decision problem with each MADM method in

the acceptable set and obtaining the ranking outcomes. The outcomes are

presented as a matrix called the rank matrix (Rk), formed by combining the

ranking outcomes )...,,2,1( K kOk given to alternative iA (i = 1, 2, ..., I) by

Method kM (k = 1, 2, ..., K) as shown in Equation (6-2).

K. ..., 2, 1, k I; ..., 2, i

rrr

rrr

rrr

R

IKII

K

K

k

,1;

...

............

...

...

21

22221

11211

(6-2)

where rik (1 ≤ rik ≤ I) represents the rank of alternative Ai (i = 1, 2, ..., I) by

using Method Mk (k = 1, 2, ..., K).

Step 2: Calculate rank correlation (RC) between ranking outcomes

The rank correlations for Method kM (k = 1, 2, ..., K) in relation to each of the

other Methods hM (h = 1, 2, ..., K; k ≠ h) are calculated by applying Equations

(6-1) and (6-2) as

h.kK; ..., 2, 1,h K; ..., 2, 1,k MMRC hkkh );,( (6-3)


81

Step 3: Calculate the rank similarity index (RSI) for each method

The rank similarity index for Method kM (k = 1, 2, ...,K) can be calculated by

taking the average of correlations calculated by Equation (6-3) as

.;...,,2,1;...,,2,1;/)(1

hk K h K k KRCRSIK

hkhk

(6-4)

Step 4: Find the largest rank similarity index (RSI+)

....,,2,1;max K k RSIRSI k (6-5)

The method with the largest rank similarity index (RSI+) is the most preferred

one for the given MADM problem.

6.3 Numerical Example

6.3.1 Methods Used in the Example

In this example, variants of the three widely used MADM methods are used,

including: (a) the simple additive weighting (SAW), (b) the technique for order

preference by similarity to ideal solution (TOPSIS), and (c) the weighted product

(WP). The SAW and TOPSIS methods have been presented in Chapter 5. The WP

method can be presented as

I. ..., 2, 1, i ; xVJ

j

Wiji

j 1

(6-6)


82

where xij (i = 1, 2, ..., I; j = 1, 2, ..., J) is performance rating in decision matrix X as

shown in Equation (3-1); Wj (j = 1, 2, ..., J) is weight for attribute Cj (j = 1, 2, ..., J)

as shown in Equation (3-2); Vi is the overall preference value for alternative Ai (i = 1,

2, ..., I).

The alternatives Ai (i = 1, 2, ..., I) are ranked according to the value of Vi. A

higher Vi value indicates a higher ranking for the alternative Ai (i = 1, 2, ..., I).

Table 6-1 shows the nine suitable MADM methods evaluated in this example.

These methods include four variants of SAW as shown in Table 5-1, four variants of

TOPSIS as shown in Table 5-2, and the WP method shown in Equation (6-6).

Table 6-1 Nine MADM methods used in the example

MADM


M1 N1: Vector normalisation SAW

M2 N2: Linear scale transformation (max-min) SAW

M3* N3: Linear scale transformation (max) SAW

M4 N4: Linear scale transformation (sum) SAW

M5** N/A WP

M6*** N1: Vector normalisation TOPSIS

M7 N2: Linear scale transformation (max-min) TOPSIS

M8 N3: Linear scale transformation (max) TOPSIS

M9 N4: Linear scale transformation (sum) TOPSIS

* M3 is the conventional SAW method

** M5 is the conventional WP method

*** M6 is the conventional TOPSIS method


83

6.3.2 The Example

To illustrate the rank similarity based method selection approach, the decision

matrix from the graduate fellowship applicants ranking case is used (Yoon and

Hwang, 1995). Table 6-2 shows the decision matrix. The attributes weights for the

decision problem are given as W = (0.3, 0.1, 0.3, 0.15, 0.15).

The methods shown in Table 6-1 produce different ranking outcomes which

are shown as a rank matrix in Table 6-3 by applying Equation (6-2). The rank

correlation coefficients for each method with respect to other methods are calculated

by applying Equation (6-3) on Table 6-3, and the results are shown in Table 6-4. The

rank similarity index is calculated by applying Equation (6-4) on Table 6-4 and the

results are shown in Table 6-5.

Table 6-2 Decision matrix used in the example

Attribute

Alternative C1 C2 C3 C4 C5

A1 690 3.1 9 7 4

A2 590 3.9 7 6 10

A3 600 3.6 8 8 7

A4 620 3.8 7 10 6

A5 700 2.8 10 4 6

A6 650 4 6 9 8


84

Table 6-3 Resultant rank matrix

MADM method

Alternative M1 M2 M3 M4 M5 M6 M7 M8 M9

A1 6 2 5 6 6 4 2 3 5

A2 5 6 6 5 4 3 6 5 3

A3 1 5 3 2 1 2 4 2 1

A4 4 4 4 4 3 5 5 4 4

A5 3 1 1 3 5 1 1 1 2

A6 2 3 2 1 2 6 3 6 6

Table 6-4 Rank correlation coefficient between MADM methods

M1 M2 M3 M4 M5 M6 M7 M8 M9

M1 1 -0.086 0.714 0.943 0.829 0.143 0.086 0.143 0.371

M2 -0.086 1 0.600 0.029 -0.543 0.086 0.943 0.429 -0.257

M3 0.714 0.600 1 0.771 0.257 0.200 0.657 0.371 0.143

M4 0.943 0.029 0.771 1 0.771 -0.086 0.143 -0.086 0.086

M5 0.829 -0.543 0.257 0.771 1 -0.200 -0.429 -0.257 0.200

M6 0.143 0.086 0.200 -0.086 -0.200 1 0.257 0.829 0.886

M7 0.086 0.943 0.657 0.143 -0.429 0.257 1 0.543 -0.086

M8 0.143 0.429 0.371 -0.086 -0.257 0.829 0.543 1 0.714

M9 0.371 -0.257 0.143 0.086 0.200 0.886 -0.086 0.714 1


85

Table 6-5 Rank similarity index for suitable MADM methods

M1 M2 M3 M4 M5 M6 M7 M8 M9

RSI 0.393 0.150 0.464 0.321 0.079 0.264 0.264 0.336 0.25

From Table 6-5, we can select the largest RSI using Equation (6-5) as

464.0)( 3 MRSIRSI . This suggests that Method M3 produces the ranking

outcome most similar to that of all other methods. Hence, this method is the most

preferred one for the given problem under the decision context considered. These

results can be used in conjunction with other decision contexts where the decision

maker is considering multiple contexts and can select a method which most satisfies

all the contexts.

In this particular example, it is observed that the conventional SAW method

(i.e. Method M3) is the best performer, and conventional TOPSIS method (i.e.

Method M6) and WP method (i.e. method M5) do not perform well. This highlights

the need for a change in the way the existing method comparison and selection

studies are conducted. These results reinforce the argument that MADM methods

considered for selection should not just include the ones originally developed or

commonly applied (such as M3, M5 and M6 in Table 6-2). Instead, the comparisons

must be done at more detail levels including normalization procedures and

aggregation techniques, wherever possible.


86


The rank similarity based MADM method selection approach developed

provides an efficient, yet simple context dependent approach for a given problem.

Although the illustrated example has used the variants of SAW, TOPSIS and WP

methods only, the approach is applicable for selecting from any set of MADM

methods capable of producing a complete ranking outcome. The importance of

applying a problem specific method selection approach for any given MADM

problem rather than the generalised selection approach has also been highlighted.

87

Chapter 7

Developments III:

An Alternatives-Oriented Method Evaluation and

Selection

7.1 Introduction

The decision-maker-oriented and the method-oriented approach have been

developed from the perspectives of the decision maker and the MADM method

respectively, as discussed in Chapter 2. In MADM problem settings where the

decision maker is not a key stakeholder, the decision alternatives as the key

stakeholders should have a greater influence in determining the decision outcome.

The significance of the role played by the alternatives in the decision making process

has led to the development of the alternatives-oriented method evaluation and

selection approach presented in this chapter.

A recent study on MBA ranking by Financial Times has well discussed the

inconsistency and bias during problem structuring (Jessop, 2009). The same decision

problem (once structured) may need to address the method selection issue as well.

When ranking MBA programmes, the Financial Times chooses a method without

considering the view of the relevant business schools on the method selection

process. The ranking outcome has great impacts on the business schools being

ranked; hence they are the major stakeholders and should have involvement in the

method selection process. Similarly, some multiattribute ranking problems such as

Chapter 7 An Alternatives-Oriented Method Evaluation and Selection

88

ranking of universities (as the decision alternatives) are based on a given set of

evaluation criteria (as the attributes). The role of the decision maker if existent is

restricted to obtain a ranking outcome and to analyse the data. The MADM method

being applied to obtain the rank is decided subjectively by the decision maker. With

other suitable methods available, the one used by the decision maker is not

necessarily produce the most preferred outcome by all the stakeholders. It is thus our

belief that decision alternatives should play the role of the decision maker (where no

decision maker is available or the decision maker is not a key stakeholder), as in this

case the alternatives are the key stakeholders of the decision problem. As a

stakeholder, an alternative will naturally have a higher preference for a method that

gives it a better ranking. The alternatives-oriented approach developed in this chapter

uses a new performance measure called the method preference level for justifying the

method selection. The preference level indicates the satisfaction or acceptability

degree of all the alternatives as a whole for each suitable MADM method, thus

providing the basis for objective comparison. In addition to the objective

performance measure, the novelty of this approach lies in its new way of addressing

the MADM method evaluation and selection problem from the perspective of the

alternatives, which makes the method selection possible even without the presence of

the decision maker.

This chapter addresses the Decision Context C outlined in Chapter 3. In

subsequent sections, the new alternatives-oriented approach is developed along with

the performance measure. A worked example is then presented to illustrate the

effectiveness of the approach.


89

7.2 The Alternatives-Oriented Approach and the Preference Level

The alternatives-oriented approach considers the preference of each decision

alternative for selecting an MADM method to solve a given multiattribute decision

problem that requires a complete ranking of the decision alternatives. Each MADM

method produces a ranking outcome for the given problem. An alternative will

naturally have a higher preference for an MADM method which gives it a higher

rank. The preference degree of each alternative for each MADM method is

determined by considering its preference over other methods. The preference degrees

of a method given individually by all the alternatives are combined to obtain the

overall preference level of each method. The preference level of an MADM method

with respect to all the alternatives indicates the level of satisfaction or acceptability it

provides to all the alternatives as a whole. The method that provides the highest level

of satisfaction to all the alternatives as a whole is the most preferred one for the

given problem. The new approach involves five steps, given below.

Step 1: Generate the rank matrix

Each MADM Method Mk (k = 1, 2, ..., K) produces individual rankings

outcomes Ok (k = 1, 2, ..., K) for each decision alternative Ai (i = 1, 2, ..., I) for

the given decision problem Φ. The rank matrix R is obtained by organising the

rank of each alternative produced by each method, as shown by Equation (6.2)

in Chapter 6.

Step 2: Obtain the preference degree

The method preference degree indicates the extent to which a decision

alternative Ai (i = 1, 2, ..., I) prefers a Method Mk (k = 1, 2, ..., K) over other


90

methods. The method which provides the alternative with the highest rank will

receive the highest degree of preference (i.e. 1). The preference degree pik (i =

1, 2, ..., I; k = 1, 2, ..., K) of each Method Mk (k = 1, 2, ..., K) generated by each

alternative Ai (i = 1, 2, ..., I) is obtained by

K. ..., 2, 1, k I; ..., 2, 1, i ; K

bKp ki

ik

(7-1)

where K is the number of methods considered; bki is the number of methods

producing better ranking than Method Mk (k = 1, 2, ..., K) for alternative Ai (i =

1, 2, ..., I).

The preference degree matrix P is generated by combining preference degrees

of all the methods with respect to each alternative as

IKII

K

K

ppp

ppp

ppp

P

...

............

...

...

21

22221

11211

(7-2)

Step 3: Calculate the scaled preference degree

Preference degrees of different decision alternatives in Equation (7-2) have

different units which require being converted into a single unit for comparison.

The highest preference degree of an alternative for any method can be 1.

Hence, the conversion of preference degrees into a unified scale should be in

such a manner that for any alternative the preference degrees are summed up to

1. This scaling can be obtained by applying Equation (5-6) in Chapter 5 and

Equation (7-2) as


91

K. ..., 2, 1, k I; ..., 2, 1, i ;

p

pu K

kik

ikik

1

(7-3)

The resultant scaled preference matrix U is given as

IKII

K

K

uuu

uuu

uuu

U

...

............

...

...

21

22221

11211

(7-4)

Step 4: Calculate the preference level

The method preference level Lk (k = 1, 2, ..., K) is the overall preference degree

for each method Mk (k = 1, 2, ..., K) by all the decision alternatives. It is

calculated as the average of the scaled preference degrees to a method for each

alternative Ai (i = 1, 2, ..., I) and presented in percentage for ease of

comparison, given as

K. ..., 2, 1, k I; ..., 2, 1, i ; IuLI

iikk

100*]/)[(1

(7-5)

Step 5: Select the most preferred method

The method Mk (k = 1, 2, ..., K) with the highest preference level Lk (k = 1, 2,

..., K) is the most preferred (or acceptable) one for the decision problem Φ

under investigation, as it best satisfies all the decision alternatives in terms of

their method preferences as a whole. The most preferred method can be

selected by finding the highest preference level kL (k = 1, 2, ..., K) as


92

K. ..., 2, 1, k ; LMaxL kk )( (7-6)


To illustrate the alternatives-oriented method selection approach, we use the

decision matrix from the graduate fellowship applicants ranking problem presented

in Yoon and Hwang (1995). Table 6-2 in Chapter 6 shows the decision matrix. The

attribute weights for the decision problem are given as W = (0.3, 0.2, 0.2, 0.15, 0.15).

We will use the nine MADM methods suitable for solving this problem shown in

Table 6-1 in Chapter 6.

The Methods (M1, M2, ..., M9) given in Table 6-2 are applied separately to

solve the decision problem given in Table 6-1, with six alternatives (A1, A2, ..., A6) to

be ranked. Table 7-1 shows the ranking outcomes obtained by each MADM method.

Table 7-2 shows the rank matrix obtained by following Step 1 with Table 7-1.

Table 7-3 shows the method preference degree, which is calculated by Equations (7-

1) and (7-2) using the data in Table 7-2.

Table 7-4 shows the scaled preference degree, obtained by Equations (7-3)

and (7-4) using data in Table 7-3. Table 7-5 shows the overall preference level for

each MADM method (M1, M2, ..., M9), which is calculated by Equation (7-5) using

data in Table 7-4.


93

Table 7-1 Ranking outcomes obtained

MADM

method Ranking

M1 A6 > A4 > A2 > A3 > A1 > A5

M2 A6 > A5 > A1 > A4 > A3 > A2

M3 A6 > A4 > A2 > A3 > A1 > A5

M4 A6 > A4 > A2 > A3 > A1 > A5

M5 A6 > A4 > A3 > A2 > A1 > A5

M6 A6 > A2 > A4 > A3 > A5 > A1

M7 A1 > A5 > A6 > A4 > A2 > A3

M8 A6 > A4 > A2 > A3 > A5 > A1

M9 A2 > A4 > A3 > A6 > A5 > A1

Table 7-2 Resultant rank matrix

MADM method


A1 5 3 5 5 5 6 1 6 6

A2 3 6 3 3 4 2 5 3 1

A3 4 5 4 4 3 4 6 4 3

A4 2 4 2 2 2 3 4 2 2

A5 6 2 6 6 6 5 2 5 5

A6 1 1 1 1 1 1 3 1 4


94

Table 7-3 The method preference degree matrix

MADM method


A1 7/9 8/9 7/9 7/9 7/9 1/3 1 1/3 1/3

A2 7/9 1/9 7/9 7/9 1/3 8/9 2/9 7/9 1

A3 7/9 2/9 7/9 7/9 1 7/9 1/9 7/9 1

A4 1 2/9 1 1 1 1/3 2/9 1 1

A5 4/9 1 4/9 4/9 4/9 7/9 1 7/9 7/9

A6 1 1 1 1 1 1 2/9 1 1/9

Table 7-4 The scaled method preference degree matrix

MADM method


A1 0.13 0.15 0.13 0.13 0.13 0.06 0.17 0.06 0.06

A2 0.14 0.02 0.14 0.14 0.06 0.16 0.04 0.14 0.18

A3 0.13 0.04 0.13 0.13 0.16 0.13 0.02 0.13 0.16

A4 0.15 0.03 0.15 0.15 0.15 0.05 0.03 0.15 0.15

A5 0.07 0.16 0.07 0.07 0.07 0.13 0.16 0.13 0.13

A6 0.14 0.14 0.14 0.14 0.14 0.14 0.03 0.14 0.02

Table 7-5 The preference level for MADM method

MADM

method M1 M2 M3 M4 M5 M6 M7 M8 M9

Preference

level L (%) 12.48 8.94 12.48 12.48 11.76 10.84 7.51 12.15 11.38


95

The highest preference level identified from Table 7-5 by Equation (7-6) as

Lk+ = 12.48% (k = 1, 2, …, K). Methods M1, M3 and M4 produce the same ranking

outcome which has the highest preference level from all the alternatives as a whole.

Hence, the same ranking outcome produced by Methods M1, M3 and M4 is most

acceptable to all the alternatives as a whole for the given decision problem, thus

making any of Methods M1, M3 and M4 be the most preferred method. This result

shows that the most preferred method is selected if it produces the most preferred

ranking outcome for all the alternatives as a whole. As shown in this example, there

may be more than one most preferred MADM method, if these methods produce the

same most preferred ranking outcome.

7.4 Application in Decision Support Systems

Figure 7-1 shows an existing typical decision support system (DSS) for

solving multiattribute decision problems. In a DSS of this kind, active participation

of the decision maker is required at Stages 1 and 2. At Stage 1, the decision maker

constructs the decision problem with a decision matrix and a weight vector. During

Stage 2, the decision maker chooses a method from a given set of suitable methods

which is used at Stage 3 to solve the problem. This system may not produce the best

outcome, because the method selected by the decision maker may not necessarily

produce the most preferred ranking outcome by all the stakeholders. Research shows

that there is no best way to select the most suitable MADM method for a given

decision problem. The high dependency of the existing DSS on the decision maker

for the preferred method selection may induce bias, depending on the decision

maker’s knowledge, expertise, experience and preference.


96

Figure 7-1 Existing DSS for MADM problems

Figure 7-2 shows a new DSS based on the alternatives-oriented approach

developed in this paper for solving the general multiattribute decision problem. This

new system uses the alternatives-oriented method selection approach to combine

END

Ranking outcome

Obtain ranking outcome

Stage 3: Problem solving

Subjectively select a preferred method

Stage 2: Method selection

Preferred method

Suitable MADM methods

Weight vector

Identify attributes

Generate the multiattribute decision problem

Identify decision alternatives

Stage 1: Problem formulation

START

Decision matrix

Problem statement


97

Stages 2 and 3 from Figure7-1 into a single stage. The system uses each method from

the suitable set of methods to obtain ranking outcomes, which are then used to find

the most preferred outcome and method. The system provides an objective way of

method selection for producing the most preferred outcome, thus eliminating the

subjective method selection dependency on the decision maker.

Figure 7-2 Alternatives-oriented DSS for multiattribute decision problems

START

Identify decision alternatives

Problem statementStage 1: Problem formulation

Identify attributes

Stage 2: Method selection and problem solving

Select preferred outcome and method

Ranking outcomes

Obtain ranking outcomes

Preferred outcome

END

Preferred method

Generate the multiattribute decision problem Weight

vector

Decision matrix

Suitable MADM methods


98

Table 7-6 shows a comparison between an existing DSS and the new

alternatives-oriented DSS for the general multiattribute decision problem.

Table 7-6 Comparison between existing DSS and alternatives-oriented DSS

Existing DSS Alternatives-oriented DSS

Input Problem statement

Suitable methods

Problem statement

Suitable methods

Method

selection

Subjective approach

Selected by the

decision maker based

on knowledge and

experiences.

Objective approach

Selected by the system

based on ranking

outcomes.

Ranking One ranking obtained

by only the chosen

method.

Multiple rankings

obtained by each of the

suitable methods.

Output Ranking by the chosen

method.

Preferred outcome,

relative to other

outcomes.

Preferred method,

relative to other methods.


99


Method selection has become a key research issue in solving a multiattribute

decision making (MADM) problem. To address this important issue, this chapter has

presented a new alternatives-oriented approach by considering the preference of the

decision alternatives as the stakeholders of the decision problem. Departure from the

decision-maker-oriented and the method-oriented approaches in method selection

research, the alternatives-oriented approach objectively selects the best MADM

method that produces a ranking outcome preferred most by all the alternatives as a

whole. The approach is efficient in calculating the total preference level for each

MADM method by considering the preference of each alternative. The approach with

its objective measure is particularly suitable for problem settings where no decision

maker is available for method selection or the decision maker is not a key

stakeholder. A numerical example has also been presented to demonstrate the

simplicity and ease of use of the new approach. Although a study of comparing

compensatory MADM methods with cardinal rankings is exemplified, the approach

is applicable to compare any set of suitable MADM methods that produce a complete

ranking. With its simplicity in concept and computation, it can be readily

incorporated into a decision support system for solving multiattribute decision

problems that require a complete ranking of the decision alternatives.

100

Chapter 8

Developments IV:

Comparisons between TOPSIS and Modified

TOPSIS Methods

8.1 Introduction


(Hwang and Yoon, 1981) is one of the most widely used MADM methods for

solving practical MADM problems. A variant of TOPSIS named modified TOPSIS

was developed with the argument about how the attribute weight should be applied

while solving MADM problems (Deng et al., 2000). Both TOPSIS and modified

TOPSIS have been applied for problem solving by various researchers. Both

methods use the same Euclidean distance measure with the exception of when the

attribute weight is to be incorporated with the solution. It is very difficult for the

decision maker to choose between these two methods due to extreme similarity

between them in their mathematical structures and their applicability to solve same

kind of MADM problems. Thus there is a need to evaluate and compare these two

methods to justify their suitability and applications.

In this chapter, the Decision Context D outlined in Chapter 3 is addressed.

Comparison studies between the TOPSIS and the modified TOPSIS methods are

conducted to justify the appropriateness of the usage of attribute weights.

Chapter 8 Comparisons between TOPSIS and Modified TOPSIS Methods

101

8.2 TOPSIS and Modified TOPSIS

The TOPSIS and the modified TOPSIS methods are explained by considering

how these methods are applied to solve the general MADM problem Φ as given in

Equation (3-3). The MADM problem Φ consists of the decision matrix X as shown in

Equation (3-1) and the attribute weight vector W given in Equation (3-2) in Chapter

3.

8.2.1 The TOPSIS Method

It has been used extensively to solve various practical MADM problems for

comprehensive mathematical concept, easy usability and simplicity, computational

efficiency and ability to measure alternative performances in simple mathematical

form (Yeh, 2003).

In TOPSIS, an index known as similarity to positive-ideal solution is defined

by combining the closeness to positive-ideal solution and remoteness to negative-

ideal solution. This index is used to rank the competing alternatives (Hwang and

Yoon, 1981; Zeleny, 1982). The TOPSIS method has been presented in detail in

Chapter 5 Equations (5-7) and (5-9) to (5-15).

8.2.2 The Modified TOPSIS Method

Modified TOPSIS incorporates the attribute weights with performance ratings

in a different manner from the TOPSIS method. The overall performance index is

calculated using the distance from positive-ideal and negative-ideal solutions. The

distance is related with the alternative weights. The modified TOPSIS proposes the


102

use of alternative weights with the Euclidean distances (Deng et al., 2000). Modified

TOPSIS inherits all the positive aspects of TOPSIS, and also rectifies the use of non-

weighted Euclidean distance in TOPSIS. The modified TOPSIS method involves the

following steps.

Step 1: Obtain normalised decision matrix

The normalised decision matrix is calculated similar to TOPSIS in Chapter 5.

The matrix can be presented as Equation (5-7) in Chapter 5.

Step 2: Identify the positive-ideal and negative-ideal solutions

The positive-ideal solution B* and the negative-ideal solution B- can be

obtained in terms of normalised performance ratings from Equation (5-7) as

*

J*2

*1 y ..., ,y ,yB * (8-1)

J21 y ..., ,y ,yB (8-2)

Where

attributecost for ; min

attributebenifit for ; max*

ij

ijj y

yy

attributecost for ; max

attributebenifit for ; min

ij

ijj y

yy

Step 3: Obtain the weighted Euclidean distance

The weighted Euclidean distances from the positive-ideal and negative-ideal

solutions for each alternative Ai (i = 1, 2, ..., I) are calculated by applying

Equations (3-2), (5-7), (8-1) and (8-2) as


103

I. ..., 2, 1,i ;yyWDJ

jjijji

1

2** )( (8-3)

I. ..., 2, 1,i ;yyWDJ

jjijji

1

2)( (8-4)

where Wj (j = 1, 2, ..., J) is weights for attributes Cj (j = 1, 2, ..., J).

Step 4. Obtain the overall performance index

The overall performance index for each alternative Ai (i = 1, 2, ..., I) is

obtained as

I. ..., 2, 1,i;DD

DV

ii

ii

)( * (8-5)

Performance index Vi (i = 1, 2, ..., I) is used to rank the competing alternatives.

A higher index value indicates a better alternative performance.

8.3 Method Comparisons

The TOPSIS and modified TOPSIS methods are compared under two

different weight settings: (a) all the attribute weights are equal and (b) the attribute

weights are not equal.

8.3.1 Comparison with Equal Weight Settings

A problem solving simulation is done with more than 1,000 MADM

problems with equal attribute weight settings. For each problem, the TOPSIS and the


104

modified TOPSIS methods produces exactly the same ranking outcome. This result

can be justified by the following mathematical proof.

The TOPSIS Equation (5-15) is expanded using Equations (5-13) and (5-14) as

I. ..., 2, 1,i ;

vvvv

vv

VJ

jjij

J

jjij

J

jjij

i

))()((

)(

1

2

1

2*

1

2

(8-6)

Equation (8-6) can be further extended by applying Equations (5-9) to (5-12) as

I. ..., 2, 1,i ;

yWyWyWyW

yWyW

VJ

jjjijj

J

jjjijj

J

jjjijj

i

))()((

)(

1

2

1

2*

1

2

(8-7)

or

I. ..., 2, 1,i ;

yyWyyW

yyW

VJ

jjijj

J

jjijj

J

jjijj

i

))()((

)(

1

22

1

2*2

1

22

(8-8)

With the equal weight settings, applying WWj to Equation (8-8)

I. ..., 2, 1,i ;

yyWyyW

yyW

VJ

jjij

J

jjij

J

jjij

i

))()((

)(

1

22

1

2*2

1

22

(8-9)

or


105

I. ..., 2, 1,i ;

yyyy

yy

VJ

jjij

J

jjij

J

jjij

i

))()((

)(

1

2

1

2*

1

2

(8-10)

Similarly, the modified TOPSIS Equation (8-5) can be expanded by using

Equations (8-3) and (8-4) as

I. ..., 2, 1,i ;

yyWyyW

yyW

VJ

jjijj

J

jjijj

J

jjijj

i

))()((

)(

1

2

1

2*

1

2

(8-11)

With the equal weight settings, applying WWj to Equation (8-11)

I. ..., 2, 1,i ;

yyWyyW

yyW

VJ

jjij

J

jjij

J

jjij

i

))()((

)(

1

2

1

2*

1

2

(8-12)

or

I. ..., 2, 1,i ;

yyyy

yy

VJ

jjij

J

jjij

J

jjij

i

))()((

)(

1

2

1

2*

1

2

(8.13)

Comparing Equations (8-10) and (8-13) it is observed that the two methods

are exactly the same. This mathematical explanation justifies the same ranking

results obtained during the simulation study. It also highlights the extreme structural

similarities between the two methods and justifies the need for further investigation

under non-equal weights.


106

8.3.2 Comparison with Non-Equal Weight Settings

A simulation study and results will be presented before providing a

mathematical comparison of the TOPSIS and modified TOPSIS methods under non-

equal weight settings.

8.3.2.1 Simulation Results

In this simulation study, the decision matrix from the graduate fellowship

applicants ranking case presented by Yoon and Hwang (1995) is used. Table 6-2 in

Chapter 6 shows the decision matrix.

The simulation is started with equal attribute weight W = (0.2, 0.2, 0.2, 0.2,

0.2) for the five attributes. With this equal weight setting, the decision problem is

solved with both the TOPSIS and modified-TOPSIS. The ranking outcomes

obtained, are exactly the same and are used as the base outcomes.

The attribute weights are then changed gradually with a step of 0.1 producing

126 distinct weight sets between the range of (0.6, 0.1, 0.1, 0.1, 0.1) and (0.1, 0.1,

0.1, 0.1, 0.6). The increment step is decided to be 0.1 because it produces significant

result variations required for this study.

For each set of weights, the MADM problem is solved using both TOPSIS

and modified TOPSIS methods. The simulation shows that 70% of the 126 weight

sets generates distinct ranking outcomes for the two methods.


107

The simulation results and the previous sections for equal weight settings

highlight the fact that the only difference between TOPSIS and modified TOPSIS is

in how the attribute weight is incorporated during calculations. A closer inspection of

expanded TOPSIS Equation (8-8) and expanded modified TOPSIS Equation (8-11)

shows that the only difference between the two methods is that in TOPSIS Wj2 is

used but in modified TOPSIS Wj is used while calculating the distances from the

positive-ideal and the negative-ideal solution. Thus, further mathematical analysis

under non-equal weight settings is required to establish the validity of these methods.

8.3.2.2 Mathematical Analysis

The modified TOPSIS method suggests that the distance between

performance ratings should be weighted rather than the performance ratings as done

in TOPSIS. Considering this argument rational and valid, the equation is derived

from the basic Euclidean distance theory (Greenacre, 2009).

A single dimension problem with two vectors P [x1] and Q [x2] shown in

Figure 8-1.

Figure 8-1 Distance in one dimensional space

The distance between P and Q is obtained as

|PQ| = | xx dx 21| (8-14)

x2

O x1

Axis 1 P [x1] Q [x2]

|x1 – x2|


108

If the dimension has any weight associated with it, then the weighted distance

can be expressed as

| xxW D 1x 21| (8-15)

Now consider the problem with two dimensions with vectors P [x1, x2] and Q

[y1, y2] as shown in Figure 8-2.

Figure 8-2 Distance in two dimensional space

Source: Adapted from Greenacre (2009)

Using the Pythagoras’ theorem for right-angled triangle, from Figure 8-2 we

can write the distance between P and Q as

|PQ|2 = (dxy)2 = (x1 – y1)

2 + (x2 – y2)2 (8-16)

or

2222

11 y x + y x = dxy (8-17)

O

x2

y1 Axis 1

P [x1, x2]

Q [y1, y2]

|x1 – y1|

Axis 2

y2

x1

|x2 – y2|

B


109

By applying Equations (8-14) and (8-15) the two dimensional weighted

Euclidean distance can be obtained from Equation (8-17) as

22222

111 | y x|W + | y x|W = Dxy (8-18)

Similarly, the Euclidean distance and the weighted Euclidean distance can be

obtained for three dimensional problems with P [x1, x2, x3] and Q [y1, y2, y3] as shown

in Equations (8-19) and (8-20) respectively.

233

222

211 )( yx y x + y x = dxy (8-19)

2333

2222

2111 |)|(|| yxW y x|W + y x|W = Dxy (8-20)

The weighted Euclidean distance for vectors P and Q with j (j = 1, 2, …, J)

dimensions can be obtained similarly as

J

jjjjxy yxWD

1

2|)|( (8-21)

or

J

jjjjxy yxWD

1

22 )( (8-22)

The mathematical derivation of Equation (8-22) proves that while calculating

the weighted Euclidean distance, squared weight should be used. The multi-

dimension used in the derivation is analogous to MADM problem solving by

TOPSIS and modified TOPSIS where the attributes are considered as dimensions.

Comparisons between Equation (8-22) and the TOPSIS Equation (8-8) and the


110

modified TOPSIS Equation (8-11) prove that the TOPSIS method applies the weight

in a correct manner.

The concept of distance weighting introduced in the modified TOPSIS is

valid and rational. The modified TOPSIS method derives objective weight using the

entropy concept (Shannon and Weaver, 1947) based on information variation in the

MADM problem (Deng et al., 2000). This objective weight shows the relative

significance of attributes in terms of their impacts on the decision outcomes. The

objective weight should be treated differently from the attribute weights provided by

the decision maker and should never be used in the process of solving the MADM

problem. The objective weight certainly can indicate the decision maker regarding

the significance of attributes so that the decision maker can be careful while solving

the problem.

On the other hand, although the TOPSIS method uses the weighting of

normalised performance rating and does not explicitly applies the distance weighting

concept, the mathematical structure of TOPSIS is implicitly the same as that of the

weighted Euclidean distance.


This chapter has provided extensive simulation and mathematical proof based

comparisons between two widely used MADM methods: the TOPSIS method and

the modified TOPSIS method. The evaluations have shown the validity of the

arguments presented for the modified TOPSIS. It has been proved that the TOPSIS

method should be used for MADM problems where both TOPSIS and modified


111

TOPSIS could be applied, as it handles the attribute weights in an appropriate

manner. This will help the decision makers who are not sure about choosing between

these two methods.

112

Chapter 9

Developments V:

Evaluation of Consensus Techniques in

Multiattribute Group Decision Making

9.1 Introduction

Multiattribute group decision making (MAGDM) problems are similar to the

multiattribute decision making (MADM) problems with the exception that there are

multiple decision makers. With multiple decision makers, the challenges in solving

such problems are significant. In addition to the challenging issues associated with

MADM problems, the major challenge in solving MAGDM problems is to find a

compromise solution that will best satisfy all the decision makers as a whole. The

decision consensus can be achieved at different stages in problem solving (Fu and

Yang, 2007). With ranking outcomes available from each of the decision makers in a

group, the conventional consensus technique to achieve the final stage consensus is

the additive Borda score technique (Hwang and Lin, 1987; Shih et al., 2001; 2004).

The additive Borda score technique is very simple and easy to use but the additive

aggregation produces a group ranking outcome which represents the central tendency

(average) of the individual ranking outcomes and not necessarily always the most

preferred one by the group. This issue highlights the need to explore other

aggregation techniques to achieve group consensus.

Chapter 9 Evaluation of Consensus Techniques in Multiattribute Group Decision Making

113

This chapter addresses the Decision Context E outlined in Chapter 3. In the

following sections, the existing group consensus techniques are discussed before

presenting a novel group consensus technique based on the concept of the Euclidean

distance based TOPSIS method (Hwang and Yoon, 1981; Yoon and Hwang, 1995).

A new group ranking outcome similarity based approach for selecting the most

preferred consensus technique is also developed before presenting a numerical

example for better illustrating the new developments.

9.2 Group Consensus Techniques

Multiattribute group decision making (MAGDM) problems with various

group decision problem settings can be solved by different approaches, as shown in

Figure 9-1. Group consensus can be achieved during any of the three stages of

MADM problem solving including (a) the initial stage, (b) the intermediate stage and

(c) the final stage (Fu and Yang, 2007).

9.2.1 Consensus during the Initial Stage

In this approach, individual decision matrices from each of the decision

makers are aggregated using some aggregation method like average, geometric

mean, etc. This process converts the group decision problem into a single decision

maker problem. The individual preferences for attribute weights are also aggregated

to generate the group weight. The decision makers then need to agree on a particular

MADM method to solve the problem. Although this approach has been used and

improved by several researchers (Parkan and Wu, 1998; Chen, 2000; Chu, 2002a; Fu


114

Figure 9-1 The group decision process in the evaluation and selection phases

Source: Adapted from Hwang and Lin (1987).

Committee agrees on criteria weight via conference

Decision to submit the recommendation to the

boss or top manager

Evaluation

Ordinal approach Cardinal approach

Scale transformation

Normalization of set

Individual Agreed criteria Individual Agreed criteria

Borda score: find the ranking of candidates

Assignment technique: find the individual preference ordering

Assignment technique: find the collective preference ordering

Borda score: find the collective preference ordering

TOPSIS: find the collective preference ordering

Borda score: find the collective preference ordering

Simple average of rating value under each criteria

Additive weighted value approach: find the collective preference ordering

TOPSIS: find the individual preference ordering

Have common criteria for committee member

Have common criteria for committee member

Have own criteria set for each individual

Have own criteria set for each individual

Committee agrees on criteria weight via conference

Committee further discusses And/ or revises


115

and Yang, 2007), it may leave the decision makers unsatisfied as they never know

the possible ranking outcomes of their individual decision matrices. The agreement

to use a particular MADM method to solve the problem is very difficult to achieve,

as some of the decision makers may have strong logical and past experience support

to use their preferred MADM method.

9.2.2 Consensus during the Intermediate Stage

This approach starts with solving individual decision matrices of each

decision maker in the group separately and applies some aggregation technique at a

later stage to obtain group ranking outcome (Shih et al., 2007). Despite providing the

importance on individual preferences, this approach does not show possible

individual ranking outcomes and the decision makers do not know the extent to

which the group ranking outcome reflects their own preferences.

9.2.3 Consensus during the Final Stage

In this approach, the individual decision matrix for each decision maker is

solved independently using TOPSIS, and then the additive Borda score (DeBorda,

1781; DeGrazia, 1953; Black, 1958; Arrow, 1963; Fishburn, 1973) is applied to

aggregate the individual ranking outcomes into the group outcome (Hwang and Lin,

1987; Shih et al., 2001 and 2004). This approach provides the decision makers with

both individual and group ranking outcomes which can be applied to find the

satisfaction level of the decision makers in the decision outcomes. The commonly

used additive aggregation technique is not necessarily the only way of achieving


116

aggregation and there is a need for comparing its performance with other aggregation

techniques.

9.3 New Consensus Technique Based on TOPSIS

The total Borda score is usually calculated by additive aggregation which is

very simple and effective. The additive aggregation always indicates the central

tendency of the group which may not be the desired solution by the group members

for any particular problem. In order to rectify this limitation, a new consensus

technique is developed based on the concept of the popular TOPSIS method (Hwang

and Yoon, 1981; Yoon and Hwang, 1995) presented in Chapter 5.

The new consensus technique is developed using the notion from the TOPSIS

method that the ideal solution has the shortest distance from the positive ideal

solution and the longest distance from the negative ideal solution. The group ideal

ranking outcome (consensus) for a given multiattribute group decision problem can

be achieved from a set of ranking outcomes Oq (q = 1, 2, ..., Q) produced by each

individual decision maker Dq (q = 1, 2, ..., Q) in the group of decision makers. The

new consensus technique involves the following steps.

Step 1: Obtain the ranking outcome

For the given decision problem Φ, obtain the ranking outcomes Oq (q = 1, 2,

..., Q) given by each decision maker Dq (q = 1, 2, ..., Q). The decision maker

Dq (q = 1, 2, ..., Q) may apply their individually preferred method to obtain the

outcome.


117

Step 2: Create the rank matrix (Rq)

The rank matrix (Rq) similar to Equation (6-2) in Chapter 6 is obtained by

arranging the ranks given to each alternative Ai (i = 1, 2, ..., I) by decision

makers Dq (q = 1, 2, ..., Q) as shown in Equation (9-1).

Q. ..., 2, 1,q I; ..., 2, 1, i ;

rrr

rrr

rrr

R

IQII

Q

Q

q

...

............

...

...

21

22221

11211

(9-1)

Step 3: Provide the Borda score

The Borda score ziq (i = 1, 2, ..., I; q = 1, 2, ..., Q) for each alternative Ai (i = 1,

2, ..., I) with respect to decision maker Dq (q = 1, 2, ..., Q) can be obtained

using Equation (9-1) as

Q. ..., 2, 1,q I; ..., 2, 1, i ;rIz iqiq (9-2)

The resultant rank score matrix Z can be given as

Q. ..., 2, 1,q I; ..., 2, 1, i ;

zzz

zzz

zzz

Z

IQII

Q

Q

...

............

...

...

21

22221

11211

(9-3)

Step 4: Identify the positive and negative ideal rank scores

The positive ideal (Z*) and negative ideal (Z-) rank scores for each decision

maker Dq (q = 1, 2, ..., Q) can be identified from Equation (9-3) as

**2

*1

* ,...,, QzzzZ (9-4)


118

QzzzZ ,...,, 21 (9-5)

where ....,,2,1;...,,2,1;max* Q q I i zz iqq

....,,2,1;...,,2,1;min Q q I i zz iqq

Step 5: Calculate the separation measures

Separation measures for each decision alternative Ai (i = 1, 2, ..., I) are

calculated using the n-dimensional Euclidean distance. The separation

(distance) of each alternative from the positive-ideal score Z* and the negative-

ideal score Z- can be obtained using Equations (9-3) to (9-5) as

.1

2** I ....., 2, 1, i; )zz(GQ

q qiqi (9-6)

.1

2 I ....., 2, 1, i; )zz(GQ

q qiqi

(9-7)

Step 6: Obtain the overall rank score

The overall rank score for each decision alternative Ai (i = 1, 2, ..., I) is

obtained by applying Equations (9-6) and (9-7) as

I. ..., 2, 1, iGG

GF

ii

ii

;* (9-8)

Step 7: Rank the alternatives

The alternatives are ranked according to the overall rank score in descending

order. The ranking outcome obtained is the group ranking outcome for the

given problem.


119

9.4 Consensus Technique Evaluation

With the availability of the traditional additive Borda score consensus

technique and the new TOPSIS based consensus technique, comparative evaluations

must be done to find out which one most satisfies all the decision makers together.

To calculate the group satisfaction, a new performance measure called the group

similarity index (GSI) is introduced. The GSI is based on ranking outcome

similarities between the group outcome and outcomes of each individual decision

maker Dq (q = 1, 2, ..., Q).

The group outcome obtained by the additive Borda score can be defines as Ob

and the group outcome obtained by the new TOPSIS aggregation technique can be

defined as Ot. The ranking outcomes obtained by the decision makers Dq (q = 1, 2,

..., Q) can be denoted as Oq (q = 1, 2, ..., Q). The consensus technique selection can

be achieved in the following steps.

Step 1: Calculate rank correlation for each group outcome

The rank correlations between each of the two group outcomes and each

outcome Oq (q = 1, 2, ..., Q) produced by each decision makers Dq (q = 1, 2,

..., Q) can be obtained by applying Equations (6-1) and (6-3) from Chapter 6 as

Q. ..., 2, 1, q ; OOORC qbqb ),()( (9-9)

Q. ..., 2, 1, q ; OOORC qtqt ),()( (9-10)


120

Step 2: Calculate the group similarity index

The group similarity index (GSI) for each of the two group outcomes is

obtained from Equations (9-9) and (9-10) by taking the average rank

correlation as

.QORCOGSIQ

qqbb /))(()(

1

(9-11)

.QORCOGSIQ

qqtt /))(()(

1

(9-12)

Step 3: Select the consensus technique

The most preferred group consensus technique for the given problem is

selected based on the value of GSI calculated in the previous step. The group

consensus technique with a higher GSI should be selected for the given

multiattribute group decision problem and its corresponding ranking outcome

will be the group outcome.


In this example, a group decision problem with eight alternatives (A1, A2, ...,

A8) is considered. The number of decision makers in the group is six (D1, D2, ..., D6).

Individual ranking outcomes from each decision maker is obtained by applying

Equation (9-1). The rank matrix Rq is generated as shown in Table 9-1. Table 9-2

shows the rank score matrix Z generated by applying Equations (9-2) and (9-3) on

Table 9-1.


121

Table 9-1 Rank matrix generated by combining individual ranking outcomes

Decision maker

Alternative D1 D2 D3 D4 D5 D6

A1 1 6 8 4 5 2

A2 2 5 6 3 7 4

A3 7 4 1 7 4 3

A4 5 8 7 5 6 1

A5 6 2 3 1 1 5

A6 8 7 4 6 3 7

A7 3 3 5 2 2 8

A8 4 1 2 8 8 6

By applying Equations (9-4) to (9-8) on Table 9-2, the overall rank score Fi (i

= 1, 2, ..., I) is calculated for each alternative Ai (i = 1, 2, ..., I). The alternatives Ai (i

= 1, 2, ..., I) are then ranked based on the overall rank score Fi (i = 1, 2, ..., I) to

obtain the group ranking outcome. Table 9-3 shows the rank score and group ranking

obtained by the new TOPSIS based consensus technique and the conventional

additive Borda score technique.

In order to select the most preferred consensus technique for this MAGDM

problem, we calculate the GSI for outcomes produced by the additive Borda score

technique and the TOPSIS based technique using Equations (9-9) to (9-12) as shown

in Table 9-4. The result shows that the new TOPSIS based consensus technique is

more appropriate for the given MAGDM problem.


122

Table 9-2 The rank score matrix

Decision maker

Alternative D1 D2 D3 D4 D5 D6

A1 7 2 0 4 3 6

A2 6 3 2 5 1 4

A3 1 4 7 1 4 5

A4 3 0 1 3 2 7

A5 2 6 5 7 7 3

A6 0 1 4 2 5 1

A7 5 5 3 6 6 0

A8 4 7 6 0 0 2

Table 9-3 The overall rank score and group ranking outcomes

TOPSIS consensus technique Additive Borda score technique

Alternative Rank score Group rank Rank score Group rank

A1 0.51637 4 22 3

A2 0.5 5 21 5

A3 0.51735 3 22 3

A4 0.41591 7 16 7

A5 0.65913 1 30 1

A6 0.3522 8 13 8

A7 0.56927 2 25 2

A8 0.47049 6 19 6


123

Table 9-4 Rank similarity index for group outcomes

Rank Correlation (RCq)

Consensus Technique O1 O2 O3 O4 O5 O6 GSI

Borda (Ob) 0.268 0.547 0.128 0.617 0.547 -0.058 0.3416

TOPSIS (Ot) 0.190 0.595 0.214 0.619 0.571 -0.119 0.3452

9.6 A simulation and Ties in Ranking Outcomes

Ranking outcomes with a tie between two or more alternatives is a common

phenomenon. This is sometimes a difficult issue for practical problem solving where

a limited number of alternatives to be selected based on the ranking outcome. A

simulation based experiment is conducted for both the additive Borda consensus

technique and the TOPSIS based consensus technique to identify which one is better

in handling the tied rank problem while producing the group ranking outcome.

The simulation is conducted by varying the rank matrix given in Table 9-1

and is then solved using both the additive Borda score and the TOPSIS based

techniques. The number of times each technique produces a tied ranking outcome is

then noted to find the ratio of tied rank. The variation in the rank matrix is achieved

by varying the importance of each decision maker considering that initially they have

equal importance.

From the simulation result, it is evident that the additive Borda consensus

technique produces around 20% more tied ranking outcomes than the new TOPSIS

based consensus technique.


124


The TOPSIS based consensus technique presented in this chapter provides a

much needed alternative consensus technique. The new technique provides the

decision makers with the opportunity to justify their consensus technique selection.

The rank similarity based consensus technique selection approach provides an

objective way to maximise the overall group satisfaction. Simulation based

experiment results highlight the superiority of the new TOPSIS based consensus

technique in producing a non-tied group ranking outcome, which is a significant

issue in various practical decision problem settings.

125

Chapter 10

Developments VI:

Comparison Based Group Ranking Outcome for

Multiattribute Group Decisions

10.1 Introduction

The objective of solving a multiattribute group decision making (MAGDM)

problem is to obtain a group decision outcome that best satisfies all the decision

makers as a whole. In order to achieve the group decision outcome, the decision

makers use various compensatory techniques to reach the group compromise

outcome (Hwang and Lin, 1987). The group compromise can be achieved at different

stages of solving an MAGDM problem (Fu and Yang, 2007). The group ranking

outcome is usually calculated by using the decision matrices provided by each

decision maker in the group (Parkan and Wu, 1998; Chen, 2000; Chu 2002; Fu and

Yang, 2007; Shih et al., 2007) or by aggregating the individual ranking outcomes

given by each of the decision makers (Hwang and Lin, 1987) (as discussed in

Chapter 9). The existing group decision making methods use a set of ranking

outcomes to achieve the group ranking outcome, limited by the number of decision

makers or by the method used. This limitation in solution space may lead to a

situation where the ranking outcome, most preferred by the group as a whole, may

not be found at all. This practical and significant issue highlights the need to develop

a method capable of finding the most preferred group ranking outcome by

Chapter 10 Comparison Based Group Ranking Outcome for Multiattribute Group Decisions

126

considering the whole solution space consisting of all the possible valid ranking

outcomes for the given MAGDM problem.

In this chapter, a new group decision method is developed. The new method

is based on the ranking outcome similarity and is capable of finding the most

preferred group ranking outcome from all the possible ranking outcomes for a given

MAGDM problem. The new method developed in this chapter addresses the

Decision Context F outlined in Chapter 3.

10.2 Methodology Development

10.2.1 Finding the Most Preferred Group Ranking Outcome

The new group decision method is based on the notion that the most preferred

outcome for an MAGDM problem, if exists, must be found if we search the whole

solution space comprising of all the possible ranking outcomes. To this end, a search

technique based on the ranking outcome similarity is developed. With alternatives Ai

(i = 1, 2, ..., I), the number of possible ranking outcomes is I!. As such, the solution

space containing all the possible ranking outcomes can be defined as β = Os (s = 1,

2, ..., S; S = I!), in which the best outcome can be found.

In a group decision setting, there are multiple decision makers Dq (q = 1, 2,

..., Q) with individual ranking outcomes Oq (q = 1, 2, ..., Q). The individual ranking

outcomes can be obtained by using an MADM method or based on the decision

maker’s own preference. The most preferred outcome for the group will be the one


127

closest to all the individual ranking outcomes. The closeness in ranking outcome is

calculated based on the outcome similarity index developed in the following section.

10.2.2 The Outcome Similarity Index

The outcome similarity index (OSI) is based on the Spearman’s rank

correlation coefficient (Spearman, 1904) as shown in Equations (6-1) and (6-3) in

Chapter 6. The OSI is the measure of the similarity of a ranking outcome Os (s = 1,

2, ..., S; S = I!) in the solution space β to each individual ranking outcomes Oq (q =

1, 2, ..., Q) given by the decision makers Dq (q = 1, 2, ..., Q). A higher value of OSI

indicates a better overall similarity. The OSI can be obtained using the following

steps.

Step 1: Obtain the individual ranking outcomes

Individual ranking outcomes Oq (q = 1, 2, ..., Q) is obtained from each of the

decision makers Dq (q = 1, 2, ..., Q). Each decision maker Dq (q = 1, 2, ..., Q) is

free to apply a preferred MADM method to obtain the ranking outcome.

Step2: Generate the solution space

The solution space β is generated by obtaining all the possible ranking

outcomes Os (s = 1, 2, ..., S; S = I!) for the set of alternatives Ai (i = 1, 2, ..., I)

to be evaluated and ranked.

Step3: Calculate rank correlations

The rank correlations between each of the ranking outcome Os (s = 1, 2, ..., S;

S = I!) in the solution space β and each of the individual ranking outcomes Oq


128

(q = 1, 2, ..., Q) given by the decision makers Dq (q = 1, 2, ..., Q) are calculated

by applying Equations (6-1) and (6-3) from Chapter 6 as

Q. ..., 2, 1, q;I! S S;..., 2, 1, s; OORC qssq ),( (10-1)

Step4: Calculate the outcome similarity index

The outcome similarity index (OSIs) for each ranking outcome Os (s = 1, 2, ...,

S; S = I!) in the solution space β is calculated by taking the average of the RCsq

(q = 1, 2, ..., Q) calculated in Equation (10-1) as

I! S S;..., 2, 1, s; QRCOSIQ

qsqs

/)(1

(10-2)

Step5: Find the highest outcome similarity index

The highest outcome similarity index sOSI (s = 1, 2, ..., S; S = I!) can be

obtained as

I! S S;..., 2, 1,s;OSIOSI ss max (10-3)

The ranking outcome corresponding to the

sOSI is the closest to all the

individual ranking outcomes Oq (q = 1, 2, ..., Q) given by the decision makers

Dq (q = 1, 2, ..., Q) and the most preferred one by all the decision makers as a

whole.


To illustrate the new method, consider a multiattribute group decision making

(MAGDM) problem where four alternatives (A1, A2, A3 and A4) are to be ranked by a


129

group of three decision makers (D1, D2 and D3). Table 10-1 shows the individual

ranking outcomes given by each decision maker by using Step 1. The solution space

β is obtained by Step 2 and is shown in Table 10-2. With four alternatives to be

ranked, the solution space will contain 4! = 24 possible ranking outcomes.

Table 10-1 Individual ranking outcomes for each decision maker

Alternatives

Decision Maker A1 A2 A3 A4

D1 1 2 3 4

D2 1 4 2 3

D3 3 2 1 4

Note that, Table 10-2 contains the ranking outcomes given by the three decision

makers in Table 10-1 as

(*) the ranking outcome given by decision maker D1;

(**) the ranking outcome given by decision maker D2;

(***) the ranking outcome given by decision maker D3.

Table 10-3 shows the OSIs (s = 1, 2, ..., 24) for each outcome Os (s = 1, 2, ...,

24) in the solution space β obtained by applying Equations (10-1) and (10-2) on

Tables 10-1 and 10-2. Using Equation (10.3) and Table 10-3, the highest outcome

similarity index can be observed as OSIs+ = 0.667, which corresponds to the outcome

O3 (A1>A3>A2>A4). Hence, the ranking A1>A3>A2>A4 is the most preferred outcome

for the given MAGDM problem.


130

Table10-2 Solution space with all the possible ranking outcomes

Alternatives

Ranking outcomes A1 A2 A3 A4

O1* 1 2 3 4

O2 1 2 4 3

O3 1 3 2 4

O4 1 3 4 2

O5** 1 4 2 3

O6 1 4 3 2

O7 2 1 3 4

O8 2 1 4 3

O9 2 3 1 4

O10 2 3 4 1

O11 2 4 1 3

O12 2 4 3 1

O13 3 1 2 4

O14 3 1 4 2

O15*** 3 2 1 4

O16 3 2 4 1

O17 3 4 1 2

O18 3 4 2 1

O19 4 1 2 3

O20 4 1 3 2

O21 4 2 1 3

O22 4 2 3 1

O23 4 3 1 2

O24 4 3 2 1


131

Table10-3 OSI for each possible outcome

Ranking

outcomes

OSI

O1 A1>A2>A3>A4 0.533

O2 A1>A2>A4>A3 0.2

O3 A1>A3>A2>A4 0.667

O4 A1>A3>A4>A2 0

O5 A1>A4>A2>A3 0.467

O6 A1>A4>A3>A2 0.133

O7 A2>A1>A3>A4 0.333

O8 A2>A1>A4>A3 0

O9 A2>A3>A1>A4 0.6

O10 A2>A3>A4>A1 -0.4

O11 A2>A4>A1>A3 0.4

O12 A2>A4>A3>A1 -0.267

O13 A3>A1>A2>A4 0.267

O14 A3>A1>A4>A2 -0.4

O15 A3>A2>A1>A4 0.4

O16 A3>A2>A4>A1 -0.6

O17 A3>A4>A1>A2 0

O18 A3>A4>A2>A1 -0.333

O19 A4>A1>A2>A3 -0.133

O20 A4>A1>A3>A2 -0.467

O21 A4>A2>A1>A3 0

O22 A4>A2>A3>A1 -0.667

O23 A4>A3>A1>A2 -0.2

O24 A4>A3>A2>A1 -0.53


132


The novel group decision method developed in this chapter for finding the

most preferred ranking outcome removes the solution space limitations in

conventional group decision making methods. The new method uses the whole

solution space rather than a partial solution space to find the most preferred outcome

which best satisfies all the decision makers as a whole. The new outcome similarity

index measures the overall ranking similarity of each of all possible group ranking

outcomes to all the individual ranking outcomes made by each of the decision

makers.

The new method with the outcome similarity index provides a simple, yet

efficient way to find the group ranking outcome. It shows a new approach to

achieving group consensus by considering all individual ranking outcomes produced

by all decision makers.

133

Chapter 11

Conclusions

11.1 Research Developments Summary

The new method evaluation approaches presented in this study are motivated

from the need for objectively comparing and selecting multiattribute decision making

(MADM) methods for a given problem under certain decision settings and decision

contexts. A major research issue has been addressed where the selection of the most

preferred MADM method is to be done from a set of suitable and acceptable MADM

method for a given decision problem. Six new developments for method evaluation

and selection have been achieved to help the decision maker(s) select the most

preferred method for a given problem. A key characteristic of these developments is

that all of them use the ranking outcomes produced by the MADM methods for the

purpose of evaluations and comparisons. The six new developments address method

evaluations in three areas of MADM research including (a) generalised method

selection, (b) single decision maker problems, and (c) group decision problems. The

developments are summarised below:

11.1.1 Developments I: A Simulation Model and Applications

The new simulation model developed and its application shown in Chapters 4

and 5 have addressed the Decision Context A where the decision maker requires a

method selection guideline. Developments I has the following advantages:

Chapter 11 Conclusions

134

(a) The model provides general guidelines for method selection under different

decision settings, including the number of attributes and the number of

alternatives that the decision problem contains and the diversity in the

decision information.

(b) The model is capable of identifying the level of sensitivity that a particular

method shows towards changes in attribute weights.

(c) The simple model can be implemented easily to develop computer based

systems to compare any number of MADM methods that can produce a

complete ranking of the decision alternatives.

(d) The application of the model justifies the use of particular normalisation

procedures with the SAW and TOPSIS methods.

11.1.2 Developments II: Rank Similarity Based Approach

In Chapter 6, a new rank similarity based method evaluation and selection

approach has been developed which has addressed the Decision Context B where the

most preferred method is to be selected from a set of suitable methods for a given

decision problem. Developments II has the following advantages:

(a) The approach can perform a problem specific comparison of MADM

methods.

(b) The approach is capable of selecting the most preferred method from a set

of suitable methods for a given decision problem.

(c) The approach applies a simple and rational objective measure to justify and

validate the evaluation and comparison of MADM methods.


135

(d) The objective measure uses the concept of outcome closeness and provides

clarity in the evaluation process.

(e) The approach is particularly applicable for evaluating MADM methods

used to solve single decision maker problems.

11.1.3 Developments III: Alternatives-Oriented Approach

The alternatives-oriented approach developed in Chapter 7 has addressed the

Decision Context C where the decision alternatives are key stakeholders. In many

practical problems often the alternatives are key stakeholders and are the ones

affected most by the decision outcome. The advantages of Developments III include

the following:

(a) The approach provides a whole new perspective to method selection.

(b) The approach provides due considerations to the decision alternatives in the

process of method selection when they are key stakeholders.

(c) The alternatives-oriented approach uses a new objective measure to

compare MADM methods by considering the preferences of the decision

alternatives.

11.1.4 Developments IV: TOPSIS and Modified TOPSIS Comparison

The comparative studies presented in Chapter 8 have addressed the

challenges for Decision Context D where the decision maker needs to select between


136

the TOPSIS and the modified TOPSIS methods. Developments IV has the following

advantages:

(a) The comparisons provide enough experimental results for the decision

maker to decide the appropriate way of using these methods.

(b) Mathematical proofs provide justification and validate the experimental

results.

11.1.5 Developments V: Group Consensus Technique

The new group consensus technique and the consensus technique comparison

approach developed in Chapter 9 have addressed the Decision Context E where the

consensus among the decision makers need to be achieved based on individual

ranking outcomes. Developments V has the following advantages:

(a) The new TOPSIS based group consensus technique may be a rational

alternative to the conventional Borda score based technique.

(b) The new technique is able to identify differences between the performances

of the decision alternatives in a finer detail, thus reducing ties in ranking

outcomes.

(c) The new consensus technique comparison compares the different consensus

techniques to find the most preferred one for a given decision problem.


137

11.1.6 Developments VI: Comparison Based Group Decision Method

The new multiattribute group decision making (MAGDM) method developed

in Chapter 10 has addressed the Decision Context F where the group outcome needs

to be found from the set of all possible solutions. Developments VI has the following

advantages:

(a) The new method provides a new perspective for solving group decision

problems.

(b) It eliminates the solution space limitations in currently used methods.

(c) The method uses the individual outcomes to find the group outcome which

provides clarity in the process, thus allowing the decision makers to validate

the outcome.

(d) The new objective measure developed is capable of measuring the level of

group satisfaction in terms of relative closeness to the group solution.

11.2 Application of the Developments

Figure 11-1 shows how the new research developments discussed in the

previous section may be used in a computer based decision support system for

method selection and problem solving. The system requires (a) a given decision

problem to be solved, (b) a set of suitable MADM method under consideration, and

(c) any specific requirements or preferences related to method selection.


138

Figure 11-1 A computer based decision support system for method selection

Suitable methods

Apply selected evaluation approach

Decision outcome

Preferred method

Decision maker(s)

Decision problem

Selection preferences

Identify decision contexts

Decision context

Developments I Developments II

Developments III Developments

Developments V Developments

Select context specific approach

Selected approach

Research developments


139

An automated module uses the decision problem settings and the preferences

of the decision maker to identify the context of the decision problem. The context

information is then used to select the context specific evaluation approach from the

set of approaches developed in this study.

The selected method evaluation approach is then applied to evaluate the set of

given methods for the given problem. This process will identify the most suitable

method for the given problem under the chosen decision context along with the

ranking outcome for the given problem.

11.3 Research Contributions

The study has significant contributions to the theoretical and practical areas

of multiattribute decision making and method evaluation. These contributions

include the following:

(a) The study introduces the idea of decision context specific method selection.

The decision context includes various decision settings and evaluation

preferences of the decision maker. The study proposes that the decision

context for method evaluation and selection needs to be identified and then

a context specific evaluation approach is to be applied.

(b) The new developments use the ranking outcomes for the purpose of

evaluation and comparisons. Often the decision makers are more concerned

about the decision outcome. The outcome based approaches are more

rationally aligned to the decision makers’ interests. The study thus provides


140

a more acceptable way of method evaluation based on the decision

outcomes.

(c) The new simulation model not only provides a set of general method

selection guidelines but also provides a general framework which can be

easily adapted to develop new method selection experiments with any set of

MADM methods. The model is efficient and can be useful in future

simulation based experiments and the development of method specific

selection guidelines.

(d) The simulation based comparison of normalisation procedures provides

useful results regarding their suitability with SAW and TOPSIS under

various decision settings. These results can be used as guidelines for their

future applications.

(e) The simulation experiments highlight a significant change necessary in the

existing method evaluation processes. Existing method evaluations compare

a method with others based on a performance measure without any further

study on internal processes of a method (such as normalisation, aggregation,

group consensus). The simulation experiments have shown that the internal

processes of an MADM method have significant impacts on the decision

outcome and for a particular method there may be multiple suitable internal

processes available. The study suggests that a method with different internal

processes should be treated as a new method and they should be evaluated

for their suitability for a given problem, instead of just evaluating the

originally proposed method.


141

(f) The rank similarity based method selection is a new way of method

evaluation based on the decision outcomes for a given problem. This

approach is simple and efficient, thus paving the way for further

developments based on the outcome similarity.

(g) A whole new perspective to method selection is discovered through the

development of the alternatives-oriented approach. Previously ignored

importance of decision alternatives as a stakeholder is duly addressed in this

new approach. This gives the method evaluation research a new dimension

“alternatives oriented” alongside the existing “decision maker oriented” and

“method oriented” studies.

(h) The new group consensus technique developed enhances the group decision

making research by providing a rational alternative to the widely used Borda

score based technique. The new comparison approach will help decision

makers choose the most appropriate consensus technique through an

objective comparison for a given problem.

(i) The new comparison based group decision method is a new addition to the

existing group decision methods in group decision analysis. The new

method provides a unique way to search for the most acceptable (to all the

decision makers as a whole) outcome from the set of all possible outcomes.

The new method handles the group consensus issue implicitly while

obtaining the group outcome.


142

11.4 Future Research

The evaluation, comparison and selection of MADM methods for a specific

decision problem are still major challenges in MADM research as well as for the

decision makers. Significant studies need to be done in this area to help the decision

makers choose the most preferred method under certain decision settings and

contexts. This study is a small stride towards that direction. The following areas

could be further explored based on the research developments in this study:

(a) The decision context specific approaches need to be extended in evaluating

fuzzy MADM methods. Many practical MADM problems are fuzzy in

nature, and to solve them fuzzy MADM methods are widely used. The

developments of comparison approaches for evaluating fuzzy MADM

methods will certainly help decision makers make rational method selection

under a fuzzy decision environment.

(b) Extensive comparative studies are needed in the area of group decision

making. The method comparison approaches and the simulation model may

be extended to address the needs in this area.

(c) Extensions of the developments are needed to address the issue of

evaluating and selecting MADM methods which do not produce a complete

ranking of the decision alternatives.

(d) Simulation is used to experiment with diverse problem scenarios for

demonstrating the general application of the new evaluation models


143

developed in this study. The research scope of this study and time

limitations have prevented the use of an empirical study for the evaluation

models. The applications of the new evaluation models to real empirical

studies are part of my future research.

144

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159

Appendix A

Notation

Ai Alternative i (i = 1, 2, ..., I).

A* Set of positive ideal solutions for weighted normalised performance ratings.

A- Set of negative ideal solutions for weighted normalised performance ratings.

bki The number of methods producing better ranking than Method Mk (k = 1, 2,

..., K) for alternative Ai (i = 1, 2, ..., I).

B* Set of positive ideal solutions for normalised performance ratings.

B- Set of negative ideal solutions for normalised performance ratings.

Cj Attribute or criteria j (j = 1, 2, ..., J).

CWn Consistency weight n.

Dq Decision maker q (q = 1, 2, ..., Q).

Dx Weighted Euclidean distance in one dimensional space.

Dxy Weighted Euclidean distance in two dimensional space.

di Difference between ranks for alternative i (i = 1, 2, ..., I).

dx Euclidean distance in one dimensional space.

dxy Euclidean distance in two dimensional space.

Di* Separation measure for alternative Ai (i = 1, 2, ..., I) from the positive ideal

solutions for normalised performance rating.

Di- Separation measure for alternative Ai (i = 1, 2, ..., I) from the negative ideal

solutions for normalised performance rating.

e Number of normalisation procedures.

Appendix A: Notation

160

Fi Overall rank score i (i = 1, 2, ..., I).

Gi* Separation measure i (i = 1, 2, ..., I) from positive-ideal rank score.

Gi- Separation measure i (i = 1, 2, ..., I) from negative-ideal rank score.

GSI Group similarity index.

h Number of MADM methods and h ≠ k (k = 1, 2, ..., K).

i Number of alternatives.

j Number of attributes.

k Number of MADM methods.

l Number of decision problems.

Lk Method preference level k (k = 1, 2, ..., K).

kL Highest method preference level Lk (k = 1, 2, ..., K).

Mk MADM method k (k = 1, 2, ..., K).

n Number of other methods that produce the same rank as Method Mk (k = 1,

2, ..., K).

Ne Normalisation procedure e (e = 1, 2, ..., E).

Ob Group outcome using Borda score.

Ok Ranking outcome k produced by Method Mk (k = 1, 2, ..., K).

Oq Ranking outcome q produced by decision maker Dq (q = 1, 2, ..., Q).

Os Ranking outcome s (s = 1, 2, ..., S). in the set of all possible solution space.

Ot Group outcome using TOPSIS based technique.

OSIs Outcome similarity index s (s = 1, 2, ..., S; S = I!).

OSIs+ Highest outcome similarity index s (s = 1, 2, ..., S; S = I!).

P Preference degree matrix.


161

pik Preference degree for Method Mk (k = 1, 2, ..., K) by alternative Ai (i = 1, 2,

..., I).

q Number of decision maker.

Rk Rank matrix for Method Mk (k = 1, 2, ..., K).

Rq Rank matrix for decision maker Dq (q = 1, 2, ..., Q).

rik Rank given to alternative Ai (i = 1, 2, ..., I) by Method Mk (k = 1, 2, ..., K).

riq Rank given to alternative Ai (i = 1, 2, ..., I) by decision maker Dq (q = 1, 2,

..., Q).

RCkh Rank correlation between ranking outcomes produced by method Mk (k = 1,

2, ..., K) and Mh where k ≠ h.

RCsq Rank correlation between ranking outcomes Os (s = 1, 2, ..., S; S = I!) and

Oq (q = 1, 2, ..., Q).

RC(Ob)q Rank correlation with ranking outcomes Ob for outcome Oq (q = 1, 2, ..., Q).

RC(Ot)q Rank correlation with ranking outcomes Ot for outcome Oq (q = 1, 2, ..., Q).

RCIk Ranking consistency index k (k = 1, 2, ..., K).

RSIk Rank similarity index k (k = 1, 2, ..., K).

RSI+ Largest rank similarity index.

s Number of ranking outcomes in the solution space β.

Si* Separation measure for alternative Ai (i = 1, 2, ..., I) from the positive ideal

solutions for weighted normalised performance rating.

Si- Separation measure for alternative Ai (i = 1, 2, ..., I) from the negative ideal

solutions for weighted normalised performance rating.

T Total number of decision problems used in the simulation run.


162

Tkn Number of times Method Mk (k = 1, 2, ..., K) produces the same ranking

outcome with n (n = 1, 2, ..., K-1) number of other methods.

U Scaled preference matrix.

uik Scaled preference degree for Method Mk (k = 1, 2, ..., K) by alternative Ai (i

= 1, 2, ..., I).

V Decision matrix consisting of weighted normalised performance ratings.

Vi Overall preference score for alternative Ai (i = 1, 2, ..., I).

vij Weighted normalised performance rating for alternative Ai (i = 1, 2, ..., I)

with respect to attribute Cj (j = 1, 2, ..., J).

vj* Positive ideal weighted normalised performance rating for attribute Cj (j =

1, 2, ..., J).

vj- Negative ideal weighted normalised performance rating for attribute Cj (j =

1, 2, ..., J).

W Weight vector consisting of attribute weights for a given problem.

Wj Weight for attribute Cj (j = 1, 2, ..., J).

Wlj The weight for attribute Cj (j = 1, 2, ..., J) for the decision problem Φl (l =

1, 2, ..., L).

Wklj The weight required for attribute Cj (j = 1, 2, ..., J) to get a ranking outcome

different from the base outcome for Method Mk (k = 1, 2, ..., K) for the

decision problem Φl (l = 1, 2, ..., L).

WSIk Weight sensitivity index k (k = 1, 2, ..., K).

kW Average change in weight for Method Mk (k = 1, 2, ..., K) for all the

attributes Cj (j = 1, 2, ..., J) of all the decision problem Φl (l = 1, 2, ..., L) in

the decision problem set Ω.


163

kljW Change in weight for attribute Cj (j = 1, 2, ..., J) of decision problem Φl (l =

1, 2, ..., L) for Method Mk (k = 1, 2, ..., K).

X Decision matrix consisting of performance ratings.

Xl Decision Matrix l (l = 1, 2, ..., L).

xij Performance rating for alternative Ai (i = 1, 2, ..., I) with respect to attribute

Cj (j = 1, 2, ..., J).

Y Decision matrix consisting of normalised performance ratings.

yij Normalised performance rating for alternative Ai (i = 1, 2, ..., I) with respect

to attribute Cj (j = 1, 2, ..., J).

yj* Positive ideal normalised performance rating for attribute Cj (j = 1, 2, ..., J).

yj- Negative ideal normalised performance rating for attribute Cj (j = 1, 2, ...,

J).

Z Rank score matrix.

Z* Set of positive-ideal rank score.

Z- Set of negative-ideal rank score.

ziq Borda score for alternative Ai (i = 1, 2, ..., I) and decision maker Dq (q = 1,

2, ..., Q).

zq* Positive-ideal rank score for decision maker Dq (q = 1, 2, ..., Q).

zq- Negative-ideal rank score for decision maker Dq (q = 1, 2, ..., Q).

Φ A multiattribute decision problem.

Φl Multiattribute decision problem l (l = 1, 2, ..., L).

Ω Set of given decision problem.

ρ Rank correlation coefficient.

β Set of all the possible ranking outcomes.

164

Appendix B

Glossary of Terms

Aggregation Technique A process to combine performance ratings

with attributes weights to get an overall

preference value.

Alternative Possible course of action.

Alternatives-Oriented Approach A method selection approach from the

perspective of the alternatives.

Attribute Characteristics or objectives to be

considered during evaluation of

alternatives.

Attribute Weight Relative importance of attributes in the

decision making process.

Consensus Technique A process to achieve unified opinion among

a group of decision makers.

Decision Alternatives Alternatives in the context of a decision

problem.

Decision Analysis A subject area devoted to decision making

issues.

Decision Contexts Specific requirements for the decision

problem and method evaluation.

Appendix B Glossary of Terms

165

Decision-Maker-Oriented Approach A method selection approach from the

perspective of the decision maker.

Decision Matrix Performance rating for each alternative with

respect to each attribute combined in a

matrix form.

Decision Settings Characteristics of a decision problem in

terms of size, data and other information.

Decision Support System A computerised system to assist the

decision maker in making rational

decisions.

Method Evaluation Criteria Specific requirements for evaluation and

comparison of MADM methods.

Group Consensus Agreement within a group of decision

makers.

Group Decision Problem Decision problem with more than one

decision maker.

Group Outcome Outcome of a group decision problem.

Method Comparison A process to compare MADM methods.

Method Evaluation A process to evaluate MADM methods

under certain performance measure.

Method-Oriented Approach A method selection approach from the

perspective of MADM methods.


166

Method Preference Level Amount to which a method is preferred

than others under certain specific

requirements.

Method Selection A process to select a method from a group

of available methods for a given decision

problem.

Multiattribute Decision Making A decision making process where multiple

alternatives are assessed based on multiple

criteria under given settings.

Negative Ideal Solution Worst possible performance rating for an

attribute over all the alternatives.

Normalisation Procedure A process to convert performance ratings

with different measurement units into a

comparable one.

Overall Preference Value A value that represents the overall

performance of an alternative with respect

to all the attributes.

Performance Ratings Performance of an alternative against an

attribute.

Positive Ideal Solution Best possible performance rating for an

attribute over all the alternatives.

Ranking Consistency A measure to indicate the level of

consistency a method shows under different

decision settings.


167

Rank Correlation Coefficient A measure to find similarity between ranks.

Rank Reversal A phenomenon where rank of two

alternatives swap irrationally with a change

in decision settings.

Solution Space Set of all possible decision outcomes.

Weight Sensitivity A measure to indicate how sensitive a

particular method is to a change in attribute

weights.

Weight Vector Set of attribute weights for an MADM

problem.

168

Appendix C

Simulation Results

Detailed results of the simulation experiments presented in Chapter 5 are given

below.

C.1 Results for SAW

C.1.1 Results for Change in Alternative Numbers

With a particular number of attributes (2, 4, ..., 20), the number of alternative

is increased from 4 to 20 in steps of 2. The effects on the ranking consistency index

(RCI) for each of the four methods can be observed in Figures (C-1) to (C-9).

Figure C-1 With 4 attributes, the effects on the ranking consistency for changes in


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Appendix C: Simulation Results

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C.1.2 Results for Change in Attribute Numbers

With a particular number of alternatives (2, 4, ..., 20), the number of

attributes is increased from 4 to 20 in steps of 2. The effects on the ranking

consistency index (RCI) for each of the four methods can be observed in Figures (C-

10) to (C-18).

Figure C-10 With 4 alternatives, the effects on the ranking consistency for changes in


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Figure C-13 With 10 alternatives, the effects on the ranking consistency for changes

in the number of attributes



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C.1.3 Results for Change in Data Range

With a particular number of alternatives and attributes (2, 4, ..., 20), the data

range is narrowed from 100% to 20% in steps of 10%. The effects on the ranking


19) to (C-24).

Figure C-19 With 4 attributes and 4 alternatives, the effects on the ranking


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C.2 Results for TOPSIS

C.2.1 Results for Change in Alternative Numbers

With a particular number of attributes (2, 4, ..., 20), the number of alternative

is increased from 4 to 20 in steps of 2. The effects on the ranking consistency index

(RCI) for each of the four methods can be observed in Figures (C-25) to (C-33).

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C.2.2 Results for Change in Attribute Numbers

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attributes is increased from 4 to 20 in steps of 2. The effects on the ranking


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C.2.3 Results for Change in Data Range

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0.03

0.035

4 6 8 10 12 14 16 18 20

Ran

kin

g C

onsi

sten

cy I

nd

ex


M1 (T)

M2 (T)

M3 (T)

M4 (T)

M1 (T)

M2 (T)

M3 (T)

M4 (T)


192





0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

100 90 80 70 60 50 40 30 20

Ran

kin

g C

onsi

sten

cy I

nd

ex

Data Range (%)

M1 (T)

M2 (T)

M3 (T)

M4 (T)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

100 90 80 70 60 50 40 30 20

Ran

kin

g C

onsi

sten

cy I

nd

ex

Data Range (%)

M1 (T)

M2 (T)

M3 (T)

M4 (T)


193





0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

100 90 80 70 60 50 40 30 20

Ran

kin

g C

onsi

sten

cy I

nd

ex

Data Range (%)

M1 (T)

M2 (T)

M3 (T)

M4 (T)

0

0.1

0.2

0.3

0.4

0.5

0.6

100 90 80 70 60 50 40 30 20

Ran

kin

g C

onsi

sten

cy I

nd

ex

Data Range (%)

M1 (T)

M2 (T)

M3 (T)

M4 (T)


194





0

0.1

0.2

0.3

0.4

0.5

0.6

100 90 80 70 60 50 40 30 20

Ran

kin

g C

onsi

sten

cy I

nd

ex

Data Range (%)

M1 (T)

M2 (T)

M3 (T)

M4 (T)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

100 90 80 70 60 50 40 30 20

Ran

kin

g C

onsi

sten

cy I

nd

ex

Data Range (%)

M1 (T)

M2 (T)

M3 (T)

M4 (T)

Documents

Decision Context Based Evaluation of Multiattribute ... · Multiattribute decision making (MADM) methods generally involve evaluating a set of decision alternatives by considering