SUPER- AND SUB-ADDITIVE ENVELOPES OF AGGREGATION FUNCTIONS: INTERPLAY BETWEEN LOCAL AND GLOBAL PROPERTIES, AND APPROXIMATION

Document Type: Research Paper

Author

Slovak University of Technology in Bratislava, Faculty of Civil Engineer- ing, Department of Mathematics and Descriptive geometry,, Radlinskeho 11, 810 05 Bratislava, Slovakia

Abstract

Super- and sub-additive transformations of aggregation functions have been recently introduced by Greco, Mesiar, Rindone and \v{S}ipeky [The superadditive and the subadditive transformations of integrals and aggregation functions, {\it Fuzzy Sets and Systems} {\bf 291} (2016), 40--53]. In this article we give a survey of the recent development regarding the existence of aggregation functions with a preassigned super- and sub-additive transformation, and address approximation of these transformations. The underpinning feature of the presented results is dependence of global properties of super- and sub-additive transformations on local properties of aggregation functions.

Keywords


[1] A. Arlotto and M. Scarsini, Hessian orders and multinormal distributions, J. Multivariate
Analysis 100 (2009), 2324{2330.
[2] G. Beliakov, A. Pradera and T. Calvo, Aggregation Functions: A Guide for Practitioners,
Springer, Berlin: Heidelberg, 2007.
[3] M. Grabisch, J. L. Marichal, R. Mesiar and E. Pap, Aggregation Functions (Encyklopedia
of Mathematics and its Applications), Cambridge University Press, 2009.
[4] S. Greco, R. Mesiar, F. Rindone and L. Sipeky, Decomposition approaches to integration
without a measure, Fuzzy Sets Syst., 287 (2016), 37{47.
[5] S. Greco, R. Mesiar, F. Rindone and L. Sipeky, The superadditive and the subadditive trans-
formations of integrals and aggregation functions, Fuzzy Sets Syst., 291 (2016), 40{53.
[6] F. Kouchakinejad and A. Siposova, A note on the super-additive and sub-additive transforma-
tions of aggregation functions: The one-dimensional case, In: Proc. Mathematics, Geometry
and their applications, STU Bratislava, ISBN 978-80-227-4611-3, (2016), 15{19.
[7] F. Kouchakinejad and A. Siposova, A note on the super-additive and sub-additive trans-
formations of aggregation functions: The multi-dimensional case, Kybernetika, 53 (2017),
129{136.
[8] F. Kouchakinejad and A. Siposova, Approximation of Super- and Sub-additive Transforma-
tions of Aggregation Functions, 6th Iranian Joint Congress on Fuzzy and Intelligent Systems
(CFIS), (2018), 319{326.
[9] F. Kouchakinejad and A. Siposova and J. Siran, Aggregation functions with given super-
additive and sub-additive transformations, Int. J. General Syst., 46 (2017), 225{234.
[10] M. Marinacci and L. Montrucchio, Ultramodular Functions, Math. Op. Res., 30 (2005) 2,
311{332.
[11] R. Mesiar, A. Kolesarova and A. Stupnanova, Quo vadis aggregation?, Int. J. General Sys-
tema, https://doi.org/10.1080/03081079.2017.1402893, (2017), 21pp.
[12] Y. Ouyang, J. Li and R. Mesiar, Relationship between the concave integrals and the pan-
integrals on fi nite spaces, J. Math. Anal. Appl., 424 (2015), 975{987.
[13] A. Siposova, A note on the superadditive and the subadditive transformations of aggregation
functions, Fuzzy Sets and Syst., 299 (2016), 98{104.
[14] A. Siposova and L. Sipeky, On aggregation functions with given superadditive and subadditive
transformations, In Proceedings of the Congress on Information Technology, Computational
and Experimental Physics (CITCEP), (2015), 199{202.
[15] A. Siposova, L. Sipeky and J. Siran, On the existence of aggregation functions with given
superadditive and subadditive transformations, Fuzzy Sets and Systems, 324 (2017), 117{126.
[16] J. Siran, Transformations of Aggregation Functions: Local Versus Global Properties and
Approximation, 6th Iranian Joint Congress on Fuzzy and Intelligent Systems (CFIS), (2018),
608{613.