sanhita paper banerjee

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Chapter 5Fuzzy Number

In most of cases in our life, the data obtained are only approximately known. In 1978, Dubois

and Prade defined any of the fuzzy numbers as a fuzzy subset of the real line. Fuzzy numbers

allow us to make the mathematical model of linguistic variable or fuzzy environment. A fuzzy

number is a quantity whose value is imprecise, rather than exact as is the case with "ordinary"

(single-valued) numbers . Any fuzzy number can be thought of as a function whose domain is

a specified set. In many respects, fuzzy numbers depict the physical world more realistically

than single-valued numbers. Fuzzy numbers are used in statistics, computer. programming,

engineering (especially communications), and experimental science. The concept takes into

account the fact that all phenomena in the physical universe have a degree of inherent

uncertainty. The arithmetic operators on fuzzy numbers are basic content in fuzzy

mathematics. Multiplication operation on fuzzy numbers is defined by the extension principle.

The procedure of addition or subtraction is simple, but the procedure of multiplication or

division is complex. The nonlinear programming, analytical method, computer drawing and

computer simulation method are used for solving multiplication operation of two fuzzy

numbers. The procedure of division is similar. In 1985 Chen further developed the theory and

applications of Generalized Fuzzy Number (GFN). Chen (1985) had also proposed the

function principle, which might be used as the fuzzy numbers arithmetic operations between

generalized fuzzy numbers. Hsieh et al.(1999) pointed out that the arithmetic operators on

fuzzy numbers presented in Chen (1985) are not only changing the type of membershipfunction of fuzzy numbers after arithmetic operations, but also they can reduce the

troublesomeness of arithmetic operations. In 1987 Dong and Shah introduced vertex method

using which the value of the functions of interval variable and fuzzy variable can be easily

evaluated. The difference between the arithmetic operations on generalized fuzzy numbers

and the traditional fuzzy numbers is that the former may deal with both non-normalized and

normalized fuzzy numbers but the later with normalized fuzzy numbers.

Definition 5.1. Fuzzy number: A fuzzy number

is an extension of a regular

number in the sense that it does not refer to one single value but rather to a connected set of

possible values, where each possible value has its own weight between 0 and 1. This weight is

known as the membership function. Thus a fuzzy number is a convex and normal fuzzy set. If is a fuzzy number, is a fuzzy convex set and if is non decreasing for and non increasing for . Definition 5.2. Triangular fuzzy number: A Trapezoidal fuzzy number (TrFN)

denoted by is defined as where the membership function


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{

Definition 2.7: Trapezoidal Fuzzy Number: A Trapezoidal fuzzy number (TrFN) denoted

by is defined as where the membership function

or, .. //

or,

..

//Definition 2.8: Generalized Fuzzy number (GFN): A fuzzy set ;,defined on the universal set of real numbers R, is said to be generalized fuzzy number if its

membership function has the following characteristics:

(i) : R[0, 1]is continuous.(ii) for all - ,(iii) is strictly increasing on [ ,] and strictly decreasing on [,].(iv)

for all

, -, where

.

Definition 2.9: Generalized Trapezoidal Fuzzy number (GTrFN): A Generalized Fuzzy

Number ;, is called a Generalized Trapezoidal Fuzzy Number if itsmembership function is given by


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or, .. / /.

Fig-2.1:Comparison between membership function of TrFN and GTrFN

Definition 2.10: Equality of two GTrFN: Two Generalized Trapezoidal Fuzzy Number

(GTrFN) = and = is said to be equal i.e. ifand only if and .Definition 2.11: A GTrFN = is said to be non negative (non positive) i.e. ( ) if and only if .Table2.1:- different types of GTrFN

Type of GTrFN

;

Conditions Rough sketch of membership

function

Symmetric ( ) or in centralform

Non symmetric type 1 ( ( )

Non symmetric type2 ( )( )

Left GTrFN( )


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Right GTrFN( )

Definition 2.12: Vertex Method [24]: When is continuous in the n-dimensional rectangular region, and also no extreme point exists in this region (including the

boundaries), then the value of interval function can be obtained by 0 .()/ .()/ 1

where

is the ordinate of the j-th vertex and

are intervals of real numbers.

Example2.1: Determine .Given ,-, ,-, ,-

The ordinate of vertices are From those ordinates, we obtain Then , - ,-Definition 2.13: Defuzzification: Let = be a GTrFN. The

defuzzification value of

is an approximated real number. There are many methods for

defuzzication such as Centroid Method, Mean of Interval Method, Removal Area Method etc.In this paper we have used Removal Area Method for defuzzification.

Removal Area Method [1]: Let us consider a real number , and a generalized fuzzynumber. The left side removal of with respect to ( ), is defined as the area

bounded by and the left side of the generalized fuzzy number. Similarly, the right sideremoval, ( ), is defined. The removal of the generalized fuzzy number with respect to

is defined as the mean of

( )

and

( )

. Thus,

( )

.

( )

()/


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( ), relative to , is equivalent to an ordinary representation of the generalized fuzzynumber.

Fig-2.2: Left removal area ( ) Fig-2.3: Right removal area ( )

( ) | | . | |/ ,( ) | | . | |/ The defuzzification value or approximated value ofi.e., ( ) .( ) ()/

Defuzzification value for GTrFN:

Let ; be a GTrFN with its membership function

and -cuts 0 1

,

, -,

Fig-2.4: Left removal area

( )Fig-2.5: Right removal area

( )


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( ) | |or, | 2 3 | ,

( ) |

|or,

| 2 3 |

The defuzzification value or approximated value ofi.e., ( ) .( ) ( ) / 3. Arithmetic operations of GTrFNsIn this section we discuss four operations (addition, subtraction, multiplication, division) for

two generalized trapezoidal fuzzy numbers based on extension principle method, interval

method and vertex method.

Let = and = be two positive generalized trapezoidalfuzzy numbers and their membership functions are

and their -cuts be , - 0 1 , , -, , - 0 1 , , -, 3.1 Addition of two GTrFNs

a) Addition of two GTrFNs based on extension principleLet where (( ) )Let

{

.. / / .. / /

(3.1.1)


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{

4.

/5 4. /5

(3.1.2)

[Note-3.1: . /

. / ]

{

(3.1.3)

The addition of two GTrFNs is another GTrFN with membership function given at equation(3.1.3)

Fig-3.1:- Rough sketch of Membership function ofb) Addition of two GTrFNs based on interval method

Let where , - , -, ,

, -


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* , - , -+

2

3

.(3.1.4)

[Note-3.2: ]

The addition of two GTrFNs

is another GTrFN

with membership function given at equation(3.1.3) and shown in Fig-3.1.c) Addition of two GTrFNs based on vertex method

Let ( ) Now the ordinate of the vertices are

.

/ .

/ . / . /

It can be shown that So [( )( )] , - 0 1


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Now following Note-3.2 we get that the addition of two GTrFNs is another GTrFN with membership function given at equation(3.1.3) and shown in Fig-3.1.

3.2 Scalar multiplication of a GTrFN

a) Scalar multiplication of a GTrFN based on extension principle method

Let where (() )Case1: When {

..

/ / .. / / (3.2.1)

(3.2.2)

{

(3.2.3)

The positive scalar (k) multiplication of a GTrFN is another GTrFN with membership function given at equation (3.2.3)


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Fig-3.2:- Rough sketch of Membership function ofCase2: When

.. / / .. / / (3.2.4)

{

(3.2.5)

(3.2.6)

The negative scalar (k) multiplication of a GTrFN is another GTrFN with membership function given at equation (3.2.6)

Fig-3.3:- Rough sketch of Membership function ofb) Scalar multiplication of a GTrFN based on interval method

Let

where

, - , -,

Case1: When


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, -

* , - , -+

2 3[Note-3.3: ]The positive scalar (k) multiplication of a GTrFN is another GTrFN with membership function given at equation (3.2.3) and shownin Fig-3.2.

Case2: When , -

* , - , -+

2 3[Note-3.4: ]The negative scalar (k) multiplication of a GTrFN is another GTrFN with membership function given at equation (3.2.6) an shown inFig-3.3.

c) Scalar multiplication of a GTrFN based on vertex methodLet ()

Now the ordinate of the vertices are


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. / , . /

. /

. /Case1: When ,

So [( )( )] , -0 1 Following Note-3.3 we get that the positive scalar (k) multiplication of a GTrFN isanotherGTrFN with membership function given at equation (3.2.3) andshown in Fig-3.2.

Case2: When , So [( )( )]

, -

Following Note-3.4 we get that the negative scalar (k) multiplication of a GTrFN isanotherGTrFN with membership function given at equation (3.2.6) anshown in Fig-3.3.

3.3 Subtraction of two GTrFNsa) Subtraction of two GTrFNs based on extension principle method

Let where (( ) )Let


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.. / / .. / /

(3.3.1)

{

4. /5 4. /5

(3.3.2)

{

[Following Note 3.1]

(3.3.3)

Thus we get that the subtraction of two GTrFNs is another GTrFN with membership function given at equation(3.3.3)

Fig-3.4:-Rough sketch of Membership function ofb) Subtraction of two GTrFNs based on interval method

Let where , - , -, , , -


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* , - , -+

2 3Following Note-3.2 we get that the subtraction of two GTrFNs is another GTrFN with membership function given at equation(3.3.3) and shown in Fig-3.4.

c) Subtraction of two GTrFNs based on vertex method

Let ( ) Now the ordinate of the vertices are . / . / . / . /

It can be shown that So [( )( )] , - 0 1

Now following Note-3.2 we get that the subtraction of two GTrFNs is another GTrFN with membership function given at equation(3.3.3) and shown in Fig-3.4.

3.4 Multiplication of two GTrFNs


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a) Multiplication of two GTrFNs based on extension principle method

Let where (( ) )Let

{ .. / / .. / /

(3.4.1)

{

(3.4.2)

Let, sup such that

* + ./ [Note-3.5: Let

* +

./

* + * +


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*+ is an increasing function in z.] * + ./ [Note-3.6: Let * +

./

* + * + *+ is a decreasing function in z.Again, and . / *+*+*+ *+*+*+ and . / *+*+*+ *+*+*+ ]


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{

(3.4.3)

Where * + , * + .

We get that the multiplication of two GTrFNs

is a generalized trapezoidal shaped fuzzy

number

with membership function given at equation (3.4.3).

Fig-3.5:-Rough sketch of Membership function ofb) Multiplication of two GTrFNs based on interval method

Let where , - , -, , , - * , - , -+

* + ./

* + ./ Now following Note-3.5 and Note-3.6 we get that the multiplication of two GTrFNs is ageneralized trapezoidal shaped fuzzy number

with

membership function given at equation (3.4.3) and shown in Fig-3.5.


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c) Multiplication of two GTrFNs based on vertex methodLet

( )


. / . / . / . /

2 3 2 3

2 3 2 3 2 3 2 3 2 3 2 3It can be shown that So

[( )( )]

, -02 3 2 3 2 3 2 31Now following Note-3.5 and Note-3.6 we get that the multiplication of two GTrFNs is ageneralized trapezoidal shaped fuzzy number withmembership function given at equation (3.4.3) and shown in Fig-3.5.

3.5 Division of two GTrFNsa) Division of two GTrFNs based on extension principle method

Let where , - , -, , and .( ) /


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{

.. / / . / .. / /

(3.5.1)

{

(3.5.2)

Let, supsuch that Similarly,

sup

[Note-3.7: *+ for

is an increasing function with z. *+ for

is an decreasing function with z.

Again, ./ ./ and ./ ./ 4 5 , - and 4 5 , - ]


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{

(3.5.3)

Thus we get that the division of two GTrFNs is a generalized trapezoidal shaped fuzzynumber . / with membership function given at equation (3.5.3)

Fig-3.6:- Rough sketch of Membership function of b) Division of two GTrFNs based on interval method

0 1 * , - , -+ } Again

Now following Note-3.7 we get that the division of two GTrFNs is a generalizedtrapezoidal shaped fuzzy number . / with membership function given atequation (3.5.3) and shown in Fig-3.6.

c) Division of two GTrFNs based on vertex method

Let ( )


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. / . / . / . /

2323 , 2323

23

23

,

23

23

It can be shown that So [( )( )] , - 62323 2323 7

Now following Note-3.7 we get that the division of two GTrFNs

is a generalized

trapezoidal shaped fuzzy number . / with membership function given atequation (3.5.3) and shown in Fig-3.6.Table-3.1:- Arithmetic operations of two Left GTrFNs = and =

Arithmetic

operations

Membership function of

i.e. Rough sketch of

Nature of

Addition[ ] Left

Generalized

Trapezoidal

Fuzzy

Number


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Subtraction[ ]

Generalized

Trapezoidal

Fuzzy

Number

Multiplication[ ] Left

Generalized

Trapezoidal

shaped

Fuzzy

Number

Division

[ ]

{

Generalized

Trapezoidalshaped

Fuzzy

Number

Remarks:- From the table-3.1 we see that the addition and multiplication of two Left

GTrFNs is a Left GTrFN and Left Generalized Trapezoidal shaped Fuzzy Number

respectively but the subtraction and the division of two Left GTrFNs is a GTrFN andGeneralized Trapezoidal shaped Fuzzy Number respectively.

Table-3.2:- Arithmetic operations of two Right GTrFNs = and = Arithmetic

operations

Membership function of

i.e.

Rough sketch of

Nature of

Addition[ ] Right

Generalized

Trapezoidal

Fuzzy

Number


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Subtraction[ ]

Generalized

Trapezoidal

Fuzzy

Number

Multiplication[ ] Right

Generalized

Trapezoidal

shaped

Fuzzy

Number

Division

[ ]

{

Generalized

Trapezoidalshaped

Fuzzy

Number

Remarks:- From the table-3.2 we see that the addition and multiplication of two Right

GTrFNs is a Right GTrFN and Right Generalized Trapezoidal shaped Fuzzy Number

respectively but the subtraction and the division of two Right GTrFNs is a GTrFN andGeneralized Trapezoidal shaped Fuzzy Number respectively.

4. Comparison among three methods based on en exampleWe consider an expression( )where more than one arithmetic operation is used.Here =, = and = be three positiveGTrFNs and their -cuts be

0 1,

0 1and

0 1

Let In vertex method, let () ( )


. / . / . / . /


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. / . /

.

/ .

/

. /. / , . / . / . /. / , . / ,

. /. / , . / . /

. /.

/

. / .

/

From the above we see that -So -cut ofi.e. [( ) ( )]

, -

0 . / . / . /. /1 -and the rough sketch of membership function of is shown in Fig-4.1.

Fig-4.1:- Rough sketch of Membership function of ( )In extension principle method


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2 0

1 0

1 0

132 0 1 0 1 0 13

Now 0 . / . / . /. /1 -and the rough sketch of membership function of is shown in Fig-4.1.In interval arithmetic method if we consider the given expression as ( ) thenwe get

0 . / . / . /. /1

-and the rough sketch

of membership function of is shown in Fig-4.1.And if we consider the expression as then its -cut 0 . /. / . /. / 1 0 . / . / . /./1

02. /. / . /. /3 2. /. / . / . /31and the rough sketch of membership function of is shown in Fig-4.2.


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Fig-4.2:- Rough sketch of Membership function of Here we get one required value of an expression in vertex method and extension principle

method while in interval method we get two possible values for the same expression. So it can

be said that vertex method or extension principle method is more useful than interval method

in the case of expressions with two or more arithmetic operations.

5. ApplicationsIn this section we have numerically solved some elementary problems of mensuration based

on arithmetic operations described in section-3.

a) Perimeter of a Rectangle

Let the length and breadth of a rectangle are two GTrFNs and , then theperimeter of the rectangle is []The perimeter of the rectangle is a GTrFN which is ageneralized fuzzy set with the membership function

Fig-5.1: Rough sketch of membership function of


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Thus we get that the perimeter of the rectangle is not less than 36cm and not greater than

48cm. The value of perimeter is increased from 36cm to 40cm at constant rate 0.2 and is

decreased from 44cm to 48cm also at constant rate 0.2. There are 80% possibilities that the

perimeter takes the value between 40cm and 44cm.

Fig-5.2: Left removal area ( ) Fig-5.3: Right removal area ( )( ) , ( ) , ( ) The approximated value of the perimeter of the rectangle is 42 cm.b) Length of a Rod

Let the length of a rod is a GTrFN =. If the length = , a GTrFN , is cut off from this rod then the remaining lengthof the rod is

The remaining length of the rod is a GTrFN

which is a

generalized fuzzy set with the membership function

{

Fig-5.4: Rough sketch of membership function ofHere we get that the remaining length of the rod is not less than 4cm and not greater than

10cm. The value of this length is increased from 4cm to 6cm at constant rate 0.35 and is

decreased from 8cm to 10cm also at constant rate 0.35. There are 70% possibilities that the

length takes the value between 6cm and 8cm.


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Fig-5.5: Left removal area ( ) Fig-5.6: Right removal area ( )( ) , ( ) , ( ) The approximated value of the remaining length of the rod is 7 cm.c) Area of a Triangle

Let the base and the height of a triangle are two GTrFNs = and =then the area of the triangle is The area of the triangle is a generalized trapezoidal shaped (concave-convex type) fuzzynumber which is a generalized fuzzy set with themembership function

{

Fig-5.7: Rough sketch of membership function ofThus we get that the area of the triangle is not less than 5sqcm and not greater than 20sqcm.

The value of area is increased from 5sqcm to 9sqcm at nonlinear increasing rate and is

decreased from 14sqcm to 20sqcm at nonlinear decreasing rate. There are 70%

possibilities that the area takes the value between 9sqcm and 14sqcm.


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Fig-5.8: Left removal area ( ) Fig-5.9: Right removal area ( )() , ()( ) The approximated value of the area of the triangle is 14 sqcm.

d)

Length of a RectangleLet the area and breadth of a rectangle are two GTrFNs = and = , then thelength of the rectangle is The length of the rectangle is a generalized trapezoidal shaped (concave-convex type) fuzzynumber which is a generalized fuzzy set with themembership function

Fig-5.10: Rough sketch of membership function ofwe get that the length of the rectangle is not less than 7cm and not greater than 17cm.The

value of length is increased from 7cm to 9cm at nonlinear increasing rate and is

decreased from 12cm to 17cm at nonlinear decreasing rate. There are 80% possibilities

that the length takes the value between 9cm and 12cm.


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Fig-5.11: Left removal area ( ) Fig-5.12: Right removal area ( )() , ()( )

The approximated value of the length of the rectangle is 11.1 cm.

e) Area of an annulusLet the outer radius and inner radius of an annulus are two GTrFNs = and = , then the area of the annulus is The area of the annulus is a generalized trapezoidal shaped (concave-convex type)fuzzy number which is ageneralized fuzzy set with the membership function

{

Fig-5.13: Rough sketch of membership function ofWe get that the area of the annulus is not less than 201.14 sqcm and not greater than 792

sqcm. The value of area is increased from 201.14 sqcm to 374 sqcm at nonlinear


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increasing rate and is decreased from 502.86 sqcm to 792 sqcm at nonlinear

decreasing rate

. There are 70% possibilities that the area takes the value

between 374 sqcm and 502.86 sqcm.

Fig-5.14: Left removal area

( )Fig-5.15: Right removal area

( )

(), ()( ) The approximated value of the area of the annulus is 378 sqcm.

6. Conclusion and future workIn this paper, we have worked on GTrFN. We have described four operations for two GTrFNs

based on extension principle, interval method and vertex method and compared three methods

with an example. We have solved numerically some problems of mensuration based on these

operations using GTrFN and we have calculated the approximated values. Further GTrFN can

be used in various problems of engineering and mathematical sciences.

Acknowledgement

The authors would like to thank to the Editors and the two Referees for their constructive

comments and suggestions that significantly improve the quality and clarity of the paper.

References

A. Kaufmann and M.M. Gupta, Introduction to fuzzy Arithmetic Theory and Application

(Van Nostrand Reinhold, New York, 1991).

A. Kaufmann and M. M. Gupta, Fuzzy Mathematical Model in Engineering and Management

Science, North-Holland, 1988.


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