Multi-Criteria Decision Making Methods: A Comparative Study
by Evangelos Triantaphyllou, Ph.D.
Probably the most perpetual intellectual challenge in science and
engineering is how to make the optimal decision in a given
situation. This is a problem as old as mankind. In some ancient
civilizations people attempted to solve complex and risky
decision problems by seeking advice from priests or the few
knowledgeable individuals. In ancient Egypt it was believed that
only the kings and the upper clergy could find what is the best
solution to a given problem. In classical Greece oracles served
a similar purpose.
Many centuries passed since then. Today mankind has replaced the
old methods with modern science and technology. The development
of scientific disciplines such as operations research, management
science, computer science, and statistics, in combination with
the use of modern computers, are nothing but aids in assisting
people in making the best decision for a given situation.
Theories such as linear programming, dynamic programming,
hypothesis testing, inventory control, optimization of queuing
systems, and multi-criteria decision making have as a common
element the search for an optimal decision (solution).
Among the previous methods, there is one class of methods which
probably has captured the attention of most of the people for
most of the time. This is multi-criteria decision making (MCDM).
That is, given a set of alternatives and a set of decision
criteria, then what is the best alternative? This problem may
come in many different forms. For instance, the alternatives or
the criteria may not be well defined, or even more commonly, the
related data may not be well defined. In many real life cases it
may even be impossible to accurately and objectively quantify the
pertinent data. Often a decision problem can be structured as a
multi-level hierarchy. Also, it is not unusual to have a case in
which all or part of the data are stochastic or even fuzzy.
The central decision problem examined in this book is how to
evaluate and rank the performance of a finite set of alternatives
in terms of a number of decision criteria. It is assumed that
the decision maker is capable of expressing his/her opinion of
the performance of each individual alternative in terms of each
one of the decision criteria. The problem then is how to rank
the alternatives when all the decision criteria are considered
In the main treatment the data are assumed to be deterministic.
In the latter part of this book we also consider the case in
which the data are fuzzy. That is, this book does not consider
stochastic or probabilistic data. Although this may sound
restricted, nevertheless it captures many real life situations,
for stochastic data are difficult to be obtained or individual
decision makers feel uncomfortable dealing with them.
The author of this book became actively involved with research in
this area of decision making when he was a graduate student at
Penn State University, more than seventeen years ago. What has
captured his attention since the early days was the
of alternative methods for solving the same MCDM problem. In
most cases the authors and supporters of these methods have
identified some weaknesses of the previous methods and then they
propose a new method claiming to be the best method. As a
result, today a decision maker has an array of methods which all
claim that they can correctly solve a given MCDM problem. The
subjectivity and the tremendous conceptual complexity involved in
many MCDM problems make the problem of comparing MCDM methods a
challenging and urgent one.
This book presents the research experiences of the author
gathered during a long search in finding which is the best MCDM
method. Although the final goal of determining the best method
seems to be unattainable and utopian, some useful lessons have
been learned in the process and are presented here in a
comprehensive and systematic manner.
A methodology has been developed for evaluating MCDM methods.
This methodology examines methods for estimating the pertinent
data and methods for processing these data. A number of
evaluative criteria and testing procedures have been developed
for this purpose. What became clear very soon is that there is
no single method which outperforms all the other methods in all
aspects. Therefore, the need which rises is how one can conclude
which one is the best method. However, for one to answer the
problem of which is the best MCDM method, he/she will first need
to use the best MCDM method! Thus, a decision paradox is
This is the main reason why a comparative approach is needed in
dealing with MCDM methods. By simply stating various MCDM
theories and methods one fails to capture the very real and
practical essence of MCDM. The present book attempts to bridge
exactly this gap. Although not every MCDM method has been
considered in this book, the procedures followed here can be
easily expanded to deal with any MCDM method which examines the
problem of evaluating a discrete set of alternatives in terms of
a set of decision criteria.
This book provides a unique perspective into the core of MCDM
methods and practice. It provides many theoretical foundations
for the behavior and capabilities of various MCDM methods. This
is done by describing a number of lemmas, theorems, corollaries,
and by using a rigorous and consistent notation and terminology.
It also presents a rich collection of examples, some of which are
extensive. A truly unique characteristic of this book is that
almost all theoretical developments are accompanied by an
extensive empirical analysis which often involved the solution of
hundreds of thousands or millions of simulated test MCDM
problems. The results of these empirical analyses are tabulated,
graphically depicted, and analyzed in depth. In this way, the
theoretical and empirical analyses presented in this book are
complementary to each other, so the reader can gain both a deep
theoretical and practical insight of the covered subjects.
Another unique characteristic of this monograph is that at the
end of almost each chapter there is description of some possible
research problems for future research. It also presents an
extensive and updated bibliography and references of all the
subjects covered. These are very valuable characteristics for
people who wish to get involved with new research in MCDM theory
and applications. Some of the findings of these comparative
analyses are so startling and counter intuitive, that are
presented as decision making paradoxes.
Therefore, this book can provide a useful insight for people who
are interested in obtaining a deep understanding of some of the
most frequently used MCDM methods. It can be used as a textbook
for senior undergraduate or graduate courses in decision making
in engineering and business schools. It can also provide a
panoramic and systematic exposure to the related methods and
problems to researchers in the MCDM area. Finally, it can become
a valuable guidance for practitioners who wish to take a more
effective and critical approach to problem solving of real life
multi-criteria decision making problems.
The arrangement of the chapters follows a natural exposition of
the main subjects in MCDM theory and practice. Thus, the first
two chapters provide an outline and background information of the
most popular MCDM methods used today. These are the weighted sum
model (WSM), the weighted product model (WPM), the analytic
hierarchy process (AHP) with some of its variants, and the
ELECTRE and TOPSIS methods.
The third chapter provides an exposition of some ways for
quantifying qualitative data in MCDM problems. This includes
discussions on the elicitation of pairwise comparisons and the
use of different scales for quantifying them. Chapters four to
seven describe some different approaches for extracting relative
priorities from pairwise comparisons and also of ways for
reducing the number of the required judgments.
Chapter eight is the longest one and it deals with a unified
sensitivity analysis approach for MCDM methods. Since no real
life decision problem can be considered completely analyzed
without a sensitivity analysis, this is a critical subject. As
with most of the chapters, this chapter provides an in depth
theoretical and empirical analysis of some key sensitivity
Chapters nine to eleven deal with the comparison of different
MCDM methods and procedures. Chapter nine presents a comparison
of different ways for processing a decision matrix. Chapter ten
presents a computational study of the AHP and the Revised AHP.
Chapter eleven presents some new cases of ranking irregularities
when the AHP and some of its additive variants are used. One can
claim that these new cases of ranking irregularities are strongly
counter intuitive. They have been analyzed both theoretically
Chapters twelve and thirteen present some fundamental concepts of
fuzzy decision making. As always, the treatments here are
accompanied with extensive comparative empirical analyses.
Finally, some conclusions and possible directions for future
research are discussed in the last chapter.
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