TJFS: Turkish Journal of Fuzzy Systems (eISSN: 1309–1190)
An Official Journal of Turkish Fuzzy Systems Association
Vol.3, No.1, pp. 16-44, 2012.
Arithmetic Operations on Generalized
Trapezoidal Fuzzy Number and its
Applications
Sanhita Banerjee*
Department of Mathematics, Bengal Engineering and Science University,
Shibpur, Howrah-711103, West Bengal, India.
Email: sanhita.banerjee88@gmail.com
*Corresponding author
Tapan Kumar Roy
Department of Mathematics, Bengal Engineering and Science University,
Shibpur, Howrah-711103, West Bengal, India.
Email: roy_t_k@yahoo.co.in
Abstract
In this paper, we have studied difuzzification method for generalized trapezoidal fuzzy numbers
(GTrFNs) and the four basic arithmetic operations of two GTrFNs based on the Zadeh's extension
principle method, interval method and vertex method. Based on these operations, some elementary
applications on mensuration are numerically illustrated with approximated values.
Keywords: Trapezoidal Fuzzy Number (TrFN), generalized fuzzy number (GFN), generalized
trapezoidal fuzzy number (GTrFN), generalized trapezoidal shaped fuzzy number.
1. Introduction
At present situation, in science and technology most of the mathematical problems are characterized
as complex process for which complete information is not always available. To handle this, the
problems need to be set up with the approximately available data. To make this possible Zadeh
(1965)[13] introduced fuzzy set theory. In recent years this subject has become an interesting branch
of pure and applied sciences. In 1985 Chen [9] further developed the theory and applications of
Generalized Fuzzy Number (GFN). Chen (1985) also proposed the function principle, which could be
used as the fuzzy numbers arithmetic operations between generalized fuzzy numbers, where these
fuzzy arithmetic operations can deal with the generalized fuzzy numbers. Hsieh et al.(1999)[12]
pointed out that the arithmetic operators on fuzzy numbers presented in Chen(1985) are not only
changing the type of membership function of fuzzy numbers after arithmetic operations, but also they
can reduce the troublesomeness of arithmetic operations. In 1987 Dong and Shah[24] introduced
Vertex Method using which the value of the functions of interval variable and fuzzy variable can be
easily evaluated. Recently GFN has also used in many fields such as risk analysis [19], maximal flow
16
[5], similarity measure [7], reliability [6,11] etc. Already there are several papers [3,6,8,10,20] on
arithmetic behaviors of GFN.
The difference between the arithmetic operations on generalized fuzzy numbers and the traditional
fuzzy numbers is that the former can deal with both non-normalized and normalized fuzzy numbers,
but the later with normalized fuzzy numbers.
In this paper, we have discussed four arithmetic operations for two GTrFNs in Section-3 based on
extension principle method, interval method and vertex method. In section-4 we have compared these
three methods based on an example. In section-5 based on these operations we have solved some
elementary problems of mensuration and have calculated required approximated values.
2. Mathematical Preliminaries
Definition 2.1: Fuzzy Set: A fuzzy set
of pairs
Here
in a universe of discourse X is defined as the following set
: X [0,1] is a mapping called the membership value of
x X in a fuzzy set
Definition 2.2: -cut of a fuzzy set:The -level set (or interval of confidence at level
the fuzzy set
of X is a crisp set
in A greater than or equal to
or -cut) of
that contains all the elements of X that have membership values
i.e.
Definition 2.3: Convex fuzzy set: A fuzzy set
is called convex fuzzy set if all
are convex sets i.e. for every element
and
and for every
.
Otherwise the fuzzy set is called non convex fuzzy set.
Definition 2.4: Interval Number: An interval number is a closed and bounded set of real
numbers.
 The addition of two interval numbers
and
denoted by
and is
defined by
 The scalar multiplication of a interval number
is denoted by kA where k is a
scalar and is defined by
 The subtraction of two interval numbers
and
 The product of two interval numbers
denoted by
and
denoted by
and is
and
denoted by
and is
defined by
where
,
 The division of two interval numbers
defined by
if
empty interval
if
17
if
if
otherwise
Definition 2.5: Extension Principle: Let
be a mapping from a set X to a set Y. Then the
extension principle allows us to define the fuzzy set
in Y induced by the fuzzy set
in X through f
as follows:
with
where
is the inverse image of y.
Definition 2.6: 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 called the membership function.
Thus a fuzzy number is a convex and normal fuzzy set. If is a fuzzy number then
is a fuzzy
convex set and if
then
is non decreasing for
and non increasing for
Definition 2.7: Trapezoidal Fuzzy Number: A Trapezoidal fuzzy number (TrFN) denoted by
defined as
where the membership function
.
is
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)
(ii)
(iii)
(iv)
:R
[0, 1]is continuous.
for all
is strictly increasing on [ , ] and strictly decreasing on [
for all
, where
.
,
].
Definition 2.9: Generalized Trapezoidal Fuzzy number (GTrFN): A Generalized Fuzzy Number
; , is called a Generalized Trapezoidal Fuzzy Number if its membership function
is given by
18
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.
if and only if
and
.
Definition 2.11: A GTrFN =
if and only if
Type of GTrFN
;
Symmetric (
is said to be non negative (non positive) i.e.
.
Table2.1:- different types of GTrFN
Conditions
Rough sketch of membership
function
) or in central
form
Non symmetric type 1 (
Non symmetric type2 (
)
19
Left GTrFN
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
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 fuzzy number .
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 side removal,
, is defined. The
removal of the generalized fuzzy number with respect to is defined as the mean of
and
. Thus,
, relative to
number.
, is equivalent to an ‘ordinary representation’ of the generalized fuzzy
20
Fig-2.2: Left removal area
Fig-2.3: Right removal area
,
The defuzzification value or approximated value of
i.e.,
Defuzzification value for GTrFN:
Let
and
;
be a GTrFN with its membership function
-cuts
,
,
Fig-2.4: Left removal area
Fig-2.5: Right removal area
or,
or,
The defuzzification value or approximated value of
i.e.,
21
,
3. Arithmetic operations of GTrFNs
In 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 trapezoidal fuzzy
numbers and their membership functions are
and their
-cuts be
,
,
,
,
3.1 Addition of two GTrFNs
a) Addition of two GTrFNs based on extension principle
Let
where
Let
…… (3.1.1)
…… (3.1.2)
[Note-3.1:
]
22
…… (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 of
b) Addition of two GTrFNs based on interval method
Let
where
,
,
….(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
23
It can be shown that
So
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)
24
Fig-3.2:- Rough sketch of Membership function of
Case2: 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 of
25
b) Scalar multiplication of a GTrFN based on interval method
Let
where
,
Case1: When
[Note-3.3:
]
The positive scalar (k) multiplication of a GTrFN is another GTrFN
with membership function given at equation (3.2.3) and shown in Fig3.2.
Case2: When
[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 in Fig-3.3.
c) Scalar multiplication of a GTrFN based on vertex method
Let
Now the ordinate of the vertices are
,
Case1: When
,
26
So
Following Note-3.3 we get that the positive scalar (k) multiplication of a GTrFN is another
GTrFN
with membership function given at equation (3.2.3) and shown
in Fig-3.2.
Case2: When
,
So
Following Note-3.4 we get that the negative scalar (k) multiplication of a GTrFN is another
GTrFN
with membership function given at equation (3.2.6) an shown in
Fig-3.3.
3.3 Subtraction of two GTrFNs
a) Subtraction of two GTrFNs based on extension principle method
Let
where
Let
…… (3.3.1)
…… (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)
27
Fig-3.4:-Rough sketch of Membership function of
b) Subtraction of two GTrFNs based on interval method
Let
where
,
,
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.
c) Subtraction of two GTrFNs based on vertex method
Let
Now the ordinate of the vertices are
It can be shown that
So
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.
28
3.4 Multiplication of two GTrFNs
a) Multiplication of two GTrFNs based on extension principle method
Let
Let
where
… … (3.4.1)
… … (3.4.2)
Let,
sup
such that
[Note-3.5: Let
is an increasing function in z.]
[Note-3.6: Let
29
is a decreasing function in z.
Again,
and
and
]
… … (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).
30
Fig-3.5:-Rough sketch of Membership function of
b) 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 a
generalized trapezoidal shaped fuzzy number
with membership
function given at equation (3.4.3) and shown in Fig-3.5.
c) Multiplication of two GTrFNs based on vertex method
Let
Now the ordinate of the vertices are
It can be shown that
So
31
Now following Note-3.5 and Note-3.6 we get that the multiplication of two GTrFNs
is a
generalized trapezoidal shaped fuzzy number
with membership
function given at equation (3.4.3) and shown in Fig-3.5.
3.5 Division of two GTrFNs
a) Division of two GTrFNs based on extension principle method
Let
where
,
,
and
… … (3.5.1)
… … (3.5.2)
Let,
sup
⇒
such that
Similarly,
sup
[Note-3.7:
for
is an increasing function with z.
for
is an decreasing function with z.
Again,
and
]
and
32
… … (3.5.3)
Thus we get that the division of two GTrFNs
is a generalized trapezoidal shaped fuzzy number
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
Again
Now following Note-3.7 we get that the division of two GTrFNs
is a generalized trapezoidal
shaped fuzzy number
with membership function given at equation (3.5.3) and
shown in Fig-3.6.
c) Division of two GTrFNs based on vertex method
Let
Now the ordinate of the vertices are
33
,
,
It can be shown that
So
Now following Note-3.7 we get that the division of two GTrFNs
is a generalized trapezoidal
shaped fuzzy number
with membership function given at equation (3.5.3) and
shown in Fig-3.6.
Table-3.1:- Arithmetic operations of two Left GTrFNs
=
Arithmetic
operations
Membership function of
i.e.
=
and
Rough sketch of
Nature of
Addition
Left
Generalized
Trapezoidal
Fuzzy Number
Subtraction
Generalized
Trapezoidal
Fuzzy Number
Multiplication
Left
Generalized
Trapezoidal
shaped Fuzzy
Number
Division
Generalized
Trapezoidal
shaped 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
34
and the division of two Left GTrFNs is a GTrFN and Generalized Trapezoidal shaped Fuzzy Number
respectively.
Table-3.2:- Arithmetic operations of two Right GTrFNs =
=
Arithmetic
operations
and
Rough sketch of
Membership function of
i.e.
Nature of
Addition
Right
Generalized
Trapezoidal
Fuzzy Number
Subtraction
Generalized
Trapezoidal
Fuzzy Number
Multiplication
Right
Generalized
Trapezoidal
shaped Fuzzy
Number
Division
Generalized
Trapezoidal
shaped 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 and Generalized Trapezoidal shaped
Fuzzy Number respectively.
4. Comparison among three methods based on an example
We consider an expression
Here
=
and their -cuts be
,
where more than one arithmetic operation is used.
=
and
35
=
be three positive GTrFNs
,
and
Let
In vertex method, let
Now the ordinate of the vertices are
,
,
,
,
From the above we see that
So -cut of
i.e.
and the rough sketch of membership function of
is shown in Fig-4.1.
Fig-4.1:- Rough sketch of Membership function of
36
In extension principle method
Now
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
then we get
and the rough sketch of membership
function of
is shown in Fig-4.1.
And if we consider the expression as
and the rough sketch of membership function of
then its -cut
is shown in Fig-4.2.
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.
37
5. Applications
In 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
, then the perimeter
and
of the rectangle is
The perimeter of the rectangle is a GTrFN
generalized fuzzy set with the membership function
which is a
Fig-5.1: Rough sketch of membership function of
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
=
rod
=
. If the length
, a GTrFN , is cut off from this rod then the remaining length of the
is
The remaining length of the rod is a GTrFN
fuzzy set with the membership function
which is a generalized
38
Fig-5.4: Rough sketch of membership function of
Here 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.
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
=
then the area
=
and
of the triangle is
The area of the triangle is a generalized trapezoidal shaped (concave-convex type) fuzzy number
which is a generalized fuzzy set with the membership
function
39
Fig-5.7: Rough sketch of membership function of
Thus 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.
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 Rectangle
Let the area and breadth of a rectangle are two GTrFNs
=
and =
, then the length
of the rectangle is
The length of the rectangle is a generalized trapezoidal shaped (concave-convex type) fuzzy number
which is a generalized fuzzy set with the membership function
Fig-5.10: Rough sketch of membership function of
40
we 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.
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 annulus
Let 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 a generalized
fuzzy set with the membership function
Fig-5.13: Rough sketch of membership function of
We 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 increasing rate
41
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 work
In 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.
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