A COMPREHENSIVE LITERATURE REVIEW OF 55YEARS OF FUZZY SET THEORY
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This work is on a “comprehensive literature review of 55years of fuzzy set theory”. Fuzzy sets have a great progress in every scientific research area. It found many application areas in both theoretical and practicalstudies from engineering area to arts and humanities, from computer science to health sciences, from life sciences to physicalsciences and also in business,administrationandaccounting. In this work, a comprehensive literature review on the fuzzy set theory is realized.In the recent years, ordinary fuzzy setshave been extended to new types and these extensions have been used in many areas such as energy, medicine, material, economicsand pharmacology sciences. This literature review also analyzes the chronological development of these extensions.
TABLE OF CONTENTS
Cover page
Title page
Approval page
Dedication
Acknowledgement
Abstract
CHAPTER ONE
1.0 Introduction
1.1 background of the study
- statement of problem
- aim and objective of the project
- definition of terms
CHAPTER TWO
2.0 LITERATURE REVIEW
- Overview of fuzzy sets
- FuzzySetTheoryandFuzzyLogic
- review of previous studies (from 1965-2020)
CHAPTER THREE
3.1 Material Used
3.2 FuzzySetTheorywithGraphicalIllustrations
3.3 SubjectAreasofFuzzyPublications
CHAPTER FOUR
4.0 Result and discussion
CHAPTER FIVE
- Conclusion
- Recommendation
- References
CHAPTER ONE
INTRODUCTION
- 1 Background of Study
According to Zadeh (1965), a fuzzy set is a class of objects whose memberships arenot precisely defined. Fuzzy sets provide a betterrepresentation of reality than the classical mathematicalbinary representation. The membership in fuzzy sets isgradual,thatmakesthetheoryinvaluabletorepresentthe limited level of precision in mental representations (Dubois, 2000).
Most of our traditional tools for formal modeling, reasoning, and computing are crisp, deterministic, and precise in character. Crisp means dichotomous, that is, yes-or-no type rather than more-or-less type. In traditional dual logic, for instance, a statement can be true or false—and nothing in between. In set theory, an element can either belong to a set or not; in optimization a solution can be feasible or not. Precision assumes that the parameters of a model represent exactly the real system that has been modeled. This, generally, also implies that the model is unequivocal, that is, that it contains no ambiguities. Certainty eventually indicates that we assume the structures and parameters of the model to be definitely known and that there are no doubts about their values or their occurrence. Unluckily these assumptions and beliefs are not justified if it is important, that the model describes well reality (which is neither crisp nor certain). In addition, the complete description of a real system would often require far more detailed data than a human being could ever recognize simultaneously, process, and understand. This situation has already been recognized by thinkers in the past. In 1923, the philosopher B. Russell referred to the first point when he wrote: ‘All traditional logic habituallyassumes that precise symbols are being employed. It is, therefore, not applicable to this terrestrial life but only to an imagined celestial existence’ according to Russell (1923). L. Zadeh referred to the second point, when he wrote: ‘As the complexity of a system increases, our ability to make precise and yet significant statements about its behavior diminishes until a threshold is reached beyond which precision and significance (or relevance) become almost mutually exclusive characteristics’ (Zadeh, 1973). For a long time, probability theory and statistics have been the predominant theories and tools to model uncertainties of reality (Zadeh, 2005). They are based—as all formal theories—on certain axiomatic assumptions, which are hardly ever tested, when these theories are applied to reality. In the meantime more than 20 other ‘uncertainty theories’ have been developed, which partly contradict each other and partly complement each other. Fuzzy set theory—formally speaking—is one of these theories, which was initially intended to be an extension of dual logic and/or classical set theory.
LotfiAskerZadeh been thefounderofthefuzzysettheorypublished his first paper on his new theory in1965. Between 1965 and 1975, Zadeh broadened thefoundation of the fuzzy set theory by establishing fuzzysimilarityrelations,linguistichedges,andfuzzydecision-making.In1970s,someresearchgroupsinJapan started to study thefuzzy settheory. In 1970,Mamdani developed the first fuzzy logic controller. In1977, some industrial applications of fuzzy logic beganin Japan and Europe. The success of fuzzy logic wasobserved in Japan at the beginning of 1980s and this ledto arevival infuzzylogic intheUSattheendof1980s.
Along with the emergence of the fuzzy set theory, manyobjectionstothetheoryhaveappeared (2008).ThosebrilliantscientistscouldnotacceptZadeh’swayofhandling the uncertainty. They so much exaggerated indecrying it that the logic of the theory was defined as“Fuzzy logic is the cocaine of science." However, thefuzzylogicproveditspowerthroughtherealtechnologyapplications.
One of the well-known applications of fuzzy logic isthatoftheSendaiSubwaysystemthatwentintooperation in 1988 in Sendai, Japan.For the control oflines, a fuzzy controller was used to run the train all daylong. Through the fuzzy controller, the line became oneof the smoothest running subway systems in the world.The commercial appliances such as heating ventilationand air conditioning (HVAC) systems have used fuzzylogic thermostats to control the heating and cooling thussaved energy by making the system more efficient andkeeping the temperature more steady than a traditionalthermostat. Fuzzy logic control systems have also beenused to control the speed of cars based on the obstaclesensed.Thesystemiscomposedofasensorinthefront panel to sense the presence of obstacles. Fuzzy logicbasedProgrammableLogicControllers(PLCs)havebeen developed by companies like Moeller, one of thelongest-establishedandmostrespectednamesintheelectrical sector. Fuzzy logic has also been used in 3Danimation systems for generating crowds as it was usedextensivelyinthemakingoftheLord oftheRings.
This work presents an extended comprehensive literature review forthe fuzzy set theory.It aims at exhibiting theexpansion areas of the fuzzy set theory in its 55th year.
1.2 Statement of the Problem
Fuzzy set theory has been used extensively in fields of complexity, cognitive sciences, and artificial intelligence, especially in numerous fields such as expert systems, knowledge discovery, information system, inductive reasoning, intelligent systems, data mining, pattern recognition, decision-making, and machine learning. Rough sets models, which have been recently proposed, are developed applying the different fuzzy generalizations. Currently, there is not a systematic literature review and classification of these new generalizations about set models for a long time, all the reviewed works have been limited on the one done from 1965 to 2015. Therefore, in this work review study was made to provide an extension of a comprehensive systematic review of methodologies and applications of recent generalisations discussed in the area of fuzzy set theory from 1965 to 2020.
1.3 Aim and Objectives
The purpose of this study is to provide an in-depth presentation of the extension and contributions of fuzzy set theory in more field of life. The objectives of this study are:
- To study the various applications of fuzzy set theory
- To exhibit theexpansion areas of the fuzzy set theory in its 55th year (that is, from 1965-2020)andpresentstheresultsofliteraturereviewwithgraphical illustrations andaclassificationwith respecttotheextensionsoffuzzysets.
- To study different publications on fuzzy set theory
1.4 Definition of terms
Terms that are frequently used in this work are defined as below:
Fuzzification: Fuzzification is the process of assigning the numerical input of a system to fuzzy sets with some degree of membership.
Fuzzy logic is a form of many-valued logic in which the truth value of variables may be any real number between 0 and 1.
Set theory:branch of mathematics which deals with the formal properties of sets as units (without regard to the nature of their individual constituents) and the expression of other branches of mathematics in terms of sets.
CHAPTER FIVE
5.1 Conclusion
Even the fuzzy set theory is a young theory, whose age is only 55; it has covered a lot of ground by today. Even there are still some researchers against it; numerous invaluable researchers delivered its present status to the world. Almost every branch of science, from engineering to social sciences, from decision sciences to computer sciences, and from physical sciences to life sciences, utilized the theory with a great success.
Extensions of fuzzy sets are the focal point of the present studies. Intuitionistic fuzzy sets take the first rank that researchers prefer utilizing them in their studies. Hesitant fuzzy sets take the second rank in this manner. Type-2 fuzzy sets take the third rank even they are older than the first two extensions. It is possible that more extensions of fuzzy sets will appear in the future, each trying to define membership functions with a different point of view.
5.2 Recommendation
The place where the fuzzy set theory has reached in its 55th year is quite far from its origin but the theory needs much time to arrive at the point it deserves.
We suggest a deeper analysis within subject areas for further research. For instance, in the subject area engineering, both theoretical works and practical works on the fuzzy set theory can be comprehensively analyzed. This will provide a better understanding of the value created by the fuzzy set theory.
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