A class of aggregation functions encompassing two-dimensional OWA operators. (English) Zbl 1205.68419

Summary: In this paper we prove that, under suitable conditions, Atanassov’s \(K_{\alpha}\) operators, which act on intervals, provide the same numerical results as OWA operators of dimension two. On one hand, this allows us to recover OWA operators from \(K_{\alpha}\) operators. On the other hand, by analyzing the properties of Atanassov’s operators, we can generalize them. In this way, we introduce a class of aggregation functions - the generalized Atanassov operators - that, in particular, include two-dimensional OWA operators. We investigate under which conditions these generalized Atanassov operators satisfy some properties usually required for aggregation functions, such as bisymmetry, strictness, monotonicity, etc. We also show that if we apply these aggregation functions to interval-valued fuzzy sets, we obtain an ordered family of fuzzy sets.


68T37 Reasoning under uncertainty in the context of artificial intelligence
Full Text: DOI


[1] Aczél, J., On Mean values, Bulletin of the American mathematical society, 54, 392-400, (1948) · Zbl 0030.02702
[2] K. Atanassov, Intuitionistic fuzzy sets, VIIth ITKR Session, Deposited in the Central Science and Technology Library of the Bulgarian Academy of Sciences, Sofia, Bulgaria, 1983, pp. 1684-1697.
[3] Atanassov, K., Intuitionistic fuzzy sets, Fuzzy sets and systems, 20, 87-96, (1986) · Zbl 0631.03040
[4] Burillo, P.; Bustince, H., Entropy on intuitionistic fuzzy sets and on interval-valued fuzzy sets, Fuzzy sets and systems, 78, 305-316, (1996) · Zbl 0872.94061
[5] Bustince, H., Construction of intuitionistic fuzzy relations with predetermined properties, Fuzzy sets and systems, 109, 379-403, (2000) · Zbl 0951.03047
[6] Bustince, H.; Barrenechea, E.; Pagola, M., Generation of interval-valued fuzzy and atanassov’s intuitionistic fuzzy connectives from fuzzy connectives and from K(alpha) operators. laws for conjunctions and disjunctions. amplitude, International journal of intelligent systems, 23, 680-714, (2008) · Zbl 1140.68499
[7] H. Bustince, J. Montero, E. Barrenechea, M. Pagola, Laws of conjunctions and disjunctions in interval type 2 fuzzy sets, in: Proceedings of the IEEE World Congress on Computational Intelligence, WCCI2008, Hong Kong, 2008, pp. 615-620.
[8] Bustince, H.; Montero, J.; Pagola, M.; Barrenechea, E.; Gómez, D., A survey on interval-valued fuzzy sets, (), (Chapter 22) · Zbl 1371.03076
[9] Calvo, T.; De Baets, B.; Fodor, J., The functional equations of Frank and alsina for uninorms and nullnorms, Fuzzy sets and systems, 120, 385-394, (2001) · Zbl 0977.03026
[10] Calvo, T.; Kolesárová, A.; Komornikova, M.; Mesiar, R., Aggregation operators: properties classes and construction methods, (), 3-104 · Zbl 1039.03015
[11] Cornelis, C.; Deschrijver, G.; Kerre, E.E., Advances and challenges in interval-valued fuzzy logic, Fuzzy sets and systems, 157, 622-627, (2006) · Zbl 1098.03034
[12] Cutello, V.; Montero, J., Hierarchical aggregation of OWA operators: basic measures and related computational problems, Uncertainty, fuzziness and knowledge-based systems, 3, 17-26, (1995) · Zbl 1232.90323
[13] Deschrijver, G.; Cornelis, C.; Kerre, E.E., On the representation of intuitionistic fuzzy t-norms and t-conorms, IEEE transactions on fuzzy systems, 12, 45-61, (2004)
[14] Fodor, J.; Marichal, J., On nonstrict means, Aequationes mathematicae, 54, 308-327, (1997) · Zbl 0907.39021
[15] Fodor, J.; Roubens, M., Fuzzy preference modelling and multicriteria decision support, (1994), Kluwer Dordrecht · Zbl 0827.90002
[16] Gómez, D.; Montero, J., A discussion on aggregation operators, Kybernetika, 40, 107-120, (2004) · Zbl 1249.68229
[17] Gorzalczany, M.B., A method of inference in approximate reasoning based on interval-valued fuzzy sets, Fuzzy sets and systems, 21, 1-17, (1987) · Zbl 0635.68103
[18] Klement, E.P.; Mesiar, R.; Pap, E., Triangular norms, (2000), Kluwer Dordrecht · Zbl 0972.03002
[19] Klir, G.J.; Folger, T.A., Fuzzy sets, uncertainty and information, (1988), Prentice Hall Englewood Cliffs, NJ · Zbl 0675.94025
[20] R. Sambuc, Fonctions \(\Phi\)-Flous, Application a l’aide au Diagnostic en Pathologie Thyroidienne, Thèse de Doctorat en Médicine, University of Marseille, 1975.
[21] Trillas, E., Sobre funciones de negación en la teorı´a de conjuntos difusos, Stochastica, III-1, 47-59, (1979), (in Spanish). English translation reprinted, in: S. Barro, A. Bugarin, A. Sobrino (Eds.), Advances in Fuzzy Logic, Universidad de Santiago de Compostela, 1998, pp. 31-43
[22] Yager, R.R., On ordered weighted averaging aggregation operators in multicriteria decision-making, IEEE transactions on systems, man and cybernetics, 18, 183-190, (1988) · Zbl 0637.90057
[23] Yager, R.R., Families of OWA operators, Fuzzy sets and systems, 59, 125-148, (1993) · Zbl 0790.94004
[24] Zadeh, L.A., Fuzzy sets, Information control, 8, 338-353, (1965) · Zbl 0139.24606
[25] Zadeh, L.A., The concept of a linguistic variable and its application to approximate reasoning - I, Information sciences, 8, 199-249, (1975) · Zbl 0397.68071
[26] Zadeh, L.A., Is there a need for fuzzy logic?, Information sciences, 178, 2751-2779, (2008) · Zbl 1148.68047
[27] Zadeh, L.A., Toward a generalized theory of uncertainty (GTU) - an outline, Information sciences, 172, 1-2, 1-40, (2005) · Zbl 1074.94021
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.