A FRAMEWORK OF DESCRIPTIVE
DECISION-MAKING UNDER UNCERTAINTY
USING DEMPSTER-SHAFER THEORY AND
PROSPECT THEORY
A Dissertation Submitted to
NAGAOKA UNIVERSITY OF TECHNOLOGY
GRADUATE SCHOOL OF ENGINEERING
INFORMATION SCIENCE AND CONTROL ENGINEERING
in Partial Fulfillment of the Requirements for the Degree of
DOCTOR OF ENGINEERING
August, 2013.
Submitted By
Elhum Nusrat
Supervised By
Professor Koichi Yamada
Copyright by Elhum Nusrat, 2013.
All Rights Reserved
ii
ABSTRACT
Decision-making (DM) is undoubtedly one of the most fundamental
activities of human life. It is an absolute fact that various problems demand decisions to
be taken even if the outcomes are uncertain or the probability of the occurrences of the
outcomes is unknown. Therefore, the necessity of a standard decision-making framework
cannot be denied. The problem of decision-making can be viewed from two aspects:
firstly, how decisions are made and secondly, the characteristics of the underlying context
i.e. decision-making under certainty, risk, uncertainty, ignorance. Decision-making was
greatly dominated for centuries by normative decision-making theories that deal with
how decisions should be made and rely on the rationalistic ideology. Empirical studies
have proved that human beings are not always rational in making decisions and often
violate the axioms of the normative decision theory. Therefore, another type of decisionmaking model has been researched in order to explain the paradoxes that violate the basic
tenets of the normative models. This stream of decision-making is known as ‘descriptive’
decision-making model. The descriptive decision-making models seek to explain and
predict how people actually make decisions. The second aspect of decision-making is the
underlying context of the decision problem that explains whether the decision-making is
being done under certainty, risk, uncertainty or ignorance. Due to the widespread
presence of risk and uncertainty in decision-making problems of various disciplines,
decision-making under risk and decision-making under uncertainty have grasped the
attention of the research communities. Yet, significant shortcomings have been observed
in the existing researches that motivated us to focus on developing a standard decisionmaking framework.
Inspired by the fundamental characteristics of descriptive decision theory,
this research hypothesizes that different attitudes of human being towards uncertainty
may have a significant impact on the decision-making paradigm. For proving the
hypothesis, a descriptive decision-making framework under uncertainty using DempsterShafer Theory (DST) and Prospect Theory (PT) has been proposed in this research which
is the major contribution of this study. The proposed framework is capable of dealing
with uncertainty present in any decision-making problem and demonstrating the decision
varieties of people in different decision attitudes. The decision-making problem is
defined using the conventional yet popular representation approach including alternative,
states of nature, outcomes and corresponding utilities. The uncertainty has been
represented using Dempster-Shafer Theory of Evidence where a basic belief assignment
(bba) is associated to each subset of the set of states of nature. This representation of
iii
uncertainty is more robust compared to the existing theories of decision-making under
uncertainty. In the process of deriving the decisions, the problem is first transformed into
a decision-making problem under risk by approximating the probabilities from bba in
three different attitudes of a human decision-maker by the use of Ordered Weighted
Averaging (OWA) operators. Later, the study demonstrates how numerous attitudes of
human can be mathematically expressed by changing the parameters of OWA operator
which was unexplored by contemporary researches. So, this is considered as one of the
contributions of the research. Moreover, another contribution of this research is that the
proposed framework solves the previously unsolved problem of lack of consideration
between the probability weighting function and that of bba.
While dealing with uncertainty, it is required to take into account the
existence of multiple information sources providing evidences for the hypothesis in
question. Evidence combination implies the aggregation of uncertain pieces of
information issued from different sources dealing with the same problem. This research
considers this issue and contributes by proposing a new method of evidence combination
based on weighted average and reliability of the information sources.
From the very onset, the objective of this research has been to demonstrate
the decision variety of people in decision-making under uncertainty by proposing a
descriptive decision-making model. To reach the ultimate goal, Prospect Theory was
applied to the decision-making problem for deriving people’s decisions under uncertainty.
For validation of our approach, it has been applied to an example problem of decisionmaking about lung cancer treatment where the patients and/or their caregivers are the
decision-makers. It has been evident from this example problem that human beings are
not always rational and their decisions often vary on the basis of their attitudes or
outlooks toward uncertainty. Compared to the contemporary DM models, our proposed
model is more robust in uncertainty expression which makes it more efficient in
explaining human attitudes and decisions. Therefore, this research is possible to be
applied to the decision support systems requiring the analysis of human attitudes and
human decision-making.
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ACKNOWLEDGMENTS
Upon the completion of the dissertation, I have incurred debts to a number
of people and institutions. First of all, I would like to express my profound gratitude to
the distinguished faculty members of Department of Management and Information
Systems Science at Nagaoka University of Technology for their helpful supervision,
critical evaluation and, appreciation of my work through the course of this research. Also,
I would like to express my deep gratitude to the jury.
Particularly I deeply owe to Professor Koichi Yamada, the main
supervisor of the research. His eminent scholarship, thought, outstanding advice, and
appreciation made this research at this standard even though, I truly believe proper use of
those invaluable comments could make this thesis much better. From the beginning of my
journey as a student of NUT until the completion of this dissertation, his untiring support
and acumen in the subject matter has not only enlightened my thesis but also enlightened
my thought process as a person. I hope I have done justice to his priceless counsel. I am
also very grateful to Assistant Professor Muneyuki Unehara for his constructive
comments, suggestions and comprehensions and the critical evaluation which eventually
develop my understanding as a researcher.
I would like to take the opportunity to express my gratitude to the
Government of Japan and the authority of MEXT scholarship for providing me with the
tremendous opportunity to study in Japan and especially at the renowned institution as
NUT. It is really like a dream come true.
In this connection I would like to convey my thanks to the Kokusai-ka for
their kind suggestions and help in time of needs and all the way of my academic life here
in Nagaoka. Here, I very much appreciate the kindness and support of my friends in and
outside the University. Words fail me to write their contribution in making my life easy
here in Japan.
Finally, my heartfelt thanks and gratitude goes to my family, for their love,
moral support and enduring encouragement for whatever I aspire to do throughout my
life, which continues all the way of accomplishment of the dissertation. Without their
sacrifice and support I would not be here today.
v
DEDICATION
This dissertation is dedicated to my mother Mrs. Jeyaun Nahar Begum
and to the God Almighty.
vi
TABLE OF CONTENTS
Abstract ............................................................................................................................. iii
Acknowledgments ..............................................................................................................v
Dedication ......................................................................................................................... vi
List of Tables ......................................................................................................................x
List of Figures ................................................................................................................... xi
Chapter 1: Introduction ....................................................................................................1
1.1 Decision-making and Human Life........................................................................1
1.2 Decision-making under Uncertainty ....................................................................3
1.3 Motivation and Objective of the Research ..........................................................5
1.4 Organization of the Dissertation...........................................................................8
Chapter 2: Literature Review .........................................................................................10
2.1 History of Decision Theory .................................................................................10
2.2 Decision-making Basics .......................................................................................12
2.3 Various forms of Decision-making .....................................................................14
2.3.1 Normative vs. Descriptive ..........................................................................14
2.3.2 Group vs. Individual Decision-making .....................................................15
2.3.3 Multicriteria Decision-making (MCDM) ..................................................16
2.3.4 Shared Decision-making .............................................................................17
2.3.5 Different Realms of Decision-making .......................................................18
2.4 Related Works ......................................................................................................18
2.4.1 Expected Utility Theory .............................................................................19
2.4.2 Rank Dependent Utility ..............................................................................23
2.4.3 Prospect Theory in Brief ............................................................................24
2.4.4 Other Decision Models of Decision-making under Uncertainty .............24
2.5 Problems of the Existing Decision-making Models...........................................27
2.6 Background Theories...........................................................................................30
2.6.1 Dempster-Shafer Theory of Evidence .......................................................30
2.6.2 Related Works on Dempster-Shafer Theory and Its Applications ........34
2.6.3 Prospect Theory ..........................................................................................37
2.6.4 Extensions of Prospect Theory and Related Works ................................43
Chapter 3: NY-DDM: Proposed Descriptive Decision-making Framework under
Uncertainty .................................................................................................................47
3.1 Overview of the NY-DDM Framework..............................................................47
3.2 Evidential Decision-making Problem.................................................................49
3.3 Converting an EDMP into DM under Risk .......................................................50
3.4 Applying Prospect Theory ..................................................................................56
3.5 Human Attitudes and Decision-making .............................................................58
3.6 Conclusion ............................................................................................................62
Chapter 4: Combination of Evidences and the Proposed Method of Evidence
Combination ...............................................................................................................63
4.1 Necessity of Evidence Combination ...................................................................63
4.2 Related works on Evidence Combination Rules ...............................................65
4.2.1 Dempster’s Rule of Combination ..............................................................66
4.2.2 Yager’s Modified Dempster’s Rule ...........................................................67
4.2.3 Dubois And Prade’s Disjunctive Consensus Rule ....................................68
4.2.4 Hau and Kashyap’s Rule ............................................................................68
4.2.5 Murphy’s Rule of Combination .................................................................69
4.2.6 Yamada’s Combination by Compromise..................................................69
4.3 Proposed Rule of Evidence Combination for Lung Cancer Diagnosis and
Staging...................................................................................................................72
4.4 Conclusion ............................................................................................................80
Chapter 5: Decision-Making of a Lung Cancer patient regarding treatment By
NY-DDM .....................................................................................................................82
5.1 Medical Decision-making: Past and Present .....................................................82
viii
5.2 Few Facts about lung cancer ...............................................................................86
5.2.1
Diagnosis and Staging ..............................................................................88
5.2.2 Treatment ....................................................................................................92
5.3 Decision-making about Treatment in Lung Cancer .........................................94
5.3.1
Problem Formulation ..............................................................................95
5.3.2
Set of States of Nature .............................................................................95
5.3.3
Set of Alternatives ....................................................................................95
5.3.4
Basic Belief Assignments (bba) ...............................................................96
5.3.5
Outcomes and Utilities .............................................................................98
5.3.6
Evaluating the Alternatives ...................................................................103
5.4 Discussion............................................................................................................104
Chapter 6: Conclusion ...................................................................................................107
6.1 Summary of the Research .................................................................................107
6.2 Major Contributions of the Research ..............................................................110
6.2 Limitations ..........................................................................................................112
6.3 Recommendations for Future Research ..........................................................112
APPENDIX A: Publication of the Research ................................................................114
APPENDIX B: Graphical Representation of the Values of the Alternatives in All
Five Cases .................................................................................................................116
References .......................................................................................................................122
ix
LIST OF TABLES
Number
Page
Table 1: Decision matrix, ai: alternatives, sj: states of nature, uij: values of outcomes
or utility............................................................................................................13
Table 2: Decision matrix for Fire Insurance ......................................................................13
Table 3: Estimates of the Loss-aversion Coefficient .........................................................45
Table 4: Evidential Decision-making Problem (EDMP) ...................................................50
Table 5: Values of λ, α, γ ...................................................................................................57
Table 6: List of Information Sources for Lung Cancer Staging and Corresponding
Reliabilities ......................................................................................................73
Table 7: Comparison of Combination Menthods...............................................................76
Table 8: Comparison of Combined Evidences using Discounting Method .......................77
Table 9: Combined Evidences using the Proposed Combination Method ........................79
Table 10: Overall Staging and TNM Equivalent of NSCLC .............................................90
Table 11: List of Treatment Alternatives ...........................................................................97
Table 12: List of Basic Belief Assignments(BBA)............................................................97
Table 13: Outcome Table and Utilities ..............................................................................99
Table 14: Decision Table with Outcomes and Utilities ...................................................100
Table 15: Probabilities in Equative Approach in Case 1 .................................................101
Table 16: Probabilities in Pessimistic Approach in Case 1 .............................................101
Table 17: Probabilities in Optimistic Approach in Case 1 ..............................................102
Table 18: Overall Values of 21 Alternatives in Case 1 ...................................................103
Table 19: Patient’s Decisions Regarding Treatment in Different Attitudes ....................105
x
LIST OF FIGURES
Number
Page
Figure 1: The hypothetical value function ........................................................................43
Figure 2: Weighting Function ...........................................................................................43
Figure 3: Overview of NY-DDM Framework ..................................................................48
Figure 4: Anatomy of Human Respiratory System...........................................................87
Figure 5: Few Diagnostic Methods ....................................................................................89
xi
CHAPTER 1: INTRODUCTION
1.1 DECISION-MAKING AND HUMAN LIFE
Decision-making (DM) is the study of identifying and choosing alternatives based on the
values and the preferences of the decision-maker. Making a decision implies that there
are more than one choice under consideration and the task of a decision-maker is to select
any one (sometimes more) alternative. Viewing it from a different perspective, decisionmaking can be thought of the process of sufficiently reducing uncertainty to allow a
reasonable choice to be selected among a set of alternatives. Let’s look back to history to
find some fascinating examples. On 6 September 1492, Christopher Columbus set off
from the Canary Islands and sailed westward in an attempt to find a new trade route and
after a five week journey across the Atlantic, land was visible. This decision of Columbus
is undoubtedly one of the most courageous decisions by an explorer. Analyzing this
historical decision problem, we can state that Columbus had two alternatives: sailing
westwards or do not sail. The outcomes of the former one could be proving their
geographical hypothesis, making them famous or it would have been to never reach land
again; indeed a terrible way to die. The consequence of the second option was to remain
in the same status. The later history is known to all readers (Peterson, 2009).
Human life is full of big and small decisions that steer a course through
the complex social world and therefore decision-making has its bold presence in every
sphere of human life. From the very crucial fields such as business, politics, economics,
1
medical treatments to the straightforward problems such as choosing a housing or school,
buying a car or any other appliances can be considered as a field of decision-making. In
making decisions, we generally rely on heuristics. Sometimes, these heuristics work and
economize our search costs and allow us to make sound and efficient decisions
(Kahneman & Tversky, 1974). However, sometimes, these heuristics consistently lead us
away from sound decision-making; those departures from rationality are biased in one
way or another. While appropriate decisions can result in success, good relationships as
well as material comfort, improper decisions can lead to regret, loss and emotional stress.
It must be stated that, most of the decisions are usually made without having certain idea
about the future outcomes. In recognition of the far-reaching importance of decisionmaking, researchers from multiple perspectives have paid increasing attention to
decision-making under uncertainty. Another aspect by which decisions are influenced is
human behavior. Behavioral scientists argue that there are important areas where
individuals appear to have inconsistent preference. When we are hungry, scared, sleepy
or too much delighted, our preference ordering adjusts accordingly. Not only has that,
human beings had different mindset of viewing uncertainty. Decision and behavior may
be the core characteristics of decision-making phenomena. They involve the process of
human thought and reaction about the external world, which include past and possible
future events and the psychological consequences to the decision-maker, of those events.
Therefore, decision-makers’ current state of mind is usually considered in defining
preference functions. Along with this, human attitude toward uncertain aspects of any
event can also persuade the act of decision-making. In the literature of social psychology,
2
there are evidences on optimism as well as pessimism on individual decision-making and
it has also been shown in numerous literatures that human beings are not always rational.
1.2 DECISION-MAKING UNDER UNCERTAINTY
Uncertainty hardly needs an introduction. It is perhaps the most inherent and the most
prevalent property of the knowledge of the world around us. Incompleteness of
information, imprecision, approximations made for the sake of simplicity, variability of
the described phenomena, contribute all to the fact that we rarely can make categorical
statements about the world. Uncertainty is a complex characterization about data or
predictions made from data that may include several concepts including error, accuracy,
validity, quality, noise and confidence and reliability. It has various interpretations in
different fields. The dual nature of uncertainty is described with the following definitions:
(Hanssen, 1994)
Aleatory Uncertainty- This type of uncertainty results from the fact that a
system can behave in random ways. It is also known as: Stochastic uncertainty, Type A
uncertainty, Irreducible uncertainty, Variability, Objective uncertainty.
Epistemic Uncertainty- This type of uncertainty results from the lack of
knowledge about a system and is a property of the analysts performing the analysis. It is
also known as: Subjective uncertainty, Type B uncertainty, Reducible uncertainty, State
of Knowledge.
Uncertainty can occur if information is incomplete, imprecise,
fragmentary, not fully reliable, vague, contradictory, or deficient in some other way. In
3
several literatures, three types of uncertainty have been mentioned such as nonspecificity
(or imprecision), fuzziness (or vagueness) and strife (discord) (Klir & Yuan, Chapter 9).
Any one or more of these uncertainties are possible to exist in different fields. So, we are
bound to live in this uncertain world and make decisions based on our uncertain
knowledge.
Decisions under uncertainty are by no means limited to our private lives.
Physicians make decisions on a day-to-day basis and these decisions may impact life and
death of their patients. Decisions of the leaders of a country impact all its citizens through
economic, educational, health, environmental, and defense policies. Voters in turn decide
whom to choose. None of these decisions are made under certainty, either about the
available information or about the consequences of actions. It would not be an
exaggeration to state that real-world decisions not involving uncertainty either do not
exist or belong to a truly limited class.
In case of more area specific examples, engineering discipline such as in
civil engineering, mechanical engineering, industrial engineering also come across lots of
uncertain circumstances. For example, the fundamental knowledge of engineers concerns
the materials employed, how they are made, shaped; how resistant they are to stress,
weather and use and how these may fail. Over the centuries, before having proper
scientific knowledge, the design decisions were taken by some subjective judgments of
the designers. Although having proper scientific knowledge, there still remains a great
deal of uncertainty of application of the knowledge to actual design problems (Klir and
Yuan, Chapter 9). Also, decisions in health care involve a complex web of diagnostic and
4
therapeutic uncertainties, patient preferences and values, and costs. Moreover, the effects
of the treatments may also be uncertain.
Theoretically, decision-making is performed under the realms of certainty,
risk, uncertainty and ignorance. In a decision-making scenario when a decision-maker
does not have a complete probability distribution about the ‘states of nature’ or the
various extraneous factors that have effects on the decision, that DM problem is
considered as
decision-making under uncertainty. For example, when a traveler is
deciding on which route to follow to reach a destination, he is aware that each route
might suffer from different disruptions at different point of time. But, it is not possible to
predict the disruptions beforehand with a complete probability distribution. This research
focuses on a decision-making problem where the information contains such uncertainty
and introduces a framework of descriptive decision-making under uncertainty and
explains the decision variety of a decision-maker in different decision-attitudes.
1.3 MOTIVATION AND OBJECTIVE OF THE RESEARCH
In DM literature, two basic approaches of decision-making models are widely discussed:
Normative or prescriptive and Descriptive decision model. A normative decision model
deals with how decisions should be made. The word ‘should’ can have different
interpretations. But most of the decision scientists agree that it refers to the prerequisites
of rational decision making. Therefore, a normative decision model is a model about how
decisions should be made in order to be rational (Hanssen, 1994). On the other hand,
Descriptive decision-making refers to what people with different attitudes actually
5
accomplish in case of decision-making. Descriptive models use cognition to explain
decision-making; however, normative theories consist of “rationalistic” tenets that show
how decision makers should decide. Due to the strictness of rationality, the normative
approach alone cannot explain diversified human decisions and various paradoxes have
been reported (Ellsberg 1961, Allais 1979). Therefore, descriptive DM model was
evolved and researched. Nonetheless, there are still certain aspects of the descriptive
model yet to be explored. In the field of medical DM, although decision support systems
are being used in different parts of the world, most of these decision support systems are
constructed to support the health professionals either by keeping patient records of
different diseases or providing the physicians suggestions of a certain ailment. However,
health professionals are increasingly encouraged to involve patients in treatment
decisions because increased patient involvement, a result of sociopolitical changes, is an
important part of health care quality improvement. Even though, patient centric decision
support systems are typically few in practice. The scarcity of researches on patientrelated decision-making under uncertainty and the association between human
psychologies and decision-making typically motivated me to choose this topic as my
doctoral research. The motivating questions that form the basis of this dissertation are:
Q1. What are the fundamental characteristics of a descriptive decision-making problem?
How does it represent human cognition?
Q2. What is the most appropriate theory to express more than one type of uncertainty?
Q3. Is it possible to assess the general applicability of the proposed DM framework? Are
we going to suggest any ‘best’ decision?
6
Q4. Can this proposed DM framework ensure increased patient involvement in treatment
decisions?
Together, the answers to the motivating questions represent the core
contributions of the dissertation, while few other questions have come up later which
were necessary and logical extensions of this core. The contributions of this research
have been summarized below.
This research proposes a descriptive DM framework under uncertainty.
We have focused on a problem where the information contains uncertainty and
introduced a descriptive DM model under uncertainty. At first a DM problem under
uncertainty has been modeled as the Evidential DM Problem (EDMP) and converted into
DM problem under risk. In our proposed framework, we actually approximated the basic
belief assignment (bba) associated to the subsets of state of nature by probability where
the approximation depends on the decision-makers’ attitude (such as equative,
pessimistic, optimistic) to uncertainty. Later on, Prospect Theory (PT) is applied to the
transformed problem and it serves the descriptive part of the decision-making framework
explaining how a decision-maker formulates decision in different attitudes. The proposed
framework has been named as NY-DDM.
By using the proposed descriptive DM framework or NY-DDM, we have
illustrated a scenario of decision-making regarding treatment alternatives of the lung
cancer victims. To the best of our knowledge, such DM problem of lung cancer treatment
alternatives from the patient-centric view is an unexplored area of research till date.
7
One of the central issues in any decision-making (DM) problem is the
representation of the knowledge about uncertain variables where external variables lead
to different payoffs. While explicating the DM problem of lung cancer treatment
alternatives, we have come across the difficulty of combining pieces of information from
multiple sources and generate more convincing as well as effective knowledge of
uncertain variables. In this regard, our research has a significant contribution in proposing
an improved method of evidence combination.
1.4 ORGANIZATION OF THE DISSERTATION
The dissertation has a pretty straightforward and traditional structure. Paying tribute to
the writers and researchers in the area which one intends to study is one of the
fundamental principles of any scientific endeavour. Knowing what has been already
accomplished prevents us from making the same mistakes, from ‘reinventing the wheel’;
it forces us to engage in a disciplined interpretation of the work of others. Chapter 2
includes the foundational concepts of this research along with the related researches. In
Chapter 2, history of decision theory, decision-making basics and models, influential
works of decision-making and uncertainty domain along with the necessary theoretical
background have been covered.
Chapter 3 illustrates the proposed descriptive DM framework (NY-DDM)
in detail. The concluding part of Chapter 2 has pointed out the shortcomings of the
methods in the literature and Chapter 3 attempts to resolve the controversial issues by
explaining the proposed framework. Together with the issue of resolving certain
8
problems, Chapter 3 also explains the novelty of this research. Chapter 4 describes the
topic of evidence combination during the presence of multiple information sources.
Along with the discussion on existing evidence combination rules, this chapter points out
the shortcomings of the existing rules for our area of application of the DM framework
and consequently illustrates the proposed rule of evidence combination.
The completion of Chapter 3 and Chapter 4 will lead to the question of
the utilization of NY-DDM. Therefore, Chapter 5 is designed to explain an example
application of the proposed DM framework. Chapter 5 contains the formulation of DM
problem of lung cancer treatment alternatives and demonstrates the decision variety of
patients due to dissimilar attitude towards uncertainty. The last part, Chapter 6,
summarizes the entire research and discusses its strength and limitations as well as
indicating the areas of future work.
Some parts of the dissertation have been published as journal papers and
the conference proceedings as written in the list of author’s publication (Appendix A).
Chapter 3 is published in the journal International Journal of Uncertainty, Fuzziness and
Knowledge-Based Systems (IJUFKS) and in the proceedings of Congress of The Shinetsu Chapter of the Institute of Electronics, Information and Communication Engineers,
IEEE Shin-etsu session. A part of Chapter 5 has been published in International Journal
of Computer Applications.
9
CHAPTER 2: LITERATURE REVIEW
The previous chapter discusses the importance of decision-making (DM) in human life
and also mentioned the motivation and objective of conducting this research. The prime
focus of this chapter is to set a background of this dissertation and to provide the readers
with an insight why this type of research is considered necessary. To achieve this goal,
the very basics of this research have been included in this chapter at first.
Most
importantly, this chapter throws light on the pioneering works along with the
contemporary researches in the domain of DM, uncertainty and several related theories.
2.1 HISTORY OF DECISION THEORY
The history of decision theory can roughly be divided onto three different phases: the Old
period, the Pioneering period and the Axiomatic period. The Old period started in ancient
Greece, however, Greece did not develop a theory of rational decision making. Ancient
Greece identified decision making as one of many areas of further exploration. Further,
Aristotle’s quote in his several books gave some hints of rational preference though
nothing was proved during the fifteen hundred years that followed the decline of the
Greek period (Peterson, 2009, Chapter 1).
The second major development phase was the Pioneering period when
Blaise Pascal and Pierre de Fermat started to exchange letters about probability theory.
Their interest was typically triggered by a question regarding the likelihood of getting at
10
least one pair of sixes if a pair of fair dice is thrown 24 times raised by a French
nobleman with a great interest in gambling. Rather than providing an approximate answer
by empirical trials, Fermat and Pascal started to work out an exact, mathematical solution.
Another major breakthrough was the 1662 publication of a book known as Port-Royal
Logic (Arnauld & Nicole, 1662/1996) which was published anonymously by an
organization belonging to the Catholic Church. This book contained the first clear
information of the principle of maximizing expected values. Even though this explanation
was reliable in many situations, it leads to counterintuitive recommendations in several
cases, for example, amount of money. In 1738 Daniel Bernoulli introduced the notion of
moral value as an improvement on the imprecise terms ‘good’ and ‘evil’ mentioned by
the authors of Port-Royal Logic. According to Bernoulli, the moral value of an outcome
is a term that refers to how good or bad that outcome is from the decision-maker’s point
of view. For example, a multi-millionaire will be indifferent between a status quo and a
gamble where the winning amount is a million (Peterson, 2009, Chapter 1).
Modern decision theory is dominated by attempts to axiomatize the
principles of rational decision making. This period has two significant points. The first
was Ramsey’s paper (Ramsey, 1926). In his paper on probability, he proposed a set of
eight axioms of how a rational decision maker should choose among uncertain
alternatives. The second point was the book Theory of games and Economic Behavior
(von Neumann and Morgenstern, 1947) presenting a set of axioms for how rational
decision makers ought to choose among lotteries. It has been said that the authors were
unaware of Ramsey’s work during the first publication of their book. But their work on
11
game theory was unique and it must be regarded as a major breakthrough. 1950s can be
called as the ‘golden period’ of decision theory that produced several influential papers
and books including the book The Foundations of Statistics (Savage, 1954) in which the
author presented yet another important axiomatic analysis of the principle of maximizing
expected utility. Since then, decision theory has grabbed the attention of researchers from
various disciplines and it has become an interdisciplinary topic to which philosophers,
economists, computer scientists, statisticians, psychologists can contribute their
knowledge.
2.2 DECISION-MAKING BASICS
Decision-making can be regarded as the cognitive process resulting in the selection of a
course of actions among several scenarios. This section portrays a general scenario of a
classic DM problem. A classic DM paradigm involves the selection of a best alternative
in which the outcomes are a function of the states of nature. The problem of decisionmaking can be captured by Table 1.
The alternatives shown in the decision matrix in Table 1 are the possible
choices open to the decision maker. The various extraneous factors that have effects on
the decision of the decision maker can be summarized into a number of cases which is
called states of nature. An outcome of a decision is the combined effect of the chosen
alternative and the states of nature. For example: someone is thinking of taking out fire
insurance on his home. Perhaps it costs $100 to take out insurance on a house worth
12
$100,000, and someone might think whether it is worthy or not. So, the decision matrix
will take the form as mentioned in Table 2.
Table 1: Decision matrix, ai: alternatives, sj: states of nature, uij: values of outcomes or
utility
s1
…………..
sj
………
a1
u11
……………
u1 j
………..
sn
u1n
:
:
:
:
:
:
ai
ui1
…………..
uij
…………
uin
:
:
:
:
:
:
aq
u q1
……………
u qj
………….
uqn
Table 2: Decision matrix for Fire Insurance1
Fire
No Fire
Take out insurance
No house and $100,000
House and $0
No insurance
No house and $100
House and $100
Each outcome of the decision matrix can be measured by its utility. In
economics, utility is a measure of desirability of consumption of, or contentment of, any
good and service or any kind of action. Economists have distinguished utility as cardinal
and ordinal utilities. Cardinal utility refers to a property of mathematical indices that
preserve preference orderings uniquely up to linear transformations. On the other hand,
when the utility of any good or service cannot be measured numerically, ordinal utility
1
M. Peterson, An Introduction to Decision Theory (Cambridge University Press, 2009)
13
theory is used in those cases. A utility function in this study maps an outcome to a real
number.
In different researches, the notations ‘value of outcome’, ‘outcome’ and
‘utility’ has been used interchangeably. In this study, we define outcome as the
consequence of a certain (alternative, state) pair. By the term ‘utility’, we mean the
assessment of an outcome after applying the utility function.
2.3 VARIOUS FORMS OF DECISION-MAKING
2.3.1 NORMATIVE VS. DESCRIPTIVE
Models of decision-making can be classified into two groups: Normative or prescriptive
and Descriptive decision models. A normative decision model deals with how decisions
should be made. Although the word ‘should’ can have different interpretations, most of
the decision scientists agree that it refers to the prerequisites of rational decision-making.
Rational behavior is typified by a decision-maker who has a “well-organized and stable
system of preferences and a skill in computation that enables him to calculate, for the
alternative courses of action that are available to him, which of these will permit him to
reach the highest attainable point on his preference scale” (Simon, 1955). Therefore, a
normative decision model is a model about how decisions should be made in order to be
rational. Extensive researches (Baron, 1994; Dawes, 1988; Evans, 1989; Evans & Over,
1996; Kahneman, Slovic, & Tversky, 1982; Kahneman & Tversky, 1979; Raiffa, 1985;
Shafir & Tversky, 1995) have proved that people are not always rational while making
decisions and violate the axioms of utility theory. Various paradoxes such as Allais
14
paradox and Ellsberg paradox etc. have been reported. Therefore, another type of DM
model was researched that explains the paradoxes that violate the basic tenets of the
normative model.
Both descriptive and normative DM theories possess individual
characteristics. Descriptive decision models seek to explain and predict how people
actually make decisions. Putting it differently, descriptive decision-making describes
what people actually accomplish in decision-making in different attitudes. This is an
empirical discipline, stemming from experimental psychology. Descriptive models use
cognition to explain decision-making; however, normative theories consist of
“rationalistic” tenets that show how decision makers should decide. Reaching any
decision in descriptive DM is influenced by psychological elements. Our study aims at
proposing a descriptive decision-making model that explains the decision variability at
different phenomenon of attitude rather than providing an optimum solution which is the
basic goal of any normative decision-making framework.
2.3.2 GROUP VS. INDIVIDUAL DECISION-MAKING
As the name implies, group decision-making is a situation when two or more individuals
collectively make a decision among the alternatives. Group decision-making has the
advantages of drawing from the experiences and perspectives of a larger number of
individuals. Hence, they have the potential to be more creative and lead to a more
effective decision. In fact, groups may sometimes achieve results beyond what they could
have as individuals. If a group is diverse, better decisions can be made. On the other hand,
15
groups may suffer from coordination problems. Anyone who has worked with a team of
individuals on a project can attest to the difficulty of coordinating members' work or even
coordinating everyone's presence in a team meeting. Furthermore, groups can suffer from
social loafing, or the tendency of some members to put forth less effort while working
within a group. Groups may also suffer from groupthink. Finally, group decision making
takes a longer time compared to individual decision making, given that all members need
to discuss their thoughts regarding different alternatives. Thus, whether an individual or a
group decision is preferable will depend on the specifics of the situation.
Apart from the early researches on group decision-making (Black, 1948),
the contribution of communication to the study of groups has been in development and
refinement of methods for the study of interaction processes (Fisher, 1970; Stasser &
Titus, 1985; Jarboe, 1988, 1996; Poole, 1985; Poole & Hirokawa, 1986; Poole & Holmes,
1995).
2.3.3 MULTICRITERIA DECISION-MAKING (MCDM)
Multiple criteria decision making or multi-criteria decision-making (MCDM) refers to
making decisions in the presence of multiple, usually conflicting criteria. In personal
context, a house or a car one buys may be characterised in terms of price, size, style,
safety, comfort, etc. Generally, there exist two distinctive types of MCDM problems due
to the different problems settings: one type having a finite numbers of alternative
solutions and the other an infinite number of alternative solutions. It is also possible to
describe a MCDM problem using a decision matrix. Usually, there are two types of
16
MCDM methods in literature: compensatory method allowing trade-offs between
attributes and non-compensatory methods not permitting trade-offs between attributes.
Saaty’s (Saaty, 1988) Analytic Hierarchy Process (AHP) is a popular MCDM technique
that falls under the category of compensatory methods. Evidential Reasoning (ER) is
another popular approach of this category. ER and its variations have been discussed by
many researchers (Yang and Xu, 2000; Yang, 2001; Yang and Singh, 1994).
At the very beginning of MCDM researches, many of the topics were
optimization-related. Goal programming, conceived by Charnes and Cooper (1961) was
an early contribution. Along with this, vector optimization algorithms for computing the
set of all non-dominated solutions of a multiple objective program attracted considerable
interest (Geoffiion 1968; Evans and Steuer 1973; Yu and Zeleny 1975; Gal 1977; Bitran
1979; Ecker, Hegner, and Kouada 1980). Due to the size of the non-dominated set, it is
difficult to single out a final solution and therefore interactive procedures (Benayoun et al.
1971; Wierzbicki 1980) moved to the centre stage in the 1980s.
2.3.4 SHARED DECISION-MAKING
Shared decision-making (SDM) between physician and patient is an idea founded in
ethics and the law in some evidence of superior health outcomes. It is a practical
compromise between the autonomy of a person (patient) and the paternalistic power of
physicians. SDM includes the notion of a medical encounter of “meeting of experts”- the
physician as an expert of the medical field and the patient as an expert of his or her life,
values and circumstances (Tuckett et al. 1985). The relevance of SDM to patient safety
17
has been highlighted by different concepts. Firstly, patient’s involvement in decisionmaking and training of health professionals for “new rules” for 21st century healthcare
that makes the patient the “source of control”. Moreover, it is believed that some errors
and adverse events in healthcare can be avoided through the involvement of patient in
healthcare decision-making (Dowell et al. 2007). Further, evidence-based medicine and
the revelation of great variations in healthcare are also the evidence of the need of SDM.
It is believed that, by adopting these, unnecessary care can be reduced as well as
providing financial savings (Wennberg et al. 2007).
2.3.5 DIFFERENT REALMS OF DECISION-MAKING
On the basis of the information known about the states of nature, DM can be performed
under certainty, risk, uncertainty and ignorance. In a DM situation involving certainty,
the decision-maker is reasonably sure about the outcome; the information is available and
is considered to be reliable. In a DM problem under risk, a decision-maker knows a
complete probability distribution on the states of nature whereas in DM under
uncertainty, we do not have a complete probability distribution about the states of nature.
On the other hand, in DM under ignorance, no probability information is available about
the states of nature. We are particularly interested in the area of DM under uncertainty
and this research proposes a descriptive decision-making model under uncertainty.
2.4 RELATED WORKS
Uncertainty involved in any problem-solving situation is a result of some information
deficiency. As mentioned in Chapter 1, information can be incomplete, imprecise,
18
fragmentary, not fully reliable, vague contradictory or deficient in some other ways and
thus result in uncertainty. This section focuses on the conventional as well as
contemporary theories and approaches that deal with decision-making under uncertainty.
2.4.1 EXPECTED UTILITY THEORY
As mentioned in Section 2.2, utility is a possible way to measure the outcomes. Expected
Utility Theory (EUT) states that the decision maker (DM) chooses between alternatives
or equivalent risky prospects, which are lists of pairs of a utility and its probability, by
comparing their expected utilities, i.e., the weighted sums obtained by adding the utilities
of outcomes multiplied by their respective probabilities. EUT was first proposed by
Daniel Bernoulli (1738) in response to an apparent puzzle surrounding what price a
reasonable person should be prepared to pay to enter a gamble. It was the conventional
wisdom at the time that it would be reasonable to pay anything up to the expected value
of a gamble, but Bernoulli presents this counterexample. A coin is flipped repeatedly
until a head is produced; if you enter the game, you receive an outcome of, say, $2n
where n is the number of the throw producing the first head. This is the so-called St.
Petersburg game. It is easy to understand that the expected monetary value is infinite, yet
Bernoulli believed most people would only be prepared to pay a relatively small amount
to enter it, and he took this intuition as evidence that the “value” of a gamble to an
individual is not, in general, equal to its expected monetary value. He proposed a theory
in which individuals place subjective values, or “utilities,” on monetary outcomes and the
value of a gamble is the expectation of these utilities. While Bernoulli’s theory—the first
19
statement of EUT—solved the St. Petersburg puzzle, it did not find much favour with
modern economists until the 1950s.
Interest in the theory was re-energized when von Neumann and
Morgenstern (1947) showed that the expected utility hypothesis could be derived from a
set of apparently appealing axioms on preference. Since then, numerous alternative
axiomatizations have been developed. Some of these seem highly attractive from the
normative point of view. The expected utility hypothesis can be derived from the
following axioms: completeness, transitivity, continuity and independence (Starmer,
2000). Completeness assumes that an individual has well defined preferences and can
always decide between any two alternatives. According to transitivity axiom, if there are
three prospects or alternatives p, q and r with pq and qr , then pr. Independence
implies when two prospects or alternatives are mixed with a third one maintain the same
preference order as when the two are presented independently of the third one. Together
the axioms of completeness, transitivity and continuity imply that preferences over
prospects or alternatives can be represented by a function U () which assigns a realvalued index to each prospect. The function U () is a representation of preference in the
sense that, an individual will choose the prospect or alternative q over the prospect r if,
and only if, the value assigned to q by U () is no less than that assigned to r. The
independence axiom of EUT places quite strong restrictions on the precise form of
preferences and it is the axiom that gives the standard theory most of its empirical content.
If all of these axioms hold, preference of an alternative q can be represented by the
following equation:
20
U (q) i pi u( xi ) ,
(1)
where xi is the i-th outcome, u(xi) is the utility of xi, and pi is the probability of xi. The
probability pi would be replaced by a weight in other theories explained below.
Empirical studies dating from early 1950s have exposed a variety of
patterns in choice behavior that appear inconsistent with EUT. One popular example of
the violation of EUT and its role in motivating the development of new theories of
decision-making has been explained here. The violations of EUT can fall under two
broad categories: the first category primarily consists of a series of observed violations of
the independence axioms of EUT; the latter of evidence that seems to challenge the
assumption that choices derive from well-defined preferences.
One
well-known
example
that
explains
decision
behavior
can
systematically violate the independence axiom was discovered by Allais (Allais, 1953)
which played an important role in stimulating the theoretical developments in non-EU
theory. These are the so-called common consequence effect and common ratio effect.
Two problems were stated as follows where there are two prospects:
“The choice between two prospects: s1 = ($1M, 1.0) or r1 = ($5M, 0.1; $1M, 0.89; 0,
0.01).
The first option gives one million U.S. dollars for sure; the second gives five million with
a probability of 0.1; one million with a probability of 0.89, otherwise nothing. What will
you choose?”
“If there are two prospects such as: s2 = ($1M, 0.11; 0, 0.89) or r2= ($5M, 0.1; 0, 0.9).
What will you choose?”
21
Allais believed that EUT was not an adequate theory of individual risk
preferences and therefore he designed these problems as a counterexample. A person
with expected utility preference would either choose both ‘s’ options, or choose both ‘r’
options across this pair of problems, because s1 is equivalent to ($1M, 0.11; $1M, 0.89),
and prospects s2, r2 are nothing but those derived by removing ($1M, 0.89) from s1 and
r2, respectively. But most of the people responded such that s1 was chosen in the first
problem and r2 was selected in the second problem. This is the famous “Allais paradox”.
Moreover, EUT was also criticized for its linear probabilities and the
monotonicity of the utility function that implies stochastic dominance preference
(Machina, 1987). As the evidences against the EUT were being discovered, consequently,
a wave of theories began to emerge at the end of the 1970s. Most of these approaches had
the spirit to seek “well-behaved” theories of preference consistent with observed
violations of independence. Later on, researchers dug into the other problems related to
EUT such as preference reversal and others (Slovic and Lichtenstein, 1968; Grether and
Charles Plott, 1979; Karni and Safra, 1987; Segal, 1988; Fishbrun, 1989). Consequently,
researchers came up with some influential papers and initiated a new era of research on
decision-making theories. Some of the well-known theories in this domain are
generalized expected utility (GEU), the rank-dependent models introduced by Quiggin
(1981) for decision under risk and by Schmeidler (1989) for decisions under uncertainties
etc.; later on rank dependence was incorporated in original prospect theory (Kahneman
and Tversky, 1979) and cumulative prospect theory (Tversky and Kahneman, 1992) was
developed.
22
2.4.2 RANK DEPENDENT UTILITY
Rank-dependent utility (RDU), developed by Quiggin (Quiggin, 1982), assumes that the
weight assigned to an outcome is a function of cumulative probabilities called rank in the
distribution of possible outcomes. Quiggin developed RDU to eliminate the violations of
stochastic dominance implied by the models like subjective expected utility and prospect
theory (Kahneman & Tversky, 1979) which transformed the probabilities of each
outcome to weights in a nonlinear and non-additive fashion. In RDU the weight given to
i-th outcome xi is defined by the next equation.
( xi ) w( pi ... p1) w( pi 1 ... p1) ,
(2)
where the outcomes are sorted in the decreasing order of their utilities as u(x1) ≥ ... ≥
u(xn), and w is a weighting function defined in [0,1] which is strictly increasing and
satisfies w(0)=0 and w(1)=1. The essence of RDU is that the decision weight in the
prospect is not just a probability or a weighted probability but is defined by cumulative
probabilities called rank with a probabilistic weighting function.
The RDU proposed by Quiggin was designed for decision-making under
risk. Schmeidler extended the idea of RDU to the realm of uncertainty (Schmeidler,
1989). RDU under uncertainty has almost the same structure of prospects as RDU under
risk, and the weight given to i-th outcome is given by the next equation.
( xi ) W ( Ei ... E1) W ( Ei 1 ... E1) ,
(3)
where the outcomes are sorted in the decreasing order of their utilities as u(x1) ≥ ... ≥
u(xn), {E1,...,En} is a partition of the states of nature S, and W is a strictly increasing
23
weighting function satisfying W(Ø)=0 and W(S)=1. Thus, the function W could be
regarded as a non-additive uncertainty measure that replaces probability.
RDU theory was incorporated in Cumulative Prospect Theory (Teversky
&Kahneman, 1992). Some other influential researches in this domain include Choquet
Expected Utility (Waker, 1990), qualitative probability theory (Chateauneuf 1996; Gilboa
1985) and others. In studies on coherent risk measures (Artzner et al. 1999), RDU with
probabilistic sophistication has received much attention (Tsukahara 2009).
2.4.3 PROSPECT THEORY IN BRIEF
Prospect theory (PT), proposed by Kahneman and Tversky (Kahneman & Tversky, 1979),
is a revolutionary theory of descriptive decision-making under risk that could incorporate
irrational behavior in an empirically realistic manner. The authors conducted extensive
experiments and showed that people systematically violate the axioms of the EUT while
making decisions in reality. Prospect theory predicts that the decision-makers tend to be
risk-averse in the domain of gain and risk-seeking in the domain of losses. The main
disagreements of PT with EUT are: i) Outcomes are defined in terms of gains and losses
relative to a reference point; ii) there is a nonlinear probability weighting function π(.)
that reflects the impact of probabilities. An extensive discussion on PT has been done in
Section 2.6.3.
2.4.4 OTHER DECISION MODELS OF DECISION-MAKING UNDER UNCERTAINTY
In the field of decision-making under uncertainty, which deals with decision-making with
incomplete knowledge of probabilities, a normative model with generalized expected
24
utility (GEU) has been proposed by Yager (Yager 1992, 2004). GEU utilized DempsterShafer Theory (DST) of Evidence (see section 2.6.1 for the details of DST) for
representing uncertainties. Here, the preference of an alternative (or a prospect) is
obtained by the following equation:
U geu (ai ) m( S k )V (ai , S k )
(4)
k
where m( S k ) is a basic belief assignment and S k is a focal element representing a set of
states of nature. V (ai , S k ) is a utility evaluation depending on ai and S k , and can be
given by a parametric function called Ordered Weighted Averaging (OWA) (Yager,
1988) operator, whose parameters could be determined by the attitude to uncertainty of
the decision-maker.
An ordered weighted averaging (OWA) operator of dimension n is a
function F : R n R that has associated with it a weighting vector W (w1 , w2 ,....wn ) ,
such
that
wi [0,1]
and
i wi 1
.
For
any
set
of
values b1,b2, ...bn ,
F (b1 , b2 ,.....bn ) w1c1 w2 c2 ....... wn cn i (wi ci ) , where c1 , c2 ,...., cn are the
argument values in descending order. With OWA operator, the utility evaluation is given
by V (ai , Sk ) F (ui1,...uih ,...uim ) , where uih is the utility of the alternative ai and state
sh S k . Various aggregation operators are found in the literature for aggregating the
information (Beliakov et al., 2007; Merigo et al., 2010). Among those, OWA is a very
common and widely used aggregation operator providing a parameterized family of
25
aggregation operators. According to the definition of OWA operator, the weighting
vector W can be determined as W= [0, 0, 0,….,1] in pessimistic attitude. Then, the
aggregation rule used in the pessimistic strategy is, F (b1, b2 ,....bn ) Min j [b j ] . For
example, if the arguments of OWA operator are (10,15,40,5), then F(10,15,40,5)=
(0*40+0*15+0*10+1*5)=5 according to the pessimistic strategy. It is to be noted that the
weights of the OWA operator are associated with the position in an ordered argument.
As we have noticed, Generalized Expected Utility (GEU) provides a
framework of decision-making under uncertainty where the optimal alternative is selected.
Although it assumes the degree of optimism or pessimism in evaluating the optimal
alternative, GEU is a model to give the solution that optimists or pessimists "should"
choose, and does not explain their real attitudes. In the sense, GEU is a normative DM
model under uncertainty.
Tamura et al. (Tamura et al., 2005, 2006, 2008) proposed a descriptive model
of decision-making under uncertainty as an expansion of the Prospect Theory. It
evaluates an alternative or an expanded prospect (sets of outcomes with bba) by the next
equation.
U Tmr (ai ) (mk )hi ( S k | )
(5)
k
where is a weighting function for bba, α is the pessimism index that represents the
(k )
decision attitude and is decided by the question for the minimum/worst value vmin
of
(k )
. The term u ij* represents gain or loss
v(uij* ) for s j S k and for the maximum/best vmax
26
of the utility for the pair ai and sj (Please refer to section 2.6.3). hi ( S k ) is determined
by the three values:
(k )
vmin
,
(k )
vmax
and
(k )
vave
s j S k
v(uij* )
Sk
as well as the pessimism index α.
Another decision-making methodology called Cautious Ordered Weighted
Averaging with Evidential Reasoning (COWA-ER) has been introduced by Tacnet and
Dezert (Tacnet & Dezert, 2011). The authors suggested a method for decision under
uncertainty based on imprecise valuation of alternatives by considering the results of two
extreme attitudes: pessimistic and optimistic while getting the weighting vector W of
Yager's approach. The authors claimed that when the expected values are imprecise since
they belong to interval (bounds are computed with extreme pessimistic and optimistic
attitudes), it is possible to obtain the best alternative by their proposed four step
methodology. The goal of this study is to choose the best alternative by neither being a
pessimist nor an optimist. Tacnet & Dezert have given emphasis to the problem when it is
uncertain which attitude (pessimistic or optimistic) to follow while considering the
expected outcomes to be imprecise as they belong to an interval.
2.5 PROBLEMS OF THE EXISTING DECISION-MAKING MODELS
The preceding sections discussed a number of widely known or recently published DM
models and theories. Classical theories of DM emphasize the rationalistic process and in
general are applicable to the DM problems that consider the optimal solutions. These are
typically called ‘normative’ DM models that describe how decisions should be made.
Expected Utility Theory (EUT), Subjective Expected Utility (SEU), Choquet Expected
27
Utility (CEU), Generalized Expected Utility (GEU) and the approach by Tacnet and
Dezert fall into this category, where the latter two are DM models under uncertainty and
the former are DM models under risk. However, human beings are not always rational in
making decisions as shown by the famous examples of Allais (decision under risk) and
Ellsberg (decisions under uncertainty) paradoxes.
In response to the limitations of normative DM models, numerous descriptive DM
models have been proposed by researchers in order to express people’s actual decisionmaking process. One of the well-known descriptive models under risk is Rank Dependent
Utility (RDU) proposed by Quiggin. Later, it was extended to RDU under uncertainty by
Schmeidler. From the viewpoint of capability of uncertainty representation, Schmeidler's
model had a limitation that the states of nature must be partitioned to {E1,...,En} as
explained in section 2.4.2, which means that it cannot deal with uncertainty defined on
different pieces of evidence that have an overlap.
Another descriptive model, Prospect Theory (PT), is the most famous
descriptive DM theory for decision-making under risk that successfully explains how
people actually make decisions. It was extended to the cumulative prospect theory (CPT)
by introducing the idea of RDU under uncertainty as explained in section 6.2.4. However,
it has also the same limitation as the RDU under uncertainty.
Tamura et al. also extended the PT to a model under uncertainty as
mentioned in the previous section. The model would have two issues when being applied
to real problems.
28
First, hi ( S k ) in equation (5) is determined only by the three
(k )
(k )
(k )
representative values vmin , vmax
, and vave
as well as α. The approach seems an
improvement of Hurwicz approach (Hurwicz, 1952) because it takes a weighted average
(k )
(k )
(k )
between vmin and vmax
, namely hi (S k ) vmin (1 )vmax
, if it is applied to the
(k )
current context. The common problem of these approaches is that the distribution of
v(uij* ) cannot be taken into account for the calculation of hi ( S k ) . An example that the
model cannot implement is the median approach, which assigns the median value of
v(uij* ) among all s j in S k to hi ( S k ) . It cannot implement other attitudes such as 2nd
optimistic/pessimistic as well.
The second drawback of this model is the lack of consideration about the
difference between the weighting function of probability and the one of bba. The
requirements of weighting function of probability will be shown afterwards that were
derived from exhaustive discussions by Kahneman and Tversky (Kahneman & Tversky,
1979). However, there is no guarantee that they could be applied to a weighting function
of bba. Rather they might be different from those of bba, thinking of the basic difference
between probability and bba. One way to prove this is to perform extensive survey over a
huge number of subjects done in the literature of Prospect theory. But as a matter of fact,
it might be almost impossible, because we must estimate far more number of weighting
S
values than the authors of PT did, i.e. 2 values of bba instead of |S| probabilistic values.
Our approach as explained in the next chapter overcomes this problem because we can
29
convert the EDMP into a Probabilistic Decision Making Problem. Therefore, the
weighting function of probability can be applied directly without any confusion.
The descriptive DM framework under uncertainty proposed in this
research resolves all of the issues mentioned earlier in this section. The application of
DST successfully represents different types of uncertainties and solves the issue of
overlapping evidences which was a shortcoming of RDU under uncertainty. Moreover,
the utilization of the OWA operators provides a solution of representing various decision
attitudes (such as: 2nd optimistic/pessimistic, median attitude etc.) which were
unexplained in the previous researches. Our approach is free from the consideration of
the difference between the weighting function of probability and the one of bba which
was one of the two problems associated with the DM model proposed by Tamura et al.
Furthermore, the application of PT to the converted probabilistic DM problems renders a
descriptive perspective to our framework which is capable of expressing how human
beings make decisions in reality.
2.6 BACKGROUND THEORIES
The proposed research is closely related to two well-known theories: Dempster-Shafer
Theory of Evidence (DST) and Prospect Theory (PT). This section emphasizes the
evolution and the basics of these two theories.
2.6.1 DEMPSTER-SHAFER THEORY OF EVIDENCE
Since probability theory cannot address different types of uncertainties, numerous
alternative approaches have been proposed to deal with uncertainty such as Dempster30
Shafer Theory (DST) (Dempster 1967; Shafer 1976), the transferable belief model (Smets,
1994), the probability of modal propositions, various nonstandard and fuzzy logics and
others. In this research we are particularly interested in DST, which is sometimes called
evidence theory.
Evidence theory is based on two dual non-additive measures: belief
measures and plausibility measures. Given a finite universal set X (holding the mutually
exclusive and exhaustive properties), a belief measure is a function
Bel : ( X ) [0,1]
(6)
such that Bel (Ø) 0, Bel ( X ) 1, and
Bel ( A1 A2 .... An ) Bel ( A j )
j
Bel ( A j Ak ) ... (1) n1 Bel ( A1 .... An )
j k
(7)
where Ω(X) is the power set of X. Due to the inequality, belief measures are called
superadditive. For each A (X ) , Bel(A) is interpreted as the degree of belief that a
given element of X belongs to the set A. From a mathematical point of view, a belief
function can be treated as a mathematical object satisfying a certain set of axioms (Klir &
Yuan, 1995).
Associated with each belief measure is the plausibility measure, Pl,
defined by the equation:
Pl ( A) 1 Bel ( A )
for all A (X ) , where A is the complement of A.
31
Similarly,
(8)
Bel ( A) 1 Pl ( A )
(9)
Plausibility measures are subadditive. For all possible subsets of X, the
following equation holds:
Pl ( A1 A2 .... An ) Pl ( A j )
j
Pl ( A j Ak ) ... (1) n1 Pl ( A1 A2 .... An ) (10)
j k
Belief and plausibility measures can conveniently be characterized by a
function called basic belief assignment (bba):
m : ( X ) [0,1]
such that m(Ø) 0 and
(11)
m( A) 1.
A( X )
The value m(A) for each A (X ) expresses the portion to which all available evidence
supports the claim that a particular element of X, belongs to the set A (Klir & Yuan,
1995).
There corresponds to each belief function one and only one basic belief
assignment. Conversely, there corresponds to each bba one and only one belief function.
They are related by the following two formulae:
Bel ( A)
m( B),
for
all A X
(12a)
Bel ( B)
(12b)
B A
m( A)
(1)
A B
B A
Another important term related to DST is focal element. Every set A for which m(A)>0 is
usually called a focal element of m. Focal elements are subsets of X on which the
32
available evidence focuses. The pair <F,m> where F and m denote a set of focal
elements and the associated bba, respectively, is called the body of evidence (Klir &
Yuan, 1995).
DST is based on two ideas: the idea of obtaining degrees of belief about a
related question, and Dempster’s rule for combining such degrees of belief when they are
based on “distinct” (Dempster, 1967) items of evidence. Evidence obtained in the same
context from two distinct sources can be combined by DST. If the two basic belief
assignments are m1 and m2 on some power set Ω(X), then the standard way of
combining evidence is expressed by the formula
m1,2 ( A)
m1 ( X )m2 (Y )
X Y A
1 K
(13)
for all A Ø, and m1,2 (Ø) 0, where
K
m1 ( X )m2 (Y )
X Y Ø
(14)
This formula is known as Dempster’s rule of combination. There are several criticisms
regarding this combination rule and a number of alternative approaches have been
proposed. In this research, we have also proposed a new rule of combination suitable for
the proposed DM model and the application area. The proposed combination rule has
been discussed in Chapter 4 of this dissertation.
The question that certainly arises in the reader’s mind is the reason of the
choice of Dempster-Shafer Theory (DST) in this research. As we have mentioned in
33
Chapter 1, there are several types of uncertainty and probability theory may not be
applicable to all cases. Clearly there is a need for robust uncertainty representation so
that scientists and policy makers can better understand and characterize the properties of
the predictions they make based on their models. The motivation for selecting DST can
be characterized by the following reasons:
a) DST has versatile expressiveness of uncertainty than other theories. It represents
uncertainty due to ignorance as well as conflict or strife, while Probability
measure represents uncertainty merely due to conflict.
b) DST contains Probability theory as a special case as well as Possibility theory that
provides the mathematical basis to Fuzzy set theory.
c) The huge number of examples of applications of DST in engineering during last
several years has proved the effectiveness of DST in presenting uncertainty in
practical areas.
2.6.2 RELATED WORKS ON DEMPSTER-SHAFER THEORY AND ITS APPLICATIONS
Since the work of Shafer, many interpretations of DST have been proposed. According to
Smets (Smets & Kennes, 1994), any model for belief has at least two components: one
static that describes the state of belief, and the other is dynamic that explains how to
update the belief when new pieces of information are given. The transferable belief
model (TBM, for short) introduced by Smets (Smets & Kennes, 1994, Smets 1994)
provides a model for the representation of quantified belief. This model is based on the
assumption that beliefs manifest themselves at two mental levels: the credal level where
34
beliefs are entertained and the pignistic level where beliefs are used to make decisions
(from credo, I believe and pignus, a bet both in Latin). Especially, the TBM justifies the
use of belief functions to model subjective, personal beliefs even in the cases where every
probability concept is absent at the credal level. Once probabilities are defined
everywhere the TBM is reduced to the Bayesian model.
Another belief based model called ‘Hint Model’ was proposed by Kohlas
and developed further by Kohlas and Monney (Kohlas & Monney, 1995). It starts with
the original structure of Dempster (Ω, P, Г, ϴ) where Ω and ϴ are two sets, P is a
probability measure on Ω and Г is a multivalued mapping from Ω into ϴ. The authors at
first assume a certain question, whose answer in unknown. The set ϴ is called the frame
of discernment containing the possible answers to the question. Ω is interpreted as the set
of possible interpretations allowed from the light of the available information. Exactly
one of the elements must be the correct interpretation, but it is unknown which
one. Furthermore, the assumption that not all possible interpretations are equally likely
induces the known probability measure P on Ω. In the simplest case, one can assume that
if ω is the correct interpretation, then the correct answer must be within some
nonempty subset ( ) of ϴ, the focal set of interpretation.
Falgin and Halpern (Falgin & Halpern, 1992) introduced a probabilistic
approach to deal with uncertainty by using the standard mathematical notions of inner
measure and outer measure induced by the probability measure. Inner measures induced
by probability measures turn out to correspond in a precise sense to DS belief functions.
This can be viewed as a special case of Dempster’s structure at least in the finite case.
35
Since the evolution of DST, various influential researches have been
performed in order to have complete understanding and to comprehend its merits as well
as demerits. These researches can mostly be divided into a number of broad categories
such as general issues, fuzzification of DST, DST in decision-making and optimization,
DST for managing uncertainty in knowledge-based systems, different application areas
and many more. Some of the pioneer researches on the general issues of DST include the
explanations of DST and contribution to further extension (Smets, 1994), measures of
uncertainty in DST of evidence (Klir, 1994), updating the beliefs in belief function theory
(Dubois & Prade, 1994), comparative beliefs (Wong et al. 1994), efficient
implementation of DST (Xu & Kennes, 1994), belief functions and parametric models
(Shafer, 2008) etc.
DM is another important area that interests the DST researchers. Strat
(Strat, 1994) offered a probabilistic interpretation of a simple assumption that
disambiguates decision problems represented with decision functions. He also showed
how the decision analysis methodology frequently employed in probabilistic reasoning
can be extended for use with belief functions. Furthermore, Shenoy (Shenoy, 1994)
described how DST of belief function can fit in the framework of valuation-based
systems (VBS) that initially served as a framework for managing uncertainty in expert
systems.
DST is also used in sensor fusion and multi-target tracking in clutter
which has been shown to be very challenging for both of track association and the
estimation of the target’s state (Dallil et al., 2013). Furthermore, DST has also been used
36
in neural network classifier (Denoeux, 2000). Also, DST has been utilized in the process
of sensor fusion in autonomous mobile robots (R. Murphy, 1998). This information
obtained from the classifier may be used to implement various decision rules allowing for
ambiguous pattern rejection and novelty detection. The outputs of several classifiers may
also be combined in a sensor fusion context, yielding decision procedures which are very
robust to sensor failures or changes in the system environment. Recently, a combination
of fuzzy inference system and DST has been proposed by Ghasemi et al. and applied to
brain Magnetic Resonance Imaging (MRI) for the purpose of segmentation where the
pixel intensity and spatial information are used as features (Ghasemi et al., 2013).
Another area of utilization of DST is in wireless networks. Fragkiadakis et al.
(Fragkiadakis et al., 2013) has presented an intrusion detection algorithm using DST to
detect physical layer jamming attacks in wireless networks. In addition to these,
numerous fields such as weighting method for context inference, service-aware
computing, image processing, remote sensing data fusion have been contributed by
researches of DST.
2.6.3 PROSPECT THEORY
Until the end of 1970s, irrational behavior was believed to be chaotic and inappropriate
for modeling. Till this period, the normative expected utility theory (EUT) was the best
suited model of descriptive behavior and decision-making was highly dominated by EUT.
At this point, Kahneman and Tversky proposed a groundbreaking theory that could
incorporate irrational behavior in an empirically realistic manner. This theory was named
‘Prospect Theory’ by the authors. It is to be noted that their work on Prospect Theory is
37
the second most cited paper in economics during the period, 1975-2000. Kahneman was
awarded the 2002 Nobel Memorial Prize in Economics for his work in Prospect theory.
Prospect Theory (PT) is a theory where psychological insights are
considered. The authors have demonstrated in numerous highly controlled experiments
that most people systematically violate all of the basic axioms of subjective expected
utility theory in their actual decision-making behavior at least some of the time. These
findings were contrary to the normative implications inherent within classical subjective
expected utility theories. Prospect theory is based on psychophysical models.
Traditionally, psychophysics investigates the precise relationship, generally expressed
mathematically, between the physical and psychological worlds. Kahneman and Tversky
applied psychophysical principles to investigate judgment and decision-making. In reality,
people’s decisions are made according to how their brains process and understand the
information and not only on the basis of the inherent utility that a certain option possesses
for a decision-maker. Therefore, Kahneman and Tversky tried to show in their work that
normative and descriptive theories cannot be combined into a single, adequate model
such as von Neumann and Morgenstern attempted to prove and the argued that normative
theories actually fail to offer a satisfactory understanding of human’s decision behavior.
The formulation of PT in Kahneman and Tversky (Kahneman & Tversky,
1979) is concerned with prospects of the form (x,p;y,q) which have two non-zero
outcomes. In such a prospect, one receives outcome x with probability p and y with q and
nothing with probability 1-p-q, where p q 1. The evolution of PT was started by
criticizing three tenets on which the EUT is based.
38
(i) The overall utility of an alternative or a prospect (outcomes of an alternative with a
probability distribution) is the expected utility of its outcomes,
(ii) The domain of utility function is final states including the initial asset rather than
gains and losses, and
(iii) Risk aversion in EUT leads to a concave utility function.
Then, the theory pointed out that EUT could not explain the following three effects due to
the tenets:
i) Certainty effect, which means people overweight outcomes that are considered
certain, relative to outcomes which are merely probable. This effect leads decisionmakers to risk aversion in the positive domain of utility. It also denies conventional
principles such as the sure-thing principle (independence axiom) and the substitution
axiom used in EUT. Independence means if an agent is indifferent between simple
prospects A and B, the agent is also indifferent between A mixed with a random simple
prospect C with probability p and B mixed with C with the same probability p. The
substitution axiom of utility theory asserts that if B is preferred to A, then any
(probability) mixture (B,p) must be preferred to the mixture (A,p).
ii) Reflection effect, which represents that the preference in the negative domain of
utility is the mirror image of the preference in the positive domain. People's behaviors in
the negative domain are "risk seeking" instead of "risk averse", and the utility function is
convex instead of concave.
iii) Isolation effect: when each alternative is composed of several components,
people tend to disregard the common components and focus on the difference. This
39
behavior may produce different choices from those by EUT, which violates the basic
supposition of EUT that choices are determined solely by the probabilities of the final
states.
Prospect Theory’s central prediction is that choices between uncertain
outcomes are determined by the combination of an outcome’s apparent value (predicted
by the value function) and the importance or weight assigned to a particular outcome
(called the weighting function). The value function v assigns a gain or loss x of each
outcome a number v(x) which reflects the subjective value of that outcome. Furthermore,
the weighting function π associates with each probability p a decision weight π(p)which
reflects the impact of p on the over-all value of the prospect. Therefore, the prospect
theory evaluates an alternative ai by
U prs (ai ) ( p j )v(uij* )
(15)
j
where p j P( s j ) is probability of state s j and uij* u (ai , s j ) u0 . u (ai , s j ) is utility
given ai and s j . u 0 is a reference point, which usually represents the current asset.
Thus, uij is a gain if it is positive, and a loss if it is negative. Clearly, it is not an
evaluation of the final state.
A critical aspect of the successful application in PT, specifically in
quantitative applications, is the form of value function and weighting function. The value
40
function mentioned by Tversky and Kahneman (1992) states, if preference homogeneity
holds, then the value function has the power form:
x , if x 0
v( x)
( x) , if x 0.
(16)
with loss aversion implying that λ >1. Here, α < 1 and x is either gain or loss in utility.
For α < 1, the value function exhibits risk aversion over gains and risk seeking over
losses. Furthermore, if λ the loss-aversion coefficient is greater than one, individuals are
more sensitive to losses than gains. Preference homogeneity is a necessary and sufficient
condition for the preferences given in Eq.(16) that gives rise to an effective value
function. It has been shown by formal proof that the probability weighting function for
losses must be the same as that for gains. These claims are proved to be consistent with
the empirical evidence (Tversky & Kahneman, 1992). Therefore, the choice of Eq. (16)
and Eq. (17) as value function and weighting function, respectively in our DM problem is
justifiable.
( p)
p
p
(1 p)
1
,
0 1.
(17)
The values of the co-efficient (α, λ, γ) of value function and weighting function used in
our research will be discussed in later chapters.
In the original paper of PT, value function and weighting function have
been shown the following properties in general.
41
The value function v(x), where x is gain or loss in utility measured from a
reference point,
Is concave for gains, and convex for losses
The value function is steepest near the point of reference:
Sensitivity to losses or gains is maximal in the very first unit of
gain or loss
Is steeper in the losses domain than in the gains domain
Suggests a basic human mechanism (it is easier to make people
unhappy than happy)
Thus, the negative effect of a loss is larger than the positive effect
of a gain.
The weighting function is monotonically increasing and continuous, but is
not defined in the vicinity of zero and one. In the other area, it should satisfy the
following:
i) π(0)=0, π(1)=1;
ii) Sub-additive, namely (rp) r (p), 0 r 1,
iii) π(p)> p when p is small,
namely ( p) (1 p) 1, 0 p 1 ,
iv) Sub-certainty,
v) Sub-proportional, namely (pq) (p) (rpq) (rp) , 0 p,q,r 1
42
Figure 1 and Figure 2 show the value function and weighting function proposed in
original Prospect theory.
Figure 1: The hypothetical value function
Figure 2: Weighting Function
2.6.4 EXTENSIONS OF PROSPECT THEORY AND RELATED WORKS
The original Prospect Theory (PT) was improved by Tversky and Kahneman (Tversky &
Kahneman, 1992). In this version, the authors first used Quiggin’s rank dependence for a
43
theoretical correction in probability weighting. Later, they extended the theory from risk
to uncertainty using Schmeidler’s rank dependence (Wakker, 2011). This improved
version was named Cumulative Prospect Theory (CPT).
As in PT, the cumulative prospect theory (CPT) also includes a value
function and a weighting function. The value function v in CPT is steeper for losses than
for gains as it was mentioned in original PT. Also v is concave above the reference point
and convex below the reference point. The last two conditions imply that the impact of
changes becomes less important with the distance from the reference point. The first
condition implies that losses loom larger than gains (Tversky & Kahneman, 1992). The
value function in CPT exhibits reference independence, diminishing sensitivity and loss
aversion like PT. However, in CPT, the authors applied the cumulative probabilities
instead of individual probabilities in order to include nonlinear preferences. This
modification gives CPT the advantage of satisfying stochastic dominance and the ability
to be applied to any number of outcomes.
Apart from CPT, research endeavors on different sections of PT have been
observed in literature. Among them, loss aversion, selection of value function and
weighting function, values of co-efficient of value function and weighting function,
measurement of utility, reference dependence received much attention of the researchers.
44
Table 3: Estimates of the Loss-aversion Coefficient2
Loss aversion is one of the reasons for which people deviate from expected utility. Many
empirical studies have found evidence of loss aversion (Tversky & Kahnemam, 1991). It
can also explain a variety of field data (Camerar, 2000). Some important examples
include equity premium puzzle, asymmetric price elasticity etc. Furthermore, to measure
loss aversion, the utility for gains and for losses have to be determined completely. The
main problem in designing a method to measure utility completely is that prospect theory
assumes that people weight probabilities and that probability weighting for gains may be
different from probability weighting for losses. Wakker and Deneffe (Wakker & Deneffe,
1996) showed how utility can be measured for gains and losses separately under prospect
2
Source: M. Abdellaoui et al., Loss aversion under prospect theory: a parameter-free measurement,
Management Science, Vol. 53 (10), 1659-1674. (2007)
45
theory. A method to measure loss aversion has been proposed without making any
parametric assumptions (Abdellaoui et al. 2007).
Apart from loss aversion, many empirical studies have confirmed the
importance of loss aversion coefficients. Table 3 gives an overview of studies that have
estimated a loss aversion co-efficient.
Different studies, besides adopting different parametric assumptions about
utility and probability weighting, used different definitions of loss aversion. The
estimated values for the coefficient of loss aversion vary and hard to compare because of
the different assumptions and definitions used, and due to the reported median values and
other mean values. Moreover, the studies related to all the coefficients or the
interdependency of the value function and weighting function coefficients are rarely
performed (Abdellaoui et al. 2007). For all these reasons, we have studied this issue in
the research and we have used 48 combinations of the coefficients (λ, α, γ) based on the
previous studies for evaluating the alternatives. Finally, the values of λ, α, γ have been
explained that closely resemble natural behavior pattern of the decision-makers in our
area of interest. This topic has been thoroughly discussed in Chapter 3.
46
CHAPTER 3: NY-DDM: PROPOSED DESCRIPTIVE
DECISION-MAKING FRAMEWORK UNDER
UNCERTAINTY
The previous chapter elucidated the state of the art theories of DM under uncertainty and
pointed out the challenges for a standard descriptive DM framework. This chapter
describes the proposed descriptive DM framework under uncertainty, NY-DDM, in detail.
The proposed framework has been named as NY-DDM. In Section 3.1, an overview of
the entire framework or the decision-making process is presented. After that, each phase
is discussed eventually from Section 3.2 to 3.4. Furthermore, Section 3.5 discusses on
defining various decision attitudes which were unexplored in the previous researches of
the similar field.
3.1 OVERVIEW OF THE NY-DDM FRAMEWORK
The proposed model is designed in such a way that it can start from the very basic stage
of problem formulation under uncertainty. The problem under discussion has been named
as ‘Evidential Decision-making Problem (EDMP)’. Uncertainty is represented by using
the belief function of DST. This framework can be partitioned in the following few
phases: (i)Problem Formulation, (ii) Combination of Evidences, (iii) Approximation of
Probability from bba, (iv) Applying Prospect Theory and (v) Decision-making. The work
flow diagram is illustrated in Figure 3.
47
PHASE 1: Formulation of EDMP:
Set of Alternatives, Set of States of nature, Set of Outcomes and
corresponding Utilities, BBA.
PHASE 2: Combination of Evidences
PHASE 3: Approximation of Probabilities from bba
Converting the EDMP into a DM under Risk by approximating
probabilities from bba.
PHASE 4: Applying Prospect Theory
Set appropriate coefficient values for value function and weighting
function and obtain the overall values of each alternative in different
attitudes of decision-maker toward uncertainty.
PHASE 5: Decision-making under Uncertainty for Different
Attitudes of Decision-maker.
Figure 3: Overview of NY-DDM Framework
First step of this entire endeavor is to closely observe the decision-making
problem, figure out the uncertainties and find out all the alternatives, states and outcomes
as well as the basic belief assignments. It is certainly possible to have more than one
sources of information for the hypothesis in question. In this case, it is required to
combine the evidences or support for the claim of the hypothesis which constitutes the
utilization of any suitable evidence-combination rule. The combination of evidences is
carried out in the second phase of the NY-DDM. If there is only one information source,
there may not be any need to combine the evidences. Therefore, it is possible to go to
Phase three directly from Phase one as shown in Figure 3.
48
The third phase deals with the approximation of probabilities from bba
which is possible to perform according to the attitudes of the decision-maker. Initially,
we have worked with three basic attitudes: Equative, Pessimistic and Optimistic. But,
later in this chapter, it has been explained how to achieve various attitudes only by
changing the parameters of OWA operators. The next phase constitutes of the application
of prospect theory. In order to obtain reliable decisions, it is essential to set the coefficient values that correctly reflect the decisions regarding treatment of the lung cancer
patients. The last phase of the process provides the decisions based on attitudes toward
uncertainty.
In this chapter, the basic phases of NY-DDM have been illustrated: Phase
1, Phase 3, Phase 4 and Phase 5 in order to give a coherent interpretation of the general
framework to the readers along with adequate proofs and examples. Phase 2, combination
of pieces of evidences, will be vividly discussed in Chapter 4.
3.2 EVIDENTIAL DECISION-MAKING PROBLEM
An Evidential Decision Making Problem (EDMP) can be defined in different ways. But
we have followed the definition that has similarity to the conventional definition of
decision-making in different literature where A= {ai| i=1….N} is the set of alternatives,
S={sj | j = 1,...,M} is the states of nature and the outcome set O {oij oij f (ai , s j )} and
the utility function uij u (oij ) . The uncertainty is defined on the states of nature
49
S : m( B) where BS. This m(B) is the basic belief assignment (bba). Any EDMP can be
explained by Table 4.
Decision-making is practically performed under the realm of certainty,
under risk, under uncertainty and under ignorance as mentioned in Section 1. Our
definition of EDMP satisfies all of these realms. If all bbas of alternatives in EDMP are
certain, EDMP shrinks to a decision-making problem with certainty. If all focal elements
of all bbas are singleton, then EDMP becomes a decision-making problem under risk. If
all bbas are vacant, EDMP is a decision-making under ignorance. Then, otherwise, it is a
decision-making problem under uncertainty.
Table 4: Evidential Decision-making Problem (EDMP)
s1
……
sj
…
sM
a1
o11
……
o1 j
…
o1M
:
:
:
……
:
ai
:
aN
oi1
:
o N1
:
……
oij
:
……
:
o Nj
:
…
:
oiM
:
o NM
3.3 CONVERTING AN EDMP INTO DM UNDER RISK
The approach we have considered to solve a DM problem under uncertainty is the
probability approximation. We have approximated the bba by probability. A seemingly
standard way of approximation from bba to probability is the one by equidistribution.
Equidistribution is a way to obtain an approximate probability distribution from a mass
50
function m. Let focal elements of m be S1....S K . Then, equidistribution derives an
approximate probability distribution Papp ( s j ) on S from m using the next equation.
Papp ( s j )
m( S k )
Sk
s j S k
(18)
When the mass function is defined on a total ordered set, or a set whose
elements have a numerical attribute as in this case, we could assign probability
distributions of the worst case and the best case which are consistent with the mass
function; the best case assigns the whole mass of S k namely m( S k ) , to the largest/best
(k )
(k )
element(s) sbest S k , and the worst case to the smallest/worst element(s) s worst S k .
(k )
(k )
Note that sbest and s worst depend on alternative ai , because utility is a function of a
pair of ai and s j .
Similarly to the generalized expected utility theory, we can consider
pessimistic/ optimistic probability distributions. These can be expressed by the following
equations:
Ppes ( s j | a j ) m pes ( s j , S k | ai )
(19)
k
Popt ( s j | a j ) mopt ( s j , S k | ai )
(20)
k
m( S k )
, if s j S k , u (oij ) Min u (oih )
m pes ( s j , S k | ai ) N pes
,
s h S k
0, otherwise.
51
(21)
m( S k )
, if s j S k , u (oij ) Max u (oih ),
mopt ( s j , S k | ai ) N opt
s h S k
0, otherwise.
where N pes
(22)
and N opt are numbers of s j S k satisfying u (oij ) Min u (oih ) and
shSk
satisfying u (oij ) Max u (oih ) , respectively . In Eq.(19) and Eq.(20), Ppes ( s j | ai ) and
shSk
Popt (s j | ai )
are used instead of Ppes ( s j ) and Popt ( s j )
respectively, because the
pessimistic/ optimistic probability distribution depends on the alternative the decision
maker chooses. Ppes ( s j | ai ) is the probability distribution in the case where we assume
(k )
that s worst happens with a probability equal to m( S k )
, when the decision-maker
N pes
chooses ai.
Then, the expected utilities using the approximated probability
distributions of Eq. (18) – Eq. (20) are given by the following equations, respectively:
U app (ai ) Papp ( s j )uij
(23)
j
U pes (ai ) Ppes ( s j ai )uij
(24)
j
U opt (ai ) Popt ( s j ai )uij
(25)
j
It can be proved that decision-making with Eq. (23), Eq. (24) or Eq.(25) is
equivalent to one with equative, pessimistic or optimistic approach of GEU and OWA,
respectively. Instead we show that any solution with OWA operator can be transformed
52
into a DM under risk below. This fact lets us discuss descriptive model of DM under
uncertainty in the framework of DM under risk.
Let us represent V (ai , S k ) in Generalized Expected Utility (GEU) by the
OWA operator. It can be written as:
V ( ai , S k )
(k )
where, wl
l 1, N k
(i , k )
wl(k ) uind
(l )
(26)
[0,1] is the weight, N k S k is the cardinality of S k , and ind(l) is the
index of s j where uij u(ai , s j ) is the l-th largest in S k . 1 ind (l ) N k ,
(i , k )
w1 ..... wN k 1. The term u ind
(l )
is the l-th largest utility among uij u(ai , s j ) ,
s j S k . By substituting Eq. (4) for V (ai , S k ) in Eq.(26), we get the following.
U geu (ai )
m(S k )
k 1, K
m( S k )
k 1, K
j 1, N k 1, K
(
wl
l 1, N k
j 1, N
( k ) (i , k )
uind (l ) )
w ( k ) 1
ind
m( S k ) w ( k ) 1
ind
( j)
( j)
uij
uij
j uij
(27)
j 1, N
where w0k 0 .
l , if j ind (l ), 1 l N k
ind 1 ( j )
0, otherwise
53
(28)
j m( S k ) w(k ) 1
ind
k
It is clear that j [0,1] and
( j)
(29)
j j 1 . This means that we can give a
probability distribution j by Eq. (29) so that Eq. (4) and Eq. (27) would have the same
values. In addition, it means that a DM problem under uncertainty with GEU and OWA
operator could be transformed into a DM problem under risk equivalent to the original
one. This fact can be explained by a simple numerical example.
We suppose the following case:
States of nature: S = {s1, s2, s3}, bba: m (S1) = 0.2, m (S2) = 0.3, m (S3) = 0.4, m (S4) = 0.1,
where S1 = {s1}, S2 = {s1, s2}, S3 = {s2, s3}, S4 = {s1, s2, s3} and the utility of alternative
a0 : u(a0 , s1 ) u01 4 , u(a0 , s2 ) u02 6 , u(a0 , s3 ) u03 10 .
Then, the weights of the OWA operator, W (w1, w2 ...wn ) where n N k S k
(k=1,2,3,or 4), is supposed to be given as: W=(1.0), W=(0.6,0.4), W=(0.6, 0.24, 0.16)
when n = 1, 2, 3 respectively. Then, V (a0 , S k ) used in GEU is obtained using Eq.(26)
as follows:
(0,1)
V (a0 , S1 ) w1(1)uind
(1) 1.0 u01 4.0
(0,2)
(2) (0,2)
V (a0 , S 2 ) w1(2)uind
(1) w2 uind ( 2) 0.6u02 0.4u01 0.6 6 0.4 4 5.2
(0,3)
(3) (0,3)
V (a0 , S3 ) w1(3)uind
(1) w2 uind (2) 0.6u03 0.4u02 0.6 10 0.4 6 8.4
54
(0,4)
( 4) (0,4)
( 4) (0,4)
V (a0 , S 4 ) w1( 4)uind
(1) w2 uind ( 2) w3 uind (3) 0.6u03 0.24u02 0.16u01
0.6 10 0.24 6 0.16 4 8.1
From Eq.(4) we get the next GEU value:
U geu (a0 ) 0.2 4.0 0.3 5.2 0.4 8.4 0.1 8.1 6.5 .
The value of U geu (a0 ) could be also obtained using Eq.(27) as proved
above. First, we calculate the values of Eq.(28). In the case of S1 where N1 1 ,
ind 1 (1) 1 and ind 1(2) ind 1(3) 0 . For S2 where N2 = 2, ind 1 (1) 2 , ind 1 (2) 1 ,
1
1
1
1
and ind (3) 0 . For S3 where N3 = 2, ind (1) 0 , ind (2) 2 , and ind (3) 1 .
1
1
1
Finally, in the case of S4 where N4 = 3, ind (1) 3 , ind (2) 2 , and ind (3) 1 .
Then, from Eq.(28) we get,
1 m( S k ) w( k ) 1
ind
k
(1)
m( S1 ) w1(1) m( S 2 ) w2( 2) m( S 3 ) w0(3) m( S 4 ) w3( 4)
0.2 1.0 0.3 0.4 0.4 0.0 0.1 0.16 0.336
2 m( S k ) w( k ) 1
ind
k
( 2)
m( S1 ) w0(1) m( S 2 ) w1( 2) m( S 3 ) w2(3) m( S 4 ) w2( 4)
0.2 0.0 0.3 0.6 0.4 0.4 0.1 0.24 0.364
3 m( S k ) w( k ) 1
ind
k
(3)
m( S1 ) w0(1) m( S 2 ) w0(2) m( S 3 ) w1(3) m( S 4 ) w1( 4)
0.2 0.0 0.3 0.0 0.4 0.6 0.1 0.6 0.300
Now
we
can
get
the
same
value
as
U geu (a0 ) j u0 j 0.336 4 0.364 6 0.300 10 6.5 .
j
55
shown
above:
From the above proof and the numerical example it is evident that any
decision attitude represented by OWA operator can be transformed into DM under risk.
In addition to this, numerous attitudes are possible to explain simply by changing the
parameters of OWA operators. Therefore, if the attitude is represented by OWA operators,
any decision attitude toward uncertainty can be dealt with by our DM framework which
is a unique characteristic of our approach.
3.4 APPLYING PROSPECT THEORY
After converting the EDMP into DM problem under risk in three example attitudes,
Prospect Theory (PT) is applied to the transformed DM problems. The selection of value
function and weighting function has been discussed earlier in Chapter 2 (Section 2.6.3).
This section discusses on the selection of the parameter values of the value function and
weighting function. The value function v(x) uses α, explaining risk attitude as well as
defining the curvature of the value function and λ, called the loss aversion coefficient and
γ measures the degree of curvature of the weighting function (Trepel, 2005). It is widely
known in different literature that α to be less than 1 whereas λ>1 resembles loss aversion.
To the best of our knowledge, there is no agreed upon value combination for these two
very important coefficients apart from the empirical examples performed in the literature
(Please refer to Table 3). Moreover, the loss aversion and risk attitudes should vary from
domain to domain. For example, we can in no way state that the loss aversion and risk
attitude of money/ finance domain match with those of health. In our research, we have
taken this important issue into account and used 48 combinations of (λ, α, γ) for
56
evaluating the alternatives and explained the values of λ, α, γ that closely resemble
natural behavior pattern of patients. The following values have been used to perform a
certain experiment in order to achieve trustworthy values of the three coefficients (λ, α, γ)
present in v(x) and π( ) for the domain of lung cancer treatment decision-making.
λ= 2.17, 2.25, 3.06, 4.8; α =0.88, 0.52, 0.32, 0.10; γ =0.25, 0.5, 0.75.
From the above mentioned values we obtained 4*4*3=48 combinations of
the coefficient trio. The resultant combination is given in Table 5.
Table 5: Values of λ, α, γ
λ
3.06
4.80
α
0.32
0.10
γ
0.25~0.75
0.25~0.75
According to PT, losses loom larger than equivalent gains. This
phenomenon is modeled by a value function with a loss limb that is steeper than the gains
limb (the parameter λ>1explains this). However, when a decision-maker is in the health
domain, it is typical that losses have at least twice (λ>2) the impact of equivalent gains
(Trepel, 2005). Our research assumes that losses have more than three times or even more
(λ>3 or λ>4) impact on a decision-maker while making decisions about treatment.
Therefore, the chosen values of λ are 3.06 and 4.80.
Furthermore, the co-efficient α measures the degree of curvature of the
value function for gains and losses. Thus, for a smaller value of α<1, the value function
for gains (losses) is increasingly concave (convex). Therefore, we have assumed that 0.32
or 0.10 as the value of α.
57
The tendency of overweighting low probabilities and underweighting
moderate to high probabilities is a characteristic of the inverse-S-shaped weighting
function of PT. This phenomenon is modeled by γ<1 in Eq. (17). Therefore, this research
has kept the value of γ within the range of 0.25~0.75. In the example application
explained later in Chapter 4, γ =0.5 has been used.
Using the values of Table 5, the alternatives have been evaluated and the
highest valued alternative has been chosen as a decision in each attitude of decision
maker.
3.5 HUMAN ATTITUDES AND DECISION-MAKING
Attitudes are defined as a mental predisposition to act that is expressed by evaluating a
particular entity with some degree of favor or disfavor. Individuals generally have
attitudes that focus on objects, people or institutions. Attitudes are also attached to mental
categories. Mental orientations towards concepts are generally referred to as values.
Attitudes are comprised of four components: cognition, affect, behavioral intentions, and
evaluation. It is no surprise that the topic of decision-making under risk and uncertainty
has fascinated observers of human behavior. From philosophers charged with providing
tactical gambling advice to noblemen, to economists charged with predicting people’s
reactions to tax changes, risky choice and the selection criterion that people seek to
optimize when making such decisions has been the object of theoretical and empirical
investigation for centuries. Numerous researches on cognition, human psychology, risk
and uncertainty have been performed for a long time. Recent researches on
58
neurobiological evidences have also improved our understanding of human valuation
under uncertainty.
PT neatly accommodates the Allais paradox as well as the fourfold risk
attitudes discussed by previous researches. The fourfold pattern of risk attitudes indicates
an individual is risk-seeking over low-probability gains and high probability losses along
with risk-averse for high-probability gains and low-probability losses.
PT explains
attitudes towards risk via distortions in the shapes of the value and weighting functions
that reflect the psychophysics of diminishing sensitivity. According to PT, people are
generally risk averse in the domain of gains and risk seeking in the domain of losses. The
value function of PT is steeper for losses than gains reflecting the ‘loss aversion’ property
and indicates losses loom larger than gains (Trepel, 2005).
Considering the importance of human attitude in decision-making, this
research has incorporated human attitude within the proposed DM framework in order to
explain decision variety depending on attitudes toward uncertainty. In the previous
section, equative, pessimistic and optimistic attitudes have been explained (Eq. 19-22).
Notably, Ppes ( s j | ai ) ( Popt ( s j | ai ) ) is the probability distribution in the case where we
(k )
(k )
assume that the state s worst ( sbest
) happens with a probability equal to m( S k ) , when the
decision-maker chooses the alternative ai . The term u (oih ) represents the utility value of
outcome oih . We can call Eq. (21) and Eq. (22) as fully pessimistic and fully optimistic
respectively when the entire value of m( S k ) is being assigned to the worst and best
elements. In a fully pessimistic case, a pessimistic decision-maker assumes that the worst
59
outcome will occur. If a pessimism index is considered within the range of [0, 1], a fully
pessimistic decision-maker holds the value of 1 in the index. Similarly, if an optimism
index is considered ranging from 0 to 1, a fully optimistic decision-maker holds the value
of 1 in the optimism index.
In reality, humans have different emotional states rather than only being
fully pessimistic, fully optimistic or being neutral and these states have immense
influence to solve DM problems. Empirical studies have even proved this fact. Moreover,
researchers have attempted to provide suggestions concerning the neural bases of PT
(Trepel, 2005). This research has also attempted to deduce human attitudes in between
pessimism and optimism by mathematical expressions. These attitudes can be named as
near-pessimistic (optimistic), half-pessimistic (optimistic) and quarter-pessimistic
(optimistic) attitudes. For the near-pessimistic (optimistic) attitude, we assume that the
decision-maker holds the value of 0.75 in the pessimism (optimism) index and thus 0.75*
m(S k ) is assigned to the least/worst (highest/ best) outcome and 0.25 is distributed
among other states of the focal element. Similarly, for the half-pessimistic (optimistic)
attitude, we assume that the decision-maker holds the value of 0.50 in the pessimism
(optimism) index and thus 0.50* m( S k ) is assigned to the least/worst (highest/ best)
outcome and the other 0.50 is distributed among other states of the focal element. The
derived equations for different pessimistic attitudes are given below if S k 2.
0.75 * m( S k ) / N pes , if u (oij ) Min u (oih )
s h S k
mnear pes ( s j , S k | ai )
0.25 m( S ) ( S N
k
k
pes ), otherwise
60
(30)
0.50 * m( S k ) / N pes , if u (oij ) Min u (oih )
sh S k
mhalf pes ( s j , S k | ai )
0.50 m( S k ) ( S k N pes ), otherwise
(31)
0.25 * m( S k ) N pes , if u (oij ) Min u (oih )
sh S k
mqtr pes ( s j , S k | ai )
0.75 m( S k ) ( S k N pes ), otherwise
(32)
The equations of different optimistic attitudes can also be derived as follows:
0.75 * m( S k ) N opt , if u (oij ) Max u (oih )
s h S k
mnear opt ( s j , S k | ai )
0.25 m( S ) ( S N ), otherwise
k
k
opt
(33)
0.50 * m( S k ) N opt , if u (oij ) Max u (oih )
sh S k
mhalf opt ( s j , S k | ai )
0.50 m( S k ) ( S k N opt ), otherwise
(34)
0.25 * m( S k ) N opt , if u (oij ) Max u (oih )
sh S k
mqtr opt ( s j , S k | ai )
0.75 m( S k ) ( Sk N opt ), otherwise
(35)
In addition, 2nd-(worst/best) attitude can be explained by assuming that the
state s 2(knd) worst ( s 2(knd) best ) happens with a probability equal to m( S k ) , when the
(k )
decision-maker chooses the alternative ai . Median attitude assumes s med
happens with a
probability equal to m( S k ). In this way, this research models different decision attitudes
toward uncertainty by using the features of ordered weighted averaging (OWA) operator.
This feature of defining various attitudes was absent in the previous researchers
61
mentioned in Chapter 2 and therefore, can be considered as one of the contributions of
this research.
3.6 CONCLUSION
In this chapter, we have discussed a unique theoretical approach of descriptive decisionmaking under uncertainty. This approach has proposed a descriptive DM framework
under uncertainty using DST and PT. We have shown that, this framework is able to
successfully express the uncertainty of the DM problem in question by bba, convert it to
a DM problem under risk by probability approximation from bba and analyze how people
make decisions in reality in varying attitudes. Through this framework, it has been made
possible to demonstrate human decision attitudes by mathematical expressions. Upon
discussing the significance of the coefficients of value function and weighting function of
PT in expressing risk attitude as well as loss aversion in DM under risk, this chapter
derives the coefficient values for the health domain. The next chapter focuses on the
aggregation of pieces of evidences stemming from multiple sources which is another
major part of NY-DDM framework. The significant features of the proposed descriptive
DM approach have allowed us to explain a practical example of NY-DDM in the area of
treatment decision-making of a lung cancer patient that has been discussed in Chapter 5.
62
CHAPTER 4: COMBINATION OF EVIDENCES AND THE
PROPOSED METHOD OF EVIDENCE COMBINATION
The prime focus of this chapter as its name implies is the approaches of evidence
combination. The question how to combine the obtained information stemming from
distinct sources is a topic of great interest in the community dealing with uncertainty.
Although, considerable research endeavors have been performed related to the rules of
evidence or information combination, this study proposes an improved combination rule
that suits the practical purpose of combining evidences in the arena of lung cancer staging.
4.1 NECESSITY OF EVIDENCE COMBINATION
Evidence combination implies the aggregation of uncertain pieces of information issued
from different sources dealing with the similar question of interest. Evidence is most
commonly thought of as a proof supporting a claim or belief. When multiple pieces of
evidence or multiple bodies of evidence can be derived from distinct sources of
information, it is essential to combine the evidences using combination rules in order to
meaningfully summarize and simplify the corpus of data. These combined evidences can
lead us to a decision for the particular question of interest. The present study investigates
the efficacy of evidence combination in the context of lung cancer staging which is one
of the most frequent causes of cancer death worldwide and has been chosen as a practical
example of decision-making under uncertainty in the research. Its global occurrence has
been steadily increasing during recent decades. Staging of lung cancer is of paramount
63
importance because decisions on treatment alternatives are greatly dependent on the stage
of the disease. Usually, staging is categorized as clinical and pathological. Clinical stage
is decided upon the accumulation of physical examination results, imaging studies and
tissue diagnosis results. This chapter has focused to identify the clinical stage of a lung
cancer patient combining the evidences collected from physical examination, imaging
and tissue diagnosis results using the evidence combination rules in order to decide the
appropriate treatment. The ‘combination of evidence’ has been mentioned as Phase 2 in
the complete NY-DDM framework.
One of the central issues in any decision-making (DM) problem is the
representation of the knowledge about uncertain variables where external variables lead
to different outcomes. Among various formalisms, probability theory, possibility theory,
Dempster-Shafer Theory of evidence (DST) and fuzzy measures are very common. These
representations are useful in different cases rather than being always competitive.
Among them, DST can be a good candidate for representing uncertainty when we have
multiple information sources as well as the pieces of information contain some ignorance
about uncertainty. The reason why DST is useful in case of multiple information sources
is that the theory provides several approaches to combine multiple uncertain pieces of
information. These approaches can be viewed from different perspectives such as set
theoretic standpoint, reliability of the information sources etc. Dempster’s rule (Dempster,
1967) and some other conjunctive combinations such as Dubois and Prade (Dubois &
Prade, 1988), Hau and Kashyap (Hau & Kashyap, 1990), Yager (Yager, 1987) and Zhang
(Zhang, 1994) assumed the complete reliability of the sources. However, Dubois and
64
Prade’s disjunctive combination and Yamada (Yamada, 2008) considered the sources not
to be completely reliable and therefore Yamada proposed combination by compromise
(CBC). In addition to this, Murphy proposed another combination method where multiple
sources are reliable equally to some extent (Murphy, 2000).
In this chapter, we are interested to investigate how evidence combination
rules/methods can facilitate the staging procedure of lung cancer in order to reach an
appropriate treatment decision because selection of treatment is highly dependent on
staging of lung cancer. By utilizing the proposed descriptive DM approach (NY-DDM)
along with evidence combination, we would like to explain how lung cancer treatment
decisions vary depending on the patient’s attitude towards uncertainty. The importance of
multiple information sources is to be considered because in reality, evidences of lung
cancer are derived from more than one medical examination procedure such as symptom
assessment, laboratory tests, imaging and/or tissue diagnosis of the suspected area. Pieces
of information (or evidences) from multiple sources are then combined and staging
information is derived after this combination. Subsequently, cancer treatment decisions
are possible to make.
4.2 RELATED WORKS ON EVIDENCE COMBINATION RULES
As stated earlier, multiple pieces of evidences stemming from distinct sources can be
combined using combination rules of evidences. These sources provide different
assumptions for the same universal set and almost all combination rules assume that these
sources are independent.
65
Let X and Y be two hypotheses having a belief mass in the same universal
set. Combining X with Y includes two issues: how to generate the new hypotheses, and
how to distribute the combined mass to the new hypotheses. From a set theoretic point of
view, these combination rules occupy a continuum between conjunction and disjunction
(Sentz, 2002). More simply, X Y should be chosen if information sources are
completely reliable; X Y if at least one of the sources is reliable. However, many other
combination rules are possible between these two extremes: such as CBC (Yamada,
2008). CBC takes an intermediate approach between X Y and X Y . In this section,
we briefly discuss several (but not exhaustive) combination rules of evidences.
4.2.1 DEMPSTER’S RULE OF COMBINATION
Dempster’s rule of combination has been demonstrated in Section 2.6.1 by Eq. (13) and
Eq. (14). The rule is commutative, associative, but not idempotent. When there are more
than
two
bodies
of
evidences,
m1 ( X )m2 (Y ) and
X Y are
replaced
by
m1 ( X1 )m2 ( X 2 ).....mn ( X n ) and X1 X 2 ... X n .
Some of the well-known criticisms of Dempster's rule include the
assignment of total mass to minority opinion and thus the combination becomes
counterintuitive (Zadeh, 1984). This unnatural concentration of total mass is due to the
normalization of Dempster's rule. Moreover, this rule excludes elements even if the
elements are contained in many focal elements but absent in only one of them when more
than three bodies of evidence are combined (loss of majority opinion). Assignment of
unwarranted mass has also been pointed out by researchers, which is a phenomenon that
66
a subset with no mass before combination may have a mass after the combination
(Voorbraak, 1991). It occurs due to the assignment of combined mass to only X Y . Due
to all these important issues, researchers have come up with new ideas of improving the
Dempster's rule as well as formulating new rules of combining evidences.
4.2.2 YAGER’S MODIFIED DEMPSTER’S RULE
Yager (Yager, 1987) proposed a combination rule that assigns the mass to the universal
set Θ which was discarded in Dempster’s rule when intersection of X and Y is empty.
Yager’s combination is given in Eq. (36):
m1 ( X )m2 (Y ), if A ,
X , Y ; X Y A Ø
mY ( A)
m1 ( X )m2 (Y ), if A ,
X ,Y ; X Y or X Y Ø
(36)
This rule is commutative, but neither associative nor idempotent. It does not need
normalization and therefore it does not involve the problem of assigning total mass to
minority opinion. However, this has the following shortcomings: this rule lacks the
advantage of mass convergence; unwarranted mass might be assigned as well as majority
opinion might be lost which are also the laggings of Dempster’s combination rule. When
major parts of focal elements are contradictory, which means X Y Ø , this rule can
produce almost insignificant mass, which might lead to incorrect decision (Yamada,
2008).
67
4.2.3 DUBOIS AND PRADE’S DISJUNCTIVE CONSENSUS RULE
According to Dubois and Prade, Dempster’s rule is not robust to inaccurate estimates of
uncertainty values and this lack of robustness is basically due to the normalization factor.
Dubois and Prade (Dubois & Prade, 1986, 1988) utilized a set-theoretic view and defined
their disjunctive consensus rule as follows:
m (C ) X Y C m1 ( X )m2 (Y ).
(37)
Since the union does not generate any conflict and does not reject any of the information
asserted by the sources, there is no need of any normalization procedure. This disjunctive
pooling is commutative, associative but not idempotent.
4.2.4 HAU AND KASHYAP’S RULE
Hau and Kashyap (Hau & Kashyap, 1990) proposed a combination rule similar to Yager's
rule, but it assigns the mass to the disjunction instead of the frame of discernment when
X Y is empty. Dubois and Prade also suggested the same combination previously
(Dubois & Prade, 1986). This can be represented by the following equation:
m1 ( X )m2 (Y )
m( A)
(38)
X ,Y ; ( X Y A)or ( X Y Ø X Y A)
where A Ø. It differs from Yager’s combination in a way that it does not abandon the
admissibility of evidence X, Y and assumes that either of these is correct in case of
contradiction. This rule is commutative, but not associative nor idempotent.
68
4.2.5 MURPHY’S RULE OF COMBINATION
Murphy (Murphy, 2000) pointed out another problem of Dempster's rule: failure to
balance multiple evidences illustrating some proposed solutions. According to Murphy,
averaging is the best solution of normalization problems because it provides an accurate
record of contributing beliefs. But it does not offer convergence toward certainty. In
order to provide convergence with an averaging method, Murphy proposed a technique
that applies the simple average method, and the obtained bba is combined with itself by
Dempster's combination rule. This method is free from too much dependence on a single
piece of conflicting evidence which might cause the disappearance of majority opinion.
Murphy's rule is commutative, but it is neither associative nor idempotent.
4.2.6 YAMADA’S COMBINATION BY COMPROMISE
Yamada (Yamada, 2008) takes an intermediate approach of between X Y and X Y .
The basic idea is to share the mass m1 ( X )m2 (Y ) among subsets included in X Y .
Three ways of sharing have been considered by the author for avoiding excessive
complexity:
The mass is shared between X and Y.
The mass is shared between X Y and X Y .
C
C
The mass is shared among X Y C , X Y X Y , and X Y YX .
For the mass assignment of the third one in the above, X Y C is counted twice
because the intersection is the crossover part of X and Y. Therefore, it takes ½ of the mass
69
of m1 ( X )m2 (Y ) and X Y and Y X take the other ½ of the combined mass. The assignment
of the combined mass to C, X Y and Y X after the combination is as follows:
m(C )
m( X Y )
m(YX )
1
m1 ( X )m2 (Y )
2
(39)
m1 ( X )
1
1 m1 ( X ) 2 m2 (Y )
m1 ( X )m2 (Y )
2
m1 ( X ) m2 (Y ) 2 m1 ( X ) m2 (Y )
m2 (Y )
1
1 m1 ( X )m2 (Y ) 2
m1 ( X )m2 (Y )
2
m1 ( X ) m2 (Y ) 2 m1 ( X ) m2 (Y )
(40)
(41)
When C Ø , the mass that should have been assigned to C is distributed to X Y and Y X
based on the weights of m1(X) and m2(Y) and when Y X Ø , the mass that should have
been assigned to Y X is allocated to C. The CBC is commutative; however it neither
satisfies the idempotent law nor the associative law.
Now let’s explain CBC for three mass functions m1 ( X ) , m2 (Y ) and
m3 (Z ) . In this case, we can make 7 partitions as follows: C X Y Z ,
C XY X Y Z ,
CYZ Y Z X ,
CZX Z X Y ,
X YZ X (Y Z ), Z XY Z ( X Y ) .The mass
YZX Y (Z X ),
m1 ( X )m2 (Y )m3 (Z ) is assigned to C,
C XY , CYZ , CZX , YZX , X YZ , Z XY . C is counted 3 times and each of C XY , CYZ , CZX is
counted twice that constitute 3/12 and 2/12*3 of the mass of m1 ( X )m2 (Y )m3 (Z )
respectively. The equations become very complex when any one or more of C, C XY ,
CYZ , CZX , YZX , X YZ , Z XY is empty.
70
In this chapter, we have put emphasis on Dempster’s rule, Hau and
Kashyap’s combination rule, and Murphy’s combination rule from the viewpoint of lung
cancer stage diagnosis within the framework of descriptive DM under uncertainty
described in the last chapter. The reason why Hau and Kashyap's rule is chosen as well as
the standard Dempster's rule is that it looks one of the most rational rules from the settheoretic view; it assigns the whole mass to one of the two extremes, X Y and X Y ,
depending on the situation whether we could rely on both of them or one of them,
respectively. Murphy's rule is chosen because it solves three major drawbacks of
Dempster's rule; total mass to minority opinion, loss of majority opinion and unwarranted
mass (Please refer to (Yamada, 2008) for the detailed comparison of the combination
rules of this section). In addition, Murphy's rule can be modified easily so that it uses a
weighted average instead of the simple average, when reliabilities of the sources are
different. Yamada's CBC has not been used, because the focal elements after the
combination become so many in the case of multiple sources that the intuitive
understanding of the results may become very difficult. Due to these reasons, we are
interested to compare three aforementioned rules in the next section and observe which
rule provides satisfactory staging information of lung cancer after combination of
evidences.
71
4.3 PROPOSED RULE OF EVIDENCE COMBINATION FOR LUNG CANCER
DIAGNOSIS AND STAGING
This section explains how to take advantage of the rules of evidence combination in lung
cancer diagnosis and staging mechanism. At first, a few scenarios in lung cancer
diagnosis have been explicated where Dempster’s rule of combination fails to generate
proper staging information of lung cancer for accelerating the treatment decision.
Afterward, the solutions of the problems are discussed.
As stated earlier, a physician at first checks the symptoms and signs of
lung cancer in course of diagnosis. Some well-known symptoms are cough, weight loss,
dyspnoea, chest pain, bone pain, clubbing, hoarseness, signs suggesting metastasis (for
example, in brain, bone, liver or skin), persistent haemoptysis in a smoker or an exsmoker older than 40 years, persistent cervical / supraclavicular lymphadenopathy. But as
a matter of fact, the symptoms and signs of lung cancer are difficult to be distinguished
from those of other lung diseases. Therefore, when one or some of these symptoms are
visible in a patient, he is suggested to have a Chest X-ray to get further details. Yet, there
may be suspicion of lung cancer even after having chest X-ray. In order to be certain,
clinical examinations, blood tests, CT scan are requested to perform by the physician.
Whenever the clinical examinations cannot produce certain decisions, further tests are
carried out such as CT scan, bronchoscopy, sputum cytology or needle biopsy etc. During
this diagnosis process, cancer staging is also carried out in order to decide treatment and
prognosis. Table 6 shows the categories of examination a patient usually goes through
during lung cancer diagnosis and staging.
72
Table 6: List of Information Sources for Lung Cancer Staging and Corresponding
Reliabilities
Categories
Diagnosis and Staging Information sources
Physical
Examination
Laboratory tests
and Imaging
Symptoms and Signs
Tissue Diagnosis
Reliability of
the sources
0.30
Sputum Cytology, Tumor markers, Chest X-ray, CT,
PET , PET-CT, SPECT, FDG- PET, Ultrasound.
0.60
Bronchoscopy,Mediastinoscopy,
thoracoscopy,Transthoracic needle aspiration
(TTNA), Fine Needle biopsy (FNA), Endoscopic
ultrasound (EUS),
VATS, Thoracoscopy.
0.95
At this point, two scenarios regarding lung cancer diagnosis are examined
as given in Example 4.1 and 4.2 that explain the drawbacks associated with Dempster’s
combination rule. Let S be the frame of discernment consisting of all of the stages of lung
cancer as well as no cancer. Therefore, S = {nc, oc, s0, s1a, s1b, s2a, s2b, s3a, s3b, s4}
where nc implies no cancer, oc implies occult stage, s0 implies stage 0 and so on. The
significance of all the stages will be discussed in detail in Chapter 5.
Example 4.1: The primary symptoms of lung cancer are hard to distinguish as many
other lung diseases (such as tuberculosis) share almost similar symptoms. Therefore,
when a patient shows several of these symptoms that match the ones with that of lung
cancer, the belief assignments provided by the physical examination are: m1({s0})=0.30,
m1({oc})=0.70. Next, the imaging results express higher belief for ‘no cancer’. So,
m2({nc})=0.70, m2(S)=0.30. Combining m1 and m2 by Dempster’s rule, we get:
m({s0}) 0.3, m({oc}) 0.7 where the information of m2 disappears. Later on, from
73
tissue diagnosis, it was suspected that the patient may not possess any cancer; even if
there is any minor presence of such, it is unlikely to be a major one. This can be stated as:
m3({nc})=0.90 and m3 ({nc, s0})=0.10. If the evidences gathered from three different
sources of information are combined, the combined mass becomes m({s0})=1.0 .
If Dempster’s rule of combination is applied in a case explained as Ex. 4.1,
a misleading scenario is occurred stating that the patient certainly has cancer ignoring the
opinion provided by the majority of evidences, m2({nc})=0.7 and m3({nc})=0.9. In other
words, loss of majority opinion occurs in the case depicted in Ex. 4.1. This is one of the
major drawbacks of Dempster’s combination rule.
Example 4.2: The physical examination of a probable lung cancer patient produces the
following mass: m1 ({s0, s1a}) =0.70, m1(S) =0.30. The imaging results reveal much
proof in favor of stage 1a (s1a) or stage 1b (s1b) than no cancer or occult stage (oc). So,
from the second information source, we obtain: m2({s1a, s1b}) =0.90 and m2({no, oc})
=0.10. Later, from evidences of tissue diagnosis, the belief of no cancer or occult stage
seem to be very low and we obtain the following mass values: m3({no, oc})=0.10,
m3({s2a})=0.90. After combining all of these evidences by Dempster’s rule, we obtain
m({no,oc})=1.0 . In this case, total mass has been assigned to the subset {no,oc} which is
a minority opinion. This total mass to minority opinion is another drawback of
Dempster’s combination rule.
This unnatural concentration of total mass is prone to produce a
misleading decision regarding the staging of lung cancer which might cause fatal results
74
since treatment is greatly dependent on staging. In Example 4.1 (hereafter, Ex. 4.1),
although imaging and tissue diagnosis put more emphasis on nc, the combined evidence
was completely different. Again, in Ex. 4.2, a minority opinion has been assigned the
total mass without considering the major evidences. As a result, a patient might have
mistakenly been diagnosed to have pericardial effusion, for instance, instead of cancer.
Due to these major problems, Dempster’s rule cannot be solely used for evidence
combination in lung cancer diagnosis. As mentioned earlier, Hau and Kashyap’s rule can
handle the problem of total mass to minority opinion and Murphy’s rule is able to solve
loss of majority opinion as well as total mass to minority. Table 7 shows how Hau and
Kashyap’s rule and Murphy’s rule perform in the scenarios explained in Ex. 4.1 and Ex.
4.2. In Table 7, the combined evidences of Hau and Kashyap’s rule for Ex. 4.1 put much
emphasis on {nc,oc} but this emphasis is not enough to come up with a correct staging
information, because the majority support nc but not oc. In Ex. 4.2, this method can avoid
the total mass to minority opinion occurred by Dempster’s rule for the similar scenario
and assigns the mass of 0.57 to the subset {s0,s1a, s1b,s2a}.
On the other hand, using Murphy’s rule in Ex. 4.1 shows that 0.73 is
assigned to {nc} and only 0.17 is assigned to {oc}. In Ex. 4.1, both imaging and tissue
diagnosis assigned more mass to {nc} than other states and Murphy’s rule reflects this in
the combined evidences. This certainly solves the loss of majority opinion occurred by
Dempster’s combination in Ex. 4.1. Moreover, if we examine Ex. 4.2, minority opinion
{nc, oc} has not been assigned much mass value let alone the total mass. Each of the
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subsets {s2a} and {s1a, s1b} has been assigned the mass of 0.26 and {s1a} receives the
mass of 0.25 which is certainly an improvement of the result obtained by Dempster’s rule.
Table 7: Comparison of Combination Menthods
Given BBA
Ex.4.1
Ex.4.2
m1({s0})=0.30,
m1({oc})=0.70,
m2({nc})=0.70,
m2(S)=0.30;
m3({nc})=0.90,
m3({nc, s0})=0.10
m1({s0,s1a}) =0.70,
m1 (S) =0.30;
m2({s1a,s1b})
=0.90,
m2({no,oc}) =0.10,
m3({no,oc})=0.10,
m3({s2a})=0.90
Dempster’s Rule
m({s0})=1.0 *
m({nc,oc)}=1.0**
Hau and Kashyap’s
Rule
m({nc,oc})=0.44,
m({s0})=0.01,
m({nc,s0})=0.21,
m({nc,oc,s0})=0.05
m(S)=0.29
m({s0,s1a,s1b,s2a})
=0.57,
m(S)=0.30,
m({nc,oc,s0,s1a,s2a})
=0.06,
m({nc,oc,s0,s1a,s1b})
=0.06
m({nc,oc,s0,s1a})=0.01
Murphy’s Rule
m({nc})=0.73
m({oc})=0.17
m({s0})=0.06
m({nc,s0})=0.01,
m(S)=0.02
m({s2a})=0.26,
m({s1a,s1b})=0.2
6,
m({s1a})=0.25,
m({s0,s1a})=0.18,
m({nc,oc})=0.03,
m(S)=0.02.
*Loss of majority opinion **Total mass to minority opinion
Therefore, Murphy’s rule is able to solve total mass to minority problem created by
Dempster’s combination rule in Ex. 4.2. From Table 7, we might conclude that Murphy’s
rule produces a better and more informative combination of evidences for lung cancer
staging.
Apart from the two mentioned issues posed by Dempster’s combination
rule, reliability of information sources is another serious concern of evidence
combination. The reliability of a source characterizes its ability to provide the almost
correct assumptions of the discussed problem. The issue of ‘unreliability of sources’ was
addressed by Shafer through a technique called ‘discounting’ where a positive weight is
to be assigned to the frame of discernment (Shafer, 1976). Conventionally, one is
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assigned to a fully reliable source and a zero to an unreliable source. Keeping this in
mind, we have also considered the reliability values of Physical Examination, Laboratory
tests and imaging and Tissue diagnosis in Table 6. Based on reliability of information
sources, the discounted bbas are calculated. For instance: the reliability factor of Physical
Examination is 0.30 and the beliefs obtained from this source in Ex. 4.1 are m1({s0})=0.3,
m1({oc})=0.7. Therefore, the bbas with reliability become, m1({s0})=0.30*0.30=0.09,
m1({oc})=0.70*0.30=0.21 and m1(S)=(1-0.30)+0=0.70. Table 8 shows combination
results by the three combination rules incorporating the discounting operation for two
cases mentioned in Ex. 4.1 and Ex. 4.2.
Table 8: Comparison of Combined Evidences using Discounting Method
Ex.4.1
Ex.4.2
Given BBA
m1({s0})=0.30,
m1({oc})=0.70,
m2({nc})=0.70,
m2(S)=0.30;
m3({nc})=0.90,
m3({nc, s0})=0.10
Dempster’s Rule
m({nc})=0.90
m({oc})=0.01
m({s0})=0.01
m({nc,s0})=0.05
m(S)=0.03
m1({s0,s1a}) =0.70,
m1 (S) =0.30;
m2({s1a,s1b})=0.90,
m2({no,oc})=0.10,
m3({no,oc})=0.10,
m3({s2a})=0.90
m({s1a})=0.02
m({s2a})=0.76
m({no,oc})=0.10
m({s0,s1a})=0.01
m({s1a,s1b})=0.06
m(S)=0.04
Hau and Kashyap’s Rule
m({nc})=0.64
m({oc})=0.01
m({s0})=0.01
m({nc,s0})=0.07
m({nc,oc})=0.08
m({nc,oc.s0})=0.01
m(S)=0.19
m({s1a})= 0.01
m({s2a})=0.27
m({no,oc})=0.04
m({s1a,s1b})=0.02
m({no,oc,s0,s1a,s2a})=0.01
m({s0,s1a,s1b,s2a})=0.10
m({no,oc,s0,s1a,s1b})=0.01
m(S)=0.54
Murphy’s Rule
m({nc})=0.64
m({oc})=0.07
m({s0})=0.03
m({nc,s0})=0.03
m(S)=0.22
m({s1a})=0.03
m({s2a})=0.39
m({no,oc})=0.06
m({s0,s1a})=0.08
m({s1a,s1b})=0.23
m(S)=0.21
As shown in Table 8, {nc} obtains the highest mass for all of the three
combination rules in Ex. 4.1 due to the high reliability of the sources assigning the initial
mass value. Compared to Table 7, it is an obvious improvement from the viewpoint of
Dempster’s combination rule as well as Hau and Kashyap’s rule. But, in the case of Ex.
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4.2, 0.76 is assigned to {s2a} by Dempster’s combination; Hau’s rule assigns 0.54 to the
frame of discernment, S and 0.27 to {s2a}; Murphy’s combination rule assigns 0.39 to
{s2a} and 0.21 to S. This result is surely not beyond criticism because the mass balance
between {s2a} and {s1a, s1b} is inappropriate in Dempster's combination considering the
reliabilities of m2 and m3, and mass of {s2a} is too small and S has too much mass in the
other two combination rules. Therefore, even though the evidence combination using
discounting method shows slight improvement over the conventional combination
methods, it might not perform the best for lung cancer staging.
This study aims at a combination method that can resolve the problems
created by Dempster’s rule along with the reflection of the reliability of sources i.e. the
evidences from the reliable sources are weighed heavily while assigning low weight to
the evidences obtained from the less reliable ones. In order to have a clinically useful
stage classification scheme for lung cancer diagnosis, the most reliable evidences are
usually obtained from tissue diagnosis; imaging results are the next one and evidences
from physical examination are considered to be the least reliable one. Therefore, this
research proposes weighted averaging instead of simple average in Murphy’s
combination rule. The weight is assigned by using the reliability of each source in staging
of lung cancer. Suppose, there are n sources of evidence having the reliability of
r1, r2 ,...rn and evidence from each source i provides a mass mi for a focal element. Then,
the weighted average mass m of the focal element is expressed by the following
equation:
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mi ri
m
i 1...n
(42)
ri
i 1...n
Table 9 shows the mass values obtained by weighted average of the given
masses as well as the combined mass values.
Table 9: Combined Evidences using the Proposed Combination Method
Given BBA
Weighted Average
Ex. 4.1
m1({s0})=0.30, m1({oc})=0.70,
m2({nc})=0.70, m2(S)=0.30;
m3({nc})=0.90,
m3({nc, s0})=0.10
Ex. 4.2
m1({s0,s1a}) =0.70,
m1 (S) =0.30;
m2({s1a,s1b})=0.90,
m2({no,oc}) =0.10,
m3({no,oc})=0.10,m3({s2a})=0.90
m({nc})=0.69
m({oc})=0.11
m({s0})=0.05
m({nc,s0})=0.05
m(S)=0.10
m({s2a})=0.46
m({no,oc})=0.08
m({s0,s1a})=0.11
m({s1a,s1b})=0.29
m(S)=0.05
Combined
Evidence
m({nc})=0.90
m({oc})=0.05
m({s0})=0.02
m({nc,s0})=0.02
m(S)=0.01
m({s1a})=0.14
m({s2a})=0.54
m({no,oc})=0.03
m({s0,s1a})=0.05
m({s1a,s1b})=0.24
For Ex. 4.1, the most significant mass of 0.90 is assigned to {nc} which
was very strongly supported by two highly reliable sources. Also, for Ex. 4.2, {s2a} is
assigned 0.54, {s1a, s1b} is assigned 0.24 and {s1a} has received the mass of 0.14. It is
noticeable in Ex.4.2 that {s2a} was very strongly assumed by the most reliable source,
tissue diagnosis. This has also been reflected in the combined evidence. In addition,
although {s1a, s1b} was very strongly assumed by the moderately reliable source
(reliability value: 0.60), it has not been assigned a very high mass upon combination due
to the effect of the source’s reliability on weighted average. In this way, the proposed
combination method is capable of solving loss of majority opinion and total mass to
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minority opinion problems. Moreover, the proposed method reflects the nearly exact
impact of the reliability of each source on the combined evidences.
4.4 CONCLUSION
The results presented in the preceding section show that the proposed rule of evidence
combination can be considered to show better performance in lung cancer staging
procedure in comparison to the three conventional rules of evidence combination as well
as the combinations using discounting method because the proposed rule of combination
satisfies the significant properties those are expected to be possessed by any effective rule
of evidence combination. Among those properties, assigning belief to a majority opinion
rather than a minority one, designating an appropriate amount of ignorance (if possible),
providing a combined belief that successfully reflects the strength and the weaknesses of
the sources that supplied initial beliefs are worth mentioning. The proposed rule of
evidence combination is based on weighted average where the weights are designated by
incorporating reliability of the sources i.e. medical examinations of three distinct
categories. It is noticeable that the averaging method of the proposed rule is proving a
record of the contributing beliefs collected from physical examination, laboratory test and
imaging and tissue diagnosis. More specifically, in the first example (Ex. 4.1), the state
nc, the majority opinion, received high bba values and this has been reflected in the
combined evidences by the proposed combination rule. Most importantly, due to the
reliability based weights, the impact of the strength or weakness of the corresponding
sources has also been observed in the combined evidences. Consideration of source
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reliability is enormously important in lung cancer staging mechanism. Depending on the
stages, the treatment decision of lung cancer may greatly vary. Therefore, we are very
much hopeful that staging mechanism will be more efficient by using the proposed
evidence combination rule. Once the staging information of a lung cancer patient is
ascertained, decision-making regarding treatment alternative by NY-DDM would become
more appropriate.
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CHAPTER 5: DECISION-MAKING OF A LUNG CANCER
PATIENT REGARDING TREATMENT BY NY-DDM
This chapter is dedicated to illustrate a practical example of the proposed framework of
descriptive DM under uncertainty. Considering the current trend of medical decisionmaking that supports patient’s involvement in treatment decision-making process, we
believe that our proposed framework can contribute to that particular area of medical
decision-making which requires knowing a patient’s preferences about treatment of a
serious disease such as lung cancer. Hence, in this chapter, we have presented an example
application of NY-DDM in the area of patient-centric decision-making about lung cancer
treatment. The first section discusses the pervious and contemporary researches on the
trend of medical decision-making. The later sections illustrate the basics of lung cancer
such as available treatments, possible outcomes, stages of lung cancer etc. associated
with the treatment decision-making procedure. Afterwards, the proposed DM model is
applied along with the evidence combination rule and the decisions of patients with
diverse attitudes toward uncertainty are analyzed.
5.1 MEDICAL DECISION-MAKING: PAST AND PRESENT
Medical science is all about taking decisions either under certainty or uncertainty.
Traditionally, clinicians, physicians used to play a paternalistic role in treatment
decisions and the duty of the patient was to abide by the instructions. But there has been a
revolutionary change in medical fields since the last few decades and the health
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professionals are more and more encouraged to involve patients in treatment decisions.
The patient’s preference is especially more important when there are several alternatives
of treatment and each alternative has its own trade-off and each patient has his/her own
viewpoint of judging the trade-offs (Brownlee, 2011). For example: a patient with
rheumatoid arthritis may face three anti-tumor necrosis factor-alpha (TNF-alpha) therapy
options: etanercept (Enbrel), adalimumab (Humira) and infliximab (Remicade). In
another study, Chilton & Colletee (2008) expresses how the therapy decisions vary
depending on the various influence factors.
Researches on patient-centric decision-making have a short history. In
early 1980s, researchers started experiments in the field of urology along with the
advanced technology. Multiple studies have shown that the combination of physicians’
knowledge and patient’s participation can improve patient satisfaction and physician time
(Wagner et al. 1995). Although the traditional school of thought would argue that
involving patients in choosing treatment can make them burdened or can be unfavorable
for them, yet no such evidence exists to support this issue (Hellenthal & Ellison, 2008).
In fact, a study (Greenfield et al. 1985) showed that patients feel more active when being
included in health-care decisions. In the similar study, the authors developed an
intervention to increase patient involvement in health-care. At first, the patients were
helped to read their medical records and trained to ask questions to physicians and
negotiate medical decisions with their physicians. Six to eight weeks after the trial, the
patients in the experimental group reported their preference to have an active role in
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medical decision-making. Further analysis suggested that the patients under experiments
were twice as effective as the other patients in obtaining information from physicians.
In addition, a US Veterans Affairs study addressed the issue of patient’s
role in decision-making for invasive medical procedures and the preferences for
disclosure of risks. Approximately 500 patients were asked (mean age 65 years, mean
length of formal education 12.6 years) whether they preferred patient-based, physicianbased or shared decision-making. Nearly the entire cohort (93%) responded that all
procedural risks should be addressed and more than two-thirds (68%) preferred that the
decision should be made jointly by physicians and patients. Only one-fifth (21%) of the
patients preferred the physician-based decision-making (Mazur & Hickam, 1997).
Apart from this, decisions of the patients regarding treatments also vary
depending on the seriousness of disease: benign or malignant. A study (Berry et al. 2003)
examined that treatment decisions for low-risk prostate cancer are usually based partly on
nonscientific data. In a study of 102 men with newly diagnosed, localized prostate cancer,
most of the men chose treatment on the basis of several factors: external
recommendations, essential treatment characteristics, patients’ own impressions and
economic consideration. Another study observes the influence of faith and spirituality in
case of serious terminal diseases such as advanced stage lung cancer. One hundred
patients and their caregivers and 257 medical oncologists were interviewed. Patients and
care givers agree on the factors that are important in deciding treatment for advanced
stage lung cancer but differ substantially from doctors. The recommendation of the
treating oncologists was of much importance however, the researchers found that faith in
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God was the second most important factor (Silvestri et al. 2003). The findings of similar
studies make us bound to think about the influence of human attitudes in decision-making
regarding treatment in different diseases.
Another important issue why patients are being encouraged to be involved
in treatment DM is that physicians or clinicians are not mind readers. They often do not
know or ask their patients about their values and preferences; or sometimes they may
assume that the patient’s values are similar to their own. As a result, physicians may
recommend treatment that is different from what their patients would have chosen if they
had been fully informed. Moreover, there are cases where clinicians also vary widely in
their opinions about the best course of treatment for any given condition. These
differences in clinicians’ opinions and patient’s personal beliefs should be considered in
medical decision-making procedures in order to provide a satisfactory service to the
patients. Consequently, the concept of ‘shared decision-making’ has been evolved in the
field of medical decision-making during the last two decades. Shared decision-making
(SDM) includes the notion of a medical encounter as a “meeting of experts”- the
physicians as an expert in medicine and the patients as the expert in his/her own life,
values and circumstances (Tuckett et al., 1985). It is a process by which patients and
providers consider outcome probabilities and patient’s preferences and reach a decision
in health care or treatment based on a mutual agreement.
For all these reasons, we are motivated to conduct a research endeavour
incorporating various factors that can be usually considered by a lung cancer patient
while making decisions about treatment. In addition, we have also assumed that patients
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may have different stance towards uncertainty which is possible to persuade their
treatment decisions in a comprehensive manner.
5.2 FEW FACTS ABOUT LUNG CANCER
Lung cancer is one of the three most common cancers diagnosed each year both in UK
and USA. Around 41,400 people are diagnosed with Lung cancer in the UK each year
whereas in USA, the estimated number of new cases of lung cancer for 2012 is 226,160
with the estimated death of 160,340 people which is the highest among all common
cancers according to American Cancer Society (Report 2012). Smoking is the most
important cause of lung cancer though there are many other factors that increase risks
such as: exposure to radon gas, exposure to certain chemicals, air pollution, previous lung
diseases, past cancer treatment etc. Although screening tests are helpful to find and treat
lung cancer early, unfortunately there is no generally accepted screening test for lung
cancer.
Before going into the cancer type and treatment details, a brief description
of human respiratory system is given. The lungs are a pair of cone-shaped breathing
organs in the chest that brings in oxygen into human body and releases carbon dioxide as
a waste product of the body’s cells. Each lung has sections called lobes. The left lung has
two lobes and the right lung has three lobes. Figure 4 shows the anatomy of the
respiratory system.
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Figure 4: Anatomy of Human Respiratory System3
There are several different types of lung cancer that can be divided into two main types:
1) Small cell lung cancer (SCLC): About 13% lung cancers are small cell lung
cancer. This tends to spread quickly.
2) Non-small cell lung cancer (NSCLC): About 87% are of this type that spreads
slowly than small cell lung cancer.
3
National Cancer Institute, USA: General Information about Non-small Cell Lung Cancer.
URL: http://www.cancer.gov/cancertopics/pdq/treatment/non-small-cell-lung/Patient
87
These are diagnosed based on how the cells look under a microscope. In this research, we
have dealt with NSCLC type of lung cancer. All of the stages, alternatives and outcomes
mentioned here are given based on the type NSCLC.
5.2.1
DIAGNOSIS AND STAGING
Tests and procedures to detect, diagnose and stage non-small cell lung cancer are often
done at the same time. According to visible symptoms, physical exams are asked to check
for general signs of health including checking for signs of disease such as lumps or
anything else that seems unusual. Chest X-ray is done to find tumors or abnormal fluid,
CT scan or spiral CT scan is done that can show a tumor, abnormal fluid or swollen lymph
nodes. The only sure way to know if lung cancer is present is by pathological tests by
collecting samples of cells or tissue. The necessary tests to collect samples are: Sputum
cytology, Thoracentesis, Bronchoscopy, Fine-needle aspiration, Thoracoscopy etc. Several
diagnostic procedures have been shown in Figure 5.
After lung cancer has been diagnosed, tests are done to find out the
presence of cancer cells within the lungs or to other parts of the body. This process is
known as staging. Staging describes the severity of a person’s cancer based on the extent
of the original (primary) tumor and whether or not cancer has spread in the body.
According to the National Cancer Institute of USA, staging is important for several
reasons: (www.cancer.gov)
Staging helps the doctor plan the appropriate treatment.
The stage can be used to estimate the person’s prognosis.
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Figure 5: Few Diagnostic Methods
(Top Left) Chest X-ray, (Top Right) Bronchoscopy and (Bottom) Fine-Needle Aspiration
Biopsy4
4
National Cancer Institute, USA: General Information about Non-small Cell Lung Cancer.
URL: http://www.cancer.gov/cancertopics/pdq/treatment/non-small-cell-lung/Patient
89
Knowing the stage is important in identifying clinical trials that may be
suitable for a particular patient.
Staging helps health care providers and researchers exchange information
about patients; it also gives them a common terminology for evaluating
the results of clinical trials and comparing the results of different trials.
The tests and scans are required to get some information about the stage.
Sometimes it is not possible to be certain about a stage of a cancer until after surgery.
Both number staging system and the TNM system are being used in staging lung cancer.
The TNM system is based on the extent of the tumor (T), the extent of spread to the
lymph nodes (N) and the presence of distant metastasis (M). Overall staging is also being
used for lung cancer staging and it is possible to relate these two types of staging systems.
Table 10 shows the overall stages of lung cancer with its TNM equivalent.
Table 10: Overall Staging and TNM Equivalent of NSCLC
Stage
IA
IB
IIA
IIB
IIIA
IIIB
IV
TNM Equivalent
T1a-1bN0M0
T2aN0M0
T1a-2aN1M0 or T2bN0M0
T2bN1M0 or T3N0M0
T3N1M0 or T1a-3N2M0 or T4N0-1M0
T4N2M0 or T1a-4N3M0
Any T Any N M1a-1b
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The overall stages of non-small cell lung cancer can be explained as follows:
Occult Stage: In occult (hidden) stage, cancer cannot be seen by imaging or
bronchoscopy. Lung cancer cells are found in sputum or bronchial washing.
Stage 0: Cancer cells found only in the innermost lining of the lung and the tumor has not
grown through this lining. Stage 0 tumor is also called carcinoma in situ.
Stage IA: Tumor has grown through the innermost lining of the lung into deeper lung
tissue. The tumor is no more than 3 cm across, surrounded by normal tissue and tumor
does not invade the bronchus. Cancer cells are not found in nearby lymph nodes.
Stage IB: Cancer has not spread to the lymph nodes and the tumor is more than 3 cm
across but not larger than 5 cm. Cancer has spread to the main bronchus and is at least 2
centimeters below where the trachea joins the bronchus or cancer has spread to the
innermost layer of the membrane that covers the lung or part of the lung has developed
pneumonitis (inflammation of the lung).
Stage IIA: Cancer has spread to lymph nodes on the same side of the chest as the tumour.
The tumour is not larger than 5 cm or cancer has spread to the innermost layer of the
membrane that covers the lung. When the tumour is within 5~7cms but not yet spread to
lymph node, the stage is given as IIA.
Stage IIB: Cancer has spread to nearby lymph nodes on the same side of the chest as the
tumor and the tumor is larger than 5 cm but not larger than 7 cm. Sometimes, the tumor
can be larger than 7 cm but cancer has not yet spread to lymph nodes.
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Stage IIIA: Cancer has spread to lymph nodes on the same side of the chest as the tumor
and the tumor may be of any size with partial lung collapse or one or more separate
tumors in the same lobe of the lung. Also, cancer may have spread either main bronchus,
or chest wall, larynx (voice box), trachea etc. Sometimes, cancer does not spread to the
lymph nodes but has spread to any of the following: heart, major blood vessels that lead
to or from the heart, trachea, esophagus, nerve that controls the larynx (voice box),
Sternum (chest bone) or backbone.
Stage IIIB: Cancer has spread to lymph nodes above the collarbone or to lymph nodes on
the same side or on the opposite side of the chest as the tumor; part of the lung or the
whole lung may have collapsed and there may be one or more separate tumors in any of
the lobes of the lung with cancer. Cancer may have spread anywhere near lung including
heart, major blood vessels and others.
Stage IV: Malignant growth may be found in more than one lobe of the same lung or in
the other; cancer cells may be found in other parts of the body, such as brain, liver or
bone.
5.2.2 TREATMENT
The treatment is planned taking into account of several issues such as the type of cancer,
the position of the cancer within the lung, general health of the patient, the stage of
cancer and results of blood tests and scans. Usually, non-small cell lung cancer can be
treated with surgery, chemotherapy, radiotherapy or a combination of these or sometimes
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with targeted therapies. New combinations of treatments are being studied in clinical
trials and patients can also take part in clinical trials by choice.
Occult NSCLC can be cured by surgery. Stage 0 NSCLC is also treated
with surgery or sometimes photodynamic therapy, electro cautery or laser surgery.
Treatment of Stage I include surgery, external radiation therapy, and clinical trial of
chemotherapy or radiation therapy following surgery, clinical trial of surgery followed by
chemoprevention. Treatment of stage II NSCLC includes surgery, Chemotherapy
followed by surgery, surgery followed by Chemotherapy, external radiation therapy and
clinical trials. Treatment of Stage III NSCLC that can be removed with surgery may
include surgery followed by chemotherapy, chemotherapy followed by surgery, surgery
followed by chemotherapy combined with radiation therapy, surgery followed by
radiation therapy. In some cases, radiation therapy, surgery or combination of these two
are also applied. Treatment of Stage IV NSCLC include combination chemotherapy,
combination chemotherapy and targeted therapy with a monoclonal antibody, targeted
therapy with a tyrosine kinase inhibitor, maintenance therapy with anticancer drug, laser
therapy etc..
Needless to say that almost all of these treatment methods have moderate
to severe side effects along with slight chance of complete recovery. In addition, the cost
of the treatment is also very high in any country of the world. Therefore, a patient might
have different mind-set regarding the treatment alternatives they are usually offered by
the physicians. The aim of this research is to analyze how a patient can choose a
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treatment among many alternatives based on three decision attitude: equative, pessimistic
and optimistic. The next section describes the problem formulation step by step.
5.3 DECISION-MAKING ABOUT TREATMENT IN LUNG CANCER
Cancer brings in an abrupt change in human life and lung cancer is no exception. Patients
as well as his/her caregivers have to face many situations during the cancer trajectory
where they need to take difficult decisions. Also, not all decision-makers (in this case,
patients) possess the similar perceptions. The difference of ethnicity, financial strength,
priorities and other concerns can influence decisions about treatment. On top of that,
there is often a degree of uncertainty when making decisions about treatment. The
uncertainty may be involved with a complex web of diagnostic and therapeutic
uncertainties; it may be related to the future outcome of a treatment procedure. Quite
often, there is no clear ‘right’ or ‘wrong’ answer of these issues. Even the physicians may
not be able to say certainly whether and how the treatment may affect the patient. It is not
surprising that there is often considerable disagreement between the physician and the
patient about the best course of action Therefore, it is highly essential to include the
patients in the process of decision-making about lung cancer treatment so that it is
possible to know the patient’s perception, desires and priorities. The proposed DM
framework makes it possible to identify the patient’s preferences about treatment
methods by incorporating his attitude toward uncertainty. Consequently, it can improve
the quality of decisions in health-care settings.
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This section aims at eliciting the decisions made by a lung cancer patient
holding different attitude towards uncertainty by applying the proposed DM framework.
At first, the entire problem of treatment decision-making has been framed as an EDMP
and the further steps of deducing decisions have been gradually followed.
5.3.1
PROBLEM FORMULATION
An EDMP includes the set of states of nature S, set of alternatives A and the set of
outcomes O for each state and alternative pair (ai , s j ) and utility values for each outcome.
5.3.2
SET OF STATES OF NATURE
In this problem, the set of states of nature include all the stages of lung cancer mentioned
in subsection 5.2.1 including the state ‘no cancer’. Therefore, the set of states of nature S
includes the following 10 elements: S= {nc, oc, s0, s1a, s1b, s2a, s2b, s3a, s3b, s4}
where nc implies no cancer, oc implies occult stage, s0 implies stage 0. This set of states
implies that any one state among these ten states is possible to be true for a patient
suspecting lung cancer.
5.3.3
SET OF ALTERNATIVES
The set of alternatives include all of the treatment options possible for lung cancer. As
mentioned earlier in this chapter, the standard treatments of lung cancer are surgery,
chemotherapy, radiation therapy, targeted therapies, and different combination of these
treatments according to the cancer stage and the general health of the patient. For
simplification and ease of understanding the application of NY-DDM to this treatment
95
decision problem, the set of treatment alternatives have been limited to five chief
categories each having four levels. In the cancer treatment scenario, the patients who
guess that the form of their current ailment is cancer do indulge in cancer detection
diagnosis. But, many times the diagnosis results disapprove cancer and hence no form of
cancer treatment is offered to the patient. Thus, in this study, we have also considered
‘No cancer treatment’ as an alternative along with the mentioned cancer treatments.
Table 11 describes 21 alternatives that have been considered in this DM problem.
5.3.4
BASIC BELIEF ASSIGNMENTS (BBA)
Diagnostic and staging decisions are often made under uncertainty. Uncertainty in lung
cancer staging may arise from erroneous observation or inaccurate recording of clinical
findings or misinterpretation of the data by the physicians. Uncertainty also arises due to
ambiguity of the data or images or variations in interpretation of the information.
Therefore, we are compelled to consider the uncertainty in lung cancer
staging due to its immense importance in making decisions about lung cancer treatment.
In EDMP, uncertainty is expressed by the basic belief assignment (bba) of DempsterShafer theory of evidence. The information about stage can be obtained from various
diagnostic methods as mentioned in subsection 5.2.1 and combination of evidences can
facilitate the staging of the patient. It is possible to have significant uncertainty to be
present during the diagnostic and staging procedures since all information sources are not
equally reliable.
96
Table 11: List of Treatment Alternatives
Alternative
No Cancer
treatment
Surgery
Chemotherapy
Radiation
Therapy
Targeted
Therapies
Clinical Trials
Sub-types
~
L1
L2
L3
L4
L1
L2
L3
L4
L1
L2
L3
L4
L1
L2
L3
L4
L1
L2
L3
L4
Description
No cancer treatment is offered to the patient.
Video-assisted thoratic surgery (VATS): early stage cancers
Segmentectomy or wedge resection:Part of the lobe is
removed.
Lobectomy: A section of the lung is removed
Pneumonectomy: An entire lung is removed.
Neoadjuvent therapy: Chemo with radiation before surgery.
Adjuvent therapy: Along with radiation after surgery.
Chemo as a main threatment
Second-line treatment with a single drug for advanced stage
cancer.
Steriotactic Radiation: Early stage cancer.
Intensity Modulated Radiation Therapy (IMRT)
3D-CRT
Internal Radiation Therapy
Drugs that target tumor blood vessel growth (Angiogenesis)
Drugs that target EGFR (Epidermal Growth Factor Receptor)
Drugs that target ALK gene
Complementary and alternative therapy
Trials on Physical and Behavioral therapy:Trial on supportive
care
Pre- and Post-surgery trials
Biomarker/Laboratory/Diagnostic Trials
Trials on Treatment
Table 12: List of Basic Belief Assignments(BBA)
Cases
Case 1
Considered BBA Values
m({nc})=0.90, m({oc})=0.05,m({s0})=0.02, m({nc,s0})=0.02, m(S)=0.01
Case 2
m({s1a})=0.14, m({s2a})=0.54, m({nc,oc})=0.03, m({s0,s1a})=0.05,
m({s1a,s1b})=0.24
Case 3
Case 4
Case 5
m({s1a,s2a})=0.75,m({s1a})=0.25.
m({s2a, s2b})=0.6, m(S)=0.4.
m({oc,s0,s1a})=0.2,m({s1a,s1b})=0.2,m({s1a,s3a,s3b,s4})=0.55,
m(S)=0.05
97
Table 6 mentioned in Chapter 4 includes the categories of examination a patient usually
goes through during lung cancer diagnosis and staging along with their reliabilities.
Evidences collected from each of these sources are then combined by the proposed
combination rule shown in Eq. (44) (Please refer to Section 4.3). Table 12 shows the list
of bbas that have been considered in this problem.
5.3.5
OUTCOMES AND UTILITIES
The outcomes of a certain treatment alternative may vary from one cancer stage to
another. As mentioned earlier, treatment decisions also depend on a patient’s moral
values, circumstances such as financial capability and others. Most importantly, treatment
efficiency and the probability of survival after taking the treatment as well as the side
effects of the treatment also influences a patient’s decision-making process. In this
research, the consideration of the outcome set for treatment decision depends on the
following four factors: cost or expense of treatment (C), treatment efficiency (T), 5-year
survival rate (S) and side effects (E) and the outcome set can be given by a quadruplet of
O. The outcomes can be defined by the Cartesian product of the following sets: O=OC x
OT x OS x OE where OC = {Supremely Expensive: 1, ……,Moderately Expensive: 5…., No
Expense incurred: 10}, OT={No Recovery:1, …..,Moderate Recovery: 5,……, Very Fast
Recovery: 9, Complete Recovery:10}, OS={5-year Survival Rate less than 10%:1, …..,
40% ~50%: 5……,100%: 10}, OE={Extreme Long Term Side Effects:1, …..,Moderate
Side Effects:5,……., No Side Effect: 10}. Therefore, the set O has 10*10*10*10=10000
possible outcomes. We have categorized each of C, T, S and E into 10 levels in order to
98
make a clear distinction among the outcomes for each pair of (ai , s j ) . Table 13 partly
shows the outcome table with corresponding utilities and Table 14 shows the Decision
table with all of the utilities for each pair of (ai , s j ) . Utility considered here is a relative
representation of preferences over a set of outcomes. It is essentially subjective and
personal. However, we assume the patient gives the utility by a multiple regression
equation according to his/her intuition. The utility formula that has been used in this
paper is: ((C + T*64 + S*16+ (E+4)*4), where T, S, E, C represent treatment efficiency,
5-year survival rate, side effects and cost or expense of treatment respectively.
Table 13: Outcome Table and Utilities
Outcome
Number
Description
Utility (C + T*64 +
S*16+ (E+4)*4)
o1,1,1,1
Supremely Expensive, No recovery, 5-year Survival rate
less than 10%, Extremely Long Term Side Effects
101
o10,10,10,10
No Expense incurred, Complete Recovery, 5-year
Survival rate 100%, No Side Effect
866
99
Table 14: Decision Table with Outcomes and Utilities
Alternative
No Cancer Treatment
Surgery:L1
Surgery:L2
Surgery:L3
Surgery:L4
Chemotherapy:L1
Chemotherapy:L2
Chemotherapy:L3
Chemotherapy:L4
Radiotherapy:L1
Radiotherapy:L2
Radiotherapy:L3
Radiotherapy:L4
Targeted Therapy
Targeted Therapy:L1
Targeted Therapy:L2
Targeted Therapy:L3
Clinical Trials:L1
Clinical Trials:L2
Clinical Trials: L3
Clinical Trials:L4
nc
O9, 7, 2, 9
541
O5, 8, 1, 8
581
O8, 6, 4, 8
504
O5, 9, 9, 7
769
O5, 4, 9, 2
429
O6, 3, 9, 3
370
O9, 4, 2, 3
325
O10, 8, 10, 3
710
O5, 3, 4, 9
313
O7, 5, 5, 7
451
O1, 4, 4, 1
341
O3, 5, 8, 5
487
O6, 1, 8, 10
254
O2, 9, 4, 7
686
O5, 2, 2, 9
217
O8, 9, 7, 5
732
O4, 4, 3, 10
364
O5, 1, 5, 9
201
O4, 5, 7, 7
480
O8, 4, 6, 2
384
O9, 4, 5, 1
365
oc
O4, 8, 9, 2
684
O2, 5, 10, 3
510
O3, 4, 2, 10
347
O5, 1, 2, 4
133
O10, 5, 7, 2
466
O5, 10, 6, 6
781
O8, 7, 1, 4
504
O8, 7, 10, 7
660
O2, 4, 1, 10
330
O3, 10, 10, 5
839
O2, 7, 1, 3
494
O2, 9, 5, 10
714
O4, 2, 3, 1
200
O10, 3, 10, 8
410
O2, 7, 4, 8
562
O3, 5, 1, 5
375
O6, 7, 8, 3
610
O3, 8, 9, 1
679
O6, 9, 3, 8
678
O7, 6, 2, 1
443
O8, 10, 3, 7
740
s0
O10, 8, 1, 1
558
O2, 7, 1, 3
494
O8, 10, 3, 9
748
O10, 1, 7, 1
206
O3, 7, 9, 10
651
O7, 6, 2, 10
479
O4, 8, 7, 3
656
O9, 9, 1, 6
641
O8, 8, 9, 10
720
O10, 2, 3, 6
226
O9, 8, 2, 9
605
O3, 6, 1, 7
447
O6, 10, 3, 8
742
O8, 1, 8, 5
236
O5, 4, 10, 10
477
O7, 3, 4, 5
299
O2, 9, 4, 2
666
O5, 4, 4, 1
345
O5, 5, 3, 7
417
O5, 3, 10, 4
389
O1, 8, 7, 1
645
s1a
O2, 2, 8, 8
306
O9, 2, 5, 5
253
O4, 5, 7, 7
480
O4, 9, 7, 3
720
O5, 7, 9, 2
621
O10, 2, 10, 7
342
O2, 3, 9, 5
374
O5, 8, 10, 2
701
O8, 9, 8, 5
748
O7, 3, 2, 2
255
O8, 8, 3, 5
604
O8, 3, 2, 8
280
O6, 9, 4, 9
698
O2, 5, 8, 10
506
O4, 6, 1, 6
444
O6, 4, 3, 4
342
O3, 6, 10, 9
599
O1, 1, 3, 3
141
O2, 7, 8, 4
610
O1, 1, 4, 2
153
O7, 7, 10, 3
643
100
s1b
O9, 6, 9, 8
585
O4, 3, 9, 1
360
O10, 9, 3, 5
670
O2, 6, 4, 4
482
O4, 3, 1, 1
232
O10, 10, 3, 3
726
O10, 5, 4, 3
422
O6, 1, 4, 8
182
O9, 1, 1, 6
129
O5, 6, 3, 3
465
O1, 8, 5, 2
617
O2, 4, 8, 7
430
O1, 7, 4, 10
569
O3, 4, 9, 3
431
O2, 10, 4, 3
734
O2, 7, 3, 7
542
O7, 4, 9, 4
439
O2, 3, 7, 5
342
O7, 4, 10, 2
447
O7, 6, 2, 2
447
O10, 4, 6, 6
402
s2a
O10, 9, 1, 1
622
O8, 9, 3, 4
664
O8, 7, 8, 10
640
O10, 9, 5, 10
722
O2, 4, 5, 10
394
O9, 8, 8, 6
689
O9, 2, 5, 4
249
O9, 2, 9, 10
337
O4, 6, 8, 10
572
O1, 7, 4, 3
541
O2, 2, 4, 10
250
O2, 7, 10, 5
646
O7, 9, 4, 10
703
O9, 6, 2, 9
477
O3, 7, 7, 3
591
O7, 3, 1, 4
247
O9, 6, 7, 1
525
O1, 5, 2, 7
397
O3, 5, 2, 4
387
O8, 7, 7, 2
592
O8, 1, 8, 10
256
s2b
O9, 10, 1, 5
701
O2, 7, 10, 5
646
O1, 4, 6, 4
385
O8, 6, 8, 10
576
O2, 9, 7, 2
714
O7, 1, 2, 8
151
O2, 10, 9, 5
822
O8, 8, 1, 10
592
O7, 2, 8, 4
295
O4, 4, 9, 2
428
O6, 3, 4, 5
298
O2, 2, 3, 1
198
O1, 4, 4, 8
369
O9, 4, 6, 8
409
O4, 5, 8, 4
484
O3, 3, 6, 7
335
O8, 6, 2, 7
468
O4, 4, 8, 9
440
O6, 1, 5, 9
202
O8, 2, 4, 10
256
O3, 2, 3, 6
219
s3a
O5, 5, 6, 10
477
O4, 5, 1, 8
388
O4, 7, 5, 8
580
O7, 1, 10, 7
275
O1, 4, 6, 2
377
O9, 8, 1, 9
589
O8, 2, 10, 5
332
O1, 9, 5, 5
693
O5, 3, 5, 9
329
O4, 6, 10, 4
580
O7, 4, 5, 6
383
O6, 4, 7, 4
406
O10, 6, 7, 4
538
O9, 2, 6, 1
253
O4, 1, 4, 1
152
O7, 7, 1, 6
511
O3, 8, 6, 4
643
O5, 8, 2, 5
585
O6, 4, 1, 5
314
O7, 8, 2, 6
591
O2, 1, 10, 10
282
s3b
O8, 1, 6, 2
192
O4, 7, 9, 8
644
O10, 2, 3, 10
242
O8, 7, 2, 9
540
O9, 8, 5, 10
657
O3, 1, 9, 4
243
O7, 1, 9, 3
243
O9, 9, 1, 9
653
O8, 4, 9, 8
456
O5, 9, 8, 3
737
O5, 7, 6, 6
589
O6, 3, 10, 4
390
O3, 4, 3, 7
351
O4, 5, 6, 4
452
O9, 9, 10, 5
781
O7, 3, 7, 2
335
O3, 6, 2, 1
439
O7, 5, 4, 1
411
O3, 5, 4, 4
419
O10, 2, 5, 9
270
O8, 1, 6, 10
224
s4
O8, 1, 4, 3
164
O3, 6, 6, 10
539
O2, 9, 8, 10
762
O8, 8, 8, 9
700
O5, 7, 6, 10
605
O6, 8, 4, 4
614
O6, 3, 7, 5
346
O10, 1, 10, 6
274
O8, 1, 4, 6
176
O4, 4, 6, 6
396
O2, 10, 7, 7
798
O8, 4, 2, 4
328
O2, 10, 4, 8
754
O8, 7, 1, 5
508
O1, 2, 5, 8
257
O8, 3, 9, 4
376
O3, 2, 7, 2
267
O2, 4, 5, 1
358
O10, 5, 7, 1
462
O10, 8, 5, 10
658
O2, 10, 4, 1
726
Table 15: Probabilities in Equative Approach in Case 1
States
P(sj)
nc
oc
s0
s1a
s1b
s2a
s2b
s3a
s3b
s4
0.911
0.051
0.031
0.001
0.001
0.001
0.001
0.001
0.001
0.001
Table 16: Probabilities in Pessimistic Approach in Case 1
States
Ppes(sj|a1 )
Ppes(sj|a2 )
Ppes(sj|a3 )
Ppes(sj|a4 )
Ppes(sj|a5 )
Ppes(sj|a6 )
Ppes(sj|a7 )
Ppes(sj|a8 )
Ppes(sj|a9 )
Ppes(sj|a10 )
Ppes(sj|a11 )
Ppes(sj|a12 )
Ppes(sj|a13 )
Ppes(sj|a14 )
Ppes(sj|a15 )
Ppes(sj|a16 )
Ppes(sj|a17 )
Ppes(sj|a18 )
Ppes(sj|a19 )
Ppes(sj|a20 )
Ppes(sj|a21 )
nc
0.92
0.90
0.92
0.90
0.92
0.92
0.92
0.90
0.92
0.90
0.92
0.90
0.92
0.90
0.92
0.90
0.92
0.92
0.90
0.92
0.92
oc
0.05
0.05
0.05
0.06
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.06
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
s0
0.02
0.04
0.02
0.04
0.02
0.02
0.02
0.04
0.02
0.05
0.02
0.04
0.02
0.05
0.02
0.04
0.02
0.02
0.04
0.02
0.02
s1a
0.00
0.01
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.01
0.00
0.01
0.00
s1b
0.00
0.00
0.00
0.00
0.01
0.00
0.00
0.01
0.01
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
101
s2a
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.01
0.00
0.00
0.00
0.00
0.01
0.00
0.00
0.00
0.00
0.00
s2b
0.00
0.00
0.00
0.00
0.00
0.01
0.00
0.00
0.00
0.00
0.00
0.01
0.00
0.00
0.00
0.00
0.00
0.00
0.01
0.00
0.01
s3a
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.01
0.00
0.00
0.00
0.00
0.00
0.00
s3b
0.00
0.00
0.01
0.00
0.00
0.00
0.01
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
s4
0.01
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.01
0.00
0.00
0.00
0.00
Table 17: Probabilities in Optimistic Approach in Case 1
States
Popt(sj|a1 )
Popt(sj|a2 )
Popt(sj|a3 )
Popt(sj|a4 )
Popt(sj|a5 )
Popt(sj|a6 )
Popt(sj|a7 )
Popt(sj|a8 )
Popt(sj|a9 )
Popt(sj|a10 )
Popt(sj|a11 )
Popt(sj|a12 )
Popt(sj|a13 )
Popt(sj|a14 )
Popt(sj|a15 )
Popt(sj|a16 )
Popt(sj|a17 )
Popt(sj|a18 )
Popt(sj|a19 )
Popt(sj|a20 )
Popt(sj|a21 )
nc
0.90
0.92
0.90
0.93
0.90
0.90
0.90
0.93
0.90
0.92
0.90
0.92
0.90
0.93
0.90
0.93
0.90
0.90
0.92
0.90
0.90
oc
0.05
0.05
0.05
0.05
0.05
0.06
0.05
0.05
0.05
0.06
0.05
0.06
0.05
0.05
0.05
0.05
0.05
0.06
0.06
0.05
0.06
s0
0.04
0.02
0.04
0.02
0.04
0.04
0.04
0.02
0.04
0.02
0.04
0.02
0.04
0.02
0.04
0.02
0.05
0.04
0.02
0.04
0.04
s1a
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.01
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
s1b
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
102
s2a
0.00
0.01
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
s2b
0.01
0.00
0.00
0.00
0.01
0.00
0.01
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
s3a
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
s3b
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.01
0.00
0.00
0.00
0.00
0.00
0.00
s4
0.00
0.00
0.01
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.01
0.00
0.01
0.00
0.00
0.00
0.00
0.00
0.00
0.01
0.00
5.3.6
EVALUATING THE ALTERNATIVES
After obtaining the utilities, we can proceed to Phase 3 of NY-DDM that is
approximation of probability from bba. The approximation has been performed primarily
in three attitudes: Equative, Pessimistic and Optimistic. The probabilities of all of these
attitudes for Case 1 have been shown in Table 15-17.
Table 18: Overall Values of 21 Alternatives in Case 1
No Cancer Treatment
Equative
1.01
Pessimistic
0.82
Optimistic
1.66
Surgery:L1
1.33
0.85
1.64
Surgery:L2
0.24
-0.49
0.33
Surgery: L3
-1.16
-1.45
-1.02
Surgery: L4
1.43
0.71
1.53
Chemotherapy: L1
-3.41
-3.73
-2.81
Chemotherapy: L2
-4.75
-4.62
-3.55
Chemotherapy: L3
1.13
0.91
1.61
Chemotherapy: L4
-6.05
-6.10
-4.99
Radiation Therapy: L1
0.25
-0.03
0.39
Radiation Therapy: L2
-3.88
-4.45
-3.40
Radiation Therapy: L3
0.96
0.78
1.46
Radiation Therapy: L4
-5.85
-5.96
-5.44
Targeted Therapy: L1
0.32
0.05
0.49
Targeted Therapy: L2
-4.59
-5.20
-4.08
Targeted Therapy: L3
-0.59
-0.65
0.33
Targeted Therapy: L4
-3.15
-3.97
-3.07
Clinical Trials: L1
-5.95
-6.16
-5.44
Clinical Trials: L2
1.21
0.76
1.44
Clinical Trials: L3
0.66
0.47
1.31
Clinical Trials: L4
-3.94
-3.95
-3.03
Subsequently, PT is applied to evaluate all of these treatment alternatives in three
different attitudes. At first, the reference point is considered for obtaining gains or loss.
103
The utility of outcome o3,5,1,5 (375) has been chosen as the reference point. It has been
assumed that the decision-maker is less concerned of the expenses the treatment incurs
rather than the treatment efficiency and side effects. All utilities over this value are
considered as gains and all utilities below this level are considered as losses to the
decision-maker. The reference point has been set subjectively and thereafter gains and
losses have been measured. By using Eq. (15), each alternative is evaluated where the
probabilities are converted to decision weights to reflect the impact of p and the utilities
of the outcomes have also been converted by v(.) that reflects the subjective value of that
outcome. It has been mentioned earlier that the values of three coefficients (λ, α, γ) of the
value function and the weighting function of PT has a major importance in evaluating the
alternatives. In this problem, λ=4.8, α=0.1, γ=0.5 have been used to evaluate the
treatment alternatives. The evaluation of 21 alternatives for Case 1 has been shown in
Table 18.
5.4 DISCUSSION
From Table 18, it can be inferred that for Case 1, Optimistic decision-maker chooses the
option ‘No Cancer Treatment’, a Pessimist chooses Chemotherapy of level 3 and an
equative attitude holder chooses Surgery of Level 4. Following the similar strategy, we
can evaluate the overall values of the 21 alternatives for the rest of the cases. The
decisions about treatment of the lung cancer patients for Case 1 to Case 5 are summarized
in Table 19. (Please refer to Appendix B for graphical representation of the overall values
of the alternatives and the decisions of Case 1~ Case 5)
104
Table 19: Patient’s Decisions Regarding Treatment in Different Attitudes
Cases
Case 1
Case 2
Case 3
Case 4
Case 5
Equative
Surgery: L4
1.43
Clinical Trial: L2
1.78
Surgery: L3
1.28
Surgery:L4
1.40
Radiation Therapy:L2
1.14
Pessimistic
Chemotherapy: L3
0.91
Clinical Trial: L2
1.69
Targeted Therapy: L4
1.23
Surgery:L1
-1.79
Radiation Therapy:L2
0.08
Optimistic
No Cancer Treatment
1.66
Surgery: L2
1.95
Chemotherapy:L4
1.81
Chemotherapy:L2
1.84
Chemotherapy:L4
1.81
From the result of Table 19, it is obvious that patients’ desired treatment
can vary according to their attitude towards uncertainty. When a patient is optimistic,
he/she expects the best outcome to occur and therefore selects more aggressive treatment
to improve the condition. However, a pessimist thinks the worst will happen. In Case 2,
an optimist chooses Surgery: L2 (overall value 1.95) and both the equative attitude holder
and a pessimist select Clinical Trial of level two. As we have seen earlier, Case 3 put
more emphasis on the state s1a. In Case 3, an optimist shows more preference to
Chemotherapy of level 4 (overall value 1.81), a pessimist selects Targeted Therapies of
level 4 (overall value 1.23) and the person with equative attitude selects level three of
Surgery (overall value 1.28). The decisions fully abide by the definitions given earlier in
this research. Furthermore, in case of the fourth case i.e. Case 4, an optimist selects
Chemotherapy of level 2 (overall value 1.84), a pessimist prefers Surgery level 1 (overall
value -1.79) and an equative attitude holder selects Surgery of level 4 (overall value 1.41).
In this case as well, an optimist’s decision shows that he is hopeful of having best result
by selecting a more aggressive or severe treatment. In the last scenario, both the equative
105
and pessimist go for Radiotherapy: L2 (overall value 1.14 and 0.08 respectively) whereas
an optimist’s decision is to choose Chemotherapy: L4. It is noticeable that in each case,
the overall value of the treatment alternative is the highest for the optimist and lowest for
the pessimist. The overall value of the alternative chosen by an equative lies in between.
The results presented in this section reveal the fact that uncertainty is an
important stimulus of the psychophysics of a human decision-maker and therefore creates
diverse impact on people’s decisions. As a result, the selection of treatment by the
patients of varying attitudes has also been different based on their outlook toward
uncertainty. We strongly believe that the proposed DM algorithm will be helpful to
implement decision support systems for the patients and their caregivers. This kind of
patient centric decision support systems can be utilized in acquiring knowledge about
patient’s desires for advanced and effective health-care decision-making. It is to be noted
that we are considering a DM framework which falls under the category of descriptive
decision-making theory. Therefore, we cannot comment that one decision is superior or
inferior to another or we cannot obtain an optimum solution which is the trait of
normative decision-making. Similarly, by our proposed descriptive DM model, it is also
not possible to declare any one solution of a treatment DM problem to be the best since
decisions vary depending on the attitudes of a patient which is the basic tenet of a
descriptive decision-making.
106
CHAPTER 6: CONCLUSION
The previous chapters described the proposed framework of descriptive decision-making
under uncertainty and an example application. This chapter ties together, integrates and
synthesizes the various issues raised in the entire research along with the motivating
questions mentioned in Chapter 1. Along with providing a summary of the entire research
endeavor, this chapter focuses on the limitations of this research as well as indicating the
promising areas of future research.
6.1 SUMMARY OF THE RESEARCH
The research on a framework of descriptive DM under uncertainty falls under the
category of a theory-based research. In this study, a unique model of descriptive DM
under uncertainty has been proposed and given the name NY-DDM. This research has
modeled such a framework which is capable of handling a decision-making problem right
from the scratch. At first, the DM problem is scrutinized and the uncertainty is
determined. Consequently, the alternatives, states of nature as well as outcomes are
sorted out and the uncertainty inherent in the DM problem is expressed with DempsterShafer Theory of evidence. In this way, we devise the DM problem as an Evidential
Decision-making Problem (EDMP). One important question that was raised during the
uncertainty representation is regarding the choice of the theory for representing
uncertainty. Since uncertainty has various forms of its own, it is quite logical to choose a
means of representation that covers the major variety of uncertainties. Gradually, we
107
discovered that DST has better and versatile expressiveness of uncertainty than the other
contenders. It properly represents partial and total ignorance. Ignorance can be quantified
by DST where low degree of ignorance means high confidence in results and high degree
of ignorance means low confidence in results. Furthermore, conflict can be quantified by
DST. Also, DST allows proper distinction between reasoning and decision-making and
does not pose any modeling restrictions. Apart from these, DST also possesses the scope
of accumulating the pieces of evidences stemming from multiple sources of information.
Therefore, it is evident that our choice of DST for uncertainty expression is satisfactory
and thus, the definition of EDMP is robust. This explanation answers the second
motivating question raised during the onset of this research.
The second phase of NY-DDM constitutes of the evidence combination
when multiple sources provide pieces of information that can be sometimes conflicting.
We have studied the existing evidence combination methods and discovered several
shortcomings. Therefore, during the course of this research, an improved method of
evidence combination based on weighted average of the given bba and reliability of the
information sources has been proposed for more accurate representation of uncertainty.
In the third phase of constructing the NY-DDM framework, the EDMP is
converted into decision-making problem under risks by approximating the probabilities
from the obtained/combined basic belief assignments (bba). This phase is one of the cores
of this research because the approximation is done by applying OWA operator by
resembling different attitudes of a human being during a decision-making process.
Primarily, we have worked with three major decision attitudes of a human decision-
108
maker: Equative, Pessimistic and Optimistic. But, we observed that human attitudes are
not limited to these three and it is possible to express various attitudes only by changing
the parameters of OWA operators. In this way, few other attitudes such as nearpessimistic (optimistic), quarter-pessimistic (optimistic), median attitude, 2nd best/worst
have been defined in this research.
At the end, Prospect theory is applied upon deriving the appropriate coefficient values of its value function and weighting function for the desired application
area. Thus, overall value of each alternative is obtained for each of the attitudes and
decision is made accordingly. As we mentioned earlier in Chapter 1, one of our
motivations behind this research endeavor was to recognize the true implication of
‘descriptive decision-making’ and its strength in representing human cognition. In quest
of the answer to this question, an application area has been chosen to apply the NY-DDM
framework. We have chosen the decision-making problem about treatment of lung cancer
by the patients and applied the approach. The detail of this process has been described in
Chapter 5. Upon completion of the application, the result shows that the decisions about
treatment are different due to the decision-makers’ attitudes towards uncertainty. The
result shown in Chapter 5 answers our third motivational question regarding the general
applicability of NY-DDM. In this regard, we would like to mention that the aim of this
research endeavor was never to suggest any ‘best’ solution which is the basic tenet of
normative decision-making. Our goal was to establish that human attitudes or behavior
plays an important role in decision-making by proposing a new type of decision-making
framework which performs better than the existing frameworks in the literature.
109
We are aware of the fact that criticism might arise due to the apparently
“lossy” transformation of bba to probabilities. From a theoretical point of view, critics
may assume that some part of information is lost in the process of approximation.
Decision Science has made it known that, with the physiological and psychological
limitations; humans are unable to process all the information perfectly while making
decisions. Therefore, human beings cannot be absolutely rational throughout his life. By
different empirical examples, PT proved that humans are not always rational. In this
research, we have accepted that humans can hardly deal with uncertain information
properly and the purpose of our study is to provide a framework/model of decisionmaking which points out the difference in decisions of a human decision-maker
according to his attitude toward uncertainty. We believe that people reduce or
approximate much information so that they can cope with the situation with their own
capability. In order to retrieve a solution of human’s incapability in dealing with
uncertain decision-making problems; approximation seems to be a natural and realistic
solution within the framework of descriptive decision-making even if it loses information.
We can claim that the proposed framework puts together two types of decision attitudes
for uncertainty; the attitude to risk (conflict) and to ignorance about risk.
6.2 MAJOR CONTRIBUTIONS OF THE RESEARCH
The major contributions of this research are summarized as follows:
1. This research proposes a descriptive decision-making framework (NY-DDM) that
is superior to the descriptive DM frameworks presented in the past. This model
110
can explain many forms of human attitudes which are observed in human life and
which were unexplored by the existing researches. These attitudes have
significant impact on the decision-making process.
2. Furthermore, NY-DDM solves the problem of the lack of consideration about the
difference between the weighting function of probability and the one of bba by
converting the EDMP into a probabilistic decision-making problem. As a result,
this simplifies the calculation.
3. The presence of multiple information sources is considered in this research and a
new method of evidence combination is proposed which is based on weighted
averaging and the reliability of the information sources. The proposed
combination method has shown improved performance than the existing evidence
combination rules for uncertainty representation in this research.
4. This research also proposes the values of three coefficients used in value function
and weighting function of PT that explains the risk attitudes of people in decisionmaking.
5. The application of NY-DDM in lung cancer treatment DM problem states that it
is efficient to utilize the proposed approach to understand the psychology of a
patient in a certain state of disease. This understanding can have a significant
impact on shared decision-making which is an emerging trend of medical
decision-making. Therefore, not only from the theoretical viewpoint, this research
also has important social contribution.
111
6.2 LIMITATIONS
One of the limitations of this research is that NY-DDM is a method of individual
decision-making rather than a group decision-making approach. In the areas where group
decisions are evaluated, NY-DDM might not be implemented without further
enhancement. Moreover, NY-DDM is a descriptive DM approach; it is not designed to
provide the ‘best’ decision of a problem. This may be considered as another limitation
when the ‘best’ decision is immensely required. Furthermore, we assumed in NY-DDM
that all of the possible outcomes are known but the probability of occurring of the
outcomes is not completely certain. The situations where both the outcomes and the states
of nature are unknown have not been considered in this research.
6.3 RECOMMENDATIONS FOR FUTURE RESEARCH
This research has several scopes of further enhancement in future. We recommend the
following extensions.
Decision Support Systems (DSS) use various algorithms to aid the
decisions of the decision-maker or the institution. Our research has the potential to be
used in such kind of DSS for understanding human cognition or human psychology
regarding any certain area of decision-making. The procedures of NY-DDM can thus be
transformed into a successful Decision Support System.
Also, researchers can closely observe the extensions of PT and assimilate
the model to those theories as well as modifying our model based on the usage
perspective. Not only that, researchers can explore the area of utility accumulation, belief
112
assignment etc. Furthermore, it is highly possible to modify the proposed evidence
combination rule to make it robust and nearly accurate.
Apart from the theoretical extensions, another significant area of further
research is the shared decision-making. The necessity of patient-centric decision-making
is being discussed since the last two decades. The approach we have proposed in this
research is very much suitable for the purpose of knowing the insight of a person because
of its descriptive decision-making characteristics. We strongly believe that the proposed
DM algorithm can be used to implement decision support systems for patients and their
caregivers. This kind of patient centric decision support systems can be very effective at
improving health care processes in acquiring knowledge to facilitate making informed
decisions. It is possible to couple the patient-based decision support system with highquality decision counseling so that patients may be allowed to weigh the benefits and
limitations among the appropriate alternatives.
113
APPENDIX A: PUBLICATION OF THE RESEARCH
Journal Publications:
E. Nusrat and K. Yamada: A descriptive decision-making model under
uncertainty: combination of Dempster-Shafer theory and Prospect theory, Int. J.
of Uncertainty, Fuzziness and Knowledge-Based Systems, Vol. 21, No.1, pp.79102 (Feb. 2013).
E. Nusrat and K. Yamada: Lung cancer treatment decision-making by descriptive
decision-making framework, International Journal of Computer Applications,
Vol. 67, No. 3, pp.1-8 (Apr. 2013).
International Conference Publications:
E. Nusrat, K. Yamada : Descriptive Decision-making of Lung Cancer Treatment
Using a DM Model with Dempster-Shafer Theory and Prospect Theory, SCISISIS 2012, pp. 359-364, Kobe, Japan (Nov. 21, 2012).
E. Nusrat, K. Yamada, S. Das: A Descriptive Decision-making Model of Patients’
Treatment Decision using Dempster-Shafer Theory and Prospect Theory, Western
Decision Sciences Institute 41st Annual Meeting (WDSI2012),pp. 302307,Hawaii (Apr. 2012).
E. Nusrat, K. Yamada: Descriptive Decision-making Model under Uncertainty
using Dempster-Shafer Theory and Prospect Theory, SCIS&ISIS2010, pp. 1504 1509, Okayama, Japan (Dec. 12, 2010).
Local Conference Publications:
E. Nusrat, K. Yamada: Different Decision Attitudes in Decision-making under
Uncertainty, Congress of The Shin-etsu Chapter of the Institute of Electronics,
114
Information and Communication Engineers, The IEEE Shin-etsu Session, 11A-5,
p186 (Oct. 8, 2011)
E. Nusrat, K. Yamada, M. Unehara: Decision Making under Uncertain
Environment with Dempster-Shafer Theory, Joint Workshop 2009 of Kanto and
Hokushin'etsu Chapters of SOFT, pp.63-66 (2009/10/11)
115
APPENDIX B: GRAPHICAL REPRESENTATION OF THE VALUES OF THE
ALTERNATIVES IN ALL FIVE CASES
1) Overall Values of Alternatives
Case 1:
The
value
116
highest
Case 2:
The
value
highest
The
value
highest
Case 3:
117
Case 4:
The
value
highest
The
value
highest
Case 5:
118
2) Graphical Representation of Decisions of the Patients
Case 1:
Case 2:
119
Case 3:
Case 4:
120
Case 5:
121
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