ranking of pentagonal fuzzy numbers applying incentre of ...ranking of pentagonal fuzzy numbers...

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

[email protected] professor,

Hindustan Institute of Technology and Science, Chennai -603 103,[email protected]

3B.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 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 .

Key Words :Fuzzy numbers [FN], Pentagonal fuzzy number[PFN],fuzzy arithmetic operations, alpha cut, Ranking, 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

International Journal of Pure and Applied Mathematics Special Issue

166

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-

sponding PFN then

1. Addition:APFN(+) BPFN = (λ1 + β1, λ2 + β2, λ3 + β3, λ4 + β4, λ5 + β5)

2. Subtraction: APFN(−) BPFN = (λ1 − β5, λ2 − β4, λ3 − β3, λ4 − β2, λ5 − β1)

3. Multiplication:APFN(∗) BPFN = (λ1∗β1∧λ1∗β5∧λ5∗β1∧λ5∗β5, λ2∗β2∧λ2∗β4∧λ4∗β2∧λ4∗β4, λ3∗β3, λ2∗β2∨λ2∗β4∨λ4∗β2∨λ4∗β4, λ1∗β1∨λ1∗β5∨λ5∗β1∨λ5∗β5)

4. Division:APFNBPFN

= (λ1β1∧ λ1

β5∧ λ5

β1∧ λ5

β5, λ2β2∧ λ2

β4∧ λ4

β2∧ λ4

β4, λ3β3, λ2β2∨ λ2

β4∨ λ4

β2∨ λ4

β4, λ1β1∨

λ1β5∨ λ5

β1∨ λ5

β5)

excluding the case β1 = 0 (or) β2 = 0 (or) β3 = 0 (or) β4 = 0 (or) β5 = 0Suppose, APFN and BPFNare the positive real number <+,then(3

′) APFN(∗) BPFN = (λ1 ∗ β1, λ2 ∗ β2, λ3 ∗ β3, λ4 ∗ β4, λ5 ∗ β5) and

(4′)APFNBPFN

= (λ1β5, λ2β4, λ3β3, λ4β2, λ5β1

)

4.2 Example Let APFN = (2, 4, 6, 8, 10)and BPFN = (3, 6, 9, 12, 15)be two PFNthen

1. Addition:APFN(+) BPFN = (5, 10, 15, 20, 25)

2. Subtraction: APFN(−) BPFN = (−13,−8,−3, 2, 7)

3. Multiplication:APFN(∗) BPFN = (6, 24, 54, 96, 150)


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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

• [Pl(α), Pu(α)] = [2α(λ2 − λ1) + λ1,−2α(λ5 − λ4) + λ5] similarly

• [Ql(α), Qu(α)] = [2(α− 12)(λ3 − λ2) + λ2,−2(α− 1)(λ4 − λ3) + λ3]

Hence α cut of PFN

PFNα =

{[2α(λ2 − λ1) + λ1,−2α(λ5 − λ4) + λ5] for α ∈ [0,0.5)

[2(α− 12)(λ3 − λ2) + λ2,−2(α− 1)(λ4 − λ3) + λ3] for α ∈ [0.5,1]

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.

5.2 Addition of two PFNs:

APFN(+) BPFN =

[2α((λ2 + β2)− (λ1 + β1)) + (λ1 + β1),−2α((λ5 + β5)− (λ4 + β4)) + (λ5 + β5) for α ∈ [0, 0.5)

[2(α− 12)(((λ3 + β3)− (λ2) + β2)) + (λ2β2),

−2(α− 1)((λ4 + β4)− ((λ3) + β3)) + (λ3 + β3)] for α ∈ [0.5, 1]In Numerical example 4.2,

α(A+B)PFN α = 0 α = 0.0.5 α = 1 PFN(10α + 5,−10α + 25) [5, 25] [10, 20] [15, 15] (5, 10, 15, 20, 25)

Figure 4: Addition of a normal PFNs A(2,4,6,8,10) and B=(3,6,9,12,15)

5.3 Subtraction of two PFNs:

APFN(−) BPFN =

[2α((λ2 + β2)− (λ1 + β1)) + (λ1 + β1),−2α((λ5 + β5)− (λ4 + β4)) + (λ5 + β5) for α ∈ [0,0.5)

[2(α− 12)((λ3 + β3)− (λ2) + β2)) + (λ2β2),

-2 (α− 1)((λ4 + β4)− ((λ3) + β3)) + (λ3 + β3)] for α ∈ [0.5, 1]In Numerical example 4.2,

α(A−B)PFN α = 0 α = 0.0.5 α = 1 PFN(10α− 13,−10α + 7) [−13, 7] [−8, 2] [−3,−3] (−13,−8,−3, 2, 7)

5.4 Multiplication of Two PFN:

APFN(∗) BPFN =

[2 α(λ2 − λ1) + λ1 ∗ 2α(β2 − β1) + β1,−2α(λ5 − λ4) + λ5 ∗ −2α(β5 − β4) + β5] for α ∈ [0, 0.5)

[2(α− 12)(λ3 − λ2) + λ2 ∗ 2(α− 1

2)(β3 − β2) + β1 + β2,

-2 (α− 1)(λ4 − λ3) + λ3 ∗ −2(α− 1)(β4 − β3) + β3]α ∈ [0.5, 1]In Numerical example 4.2,


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Figure 5: Subtraction of a normal PFNs A(2,4,6,8,10) and B=(3,6,9,12,15)

α(A ∗B)PFN α = 0 α = 0.0.5 α = 1 PFN(24α2 + 24α + 6, 24α2 − 120α + 150) [6, 150] [24, 96] [54, 54] (−6, 24, 54, 96, 150)

Figure 6: Multiplications of a normal PFN A(2,4,6,8,10) and B=(3,6,9,12,15)

5.5 Division of Two PFN:

[APFNBPFN

]}

=

{[[ 2α(λ2−λ1)+λ1−2α(β5−β4)+β5 ], [−2α(λ5−λ4)+λ5

2α(β2−β1)+β1 ]] for α ∈ [0, 0.5)

[[2(α− 1

2)(λ3−λ2)+λ2

−2(α−1)(β4−β3)+β3 ], [−2(α−1)(λ4−λ3)+λ32(α− 1

2)(β3−β2)+β2 ]] for α ∈ [0.5, 1]

In Numerical example 4.2,α[AB

]PFN α = 0 α = 0.0.5 α = 1 PFN

4α+2−6α+15

[0.13, 3.33] [0.33, 1.33] [0.67, 0.67] (0.13, 0.33, 0.67, 1.33, 3.33)

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,


171

step 1 :Find αAP , βAP , γAP and αBP , βBP , γBPstep 2 :Find IAP (x0, y0) and IBP (x0, y0)

step 3 : Find RAP=√

(x02 + y0

2). and RBP=√

(x02 + y0

2).

and by using the ranking of fuzzy numbers i.e.,)i) If RAP

> RBPthen AP > BP

ii) If RAP< RBP

then AP < BP

iii) If RAP≈ RBP

then AP ≈ BP

6.1 Numerical example : Let AP = (2, 4, 6, 8, 10)and BP=(3,6,9,12,15) be twonormalized fuzzy numbers

step 1 : Find αAP=2.027,βAP=4.0000,γAP=2.0276 and αBP=3.0185,βBP=6.0000,γBP=3.0185

step 2 : Find IAP = [6.0000, 0.3322]andIBP = [9.0000, 0.3328]

step 3 : Find RAP=6.0092 and RBP

=9.0062 soR AP< RBP

then AP < BP

7 Conclusion

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.

References

[1] L.A.Zadeh, Inform. Sci., 8 (1975), 199-249, 301-357; 9 (1976). 43-80.

[2] J.G. Dijkman, H. Van Haeringen,And S. J. De Lange, Fuzzy Numberss, CJour-nal Of Mathematical Analysis And Applications, 92 (1983), 301-341.

[3] T. Pathinathan and K. Ponnivalavan, Pentagonal fuzzy numbers, Internationaljournal of computing algorithm, 3 (2014), 1003-1005.

[4] Guanrong Chen,Trung Tat Pham, Introduction to Fuzzy Sets, Fuzzy Logic, andFuzzy Control Systems, CRC Press,(2001).

[5] Timothy J. Ross, Fuzzy Logic With Engineering Applications, John Wiley SonsLtd,(2004).

[6] Masaharu Mizumoto And Kokichi Tnaka, Some Properties Of Fuzzy Numbers,Advaces In Fuzzy Set Theory And Applications,North Holland Publishing Com-pany,(1979).

[7] Klir. G.J and Bo Yuan,Fuzzy Setsand Fuzzy logic, Prentice Hall of India PrivateLimited, (2005).


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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.

[9] P. Selvam,A.Rajkumar,J.Sudha Easwari,Dodecagonal Fuzzy Number, Interna-tional Journal of Control Theory and Applications, International SciencePress,9(28) (2016), 499-515.


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ranking of pentagonal fuzzy numbers applying incentre of ...ranking of pentagonal fuzzy numbers...

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