Download 9. Construction of a Fuzzy Number with Fuzziness Defined Around an Interval

Int. J. of Mathematical Sciences and Applications,
Vol. 1, No. 1, January 2011
Copyright  Mind Reader Publications
www.ijmsa.yolasite.com
Construction of a Fuzzy Number with
Fuzziness Defined Around an Interval
Rituparna Chutia, Supahi Mahanta and Hemanta K. Baruah
Department of Mathematics,
Gauhati University,
Guwahati-781014, Assam, India.
Email: [email protected].
Department of Statistics,
Gauhati University,
Guwahati-781014, Assam, India.
Email: [email protected].
Professor
Department of Statistics,
Gauhati University,
Guwahati-781014, Assam, India.
Email: [email protected].
Abstract
Fuzziness defined around a point is a very commonly used concept.
However in some situations, fuzziness defined around an interval
looks more practical. For the Gaussian Plume Model of atmospheric
dispersion for example, if certain parameters are assumed to be
fuzzy, then the fuzziness in any individual case should actually be
defined around an interval, and not around a point. In this article,
we would discuss how to construct the membership function of a
fuzzy number with unit possibility assigned to an interval.
Keywords: Gaussian Plume Model, Glivenko-Cantelli
Superimposition , Trapezoidal Fuzzy Number.
Theorem,
Set
1. Introduction
Usually in a fuzzy number fuzziness is defined around a point. But this may not
always be a practical idea in the sense that unit possibility may actually be in an
interval, and in that case we would have to fuzzify the information around that
interval. In this article, we are going to describe how to construct a fuzzy number,
using a measure theoretic standpoint, in which fuzziness is defined around an
interval.
It has been shown by Baruah (2010a) that two probability laws are needed to
define a normal fuzzy number with one probability law leading to the membership
function on the left of the point of maximum possibility, and another probability
law leading to the membership function on the right of the point of maximum
possibility. The probability laws mentioned here are not necessarily associated to
some chance factors; they are indeed used in the measure theoretic sense of the
usage of the term probability in defining a probability measure. This concept of
two probability laws leading to defining a normal fuzzy number has recently been
discussed by Baruah (2010b). The concept has also been applied successfully in
finding the membership function fuzzy numbers without using the classical
method of α-cuts, and in dealing with the arithmetical operations on fuzzy
numbers (Chutia et. al. (2010)), (Mahanta et.al. (2010).)
As for a practical situation in which defining fuzziness around a point, we are
citing the following example. The simple Gaussian Plume Model for defining
atmospheric dispersion at the ground level is given by
( , , 0) =
.
with ( ) =
and ( ) =
+ , where Q is the quantity of the
pollutant released, u is the wind parameter, h is the stack height and the plume
spread parameters are ( ) and ( ). Any of these parameters, Q for example,
need not necessarily be fuzzy around just one single point. In fact, for any plume
an assumption that the spread is around just one single point on a projection
perpendicular to the plume is an oversimplification of the practical situation. It
would indeed be more practical to presume that fuzziness here should be
hypothesized to occur around an interval. Dealing with fuzzy data initially in the
form of triangular fuzzy numbers is a common practice. In the case of a triangular
fuzzy number, we presume that around the point of maximum possibility there are
two uniform probability laws in action, one to the left of the point and one to the
right (Baruah (2011, 2010)). In the case of defining fuzziness around an interval,
we would hypothesize that around an interval of unit possibility, there are two
probability laws, one to the left of the interval and one to the right. In particular, if
the two probability laws are uniform, what we would get is a trapezoidal fuzzy
number. We insist that instead of using fuzzy data initially of the triangular type,
it would be more practical to use fuzzy data of the trapezoidal type.
In what follows, we shall first define in short a set operation called
superimposition. Thereafter use of that concept would be shown in short to define
fuzziness around a point. Construction procedure of a fuzzy number around an
interval would then follow. Finally, we would show with a numerical example of
visualizing fuzziness of that type.
2. Superimposition of Sets
The operation S of superimposition of a set A over a set B (Baruah (1999)) is
defined as:
( ) = ( − ) ∪ ( ∩ )( ) ∪ ( − )
(2.1)
where S represents the operation of superimposition, and (A∩B) (2) represents the
elements of (A∩B) occurring twice. It can be seen that for two intervals [ , ]
and [ ,
where
] gives[ , ]( )[ , ] = [ ( ) , ( ) ] ∪ [ ( ) ,
( , ), ( ) = ( , ), ( ) = ( ) = ( )]
( )
∪ [ ( ) , ( ) ]
( , ),
and
=
( , ). Accordingly, if [ , ])( ) and [ , ])( ) represent two
uniformly fuzzy intervals both with membership equal to half everywhere in the
intervals, superimposition of the fuzzy intervals [a1, b1] (1/2) and [a2, b2] (1/2) would
give rise to
( )
[ ,
]( ) ( )[
]
,
=
So for n fuzzy intervals [ ,
equal to
[ ,
]
…∪ [
(
[
), (
( ),
]
[
( )[
(
)
), ( )]
where, for example, [
( ) , ( ) ]
values of
( )]
( ) … ( )[
(
( ), ( )]
),
]
,
( ), ( )]
[
]
,
( ), ( )
…[
( )
∪
]
,
( ), ( )
. (2.2)
all with membership
everywhere, we shall have
∪[
(
∪
( )
)]
( )
∪[
with membership
,
, …,
,
]
( )
∪[
( )
(
=[
( ),
( ) , ( ) ]
(
( ) ]
)
( )
∪[
( ),
( ) ]
( )
∪
∪… ∪
,
(2.3)
)
represents the uniformly fuzzy interval
in the entire interval,
( ),
( ) ,…,
( )
being
arranged in increasing order of magnitude, and
( ),
(2),…,
( )
being values of
,
, …,
n
again arranged in increasing order of
magnitude.
3. Definition of Fuzziness from Our Standpoint
Define a random vector = ( , , … , ) as a family of , = 1, 2, … , ,
with every
inducing a sub-σ field so that is measurable. Let ( , … ) be
a particular realization of , and let
realize the value
( )
( )
where ( ), ( ),…, ( ) are ordered values of , …
in increasing order of
magnitude. Further let the sub-σ fields induced by
identical. Define now
0,
Φ ( )=
,
(
)
≤
≤
( ) ,
be independent and
<
( )
= 2,3, … ,
(2.4)
1, ≥ ( )
Φ ( ) here is an empirical distribution function of a hypothesized distribution
function Ф( ). As there has to be a one to one correspondence between a
Lebesgue-Stieltjes measure and a distribution function, we would have
( , ) = Ф( )– Ф( )
(2.5)
where is a measure in (Ω, , ), being the σ- field common to every .
Now the Glivenko-Cantelli theorem (see e.g. Loeve (1977), pp-20) states that as n
becomes infinite Φ ( ) converges to Ф( ) uniformly in . This means,
│ ( ) − Ф( )| → 0.
(2.6)
Observe that (r-1)/n in (2.4), for ( ) ≤ ≤ ( ), are indeed memberships of
(
)
[ ( ) , ( ) ]( ) and [ (
in (2.3), for = 2, 3, … , . These
), (
)]
membership values converge to a distribution function on the left of the point of
maximum possibility and to a complementary distribution function on the right of
that point. In essence, this is how a fuzzy number originates.
4. Construction of Fuzzy Number Around an Interval
Consider now two spaces (Ω , , Π ) and (Ω , , Π ) where Ω and Ω are real
intervals [ , ] and [ , ] respectively with < . Let
, ,…,
and , , … , , realizations in [ , ] and [ , ] respectively. Recall (2.3) now for
the fuzzy intervals[ ,
])( ) , [
,
])( ), … , [
,
]( ), all with constant
membership . The values of membership of the superimposed fuzzy intervals
are , , … ,
, 1,
,…,
and . These values of membership considered in
halves as 0, , , … ,
, 1 and(1,
, … , , , 0), would suggest that they
can define an empirical distribution and a complementary empirical distribution
on , , … ,
and , , … ,
respectively. In other words, for realizations of
the values of ( ) , ( ) , … , ( ) in increasing order and of ( ) , ( ) , … , ( ) again
in increasing order, we can see that if we define
0, < ( )
Ψ ( )=
,
(
)
≤
≤
( ),
= 2,3, … ,
1, ≥
1, ≤
Ψ ( )= 1−
,
(
)
≤
≤
( ),
(3.1)
( )
( )
= 2,3, … ,
0, ≥
(3.2)
( )
Then (2.6) assures that
whereΠ [ , ],
≤
( ) → Π [ , ], ≤ ≤
( ) → (1 − Π [ , ]), ≤
≤ , and [ , ], ≤ ≤
have thus seen that existence of two densities
≤ ≤
and
≤
variable[ , , , ].
≤
(3.3)
≤
(3.4)
are defined as in (2.5). We
(
( ))
and
(
( ))
for
are sufficient condition to construct a fuzzy
5. A Numerical Example
Let us now consider the minimum and maximum price of a predicable commodity
in the market for 10 consecutive days of particular month and we try to
demonstrate our idea through the following example.
Table 1: Maximum and Minimum Price
Minimum
Maximum
Price
Price
14.00
16.00
13.50
17.50
11.00
17.00
13.25
17.25
13.00
19.00
12.25
18.50
11.25
19.50
12.50
16.75
11.50
18.25
10.00
20.00
Accordingly, the fuzzy membership values of fuzzy price of the predicable
commodities x in the market of the ten consecutive days would be given
1
, 10 ≤ < 11, 19.5 < ≤ 20
⎧
⎪ 10
⎪ 2 , 11 ≤ < 11.25, 19 < ≤ 19.5
⎪ 10
⎪ 3
⎪ 10 , 11.25 ≤ < 11.5, 18.5 < ≤ 19
⎪ 4
⎪ , 11.5 ≤ < 12.25, 18.25 < ≤ 18.5
10
⎪
5
⎪ , 12.25 ≤ < 12.5, 17.5 < ≤ 18.25
( ) = 10
⎨ 6 , 12.5 ≤ < 13, 17.25 < ≤ 17.5
⎪ 10
⎪7
⎪10 , 13 ≤ < 13.25, 17 < ≤ 17.25
⎪ 8
⎪
, 13.25 ≤ < 13.5, 16.75 < ≤ 17
⎪ 10
⎪ 9 , 13.5 ≤ < 14, 16 < ≤ 16.75
⎪ 10
⎪ 10
⎩ 10 , 14 ≤ ≤ 16
Now fitting a probability distribution and a complementary probability
distribution, possibly with more data, can be a statistical exercise ultimately to get
a fuzzy membership function. However, in a discrete form this is how one can
construct a fuzzy number.
Let
( ),
( ), … ,
( )
with corresponding membership values
respectively be denoted by points ( , ), = 1,2, … ,
and
= ( )=
where
, ,…,
=
( )
. Thus a suitable function for the hypothesized distribution
function Ψ ( ) is expressible in terms of the Lagrange Interpolation formula.
Thus
Ψ ( )=∑
( ), ( ), … , ( )
( ), where ( ) =
with
corresponding
∏
. Similarly for
,
membership
respectively, may be denoted by points ( , ), = 1,2, … ,
and
= ( )=1−
.
Thus
a
suitable
,…, ,
values
function
so that
the
=
()
hypothesized
complementary distribution function Ψ ( ) is expressible in terms of the
( ),
Lagrange Interpolation formula. Thus we haveΨ ( ) = ∑
where ( ) =
∏
. This would lead us to constructing the
,
distribution function Ψ ( ) and the complementary distribution functionΨ ( ).
Thus we can construct a fuzzy number around an interval with unit membership in
the interval ( ) , ( ) , increasing from ( ) , ( ) from zero to one, and
decreasing from ( ) , ( ) from one to zero.
Thus the membership function would be of the form
Ψ ( ), ( ) ≤ ≤
1,
( )≤ ≤
( )=
⎨Ψ ( ), ( ) ≤ ≤
⎩ 0, ℎ
⎧
( )
( )
( )
6. Construction of Trapezoidal Fuzzy Number
If Ψ ( )/
and (1 − Ψ ( ))/
are two density functions in [ , ] and
[ , ] respectively, then
Ψ ( ), ≤ ≤
1,
≤ ≤
( )=
Ψ ( ), ≤ ≤
0, ℎ
can be the membership function of a fuzzy number = [ , , , ]. For the
uniform density function
given by ( ) = ∫
( )=
–
( )
( )=
=
–
–
, ≤ ≤ , the distribution function is
. Similarly, for the uniform density function
, ≤ ≤ , the distribution function is given by
( )=
–
.
It can be seen that ( ) here is the distribution function and (1– ( )) here is
the complementary distribution function of the trapezoidal fuzzy number
[ , , , ] with membership
–
, ≤ ≤
⎧
⎪1, ≤ ≤
( )=
⎨ 1 − – . , ≤ ≤
⎪
⎩ 0, ℎ
Thus we have seen that assumption of existence of two uniform densities, the
simplest form of all densities, in [ , ] and [ , ],are sufficient for the
construction of a trapezoidal fuzzy number[ , , , ].
7. Conclusions
We have shown how fuzziness can arise around an interval. A special case of such
a fuzzy number is the trapezoidal number. On an assumption that around the
interval of maximum possibility there are two uniform probability laws in action,
one to the left of the interval of unit fuzziness and one to the right, we arrive at the
trapezoidal fuzzy number. Such an assumption can be made without loss of any
generality, and hence use of such fuzzy numbers seems to be very logical. We
have in this article described the mathematical basis of using the trapezoidal fuzzy
numbers in uncertainty analysis.
Acknowledgement
This work has been funded by a BRNS Research Project, Department of Atomic
Energy. Government of India.
References
[1] Hemanta K Baruah, “Set Superimposition and Its Applications to the Theory
of Fuzzy Sets”, Journal of the Assam Science Society, 40(1) (1999) 25-31.
[2] Hemanta K. Baruah, “Construction of The Membership Function of a Fuzzy
Number”, ICIC Express Letters (To appear in February 2011).
[3] Hemanta K. Baruah, “The Randomness-Fuzziness Consistency Principle”, Int.
J. of Energy, Information & Communication, 1(1) (2010) 37-48.
[4] Rituparna Chutia, Supahi Mahanta & Hemanta K. Baruah, “An Alternative
Method of Finding The Membership of Fuzzy Number”, Int. J. of Latest
Trends in Computing, 1(2) (2010) 69-72.
[5] Supahi Mahanta, Rituparna Chutia & Hemanta K. Baruah, “Fuzzy Arithmetic
Without Using The Method of Alpha-cuts”, Int. J. of Latest Trends in
Computing, 1(2) (2010) 73-80.
[6] M. Loeve, Probability Theory, Springer Verlag, New York (1977).