Ranking of pentagonal fuzzy numbers applying incentre of centroids P.Selvam 1 , A.Rajkumar 2 and J.Sudha Easwari 3 1 Research scholar , Hindustan Institute of Technology and Science, Chennai -603 103,India and Assistant Professor, Adithya Institute of Technology,Coimbatore-641 107,India [email protected]2 Assistant professor, Hindustan Institute of Technology and Science, Chennai -603 103,India. [email protected]3 B.T Assistant, Government Higher Secondary School, Tirupur- 638 701,India. [email protected]Abstract Fuzzy numbers are based on membership function which have been clas- sified into shape of triangle,trapezoidal, bell etc., even in various different points in real numbers. The human judgment data preference are repeatedly unclear. So that the crisp values are insufficient by using then using uncer- tain numbers such as triangular,trapezoidal. Even they are not suitable in few case whereas uncertainties arises in more than four points. In such case pentagonal fuzzy number i.e., five points can be used to solve the problems. Our main concept of in this paper we introduced the five different points of pentagonal fuzzy numbers and the new operations addition, subtraction, multiplication, division. It also introduces the ranking of Pentagonal fuzzy numbers and applying incentre of centroid . Key Words :Fuzzy numbers [FN], Pentagonal fuzzy number[PFN], fuzzy arithmetic operations, alpha cut, Ranking, Centroid. International Journal of Pure and Applied Mathematics Volume 117 No. 13 2017, 165-174 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu Special Issue ijpam.eu 165
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Ranking of pentagonal fuzzy numbers applyingincentre of centroids
P.Selvam1 , A.Rajkumar2 and J.Sudha Easwari3
1Research scholar ,Hindustan Institute of Technology and Science, Chennai -603 103,India
and Assistant Professor,Adithya Institute of Technology,Coimbatore-641 107,India
Fuzzy numbers are based on membership function which have been clas-sified into shape of triangle,trapezoidal, bell etc., even in various differentpoints in real numbers. The human judgment data preference are repeatedlyunclear. So that the crisp values are insufficient by using then using uncer-tain numbers such as triangular,trapezoidal. Even they are not suitable infew case whereas uncertainties arises in more than four points. In such casepentagonal fuzzy number i.e., five points can be used to solve the problems.Our main concept of in this paper we introduced the five different pointsof pentagonal fuzzy numbers and the new operations addition, subtraction,multiplication, division. It also introduces the ranking of Pentagonal fuzzynumbers and applying incentre of centroid .
International Journal of Pure and Applied MathematicsVolume 117 No. 13 2017, 165-174ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version)url: http://www.ijpam.euSpecial Issue ijpam.eu
165
1 Introduction
L.A.Zadeh[1] was introduced the concept of fuzzy numbers and fuzzy arithmetic.Masaharu Mizumoto and Kokichi Tnaka [6] have investigated the algebraic prop-erties of fuzzy numbers under addition, subtraction, multiplication, division, jointand meet operations . J.G. Dijkman, H. Van Haeringen and S. J. De Lange [2] haveinvestigated nine operations of fuzzy numbers for addition . Triangular fuzzy num-bers are frequently used in application. In some cases triangular fuzzy numbers isnot suitable whereas uncertainties arises in more than four points. So the importantcontributes to the theory of fuzzy numbers have five different points by numerousresearchers with triangular shape fuzzy numbers. In this paper four different oper-ations like addition, subtraction, multiplication and division have been introducedusing alpha cut principle and as new approach for ranking with incentre of centroidusing pentagonal fuzzy numbers.
2 Preliminaries and Notations
2.1 Definition(FN):A fuzzy set A is defined on the set of real line, R is said to bea fuzzy number if its membership function µA : < → [0, 1] satisfies
• Convex and Normal of fuzzy set.
• A is piecewise continuous.
2.1 Definition(TFN): A fuzzy numbers A is called a triangular fuzzy number(TFN) is a particular case of semi symmetric L-R fuzzy number and if its member-ship function µA is given by
µA(x) =
x−λ1λ2−λ1 for λ1 ≤ x ≤ λ2λ3−xλ3−λ2 for λ2 ≤ x ≤ λ3
0 otherwise
Figure 1: A triangular fuzzy number A = (λ1, λ2, λ3)
3 New PFNs
3.1 Definition(PFN): A fuzzy number APFN is a PFN denoted by APFN =(λ1, λ2, λ3, λ4, λ5) whereas λ1, λ2, λ3, λ4, λ5 are real numbers and its membershipfunction
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µAPFN(x) =
12( x−λ1λ2−λ1 ) for λ1 ≤ x ≤ λ2
12
+ 12( x−λ2λ3−λ2 ) for λ2 ≤ x ≤ λ3
1− 12( x−λ3λ4−λ3 ) for λ3 ≤ x ≤ λ4
12( λ5−xλ5−λ4 ) for λ4 ≤ x ≤ λ5
0 for x < λ1 and x > λ5
Figure 2: Graphical repesentation of a normal new PFN for x ∈ [0,1]
4 Operations of PFNs:
4.1 DefinitionLet APFN = (λ1, λ2, λ3, λ4, λ5) and BPFN = (β1, β2, β3, β4, β5) be their corre-
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Figure 3: Graphical repesentation of a normal PFN A(2,4,6,8,10) andB=(3,6,9,12,15)
4. Division:APFNBPFN
= (0.13, 0.33, 0.67, 1.33, 3.33)
4.3 Definition A PFN can be defined as PFN= Pl(t), Ql(u), Pu(t), Qu(u), t ∈[0, 0.5],u ∈ [0.5, 1.0] whereas
Pl(t) = 12( x−λ1λ2−λ1 ) Pu(t) = 1
2( λ5−xλ5−λ4 )
Ql(t) = 12
+ 12( x−λ2λ3−λ2 ) Qu(t) = 1− 1
2( x−λ3λ4−λ3 )
Pl(t), Ql(u) is monotonic ascending with bounded under [0,0.5] and [0.5,1.0].Pu(t), Qu(u) is monotonic descending with bounded under [0,0.5] and [0.5,1.0].
4.4 Definition The alpha cut of the PFN in the set of elements in X is definedas
PFNα = {x ∈ X/µAPFN(x) ≥ α} =
{[Pl(α), Pu(α)] for α ∈ [0, 0.5) and
[Ql(α), Qu(α)] for α ∈ [0.5, 1] where as α ∈ [0, 1]
4.5 Definition If Pl(x) = α and Pu(x) = α, then α- cut operations intervalPFNα is obtained as
Note:The triangular fuzzy numbers and PFNs are same for the points have theequal intervals and distinct for the points have the unequal intervals.
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5 A new operation for Addition, Subtraction , Multiplication, Divisionon PFN
5.1 Definition Let APFN=(λ1, λ2, λ3, λ4, λ5) and BPFN=(β1, β2, β3, β4, β5) , for allx,λ1, λ2, . . . , λ5,β1, β2, . . . , β5 ∈ <, λ1 ≤ λ2 ≤ . . . ≤ λ5,β1 ≤ β2 ≤ . . . ≤ β5 be their correspondingPFN then for all α ∈ [0,1]. let us take the membership function on the basis of α-cut α A and α B of APFN and BPFN by using interval arithmetic.
Figure 7: Division of a normal PFN A(2,4,6,8,10) and B=(3,6,9,12,15)
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Figure 8: Graphical repesentation of a normal PFNs with incenter of centroids
6 Proposed ranking method of PFN :
In the PFN having the five different points they are A,M, O,N and E. The M and Npoints meets at points B and D. Also, join the points BO and DO . Now the normalpentagonal has been divided into two triangles and one quadrilateral ABO, ODE andOBCD respectively. Let the three plain figures are G1, G2and G3 with the centroid.To define the ranking of normalized PFN in the incentre of centroid G1, G2and G3
is taken as the point to consider. Let a normalized PFN APFN = (λ1, λ2, λ3, λ4, λ5).These three plane figures centriod are
G1 ={[
λ1+λ2+λ33
], 16
}, G2 =
{[λ2+2λ3+λ4
4
], 12
}and G3 =
{[λ3+λ4+λ5
3
], 16
}
respectively.We define the incentre
IAP (x0, y0)=
[[αAP[λ1+λ2+λ33 ]
+βAp[λ2+2λ3+λ4
4 ]+γ
AP[λ3+λ4+λ53 ]αAP+βAP+γAP
],
[αAP [ 16 ]+βAP [ 12 ]+γAP [ 16 ]
αAP+βAP+γAP
]]
Whereas αAP =
√λ4+4λ5−2λ3−3λ2)2+16
12, βAP =
√λ1+λ2−λ4−λ5)2
3and
γAP =
√λ2+4λ1−2λ3−3λ4)2+16
12
The ranking function of the normalized PFN APFN = (λ1, λ2, λ3, λ4, λ5).A set of real
numbers which map the set of all fuzzy numbers is defined as:RAP=√
(x02 + y0
2).
This is the incentre of the centroid with Euclidean distance. In the followingsteps expresses the sum and the incentre of the centroids are the using the rankof two fuzzy numbersAPFN and BPFN . Let APFN = (λ1, λ2, λ3, λ4, λ5) and BPFN =(β1, β2, β3, β4, β5)be two normalized PFN then,
International Journal of Pure and Applied Mathematics Special Issue
In this research paper we introduced a new membership function of PFN has beenintroduced with operations of addition ,subtraction,multiplication and division andalso illustrate with numerical examples. Hence this paper provides a very simplemethod to find the ranking of PFNs using the incentre of centroids.
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[8] P. Selvam,A.Rajkumar,J.Sudha Easwari,A New Method To Find OctagonalFuzzy Number, International Journal of Control Theory and Applications, In-ternational Science Press,9(28) (2016), 447-461.
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