sanhita paper banerjee
TRANSCRIPT
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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
( )
Now the ordinate of the vertices are
. / . / . / . /
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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Now the ordinate of the vertices are
. / . / . / . /
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 () ( )
Now the ordinate of the vertices are
. / . / . / . /
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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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