ABSTRACT
Fuzzy numbers have been studied extensively in mathematics along with its applications in diverse fields, especially in solving many real-life problems. This research work based on an extensive survey of the related existing works, formulates a ranking method for generalized trapezoidal fuzzy numbers by computing section-wise perimeters of the trapezoid representing a generalized fuzzy number. The perimeter of trapezoid representing a generalized fuzzy number is viewed as comprising three segments:- left and right divergences which are right angled triangles and the centre as a rectangle. In order to evaluate its degree of robustness, new similarity measures are computed and finally compared with the related results obtained under other existing similarity measure approaches by way of providing a comparison table.
TABLE OF CONTENTS
DECLARATION …………………………………………………………………………………………………………………..i
CERTIFICATION ……………………………………………………………………………………………………………….. ii
DEDICATION ……………………………………………………………………………………………………………………. iii
ACKNOWLEDMENT ………………………………………………………………………………………………………… iv
ABSTRACT ………………………………………………………………………………………………………………………… v
CHAPTER ONE
GENERAL INTRODUCTION
1.1 Background………………………………………………………………………………………………………………………………. 1
1.2 Statement of the Research Problem ……………………………………………………………………………… 2
1.3Aim and Objectives ……………………………………………………………………………………………………… 2
1.4Methodology……………………………………………………………………………………………………………….. 3
1.5Significance of the Study …………………………………………………………………………………………………….…5
1.6 Definition of Terms……………………………………………………………………….. 5
CHAPTER TWO
LITERATURE REVIEW
2.1 Methods of Ranking Fuzzy Numbers Using FuzzyIndex………………………………………..10
2.2 Methods of Ranking Fuzzy Numbers Using Area Representation……………………………12
2.3 Methods of Ranking Fuzzy Numbers Using Centroid Approach………………………….13
2.4 Methods of Ranking Fuzzy Numbers Using Preference Function…………………………14
2.5 Methods of Ranking Fuzzy Numbers Using Maximizing and Minimizing Sets…….……15
2.6 Method of Ranking Fuzzy Numbers Using Fuzzy Risk analysis………………………….15
2.7 Methods of Ranking Fuzzy Numbers Using L-R Representation…………………………15
2.8 Overview on Fuzzy Arithmetic Operations…………………………………..……………16
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CHAPTER THREE
FUNDAMENTALS OF FUZZY NUMBERS
3.1 Fuzzy Numbers …………………………………………………………………………………………………………..18
3.2 Representation of Generalized Triangular Fuzzy Numbers ………………………………………………20
3.2.1 Operations of Generalized Triangular Fuzzy Numbers and Their Properties …………………..22
3.3 Representation of Generalized Trapezoidal Fuzzy Numbers …………………………………………23
3.3.1Operations on Generalized Trapezoidal Fuzzy Numbers and Their Properties ………………23
3.4 Operations of Fuzzy Numbers Using Intervals…………………….……………………23
3.5 Operations of Fuzzy numbers Using𝜶−𝒄𝒖𝒕 Intervals …………………………………………………..25
3.6 Operations of Fuzzy Numbers Using Membership Function ………………………………………….26
3.7 The Extension Principle ………………………………………………………………………………………………30
3.8 Convex Fuzzy Numbers ……………………………………………………………………………………………..31
3.8.1 𝛼 -Level Sets of Fuzzy Numbers ………………………………………………………………………………..31
3.8.2 Support of a Fuzzy Number …………………………………………………………………………………………32
3.8.3 Operations of Fuzzy Numbers Interms of 𝜶–Level Sets. ……………………………………………..32
3.9 Algebraic Properties of Fuzzy Numbers……………………………………………………………………….32
CHAPTER FOUR
METHODS OF RANKING FUZZY NUMBERS
4.1 Liou and Wang‟s Ranking Method ……………………………………………………………………………35
4.2 Chen‟s Maximizing Set and Minimizing Set Ranking Method …………………………………….38
4.3 Chen and Tang‟s Method ofRanking Non- Normal Triangular and Trapezoidal Fuzzy Numbers ………………………………………………………………………………………………………………………….40
4.4 Lee and Li‟s Method of Ranking Fuzzy Numbers …………………………………………………………43
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4.5 Methods of Ranking Fuzzy Numbers by Computing The Centroid of a Fuzzy Number ..44
4.5.1 Cheng‟sDistance Method (Centroid index) ………………………………………………………………….44
4.5.2 Chu and Tsao‟s Method of Ranking by Computing The Centroid of a Fuzzy Number ……………………… 454.5.3Luu‟s Ranking Method Based on The Centroid Index of Fuzzy Numbers …………………………………………………………………………………………………………………………………………..47
4.5.4Babu‟s Method of Ranking Generalized Fuzzy Numbers Using Centroid of Centroids …….51
4.5.5 Rezvani‟s Method of Ranking Generalized Trapezoidal fuzzy Numbers with Euclidean Distance Between the Incentre of Centroids ………………………………………………………………………….53
4.6 Abbasbandy and Hajjari‟s Method of Ranking Generalized Trapezoidal Fuzzy Numbers in Parametric Form …………………………………………………………………………………………………………..54
4.7Abbasbandy and Asady‟s Method of Ranking Fuzzy Numbers Using Sign Distance ……….55
4.8Ezzati and Saneifard‟s Method of Ranking Fuzzy Numbers with Continuous Weighted Quasi-Arithmetic Means …………………………………………………………………………………………………..56
4.9 Methods of Ranking Fuzzy Numbers Using L-R Representation …………………………………..58
4.9.1Allahviranloo‟s Method of Ranking Fuzzy Numbers Using Weighted Distance……………….58
4.9.2 Chen and Lu‟s Method of Ranking Based on Left and Right Dorminance …………………….60
4.9.3 Luu‟sRanking Method Using Left and Right Indices ………………………………………………….61
4.9.4 Wang‟s Method of Ranking L-R Fuzzy Numbers Using Deviation Degree …………………63
4.9.5Najad and Mashinchi‟s Method of Ranking Fuzzy Numbers Based on L-R sides of Fuzzy Numbers ……………………………………………………………………………………………………………………………..65
4.9.6 Parandi‟s Method of Ranking Normal and Non-normal Fuzzy Numbers Based on L-R Areas ………………………………………………………………………………………………………………………………….65
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CHAPTER FIVE
PROPOSED METHOD OF RANKING FUZZY NUMBERS
5.1 New Method of Ranking of Generalized Trapezoidal Fuzzy Numbers and Related Results. ……………………………………………………………………………………………………………………………………….68
5.2 Similarity Measures under New Approach …………………………………………………………………..69
5.3 Results and Comparison ……………………………………………………………………………………………..71
5.4 Application of the New Method to Risk Analysis …………………………………………………………75
CHAPTER SIX
SUMMARY, CONCLUSIONS AND RECOMMENDATIONS
6.1 Summary ……………………………………………………………………………………………………………………82
6.2 Conclusions ……………………………………………………………………………………………………………….83
6.3 Recommendations ………………………………………………………………………………………………………83
References…………………………………………………………………………………………84
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CHAPTER ONE
ENERAL INTRODUCTION
1.1 Background to the Study
Zadeh (1965) formulated fuzzy set theory as a powerful tool to deal with problems involving imprecision for which other extant approaches were found incapacitated. In course of time, the concept of fuzzy sets has made a perceptible advance and a number of important concepts related to modeling complex fuzzy situations has emerged, investigated and applied. In all this, the development of fuzzy arithmetic has occupied a central position. Fuzzy arithmetic include describingfuzzy numbers, operation of fuzzy numbers, and ordering of fuzzy numbers (Dubois and prade, 1980),(Karfmann and Gupta, 1985) and (Zimmerman, 1996) for various details.
In order to seek a model in the set of real numbers 𝑅, special fuzzy sets characterized by their membership functions of the form 𝜇𝐴 : 𝑅⟶[0,1] were defined which gave rise to the concept of fuzzy numbers. A fuzzy number is a fuzzy set representing a real line interval with fuzzy boundaries (Zadeh, 1965).
Fuzzy numbers can appear in different shapes such as bell shape, triangular, trapezoidal, etc. It is known that triangular fuzzy numbers being simplest in the form have most frequently been explored in application systems. However, as observed in (Zimmermann, 1996),from computational efficiency point of view, the trapezoidal fuzzy numbers are found to have an edge. Fuzzy numbers have found applications in risk analysis, decision making, approximate reasoning; fuzzy control, optimization, forecasting, etc.(Zimmermann, 1996), (Zadeh, 1965). In most of these applications, it becomes imperative to define an ordering of fuzzy numbers and, in turn, that of an
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efficient ranking function ℛ∶𝑃 ℝ ⟶ℝor 𝑅∶𝑃(𝐼)⟶ℝ where 𝐼 is the unit interval [0, 1],𝑃(ℝ) is a set of fuzzy sets in , and 𝑃(𝐼) is the set of fuzzy sets in 𝐼. It may be noted that there is no loss of generality in restricting the domain of definition of fuzzy subsets from to 𝐼 (Bortolan&Degani, 1985).
The method of ranking fuzzy numbers was first proposed by (Jain, 1976) for decision making in fuzzy environment by way of representing ill-defined quantities as a fuzzy set. In decision theory, ranking of fuzzy numbers is, found uniquely important, for example, the concept of optimum or best choice is completely based on rankingor comparison. In course of time, a number of methods of ranking fuzzy numbers have been proposed. In spite of the existence of avariety of methods none has been found fully satisfactory.
1.2Statement of the Research Problem
Ranking fuzzy numbers play a very significant role in decision making, optimization,and data representation, just to mention a few. Many ranking methods have been proposed so far, however, there are yet no method that can give a satisfactory solution to every situation. Some of the methods are found counterintuitive, some are not discriminating,some use only the local information of fuzzy values and some cannot rank crisp numbers. This gives a room for proposing, developing and revising different ranking methods of fuzzy numbers that provide results close to intuition and other researches in the literature.
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1.3 Aim and Objectives
The aim of this research is to investigate ranking methods of fuzzy numbers and develop a new competing approach. In order to achieve this goal, the objectives are to:
(i) conduct a critical study of existing ranking methods of fuzzy numbers,
(ii) developa new approach for ranking generalized trapezoidal fuzzy numbers,
(iii) developa similarity measure using the new approach, and
(iv) apply it to risk analysis.
1.4 Methodology
We critically study some major ranking methods of fuzzy numbers: method for ranking fuzzy numbers by considering both the mean and dispersion of alternatives (Lee and Li 1988), ranking method based on integral value index (Liou and wang, 1992), ranking method of fuzzy numbers based on the concept of existence (Cheng and Lee, 1994), ranking fuzzy numbers using the distance method (Cheng, 1998), approximate approach for ranking fuzzy numbers based on left and right dominance (Chen and Lu, 2001), method of ranking fuzzy numbers based on the area between the centroid point and original point (Chu and Tsao, 2002), method of ranking fuzzy numbers with preference weighting function expectations (Liu and Han, 2005), method for ranking fuzzy numbers based on adapting two dimensions dominance (Chang et al., 2006), method for ranking p-norm trapezoidal fuzzy numbers (Chen and Tang, 2008), method of ranking of trapezoidal fuzzy numbers based on left and right spreads (Abbasbandy and Hajjari, 2009), method for ranking generalized trapezoidal fuzzy numbers based on rank, mode, divergence and spread (Kumar et al., 2011a), method for ranking in perimeters of two generalized trapezoidal fuzzy numbers (Rezvani, 2012). In this dissertation we propose to develope a new method for ranking fuzzy numbers that
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considers the section-wise perimeters of a generalized trapezoidal fuzzy number and compute similarity measures to strengthen our approach.
1.5 Significance of the study
Methods for ranking fuzzy numbers have many applications in different fields of study. The role of ranking in decision making, optimization and engineering cannot be over emphasized. As there is no general single method developed so far, it is important to explore new techniques for ranking fuzzy numbers
1.6Definition of terms
The following definitions are largely adopted from(Chang, 1981),(Zimmermann, 1996)(Asady, 2010), (Lakashmana et al., 2011), (Kumar et al., 2011), (Rezvani, 2012), etc. (a) Fuzzy Set
Let X bea universal set. Then the fuzzy subset 𝐴 ofX is defined by its membership function 𝜇𝐴 : 𝑋⟶[0,1] which assigns a real number 𝜇𝐴 (𝑥)in the interval [0, 1], to each element x∈X, where the value of 𝜇𝐴 (𝑥)at x shows the grade of membershipof xin𝐴 . (b) Complement of a Fuzzy Set
The complement 𝜇𝑐 of a fuzzy subset 𝜇 of a set X is a fuzzy subset given by: 𝜇𝑐 𝑥 =1−𝜇 𝑥 , 𝑥∈𝑋.
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(c) α–cut of a Fuzzy Set
Given a fuzzy set 𝐴 in X and any real number α ∈ [0, 1], then the α -cut or α -level or cut worthy set of 𝐴 , denoted by 𝐴 𝛼 is the crisp set 𝐴 𝛼= {x ∈X:𝜇𝐴 𝑥 ≥ α }.
The strong a-cut, denoted by 𝐴 𝛼+is the crisp set 𝐴 𝛼+= {x∈ X: 𝜇𝐴 𝑥 >α}. (d) Normal Fuzzy Set
A fuzzy subset 𝐴 of universal set U is normal if and only if𝑠𝑢𝑝𝑥∈𝑈𝜇𝐴 x =1.
(e) Convex Fuzzy Set A fuzzy subset 𝐴 of Universe set U is convex if and only if 𝜆𝑥+ 1−𝜆 𝑦≥(𝜇𝐴 x 𝛬𝜇𝐴 y ),∀𝑥,𝑦∈𝑈,∀𝜆∈[0,1], where 𝛬 denotes the minimum operator. (f)Fuzzy Number
A fuzzy set 𝐴 defined on the universal set of a real number is said to be a fuzzy number if its membership function has the following characteristics:
(i) 𝜇𝐴 ∶ ℝ→[0,1]is continuous.
(ii) 𝜇𝐴 𝑥 =0 𝑓𝑜𝑟𝑎𝑙𝑙𝑥∈ −∞,𝑎 ∪ 𝑑,∞ .
(iii) 𝜇𝐴 𝑥 is strictly increasing on [𝑎,𝑏] and strictly decreasing on 𝑐,𝑑 .
(iv) 𝜇𝐴 𝑥 =1 𝑓𝑜𝑟𝑎𝑙𝑙𝑥∈ 𝑏,𝑐 , where 𝑎≤𝑏≤𝑐≤𝑑.
(g) Triangular Fuzzy Number
A triangular fuzzy number 𝐴 is a fuzzy number with a piece wise linear membership function 𝜇𝐴 defined by
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𝜇𝐴 𝑥 = (𝑥−𝑎)(𝑏−𝑎), 𝑎≤𝑥≤𝑏 𝑐−𝑥 𝑐−𝑏 ,𝑏≤𝑥≤𝑐0, 𝑜𝑡𝑒𝑟𝑤𝑖𝑠𝑒.
written as a triplet (𝑎,𝑏,𝑐) where 𝑎≤𝑏≤𝑐;𝑎 and 𝑐 stand for the lower and upper values of the support of the fuzzy number 𝐴 , respectively, and b for the modal value.
Now,(𝑥−𝑎)(𝑏−𝑎) and 𝑐−𝑥 𝑐−𝑏 are known as left and right legs of the triangular fuzzy number 𝑎,𝑏,𝑐 . (h) Complement of a Triangular Fuzzy Number
Let 𝐴 = 𝑎,𝑏,𝑐 be a triangular fuzzy number.The complement 𝐴 𝑐 of 𝐴 is defined by 𝜇𝐴 𝑐 𝑥 =1−𝜇𝐴 𝑥 . Hence the membership function 𝜇𝐴 𝑐 is defined by 𝜇𝐴 𝑐 𝑥 = (𝑎−𝑥)(𝑏−𝑎), 𝑎≤𝑥≤𝑏 𝑥−𝑏 𝑐−𝑏 , 𝑏≤𝑥≤𝑐1, 𝑜𝑡𝑒𝑟𝑤𝑖𝑠𝑒. (i) Trapezoidal Fuzzy Number
A fuzzy number 𝐴 =(𝑎,𝑏,𝑐.𝑑) is said to be a trapezoidal fuzzy number if its membership function is given by 𝜇𝐴 𝑥 = (𝑥−𝑎)(𝑏−𝑎), 𝑎≤𝑥≤𝑏1, ,𝑏≤𝑥≤𝑐 𝑥−𝑑 𝑐−𝑑 ,𝑐≤𝑥≤𝑑.
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Alternatively,
Since both L and R of trapezoidal fuzzy numbers are linear, it can be represented by quadruple (𝑎,𝑏,𝑐,𝑑)∈𝑅4 and is completely characterized by 𝑎≤𝑏≤𝑐≤𝑑. In this case, the r-level sets are given by 𝑢𝑟=𝑎+𝑟 𝑏−𝑎 and 𝑢𝑟=𝑑+𝑟 𝑑−𝑐 .If we have 𝑏=𝑐, the quadruple reduces to a triple (𝑎,𝑏,𝑐) which is a triangular fuzzy number. (j) Generalized Fuzzy Number
A fuzzy set 𝐴 defined on the universal set of real numbers is said to be a generalized fuzzy number if its membership function has the following characteristics:
(i) 𝜇𝐴 ∶ ℝ→[0,1] is continuous,
(ii) 𝜇𝐴 𝑥 =0 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥∈ −∞,𝑎 ∪ 𝑑,∞ ,
(iii) 𝜇𝐴 𝑥 is strictly increasing on [𝑎,𝑏] and strictly decreasing on 𝑐,𝑑 ,
(iv) 𝜇𝐴 𝑥 =𝑤 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥∈ 𝑏,𝑐 , where 0<𝑤≤1.∀ 𝑎,𝑏,𝑐,𝑑∈ℝ.
Generalized fuzzy number can be represented alternatively by 𝜇𝐴 (𝑥)= 𝜇𝐴 𝐿 𝑥 , 𝑎≤𝑥≤𝑏𝑤, 𝑏≤𝑥≤𝑐𝜇𝐴 𝑅 𝑥 , 𝑐≤𝑥≤𝑑 0 , 𝑜𝑡𝑒𝑟𝑤𝑖𝑠𝑒,
where𝜇𝐴 𝐿 𝑥 : 𝑎,𝑏 →[0,𝑤] and 𝜇𝐴 𝑅 𝑥 : 𝑐,𝑑 →[0,𝑤] are two strictly monotonic and continuous functions from ℝ to the closed interval [0, w]. A fuzzy number𝐴 iscalled normal if 𝑤=1, otherwise 𝐴 is said to be a generalized or non-normal fuzzy number.The image −𝐴 of 𝐴 can be expressed by −𝑎,−𝑏,−𝑐,−𝑑;𝑤 .
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(k) Generalized Trapezoidal Fuzzy Number
A fuzzy number𝐴 is called a generalized trapezoidal fuzzy number,denoted 𝐴 = 𝑎,𝑏,𝑐,𝑑;𝑤 , if its membership function is given by 𝜇𝐴 𝑥 = 𝑤 𝑥−𝑎 (𝑏−𝑎), 𝑎≤𝑥≤𝑏𝑤, , 𝑏≤𝑥≤𝑐 𝑤(𝑥−𝑑)(𝑐−𝑑) ,𝑐≤𝑥≤𝑑.
(l) Normal and Non-normal Trapezoidal Fuzzy Number
A trapezoidal fuzzy number 𝐴 =(𝑎,𝑏,𝑐,𝑑;𝑤) is said to benormal if 𝑤=1 while
itis called non-normal if 𝑤≠1 and its membership function is given by 𝜇𝐴 𝑥 = 0, −∞<𝑥≤𝑎𝑤(𝑥−𝑎)(𝑏−𝑎), 𝑎≤𝑥≤𝑏𝑤, , 𝑏≤𝑥≤𝑐 𝑤 𝑥−𝑑 𝑐−𝑑 ,𝑐≤𝑥≤𝑑0, 𝑑≤𝑥<∞.
Figure 3.5: A generalized trapezoidal fuzzy number
R
a
c
W
b
d
0 0
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(m) Non-normal p-norm Trapezoidal Fuzzy Number
A non- normal fuzzy number 𝐴 =(𝑎,𝑏,𝑐,𝑑;𝑤) is said to be a non- normal p-norm trapezoidal fuzzy number, denoted by 𝐴 =(𝑎,𝑏,𝑐,𝑑;𝑤)𝑝, if its membership function is given by 𝜇𝐴 𝑥 = 0, −∞<𝑥≤𝑎𝑤(1− 𝑥−𝑏𝑎−𝑏 𝑝)1𝑝 , 𝑎≤𝑥≤𝑏𝑤, 𝑏≤𝑥≤𝑐 𝑤 (1− 𝑥−𝑐𝑑−𝑐 𝑝)1𝑝 ,𝑐≤𝑥≤𝑑0, 𝑑≤𝑥<∞ where p is a positive integer.
(n) Positive and Negative Fuzzy Numbers
A fuzzy number 𝐴 is said to be positive if 0<𝑎1≤𝑎2 holds for the support
𝛤𝐴 =[𝑎1 ,𝑎2]of𝐴 , that is, 𝛤𝐴 is the positive real line. ∀ 𝑎1,𝑎2∈ℝ.
Similarly,𝐴 is called negative if 𝑎1≤𝑎2≤0 , and 𝐴 is called zero if 𝑎1≤0≤𝑎2.
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