Abstract
A fuzzy real number [α, β, γ] is an interval around the real number β with the elements in the interval being partially present. Partial presence of an element in a fuzzy set is defined by the name membership function. According to the Randomness-Fuzziness Consistency Principle, two independent laws of randomness in [α, β] and [β, γ] are necessary and sufficient to define a normal fuzzy number [α, β, γ]. In this article, we have shown how to construct normal fuzzy number using daily temperature data.
Highlights
Construction of normal fuzzy number has been discussed in [Baruah, 2011b, 2012] based on the Randomness – Fuzziness Consistency Principle deduced by Baruah [Baruah, 2010, 2011a, 2011b, 2011c, 2012]
Sn (x) = 0, if x < X (1), = k / n, if X (k) ≤ x < X(k+1), k = 1, 2, ..., (n – 1), = 1, if x ≥ X(n) according to Baruah’s randomness-fuzziness consistency principle, the theoretical distribution function of the minimum temperature in [5.6, 15.8] and the theoretical complementary distribution function of the maximum temperature in [15.8, 28.6] together define a normal fuzzy number [5.6, 15.8, 28.6], which is clear from the diagram given below: X here being random, so would be Sn (X)
We have found that the estimated membership values of the left reference function for the variables 15.8, 17.8 and 19.5 are
Summary
Construction of normal fuzzy number has been discussed in [Baruah, 2011b, 2012] based on the Randomness – Fuzziness Consistency Principle deduced by Baruah [Baruah, 2010, 2011a, 2011b, 2011c, 2012]. We need to understand that if a variable X can assume values in an interval [L, U] where L follows a law of randomness in the interval [α, β] while U follows another law of randomness in the interval [β, γ], we are in a situation defining fuzzy uncertainty, with randomness defined in the measure theoretic sense (2011a) In such a case, Baruah’s principle of consistency between randomness and fuzziness states that the distribution function of L, which is known as the left reference function with reference to fuzziness, in the interval [α, β] together with the complementary distribution function of U which is known as the right reference function in the interval [β, γ], would give us the membership function of a normal fuzzy number [α, β, γ]. It should be noted here that the temperature of a particular place at a particular time everyday is necessarily probabilistic, but the daily temperature in the same place is fuzzy
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