## 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} (αx_{1} + (1-α) x_{2}) ≥ min (µ_{A}(x_{1}), µ_{A}(x_{2})) É x_{1} ,x_{2}R É 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, ð›‚) 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 ð›‚-cut using the example of dual trapezoidal fuzzy number.

### CONVERGENCE OF α-CUT

Let A_{DT} = {a, b, c, d, (ð›‚) } be a dual trapezoidal fuzzy number whose membership function function is given as

µ_{DT}(x) =

To find ð›‚-cut of A_{DT} .We first set ð›‚ [0,1] to both left and right reference functions of A_{DT.} Expressing X in terms of ð›‚ which gives ð›‚-cut of A_{DT}.

ð›‚=

â‡¨ x ^{l}= a+ (b-a) ð›‚

ð›‚=

â‡¨ x ^{r} =d-(d-c) ð›‚

â‡¨ A_{ð›‚}_{DT}= [a+ (b-a) ð›‚, d-(d-c) ð›‚]

In ordinary to find ð›‚-cut, we give ð›‚ values as 0 or 0.5 or 1 in the interval [0, 1] .Instead of giving these values for ð›‚. we divide the interval [0,1] as many continuous subinterval. If we give very small values for ð›‚, the ð›‚-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

ð›‚ = (x ^{l} + 6)/2

- X
^{l}= 2ð›‚-6

ð›‚ = (6 – x^{r})/3

- X
^{r}= 6-3ð›‚

- A
_{DT}_{ð›‚}=[ 2ð›‚-6, 6-3ð›‚ ]

When ð›‚=1/10 then A_{DT}_{ð›‚} = [-5. 8 , 5.7]

When ð›‚=1/10^{2} then A_{DT } =[-5.98 , 5.97]

When ð›‚=1/10^{3} then A_{DT}_{ð›‚} = [-5.998 , 5.997]

When ð›‚=1/10^{4} then A_{DT}_{ð›‚}=[-5.9998 , 5.9997]

When ð›‚=1/10^{5} then A_{DT}_{ð›‚}=[-5.99998 , 5.99997 ]

When ð›‚=1/10^{6} then A_{DT}_{ð›‚}=[ -5.999998 , 5.999997 ]

When ð›‚=1/10^{7} then A_{DT}_{ð›‚}=[ -5.9999998 , 5.9999997, ]

When ð›‚=1/10^{8} then A_{DT}_{ð›‚}=[ -5.99999998 , 5.99999997 ]

When ð›‚=1/10^{9} then A_{DT}_{ð›‚}=[ -5.999999998 , 5.999999997]

When ð›‚=1/10^{10} then A_{DT}_{ð›‚}=[-6 , 6]

When ð›‚=1/10^{11} then A_{DT}_{ð›‚}=[-6 , 6]

When ð›‚=1/10^{12} then A_{DT}_{ð›‚} =[-6 , 6]

When ð›‚=1/10^{13} then A_{DT}_{ð›‚} =[-6 , 6 ]

…………………………..etc

When ð›‚=1/10^{n} as n →∞ then the ð›‚-cut converges to A_{DT}_{ð›‚}=[-6, 6 ]

**Figure:** Graphical Representation of convergence of ð›‚-cut

When ð›‚=2/10 then A_{DT}_{ð›‚}= [ -5.6,5.4 ]

When ð›‚=2/10^{2} then A_{DT}_{ð›‚}= [ -5.96,5.94 ]

When ð›‚=2/10^{3} then A_{DT}_{ð›‚}=[ -5.996,5.994 ]

When ð›‚=2/10^{4} then A_{DT}_{ð›‚}=[ , -5.9996,5.9994 ]

When ð›‚=2/10^{5} then A_{DT}_{ð›‚}=[ , -5.99996,5.99994 ]

When ð›‚=2/10^{6} then A_{DT}_{ð›‚}=[ , -5.999996,5.999994 ]

When ð›‚=2/10^{7} then A_{DT}_{ð›‚}=[-5.9999996, 5.9999994 , ]

When ð›‚=2/10^{8} then A_{DT}_{ð›‚}=[ , -5.99999996,5.99999994 ]

When ð›‚=2/10^{9} then A_{DT}_{ð›‚}=[ , -5.999999996,5.999999994 ]

When ð›‚=2/10^{10} then A_{DT}_{ð›‚}=[ , -6,6 ]

When ð›‚=2/10^{11} then A_{DT}_{ð›‚}=[ -6,6 ]

When ð›‚=2/10^{12} then A_{DT}_{ð›‚}=[ -6,6 ]

When ð›‚=2/10^{13} then A_{DT}_{ð›‚}=[ -6,6]

…………………………………etc

When ð›‚=2/10^{n} as n →∞ then the ð›‚-cut converges to A_{DT}_{ð›‚}=[ -6,6 ]

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

From the above example we conclude that , In general we have { K/10^{n}} if we give different values for K as n- varies upto to ∞ if we give as n tends to ∞ then the values of A_{DT}_{ð›‚} 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.

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*.*

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