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A brief survey of this study is to identify the ranking formulas for various fuzzy numbers derived from research papers published over the past few years. This paper presents the latest results of fuzzy ranking applications very clearly and simply, as well as highlighting key points in the use of fuzzy numbers. This paper discusses the importance of pointing out the concepts of fuzzy numbers and their formulas for ranking.

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A brief survey of this study is to identify the ranking formulas for various fuzzy numbers derived from research papers published over the past few years. This paper presents the latest results of fuzzy ranking applications very clearly and simply, as well as highlighting key points in the use of fuzzy numbers. This paper discusses the importance of pointing out the concepts of fuzzy numbers and their formulas for ranking.

- 1. http://www.iaeme.com/IJARET/index.asp 73 editor@iaeme.com International Journal of Advanced Research in Engineering and Technology (IJARET) Volume 8, Issue 5, September - October 2017, pp. 73–82, Article ID: IJARET_08_05_009 Available online at https://iaeme.com/Home/issue/IJARET?Volume=8&Issue=5 ISSN Print: 0976-6480 and ISSN Online: 0976-6499 © IAEME Publication VARIOUS FUZZY NUMBERS AND THEIR VARIOUS RANKING APPROACHES Lavanya P Lecturer in Mathematics, Government Polytechnic College, Arakandanallur, Villupuram, Tamil Nadu, India ABSTRACT A brief survey of this study is to identify the ranking formulas for various fuzzy numbers derived from research papers published over the past few years. This paper presents the latest results of fuzzy ranking applications very clearly and simply, as well as highlighting key points in the use of fuzzy numbers. This paper discusses the importance of pointing out the concepts of fuzzy numbers and their formulas for ranking. Key words: Fuzzy Set, Fuzzy Numbers, Various Ranking Methods Cite this Article: Lavanya P, Various Fuzzy Numbers and their Various Ranking Approaches, International Journal of Advanced Research in Engineering and Technology, 8(5), 2017, pp 73–82. https://iaeme.com/Home/issue/IJARET?Volume=8&Issue=5 1. INTRODUCTION In 1965 Lotfi. A. Zadeh introduced the concept of Fuzziness. An approximation of analytical functions by Dubois, Prade, and Yager (1978), involves the division of the membership functions of algebraic operations into a left side and a representation of the right side by a simple analytical form. Due to the ascent within the study of Fuzzy sets, we have different types of Fuzzy numbers, namely Triangular, Trapezoidal, Pentagonal, Hexagonal, Heptagonal fuzzy numbers. Well, S.H. Chen was studying operations on fuzzy numbers with real-valued functions in 1985, as well as Klement. In 2004 Michael Hanss introduced the Triangular Fuzzy number. In many applications, fuzzy numbers are used, such as control theory, signal processing, and approximation theory. We also discussed the basic definitions of different Fuzzy numbers and their numerous fuzzy ranking approaches in this survey article. This survey alone would make it easy to consider their review of the effects of the ranking methods and their future development in the future. 2. FUZZY NUMBERS 2.1. Introduction In 1975, Hutton, B [HU] & Rodabaugh, SE [Rod] introduced a fuzzy number. A fuzzy number is the fundamental precept of the fuzzy set theory we typically use. It is chosen from the default fuzzy set of all real numbers. Like standard numbers, fuzzy sets have been either positive or
- 2. Various Fuzzy Numbers and their Various Ranking Approaches http://www.iaeme.com/IJARET/index.asp 74 editor@iaeme.com negative, where the whole space is symmetrically empty. The linguistic form is often selected to address the fuzzy number, which includes slightly, quietly. Calculations with fuzzy numbers allow parameters, properties, geometry, and initial conditions to be inserted into uncertainty. In the literature on fuzzy sets, Zadeh (1965) notes that granulation plays a part in human cognition. Membership functions are structured to represent individual and subjective human experiences as part of a fuzzy set. A fuzzy set has several functions of membership, known as operations from a well-defined universe. X with an interval between units, 0 to 1, as seen in the following equation: : [0,1] A X → (1) The degree of notification for a vague class with an infinite set of range values between 0 and 1. The notification level for fuzzy numbers with an infinite set of range values between 0 and 1 is specified by the membership function. Fuzzy numbers play a crucial role in many fields in computation, communications products engineering, scientific testing, decision-making, approximate reasoning, and optimization. Definition (Fuzzy Numbers): A fuzzy set N is said to be a fuzzy number based on the real number R : A fuzzy set N is concave. A fuzzy set N is normal. i.e., 0 0 such that ( ) 1. N x R x = ( ) N x , is piecewise continuous. N , must be closed interval for every [0,1]. is bounded. Supp N 3. TRIANGULAR FUZZY NUMBER Definition: ( ) , , N a b c = be the fuzzy number is known as a triangular fuzzy number if its membership functions : 0,1 R → are equal to , 1 ( ) , 0 N x a if x a b b a x b x c x if x b c c b otherwise − − = = − − Where( ) , , a b c , this fuzzy number is denoted by( ) , , a b c . Figure 1 Triangular Fuzzy Number
- 3. Lavanya P http://www.iaeme.com/IJARET/index.asp 75 editor@iaeme.com 3.1. Ranking Methods for Triangular Fuzzy Numbers Many types of ranking procedures have 'Triangular fuzzy numbers.' All the rankings listed here have been discovered over the past few years for 'Triangular fuzzy numbers' and compiled from studying various research papers. Only the most important of them are listed here. They are, “ − Cut”for Triangular Fuzzy Numbers[24]: ( ) ( ) ( ) L a , , = − + − − U a b a a c c b “Yeager’s ranking” method for Triangular Fuzzy Numbers[29]: ( ) 1 ( , , ) , , 2 = L U Y a b c a a d ( ) ( ) ( ) L a , , = − + − − U where a b a a c c b “Sub interval Average” method for Triangular Fuzzy Numbers[3]: ( ) ( ) 4 , , 12 + + = a b c R a b c “Sub interval Addition” method for Triangular Fuzzy Numbers[1]: 4( ) ( , , ) 6 + + = a b c R a b c “Pascal Triangular Graded Mean” for Triangular Fuzzy Numbers[24]: ( ) 2 , , 4 + + = a b c P a b c “Magnitude Ranking” for Triangular Fuzzy Numbers[24]: ( ) 1 0 1 ( , , ) 3 2 = + − Mag a b c c a b rdr “Centroid approach” for Triangular Fuzzy Numbers[10]: ( ) 2 14 2 7 , , ; 6 6 + + = a b c w C a b c w 4. TRAPEZOIDAL FUZZY NUMBER Definition. Let ( ) , , , = N a b c d be a fuzzy number, is known as a trapezoidal fuzzy number if its membership function , 1 , ( ) , 0 − − = − − N x a if x a b b a x b c x d x if x c d d c otherwise The trapezoidal fuzzy number N is denoted by quadruplet ( ) , , , = N a b c d and has the shape of a trapezoid.
- 4. Various Fuzzy Numbers and their Various Ranking Approaches http://www.iaeme.com/IJARET/index.asp 76 editor@iaeme.com Figure 2 Trapezoidal Fuzzy Number 4.1. Ranking Methods for Trapezoidal Fuzzy Numbers There are 'Trapezoidal fuzzy numbers' for several forms of ranking procedures. Even the most important of them here Over the past few years, all the rankings listed here have been found for 'Trapezoidal fuzzy numbers' and collected by reviewing different research papers. “ − Cut” for Trapezoidal Fuzzy Numbers[8]: ( ) ( ) ( ) L a , , = − + − − U a b a a d d c “Yeager’s ranking” method for Trapezoidal Fuzzy Numbers[30]: 1 2 1 4 2 ( ) 2 5 3 = + − + L U R A a a “Sub interval Average” method for Trapezoidal Fuzzy Numbers[3]: ( ) ( ) 5 , , , 20 + + + = a b c d R a b c d “Sub interval Addition” method for Trapezoidal Fuzzy Numbers[1]: 5( ) ( , , , ) 10 + + + = a b c d R a b c d “Pascal Triangular Graded Mean” for Trapezoidal Fuzzy Numbers[15]: ( ) 3 3 , , , 8 + + + = a b c d P a b c d “Magnitude Ranking” for Trapezoidal Fuzzy Numbers[16]: 5 5 ( , , , ) 12 + + + = a b c d Mag a b c d “Centroid approach” for Trapezoidal Fuzzy Numbers[10]: ( ) 2 7 7 7 , , , ; 6 6 + + + = a b c d w C a b c d w
- 5. Lavanya P http://www.iaeme.com/IJARET/index.asp 77 editor@iaeme.com 5. PENTAGONAL FUZZY NUMBER Definition. A pentagonal fuzzy number ( ) , , , , = p N a b c d e , should satisfy the following condition In the interval 0,1 , ( ) p N x is a continuous function. , and b,c a b is a continuous function and ( ) p N x is strictly increasing. , and d,e c d is a continuous function and ( ) p N x is strictly decreasing. Figure 3 Pentagonal Fuzzy number 5.1. Ranking Methods for Pentagonal Fuzzy Numbers They are many 'Pentagonal Fuzzy Numbers' ranking procedures. All the rankings listed here have been developed for 'Pentagonal Fuzzy Numbers' over the last few years and collected from the consideration of different research papers. Just the most important of them are listed here. They're, “ − Cut” for Pentagonal Fuzzy Numbers[26]: 1 2 1 2 [ ( ), ( )], [0,0.5]}{[ ( ), ( )], [0.5,1] P N P P for Q Q for = ( ) ( ) ( ) ( ) 2 , 2 for 0,0.5 2 2 ,2 2 for 0.5,1 P b a a e d e N c b b c d c d c − + − − + = − + − − − + “Robust ranking” method for Pentagonal Fuzzy Numbers[4]: 1 0 1 ( ) ( ) , ( ) 2 = − + − − p R N b a a e e d d “Sub interval Average” method for Pentagonal Fuzzy Numbers[3]: ( ) ( ) 6 , , , , 30 + + + + = a b c d e R a b c d e “Sub interval Addition” method for Pentagonal Fuzzy Numbers[1]: 6( ) ( , , , , ) 15 + + + + = a b c d e R a b c d e
- 6. Various Fuzzy Numbers and their Various Ranking Approaches http://www.iaeme.com/IJARET/index.asp 78 editor@iaeme.com “Pascal Triangular Graded Mean” for Pentagonal Fuzzy Numbers[23]: ( ) 4 6 4 , , , , 16 + + + + = a b c d e P a b c d e “Magnitude Ranking” for Pentagonal Fuzzy Numbers[28]: 1 0 1 ( ( ) ( ) ) , [0,1] 6 = + + + + p L U Mag N A A b c d A for “Centroid approach” for Pentagonal Fuzzy Numbers[6]: ( ) 3 4 3 6 2 4 , , , , ; 18 9 + + + + = a b c d e w R a b c d e w 6. HEXAGONAL FUZZY NUMBER Definition. Let N is a hexagonal fuzzy number denoted by ( ) , , , , , Hex N a b c d e f = , and its membership function ( ) Hex N x is given as, 1 2 1 1 2 2 1 ( ) 1 1 2 1 2 0 . Hex N x a for a x b b a x b for b x c c b for c x d x x d - for d x e e d f x for e x f f e otherwise − − − + − = − − − − Figure 4 Hexagonal Fuzzy Number
- 7. Lavanya P http://www.iaeme.com/IJARET/index.asp 79 editor@iaeme.com 6.1. Ranking Methods for Hexagonal Fuzzy Numbers “Hexagonal Fuzzy Numbers” also have many types of ranking procedures. Over the past several years, all ranking listed here have been found for 'Hexagonal Fuzzy Numbers' and collected by reviewing different research papers. The most relevant of these have been here. They are the, “ − Cut” for Hexagonal Fuzzy Numbers[14]: ( ) ( ) ) ( ) ( ) ) 2 , 2 ] 0,0.5 2 2 , 2 2 0.5,1 Hex b a a f e f for N c b c b e d e d for − − + − − + = − − + − − + − “Yeager’s ranking” method for Hexagonal Fuzzy Numbers[5]: 1 0 ( ) (0.5) ( ) Hex U L Y N A A d = + “Sub interval Average” method for Hexagonal Fuzzy Numbers[3]: 7( ) ( ) 42 Hex a b c d e f R N + + + + + = “Sub interval Addition” method for Hexagonal Fuzzy Numbers[1]: 7( ) ( ) 21 Hex a b c d e f R N + + + + + = “Pascal Triangular Graded Mean” for Hexagonal Fuzzy Numbers[21]: 5 10 10 5 ( ) 32 Hex a b c d e f P N + + + + + = “Magnitude Ranking” for Hexagonal Fuzzy Numbers[19]: 2 3 4 4 3 2 18 Hex a b c d e f Mag N + + + + + = “Centroid approach” for Hexagonal Fuzzy Numbers[18]: 2 3 4 4 3 2 ) 5 ( ) 18 18 Hex a b c d e f w R N + + + + + = 7. HEPTAGONAL FUZZY NUMBER Definition. A Heptagonal fuzzy numbers ( ) , , , , , , Hep N a b c d e f g = , is a fuzzy number and its membership function ( ) ( ) ( ) ( ) 2 1 2 1 2 ( ) 1 2 1 2 2 0 Hep N x a a x b b a b x c x d c x d d c x d x d x e e d e x f g x f x g g f otherwise − − − + − = − + − − −
- 8. Various Fuzzy Numbers and their Various Ranking Approaches http://www.iaeme.com/IJARET/index.asp 80 editor@iaeme.com Figure 5 Heptagonal Fuzzy Number 7.1. Ranking Methods for Heptagonal Fuzzy Numbers In so many types of rank procedures, there are many heptagonal fuzzy numbers. Many of the rankings listed have been discovered for 'Heptagonal Fuzzy Numbers' over the past few years and gathered by analyzing numerous research papers. Only the most significant of them are listed here. They are, “ − Cut” for Heptagonal Fuzzy Numbers[26]: 2 ( ), 2 ( ) [0,0.5] 2( 1)( ) , 2( 1)( ) [0.5,1] Hep a b a g g f N d c d d e d − + − − − = − − + − − − “Robust ranking” method for Heptagonal Fuzzy Numbers[27]: ( ) 1 0 1 ( ) , 2 L U Hep Hep Hep R N a a d = “Sub interval Average” method for Heptagonal Fuzzy Numbers[3]: 8( ( ) 56 Hep a b c d e f g R N + + + + + + = “Sub interval Addition” method for Heptagonal Fuzzy Numbers[1]: 8( ( ) 28 Hep a b c d e f g R N + + + + + + = “Pascal Triangular Graded Mean” for Heptagonal Fuzzy Numbers[9]: ( ) 6 15 20 15 6 64 Hep a b c d e f g P N + + + + + + = “Centroid approach” for Heptagonal Fuzzy Numbers[13]: 0 0 ( ) ( , ) 2 7 7 22 7 7 2 11 54 54 Hep R N R x y a b c d e f g w = + + + + + + =
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