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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).