Dual Trapezoidal Fuzzy Number and Its Applications

Abstract

In this paper, we introduce Convergence of α-Cut. We define at which point the α-Cut converges to the fuzzy numbers it will be illustrated by example using dual trapezoidal fuzzy number and Some elementary applications on mensuration are numerically illustrated with approximated values.

KeyWords: Fuzzy number, α-Cut, Dual trapezoidal fuzzy number, Defuzzification.

Introduction

Fuzzy sets have been introduced by Lotfi. A. Zadeh (1965). Fuzzy numbers were first introduce by Zadeh in 1975.There after theory of fuzzy number was further studied and developed by Dubois and Prade, R.Yager Mizomoto, J.Buckly and Many others. Since then many workers studied the theory of fuzzy numbers and achieved fruitful results. The fuzziness can be represented by different ways one of the most useful representation is membership function. Also depending the nature and shape of the membership function the fuzzy number can be classified in different forms, such as triangular fuzzy number, trapezoidal fuzzy number etc. A fuzzy number is a quantity whose values are imprecise, rather than exact as is the case with single valued number. Fuzzy numbers are used in statistics computer programming, engineering and experimental science. So far fuzzy numbers like triangular fuzzy number, trapezoidal fuzzy numbers, pentagonal, hexagonal, octagonal pyramid and diamond fuzzy numbers have been introduced with its membership functions. These numbers have got many applications like non-linear equations, risk analysis and reliability. In this paper, we introduce Dual trapezoidal fuzzy numbers with its membership functions and its applications. Section one presents the introduction, section two presents the basic definition of fuzzy numbers section three presents Dual trapezoidal fuzzy numbers and its applications and in the final section we give conclusion.

Basic Definitions

Fuzzy set

A fuzzy set A in a universe of discourse X is defined as the following set of pairs A= {(x, µ_A(x)): xX} Here µ_A(x) : x is a mapping called the degree of membership function of the fuzzy set A and µ_A(x) is called the membership value of xX in the fuzzy set These membership grades are often represented by real numbers ranging from [0, 1].

Fuzzy Number

A fuzzy set A defined on the universal set of real number R is said to be a fuzzy number if its membership function has satisfy the following characteristics.

( i) μ_A (x) is a piecewise continuous

(ii) A is convex, i.e., µ_A (αx₁ + (1-α) x₂) ≥ min (µ_A(x₁), µ_A(x₂)) ɏ x₁ ,x₂R ɏ α[0,1]

(iii) A is normal, i.e., there exist x_o R such that µ_A (x_o)=1

Trapezoidal Fuzzy Number

A trapezoidal fuzzy number represented with four points as A = (a b c d), Where all a, b, c, d are real numbers and its membership function is given below where a≤ b≤ c≤ d

µ_A(x)=

DUAL TRAPEZOIDAL FUZZY NUMBER

Dual Trapezoidal Fuzzy Number

A Dual Trapezoidal fuzzy number of a fuzzy set A is defined as A_DT= {a, b, c, d (α)} Where all a, b, c, d are real numbers and its membership function is given below where a≤b≤c≤d

µ_DT(x) =

where α is the base of the trapezoidal and also for the inverted reflection of the above trapezoidal namely a b c d

Figure: Graphical Representation of Dual Trapezoidal fuzzy Number

DEFUZZIFICATION

Let A_DT= (a, b, c, d, 𝛂) be a dual trapezoidal fuzzy number .The defuzzification value of A_DT is an approximate real number. There are many method for defuzzification such as Centroid Method, Mean of Interval Method , Removal Area Method etc. In this Paper We have used Centroid area method for defuzzification .

CENTROID OF AREA METHOED

Centroid of area method or centry of gravity method. It obtains the centre of area (X^*)

occupied by the fuzzy sets.It can be expressed as

X^* =

Defuzzification Value for dual trapezoidal fuzzy number

Let A_DT= {a, b, c, d (α)} be a DTrFN with its membership function

µ_DT(x) =

Using centroid area method

+dx+++dx

= + + +

+ +

₌

₌

= ++ dx+++dx

=

=

_{= c + d – a – b}

Defuzzification ₌

₌

=

APPLICATION

In this section. We have discussed the convergence of 𝛂-cut using the example of dual trapezoidal fuzzy number.

CONVERGENCE OF α-CUT

Let A_DT = {a, b, c, d, (𝛂) } be a dual trapezoidal fuzzy number whose membership function function is given as

µ_DT(x) =

To find 𝛂-cut of A_DT .We first set 𝛂 [0,1] to both left and right reference functions of A_DT. Expressing X in terms of 𝛂 which gives 𝛂-cut of A_DT.

𝛂=

⇨ x ^l= a+ (b-a) 𝛂

𝛂=

⇨ x ^r =d-(d-c) 𝛂

⇨ A_𝛂DT= [a+ (b-a) 𝛂, d-(d-c) 𝛂]

In ordinary to find 𝛂-cut, we give 𝛂 values as 0 or 0.5 or 1 in the interval [0, 1] .Instead of giving these values for 𝛂. we divide the interval [0,1] as many continuous subinterval. If we give very small values for 𝛂, the 𝛂-cut converges to a fuzzy number [a, d] in the domain of X it will be illustrated by example as given below.

Example

A_DT = (-6,-4, 3, 6) and its membership function will be

µ_DT(x) =

α- cut of dual Trapezoidal fuzzy Number

𝛂 = (x ^l + 6)/2

• X^l = 2𝛂-6

𝛂 = (6 – x^r)/3

• X^r = 6-3𝛂

• A_DT𝛂=[ 2𝛂-6, 6-3𝛂 ]

When 𝛂=1/10 then A_DT𝛂 = [-5. 8 , 5.7]

When 𝛂=1/10² then A_DT =[-5.98 , 5.97]

When 𝛂=1/10³ then A_DT𝛂 = [-5.998 , 5.997]

When 𝛂=1/10⁴ then A_DT𝛂=[-5.9998 , 5.9997]

When 𝛂=1/10⁵ then A_DT𝛂=[-5.99998 , 5.99997 ]

When 𝛂=1/10⁶ then A_DT𝛂=[ -5.999998 , 5.999997 ]

When 𝛂=1/10⁷ then A_DT𝛂=[ -5.9999998 , 5.9999997, ]

When 𝛂=1/10⁸ then A_DT𝛂=[ -5.99999998 , 5.99999997 ]

When 𝛂=1/10⁹ then A_DT𝛂=[ -5.999999998 , 5.999999997]

When 𝛂=1/10¹⁰ then A_DT𝛂=[-6 , 6]

When 𝛂=1/10¹¹ then A_DT𝛂=[-6 , 6]

When 𝛂=1/10¹² then A_DT𝛂 =[-6 , 6]

When 𝛂=1/10¹³ then A_DT𝛂 =[-6 , 6 ]

…………………………..etc

When 𝛂=1/10ⁿ as n →∞ then the 𝛂-cut converges to A_DT𝛂=[-6, 6 ]

Figure: Graphical Representation of convergence of 𝛂-cut

When 𝛂=2/10 then A_DT𝛂= [ -5.6,5.4 ]

When 𝛂=2/10² then A_DT𝛂= [ -5.96,5.94 ]

When 𝛂=2/10³ then A_DT𝛂=[ -5.996,5.994 ]

When 𝛂=2/10⁴ then A_DT𝛂=[ , -5.9996,5.9994 ]

When 𝛂=2/10⁵ then A_DT𝛂=[ , -5.99996,5.99994 ]

When 𝛂=2/10⁶ then A_DT𝛂=[ , -5.999996,5.999994 ]

When 𝛂=2/10⁷ then A_DT𝛂=[-5.9999996, 5.9999994 , ]

When 𝛂=2/10⁸ then A_DT𝛂=[ , -5.99999996,5.99999994 ]

When 𝛂=2/10⁹ then A_DT𝛂=[ , -5.999999996,5.999999994 ]

When 𝛂=2/10¹⁰ then A_DT𝛂=[ , -6,6 ]

When 𝛂=2/10¹¹ then A_DT𝛂=[ -6,6 ]

When 𝛂=2/10¹² then A_DT𝛂=[ -6,6 ]

When 𝛂=2/10¹³ then A_DT𝛂=[ -6,6]

…………………………………etc

When 𝛂=2/10ⁿ as n →∞ then the 𝛂-cut converges to A_DT𝛂=[ -6,6 ]

Simillarly, 𝛂=3/10ⁿ,4/10ⁿ,5/10ⁿ,6/10ⁿ,7/10ⁿ,8/10ⁿ,9/10ⁿ,10/10ⁿ upto these value n varies from 1to ∞ after 11/10ⁿ,12/10ⁿ…………………………………..100/10ⁿ as n varies from 2 to ∞ and101/10ⁿ,102/10ⁿ…………………………………. as n varies from 3 to ∞ and the process is goes on like this if we give the value for 𝛂 it will converges to the dual trapezoidal fuzzy number[-6,6]

From the above example we conclude that , In general we have { K/10ⁿ} if we give different values for K as n- varies upto to ∞ if we give as n tends to ∞ then the values of A_DT𝛂 converges to the fuzzy number[a,d] in the domain X.

APPLICATIONS

In this section we have numerically solved some elementary problems of mensuration based on arithmetic operation using defuzzified centroid area method

Perimeter of Rectangle

Let the length and breadth of a rectangle are two positive dual trapezoidal fuzzy numbers A_DT = (10cm, 11cm, 12cm,13cm) and B_DT = (4cm, 5cm,6cm,7cm) then perimeter C_DT of rectangle is 2[A_DT+B_DT]

Therefore the perimeter of the rectangle is a dual trapezoidal fuzzy number C_DT = (28cm, 32cm,36cm,40cm) and its membership functions

µ_DT(x) =

The Perimeter of the rectangle is not less than 28 and not greater than 40 .The perimeter value takes between 32 to 36.

Centroid area method:

X^* =

=

=

=

= 34

The approximate value of the perimeter of the rectangle is 34 cm.

2.Length of Rod:

Let length of a rod is a positive DTrFN A_DT = (10cm,11cm,12cm, 13cm). If the length B_DT = (5cm, 6cm , 7cm, 8cm), a DTrFN is cut off from this rod then the remaining length of the rod C_DT is [A_DT(-)B_DT]

The remaining length of the rod is a DTrFN C_DT = (2cm, 4cm, 6cm, 8cm) and its membership function

µ_DT(x) =

The remaining length of the rod is not less than 2cm and not greater than 8cm.The length of the rod takes the value between 4cm and 6cm.

Centroid area method:

X^* =

=

=

=

= 5

The approximate value of the remaining length of the rod is 5cm.

Length of a Rectangle

Let the area and breadth of a rectangle are two positive dual trapezoidal fuzzy numbers A_DT=(36cm,40cm,44cm,48cm) and B_DT=(3cm,4cm,5cm,6cm) then the length C_DT of the rectangle is is A_DT(:)B_DT.

Therefore the length of the rectangle is a dual trapezoidal fuzzy number C_DT=(6cm,8cm,11cm,16cm) and its membership functions

µ_DT(x) =

The length of the rectangle is not less than 6cm and not greater than 16cm .The length of the rectangle takes the value between 8cm and 11cm.

Centroid area method:

X ^* =

=

=

= 10.38

The approximate value of the length of the rectangle is 10.38cm.

Area of the Rectangle

Let the length and breadth of a rectangle are two positive dual trapezoidal fuzzy numbers A_DT=(3cm,4cm,5cm,6cm) and B_DT=(8cm,9cm,10cm,11cm) then the area of rectangle is A_DT(.) B_DT

Therefore the area of the rectangle is a dual trapezoidal fuzzy number C_DT= (24cm, 36cm, 50cm, 66cm) and its membership functions

µ_DT(x) =

The area of the rectangle not less than 24 and not greater than 66.The area of the reactangle takes the value between 36 and 50.

Centroid area method:

X ^* =

=

=

= 44.167sq.cm

CONCLUSION

In this paper, we have worked on DTrFN .We have define the Convergence of α-Cut to the fuzzy number. We have solved numerically some problems of mensuration based on operations using DTrFN and we have calculated the approximate values. Further DTrFN can be used in various problem of engineering and mathematical science.

References

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