A Comprehensive Literature Review Of 55Years Of Fuzzy Set Theory

ABSTRACT

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 practical studies from engineering area to arts and humanities, from computer science to health sciences, from life sciences to physical sciences and also in business, administration and accounting. In this work, a comprehensive literature review on the fuzzy set theory is realized. In the recent years, ordinary fuzzy sets have been extended to new types and these extensions have been used in many areas such as energy, medicine, material, economics and 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
• Fuzzy Set Theory and Fuzzy Logic
• review of previous studies (from 1965-2020)

CHAPTER THREE

3.1      Material Used

3.2      Fuzzy Set Theory with Graphical Illustrations

3.3     Subject Areas of Fuzzy Publications

CHAPTER FOUR

4.0      Result and discussion

CHAPTER FIVE

• Conclusion
• Recommendation
• References

 CHAPTER ONE

INTRODUCTION

• Background of Study

According to Zadeh (1965), a fuzzy set is a class of objects whose memberships are not precisely defined. Fuzzy sets provide a better representation of reality than the classical mathematical binary representation. The membership in fuzzy sets is gradual,that makes the theory invaluable to represent the 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 habitually assumes 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.

Lotfi Asker Zadeh been the founder of the fuzzy set theory published his first paper on his new theory in 1965. Between 1965 and 1975, Zadeh broadened the foundation of the fuzzy set theory by establishing fuzzy similarity relations, linguistic hedges, and fuzzy decision-making. In 1970s, some research groups in Japan started to study the fuzzy set theory. In 1970, Mamdani developed the first fuzzy logic controller. In 1977, some industrial applications of fuzzy logic began in Japan and Europe. The success of fuzzy logic was observed in Japan at the beginning of 1980s and this led to a revival in fuzzy logic in the US at the end of 1980s.

Along with the emergence of the fuzzy set theory, many objections to the theory have appeared (2008). Those brilliant scientists could not accept Zadeh’s way of handling the uncertainty. They so much exaggerated in decrying it that the logic of the theory was defined as “Fuzzy logic is the cocaine of science.” However, the fuzzy logic proved its power through the real technology applications.

One of the well-known applications of fuzzy logic is that of the Sendai Subway system that went into operation in 1988 in Sendai, Japan. For the control of lines, a fuzzy controller was used to run the train all day long. Through the fuzzy controller, the line became one of the smoothest running subway systems in the world. The commercial appliances such as heating ventilation and air conditioning (HVAC) systems have used fuzzy logic thermostats to control the heating and cooling thus saved energy by making the system more efficient and keeping the temperature more steady than a traditional thermostat. Fuzzy logic control systems have also been used to control the speed of cars based on the obstacle sensed. The system is composed of a sensor in the front panel to sense the presence of obstacles. Fuzzy logic based Programmable Logic Controllers (PLCs) have been developed by companies like Moeller, one of the longest-established and most respected names in the electrical sector. Fuzzy logic has also been used in 3D animation systems for generating crowds as it was used extensively in the making of the Lord of the Rings.

This work presents an extended comprehensive literature review for the fuzzy set theory. It aims at exhibiting the expansion areas of the fuzzy set theory in its 55th year.

• 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.

• 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:

1. To study the various applications of fuzzy set theory
2. To exhibit the expansion areas of the fuzzy set theory in its 55th year (that is, from 1965-2020) and presents the results of literature review with graphical illustrations and a classification with respect to the extensions of fuzzy

• To study different publications on fuzzy set theory
  • 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.