UDC 519.766.2
RAPID CALCULATION OF CONSEQUENTS FUZZINESS
INDICES AFTER MAMDANI IMPLICATION
AND FURTHER AGGREGATION
K. V. Bukhensky, Ph.D. (Phys. and Math.), associate professor, Head of the Department of Higher
Mathematics, RSREU, Ryazan, Russia;
orcid.org/0000-0003-2602-2112, e-mail: This email address is being protected from spambots. You need JavaScript enabled to view it.
A. N. Konyukhov, Ph.D. (Ped.), associate professor, RSREU, Ryazan, Russia;
orcid.org/0000-0002-1523-7110, e-mail: This email address is being protected from spambots. You need JavaScript enabled to view it.
K. V. Anisimov, fourth year student of the Faculty of Computer Science, RSREU, Ryazan, Russia;
orcid.org/0000-0001-8889-0818, e-mail: This email address is being protected from spambots. You need JavaScript enabled to view it.
The aim is to – 1) obtain the analytical formula for consequent index of fuzziness after Mamdani implication
(rule); 2) derive the analytical formula for fuzziness index of a fuzzy set (FS) gained after aggregation
of arbitrary number subnormal consequents of Mamdani implication. This work is the proceeding of former
research on the project “Fuzziness transformations in fuzzy inference systems (FIS)”. As a measure of math
fuzziness a Yager’s index with linear Hamming metric was used. The analytical formulas for integral functions
of fuzziness index for some shape functions (IFFI SF) of LR-types of fuzzy numbers (FNs) were derived.
IFFI SF allows us to calculate FS’s index of fuzziness obtained from FNs via logical operations. Common
and partial formulas for LR-type FNs with linear and clipped parabola shape functions were represented. In
order to test theoretical results several numerical experiments were accomplished. The expressions obtained
allow calculating the index of fuzziness for FSs after Mamdani implication and further consequents aggregation
without integration procedure thus reducing calculation time dramatically. Formulas may be further
analyzed in order to search the optimal conditions of Mamdani type inference in the terms of fuzziness.
Key words:fuzzy set, LR-type fuzzy number, membership function, shape function, index of fuzziness, integral
functions of fuzziness index, antecedent, consequent, Mamdani implication, linguistic variable term,
cross-factor.