symmetric pentagonal fuzzy numbers · 2018-03-27 · a fuzzy number is a quantity, whose value is...

Symmetric Pentagonal Fuzzy NumbersJ. Jesintha Rosline1 and E. Mike Dison2

1Department of Mathematics,SRM University, Chennai, Tamil Nadu, India

[email protected]. Research Department of Mathematics,

Loyola College, Chennai - 34, Tamil Nadu, [email protected]

Abstract

Fuzzy number is a tool used to quantify the vaguelydefined natural phenomenon. In this paper, we introducethe concept of symmetric pentagonal fuzzy number andquadratic pentagonal fuzzy number along with its geomet-rical illustration. In addition to that, we define the basicarithmetic operations such as addition and subtraction oftwo symmetric pentagonal fuzzy numbers with a numericalexample.

AMS Subject Classification:03E72

Keywords: Pentagonal fuzzy number, symmetric PFN, generalizedsymmetric PFN, quadratic PFN.

1 IntroductionThe data obtained for decision making in real life situation are often

approximately known. In 1965, Zadeh [5] introduced the concept offuzzy set theory to meet the problems caused by approximation andgeneralization. In 1978, Dubois and Prade [1] defined fuzzy numbersas a fuzzy subset of the real line. Fuzzy numbers allow us to make amathematical model to study linguistic variables in a fuzzy environment.

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International Journal of Pure and Applied MathematicsVolume 119 No. 9 2018, 245-253ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version)url: http://www.ijpam.euSpecial Issue ijpam.eu

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A fuzzy number is a quantity, whose value is imprecise, rather thanexact as in the case with "ordinary" numbers. Any fuzzy number canbe thought of as a function whose domain is a specified set. In manyrespects, fuzzy numbers depict the physical world more realistically thansingle-valued numbers. The fuzzy numbers and fuzzy values are widelyused in engineering applications and experimental sciences because oftheir suitability for representing uncertain information.

In 2014, T. Pathinathan and K. Ponnivalavan introduced a newfuzzy number named pentagonal fuzzy number [3]. Also they developedthe generalized concepts of pentagonal fuzzy number in 2015 along withthe set theoretic operations [4]. Rajkumar and T. Pathinathan usedpentagonal fuzzy number to sieve out the poor in the Nalanda District,Bihar [2]. The main objective of this paper is to brief out the concep-tual theory behind the pentagonal fuzzy number with the help of reallife phenomenon. In this paper, we introduce a new type of pentag-onal fuzzy numbers such as symmetric pentagonal fuzzy number andquadratic pentagonal fuzzy number along with their geometrical repre-sentation.

This paper is organized as follows. Section two presents the basicpreliminaries and definitions on fuzzy sets and fuzzy numbers. Sectionthree provides the definition of pentagonal fuzzy number and generalizedpentagonal fuzzy number with illustration. Section four introduces thetwo new definitions on pentagonal fuzzy number with examples followedby conclusion in the section five.

2 Basic PreliminariesIn this section, we present the basic definition of generalized pentagonalfuzzy number.

2.1 Generalized Pentagonal Fuzzy Number(GPFN)

A generalized pentagonal fuzzy number A denoted by AP is definedas AP=(a1, a2, a3, a4, a5;w) where a1, a2, a3, a4, a5 are the real numbersand w be the new maximum core of the fuzzy number other than 1.

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Then the membership function is given by,

FA(x) =

0 x < a1(x−a1)(a2−a1) a1 ≤ x ≤ a2w2

(x−a2)(a3−a2) a2 ≤ x ≤ a3

w x = a3w2

(a4−x)(a4−a3) a3 ≤ x ≤ a4

(a5−x)(a5−a4) a4 ≤ x ≤ a50 x > a5

with a1 ≤ a2 ≤ a3 ≤ a4 ≤ a5 and 0<w<1. If w=1, then the generalizedpentagonal fuzzy number become pentagonal fuzzy number.

3 Symmetric Pentagonal FuzzyNumber (SPFN)

This section introduces the two new definitions of pentagonal fuzzy num-ber with geometrical illustration. Also this section discusses the variousarithmetic operations for the new pentagonal fuzzy numbers.

3.1 Symmetric Pentagonal Fuzzy NumberASP=(a1, b, a2;m,n, n,m) is said to be a symmetric pentagonal fuzzynumber, if it has the membership function as follows;

F ˜ASP (x) =

0 x < a1 −m12

(x−(a1−m))m a1 −m ≤ x ≤ a1

(b−a1)2 a1 ≤ x ≤ b, 0 ≤ α ≤ 0.5

(x−(b−n))n a1 ≤ x ≤ b, 0.5 ≤ α ≤ 1

1 x = b(a2−b)

2 b ≤ x ≤ a2, 0 ≤ α ≤ 0.5((b+n)−x)

n b ≤ x ≤ a2, 0.5 ≤ α ≤ 112

((a2+m)−x)m a2 ≤ x ≤ a2 + m

0 x > a2 + m

and a1 ≤ b ≤ a2 with a1, b, a2 ∈ R. (m, m) and (n, n) are thelength of the intervals [a1 − m, a1], [a2, a2 + m] and [a1, b], [b, a2]respectively with m,n>0. When b = 1

2 , then the symmetric pen-tagonal fuzzy number will be reduced to symmetric trapezoidalfuzzy number. Also when a1 = a2, then the symmetric pentagonal

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fuzzy number will be reduced to symmetric triangular fuzzy num-ber. The geometrical illustration of the fuzzy number presentedin Figure 1 shows the symmetric representation of the pentagonalfuzzy number with m + n length in both sides of the function.

Figure 1: Symmetric Pentagonal Fuzzy Number

3.1.1 Arithmetic operations on symmetric pentagonal fuzzynumber

This section presents the various arithmetic operations on sym-metric pentagonal fuzzy numbers such as addition and subtractionalong with the geometrical illustration. The α level representationof the pentagonal fuzzy number is defined as follows;

AP = [a1+α(a2−a1); a2+2α(a3−a2); a4+2α(a3−a4); a5+α(a4−a5)](1)

Based on α-level representation of pentagonal fuzzy number, thebasic arithmetic operations such as addition and subtraction aredefined in the following sections.

3.1.2 Addition of two SPFN

Let ASP=(a1, b, a2;m,n, n,m) and BSP=(a′1, c, a

′2;m

′, n

′, n

′,m

′) betwo symmetric pentagonal fuzzy numbers. Then addition of twosymmetric pentagonal fuzzy numbers is given by:

ASP + BSP = (a1, b, a2;m,n, n,m) + (a′1, c, a

′2;m

′, n

′, n

′,m

′) (2)= (a1 + a

′1, b + c, a2 + a

′2;m + m

′, n + n

′, n + n

′,m + m

′)(3)

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3.1.3 Subtraction of two SPFN

Let ASP=(a1, b, a2;m,n, n,m) and BSP=(a′1, c, a

′2;m

′, n

′, n

′,m

′) betwo symmetric pentagonal fuzzy numbers. Then subtraction of twosymmetric pentagonal fuzzy numbers is given by:

ASP − BSP = (a1, b, a2;m,n, n,m)− (a′1, c, a

′2;m

′, n

′, n

′,m

′) (4)= (a1 − a

′2, b− c, a2 + a

′1;m + m

′, n + n

′, n + n

′,m + m

′)(5)

3.1.4 An illustrative example

Let ASP and BSP be two pentagonal fuzzy numbers defined asASP=(0.3, 0.5, 0.7, 0.2, 0.2, 0.2, 0.2) and BSP = (0.4, 0.6, 0.8, 0.2,0.2, 0.2, 0.2). Then the basic arithmetic operations such as additionand subtraction is calculated as follows:ASP + ASP = (0.7, 1.1, 1.5, 0.4, 0.4, 0.4, 0.4).ASP - ASP = (-0.1, -0.1, -0.1, 0.4, 0.4, 0.4, 0.4).

The geometrical representation of addition and subtraction oftwo symmetric pentagonal fuzzy number is shown as below:

Figure 2: Addition and Subtraction Pentagonal Fuzzy Number

3.2 Generalized Symmetric Pentagonal FuzzyNumber (GSPFN)

A fuzzy number AP=(a1, b, a2;m,n, n,m;w1, w2) is defined to begeneralized symmetric pentagonal fuzzy number, if its member-ship function has the following form;

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FA(x) =

0 x < a1 −m

w1(x−(a1−m))

ma1 −m ≤ x ≤ a1

(b−a1)2 a1 ≤ x ≤ b, 0 ≤ α ≤ 0.5

w2(x−(b−n))

na1 ≤ x ≤ b, 0.5 ≤ α ≤ 1

w2 x = b(a2−b)

2 b ≤ x ≤ a2, 0 ≤ α ≤ 0.5w2

((b+n)−x)n

b ≤ x ≤ a2, 0.5 ≤ α ≤ 1w1

((a2+m)−x)m

a2 ≤ x ≤ a2 + m0 x > a2 + m

and a1 ≤ b ≤ a2 with a1, b, a2 ∈ R. (m, m) and (n, n) arethe length of the intervals [a1−m, a1], [a2, a2 +m] and [a1, b], [b, a2]respectively with m,n>0. w1 and w2 are the different α-levels whichtakes value between 0<w<1 with 0<w1+w2≤1. The geometricalillustration of the generalized symmetric pentagonal fuzzy numberis presented in the Figure 3 as follows:

Figure 3: Generalized Symmetric Pentagonal Fuzzy Number

3.3 Quadratic Pentagonal Fuzzy Number (QPFN)A quadratic pentagonal fuzzy number AP is a special kind ofpentagonal fuzzy number constructed by using parabolic curves.More specifically, the quadratic pentagonal fuzzy number has itsroots adopted from Lotfi A. Zadeh’s concept of concentration anddilation [6]. The concentration and dilation are the two opera-tors which makes certain changes in the magnitude of the grade ofmembership. The quadratic pentagonal fuzzy number is defined

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as AP=(a1, a2, a3, a4, a5) where a1, a2, a3, a4, a5 are the real numbersand its membership function is given by,

FA(x) =

0 x < a1(x−a1)2(a2−a1)2 a1 ≤ x ≤ a212

(x−a2)2(a3−a2)2 a2 ≤ x ≤ a3

1 x = a312

(a4−x)2(a4−a3)2 a3 ≤ x ≤ a4

(a5−x)2(a5−a4)2 a4 ≤ x ≤ a5

0 x > a5

Figure 4: Quadratic Pentagonal Fuzzy Number with the effect ofconcentration and dilation

4 ConclusionIn this paper, we analysed the various kinds of pentagonal fuzzynumber such as symmetric pentagonal fuzzy number, quadraticpentagonal fuzzy number with their generalized form. Also thegeometrical representation of the newly defined pentagonal fuzzynumbers along with their basic arithmetic operations were defined.

References[1] D. Dubois and H. Prade, Fundamentals of Fuzzy Sets, Springer

Science+Buisness Media, New York (2000).

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[2] Rajkumar and T. Pathinathan, Sieving out the Poor usingFuzzy Decision Making Tools, indian Journal of Science andTechnology, 8, 22, (2015), 1-16.

[3] T. Pathinathan and K. Ponnivalavan, Pentagonal Fuzzy Num-ber, International Journal of Computing Algorithm, 3, (2014),1003-1005.

[4] T. Pathinathan, K. Ponnivalavan and E. Mike Dison, DifferentTypes of Fuzzy numbers and Certain Properties, Journal ofComputer and Mathematical Sciences, 6, 11, (2015), 631-651.

[5] L. A. Zadeh, Fuzzy Sets, Information and Control, 8, (1965),338-353.

[6] L. A. Zadeh, A Fuzzy-Set-Theoretic Interpretation of Linguis-tic Hedges, Journal of Cybernetics, 2, 3, (1972), 4-34.

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symmetric pentagonal fuzzy numbers · 2018-03-27 · a fuzzy number is a quantity, whose value is...

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