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SPRINGER 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 and Technology Computational Intelligence Series editor Janusz Kacprzyk, Warsaw, Poland About this Series The series “Studies in Computational Intelligence” (SCI) publishes new develop- mentsandadvancesinthevariousareasofcomputationalintelligence—quicklyand with a high quality. The intent is to cover the theory, applications, and design methods of computational intelligence, as embedded in the fields of engineering, computer science, physics and life sciences, as well as the methodologies behind them. The series contains monographs, lecture notes and edited volumes in computational intelligence spanning the areas of neural networks, connectionist systems, genetic algorithms, evolutionary computation, artificial intelligence, cellular automata, self-organizing systems, soft computing, fuzzy systems, and hybrid intelligent systems. Of particular value to both the contributors and the readership are the short publication timeframe and the world-wide distribution, which enable both wide and rapid dissemination of research output. More information about this series at http://www.springer.com/series/10618 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 LibraryofCongressControlNumber:2015956365 ©TheAuthor(s)2016 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilarmethodologynowknownorhereafterdeveloped. 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Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAGSwitzerland 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 cognition 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 membership 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