Abstract
This paper investigates some fundamental questions involving additions of interactive fuzzy numbers. The notion of interactivity between two fuzzy numbers, say A and B, is described by a joint possibility distribution J. One can define a fuzzy number \(A +_J B\) (or \(A -_J B\)), called J-interactive sum (or difference) of A and B, in terms of the sup-J extension principle of the addition (or difference) operator of the real numbers. In this article we address the following three questions: (1) Given fuzzy numbers B and C, is there a fuzzy number X and a joint possibility distribution J of X and B such that \(X +_J B = C\)? (2) Given fuzzy numbers A, B, and C, is there a joint possibility distribution J of A and B such that \(A +_J B = C\)? (3) Given a joint possibility distribution J of fuzzy numbers A and B, is there a joint possibility distribution N of \((A +_J B)\) and B such that \((A +_J B) -_N B = A\)? It is worth noting that these questions are trivially answered in the case where the fuzzy numbers A, B and C are real numbers, since the fuzzy arithmetic \(+_J\) and \(-_N\) are extension of the classical arithmetic for real numbers.
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Acknowledgements
This work was partially supported by FAPESP under grant no. 2016/26040-7 and CNPq under grants no. 306546/2017-5 and 142414/2017-4.
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Esmi, E., de Barros, L.C., Wasques, V.F. (2019). Some Notes on the Addition of Interactive Fuzzy Numbers. In: Kearfott, R., Batyrshin, I., Reformat, M., Ceberio, M., Kreinovich, V. (eds) Fuzzy Techniques: Theory and Applications. IFSA/NAFIPS 2019 2019. Advances in Intelligent Systems and Computing, vol 1000. Springer, Cham. https://doi.org/10.1007/978-3-030-21920-8_23
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DOI: https://doi.org/10.1007/978-3-030-21920-8_23
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