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Construction of a fuzzy number with fuzziness defined around an interval
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.
... ), x ∈ [ξ 2 , ξ 3 ] and using the set operations namely superimposition lies between the distribution and fuzzy membership functions defined by (Baruah, H.K., 1999(Baruah, H.K., , 2010[2] [8]. By using this concept, the triangular fuzzy membership function of the n √ x, e x , x −1 and trapezoidal fuzzy membership function are defined by (Chutia, R et al., 2010(Chutia, R et al., , 2011[9] [6]. The traditional ways of using crisp values are inadequate. ...
Fuzzy numbers are based on membership function such as the trapezoidal shape of the trapezoidal fuzzy number have one interval with two uncertain intervals. A crisp value of trapezoidal fuzzy number is not sufficient when uncertainty arises in more than four intervals such as the hexagonal, octagonal, decagonal, dodecagonal, etc., In this paper we take Dubois Prade left and right reference function with distributive function and complementary function respectively and find the generalization of the membership function of n-trapezoidal shapes uncertain numbers in a very simple way. Also illustrated with few examples also.
Since the beginning, for the last forty five years, it has been accepted that fuzziness and randomness are independent concepts. Workers in fuzzy mathematics have been trying to link randomness with fuzziness without any success. In this article, we would establish that trying to impose one probability space over an interval defining a possibility space is not correct. We need two probability spaces to define a possibility space. On the other hand, given a possibility space, it can be defined by two probability spaces. Hence the measure theoretic matters of possibility must be geared to forming two probability measures. The correct randomness -fuzziness consistency principle is: Poss [x] = θ Prob [a ≤ y ≤ x] + (1 – θ) {1 – Prob [b ≤ y ≤ x]}, where θ is 1, if a ≤ x ≤ b, and is 0, if b ≤ x ≤ c. In other words, fuzziness is rooted at randomness.
In this article, an alternative method to evaluate the arithmetic operations on fuzzy number has been developed, on the assumption that the Dubois-Prade left and right reference functions of a fuzzy number are distribution function and complementary distribution function respectively. Using the method, the arithmetic operations of fuzzy numbers can be done in a very simple way. This alternative method has been demonstrated with the help of numerical examples.
In this article, it has been shown that the Dubois-Prade left and right reference functions of a fuzzy number viewing as a distribution function and a complementary distribution function respectively, leads to a very simple alternative method of finding the membership of any function of a fuzzy number. This alternative method has been demonstrated with the help of different examples.
In this article, it has been established that the concepts of fuzziness and randomness are linked. A set operation called superimposition and the Glivenko-Cantelli theorem on convergence of empirical probability distributions have been applied to arrive at the result. It has been shown how two distribution functions with reference to two independent probability measures can be used to construct the membership function of a normal fuzzy number. ICIC International