Table Of ContentSPRINGER BRIEFS IN APPLIED SCIENCES AND
TECHNOLOGY COMPUTATIONAL INTELLIGENCE
Ranjit Biswas
Is ‘Fuzzy Theory’
an Appropriate
Tool for Large
Size Problems?
123
SpringerBriefs in Applied Sciences
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Computational Intelligence
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Ranjit Biswas
‘ ’
Is Fuzzy Theory
an Appropriate Tool
for Large Size Problems?
123
Ranjit Biswas
Faculty of Engineering andTechnology,
Departmentof Computer Science
andEngineering
Jamia Hamdard University
NewDelhi
India
ISSN 2191-530X ISSN 2191-5318 (electronic)
SpringerBriefs inApplied SciencesandTechnology
ISBN978-3-319-26717-3 ISBN978-3-319-26718-0 (eBook)
DOI 10.1007/978-3-319-26718-0
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Contents
Is ‘Fuzzy Theory’ an Appropriate Tool for Large
Size Problems? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Two Hidden Facts About Fuzzy Set Theory (and, About
Any Soft Computing Set Theory) . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.1 Understanding the ‘Discovery of CIFS’ by an Example . . . . . . 3
3 Cognitive Intuitionistic Fuzzy System (CIFS) . . . . . . . . . . . . . . . . . . 5
3.1 Atanassov Trio Functions and Atanassov Constraint . . . . . . . . 5
3.2 Atanassov Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.3 Impossible Types of Initialization . . . . . . . . . . . . . . . . . . . . . 7
3.4 Atanassov Trio Bags . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.5 Atanassov Processing Time (APT). . . . . . . . . . . . . . . . . . . . . 8
3.6 ‘Atanassov Speed of Convergence’ (ASC)
for an Element x. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.7 Categorizing the ‘Quality’ of a Decision Maker. . . . . . . . . . . . 14
3.8 Speed of Evaluation (Trio Speeds). . . . . . . . . . . . . . . . . . . . . 15
3.9 Decision Trajectory Curve and Atanassov Point
of Convergence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.10 Atanassov Absolute Triangle. . . . . . . . . . . . . . . . . . . . . . . . . 19
3.11 ‘Decision Line-Segments’ and ‘Atanassov
Line-Segment’ for an Element x . . . . . . . . . . . . . . . . . . . . . . 19
3.12 ‘Atanassov Triangle’ for the Element x . . . . . . . . . . . . . . . . . 21
3.13 ‘General IF Case’ and ‘Five Special IF Cases’
in Theory of CIFS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4 Is ‘Fuzzy Theory’ an Appropriate Tool for Large
Size Problems?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
5 Ordering (or Ranking) of Elements in an IFS
on the Basis of Their Amount of Belongingness . . . . . . . . . . . . . . . . 30
5.1 An Example Showing Application Potential
of ‘Ordering (or Ranking) of Elements’ . . . . . . . . . . . . . . . . . 30
v
vi Contents
6 An Application Domain to Understand the Potential
of Intuitionistic Fuzzy Theory Over Fuzzy Theory. . . . . . . . . . . . . . . 32
6.1 Object Recognition: An Impossible Task Without CIFS. . . . . . 32
6.2 Explaining the Method by an Instance . . . . . . . . . . . . . . . . . . 36
7 An Example of Application Domain to Understand
the Potential of Fuzzy Theory Over Intuitionistic Fuzzy
Theory in Some Cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
7.1 Penalty Shootout. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
7.2 CESFM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
7.3 “Fuzzy Pocket Machine” M for the Referee . . . . . . . . . . . . . . 45
7.4 Three Algorithms Algo-1, Algo-2 and Algo-3
for De-Fuzzification. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
7.5 CS Values of Other Parameters in CESFM. . . . . . . . . . . . . . . 48
7.6 ‘CS Score’ of a Team . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
7.7 A Hypothetical Example of a Football Game . . . . . . . . . . . . . 51
7.8 Comparing CESFM with Obsolete FIFA Rules. . . . . . . . . . . . 54
8 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
8.1 Theory of CISF. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
8.2 Pattern Recognition or Object Recognition: An Impossible
Task Without CISF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
8.3 Unique Application of Fuzzy Theory in Sports . . . . . . . . . . . . 58
9 Future Research Directions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
Abbreviations and Their Significance. . . . . . . . . . . . . . . . . . . . . . . . . 63
Abstract
To the world’s scientists and researchers, the most popular soft computing set
theories are as follows: fuzzy set theory, intuitionistic fuzzy set theory (vague sets
arenothingbutintuitionisticfuzzysets,asjustifiedandreportedbymanyauthors),
i–v fuzzy set theory, i–v intuitionistic fuzzy set theory, L-fuzzy set theory, type-2
fuzzysettheory,roughsettheory,softsettheory,etc.Insuchtheories,thevalueof
µ(x) is proposed by the concerned decision maker by his best possible judgment.
Whileahungrytigerfindshisfoodlikeonecoworonebuffalooronedeer(orany
other animal of his own list) in the forest, he decides a lot by his best possible
judgment on a number of significant parameters before he starts to chase and also
during the real-time period of his chasing. The decision makers are human being,
animal/bird, or any living thing which has brain (we do not consider software or
robots which have artificial intelligence). The most important (but yet an open
unsolved problem) question in the theory of soft computing is: How does a cog-
nition system of human/animal evaluate the membership value µ(x)? This work
unearths a ground-level reality about the ‘Progress’ of decision-making process
in the human/animal cognition systems while evaluating the membership value
µ(x) by proposing the Theory of CIFS. It is finally justified and concluded that
‘fuzzy theory’ may not be an appropriate tool to deal with large-sized problems,
while in pursuance of excellent results. But at the end, two examples of
‘decision-making problems’ with solutions are presented, out of which one will
show the dominance of the application potential of intuitionistic fuzzy set theory
over fuzzy set theory and the other will show the converse, i.e., the dominance
oftheapplicationpotentialoffuzzysettheoryoverintuitionisticfuzzysettheoryin
some cases.
(cid:1) (cid:1) (cid:1)
Keywords CIFS Atanassovtriofunctions Atanassovinitialization Atanassov
(cid:1) (cid:1) (cid:1) (cid:1) (cid:1)
constraint Atanassov trio bags h-bag m-bag n-bag Atanassov speed of
(cid:1) (cid:1) (cid:1)
convergence Atanassov processing time (APT) Atanassov absolute triangle
(cid:1) (cid:1)
Decision trajectory curve Atanassov point of convergence Atanassov
vii
viii Abstract
(cid:1) (cid:1) (cid:1)
line-segment Zadeh line-segment Atanassov triangle Intuitionistic
(cid:1) (cid:1) (cid:1) (cid:1)
µ-index (iµi) Atanassov indicator CESFM Positive parameters Negative
(cid:1) (cid:1) (cid:1) (cid:1)
parameters CES-score (CS) Fuzzy pocket machine Punishment fuzzy set
(cid:1) (cid:1) (cid:1)
CPS PCS NCS CESFM-software
‘ ’
Is Fuzzy Theory an Appropriate Tool
for Large Size Problems?
1 Introduction
The most important question (but yet an open unsolved problem) in the theory of
soft-computing is as mentioned below:
HowDoestheCognitionSystemofaHumanorAnimal(orofanylivingthing
which has brain) Evaluate the Membership Value µ(x)?
The work in this book is based on philosophical as well as logical views on the
subject of decoding the ‘progress’ of decision making process in the Human/
Animal cognition systems while evaluating the membership value µ(x) in a fuzzy
set or in an intuitionistic fuzzy set or in any such soft computing set model or in a
crisp set, developing a new theory called by “Theory of CIFS”. By ‘cognition
system’weshallmeanthroughoutherethecognitionsystemofahumanbeingorof
a living animal or of a bird or of any living thing which has brain (we do not
consider robots or software which have artificial intelligence). At the end of this
work one could realize the final outcome which is: “at the ground reality of
cognition system while evaluating a fuzzy membership value µ(x) or a mem-
bership value µ(x) corresponding to any soft computing set theory model or
even crisp membership value (0 or 1), it is not possible to apply the fuzzy
theoryoranysoftcomputingsettheoryorcrisptheoryinanydecisionmaking
process without intuitionistic fuzzy theory”.
Quite naturally, three immediate questions arise in mind: (1) whether a crisp
decision maker or a fuzzy decision maker or any soft computing decision maker
needstohaveknowledgeabout‘IntuitionisticFuzzySetTheory’?(2)Cantheynot
make any decision if they are not aware/knowledgeable of intuitionistic fuzzy
theory? and (3) What’s about animals, birds or other living things who are also
decision makers? For example, a hungry lion thinks and decides a lot before
chasing a buffalo and also during the period of chasing a buffalo! Even in many
situationsthey decidewhether itisappropriatetochase, orevenafterchasingthey
©TheAuthor(s)2016 1
R.Biswas,Is‘FuzzyTheory’anAppropriateToolforLargeSizeProblems?,
SpringerBriefsinComputationalIntelligence,DOI10.1007/978-3-319-26718-0_1
