A simple method for computing the entropy of the product of general fuzzy intervals (Q5952565)

This paper discusses a measure of fuzziness (here called the entropy) of the product of two LR-fuzzy numbers. The main result gives a formula to compute the entropy of the product of two trapezoidal (triangular) fuzzy numbers avoiding an explicit computation of that product. In fact, the output entropy depends on the applied entropy function \(h\) and of the incoming spreads of multiplied trapezoidal fuzzy numbers only. An approximate result for more complicated shaped fuzzy numbers is included, too, as well as some numerical examples.NEWLINENEWLINENEWLINEObserve that the mentioned result is not correct, in general, in so far as the discussed product may lie outside of the frame \(X\) that the authors deal with (from density \(p(x)= p=\text{const}\) follows the boundedness of the real subset \(X\)). Moreover, precise general results extending the authors' results to the case of arbitrary shapes \(L\) and \(R\) can be found in [\textit{A. Kolesárová} and \textit{D. Vivona}, Entropy of \(T\)-sums and \(T\)-products of \(L\)-\(R\) fuzzy numbers, Kybernetika 37, 127-145 (2001)], and hence the approximate formula in Proposition 3 is now superfluous. Note that there is no known direct extension of the results presented to the product of 3 or more fuzzy numbers. However, this topic does not seem to be of great importance.