Abstract
Aggregation (fusion) of several input values into a single output value is an indispensable tool not only of mathematics or physics, but of majority of engineering, economical, social and other sciences. The problems of aggregation are very broad and heterogeneous, in general. Therefore we restrict ourselves in this contribution to the specific topic of the aggregation of finite number of real inputs only. Closely related topics of aggregating infinitely many real inputs [23,109,64,52,43,42,44,99], of aggregating inputs from some ordinal scales [41,50], of aggregating complex inputs (such as probability distributions [107,114], fuzzy sets [143]), etc., are treated, among others, in the quoted papers, and we will not deal with them. In this spirit, if the number of input values is fixed, say n, an aggregation operator is a real function of n variables. This is still a too general topic. Therefore we restrict our considerations regarding inputs as well as outputs to some fixed interval (scale) I = [a, b] ⊑ [-∞, ∞]. It is a matter of rescaling to fix I = [0,1].
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
N. H. Abel: Untersuchungen der Funktionen zweier unabhängigen veränderlichen Grössen x und y wie f (x, y), welche die Eigenschaft haben, dass f (z, f (x, y)) eine symmetrische Funktion von x, y und z ist. J. Reine Angew. Math. 1 (1826) 1115.
J. Aczél: On mean values. Bulletin of the American Math. Society 54 (1948) 392–400.
J. Aczél: Lectures on Functional Equations and their Applications. Academic Press, New York, 1966.
J. Aczél and C. Alsina: Characterization of some classes of quasilinear functions with applications to triangular norms and to synthesizing judgements. Methods Oper. Res. 48 (1984) 3–22.
P. Benvenuti and R. Mesiar: Integrals with respect to a general fuzzy measure. In: M. Grabisch, T. Murofushi and M. Sugeno, eds. Fuzzy Measures and Integrals. Theory and Applications. Physica- Verlag, Heidelberg, 2000, pp. 205–232.
P. Benvenuti and R. Mesiar Pseudo-arithmetical operations as a basis for integration with respect to a general fuzzy measure. Inform. Sc.,to appear.
B. Bouchon-Meunier, ed.: Aggregation and Fusion of Imperfect Information. Physica-Verlag, Heidelberg, 1998.
T. Calvo, J. Martin, G. Mayor and J. Torrens: Balanced discrete fuzzy measures. Int. J. Uncertainty, Fuzziness and Knowledge-Based Systems 8 (2000) 665–676.
T. Calvo and B. De Baets: On a generalization of the absorption equation. Int. Fuzzy. Math. Publ. 8 (2000) 141–149.
T. Calvo, B. De Baets and J.C. Fodor: The functional equations of Alsina and Frank for uninorms and nullnorms. Fuzzy Sets and Systems 120 (2001) 15–24.
T. Calvo and G. Mayor: Remarks on two types aggregation functions. Tatra Mount. Math. Publ. 16 (1999) 235–254.
T. Calvo, G. Mayor, J. Torrens, J. Suíier, M. Mas and M. Carbonell: Generation of weighting triangles associated with aggregation functions. Int. J. Uncertainty, Fuzziness and Knowledge-Based Systems 8 (2000) 417–451.
T. Calvo and R. Mesiar: Weighted means based on triangular conorms. Int. J. of Uncertainty, Fuzziness and Knowledge-Based Systems 9 (2001).
T. Calvo and R. Mesiar: Criteria importances in median-like aggregation. IEEE Transactions on Fuzzy Systems,to appear.
T. Calvo and R. Mesiar: Generalized medians. Fuzzy Sets and Systems, to appear.
T. Calvo and R. Mesiar: Continuous generated associative aggregation operators. Fuzzy Sets and Systems,to appear.
T. Calvo and R. Mesiar: Stability of aggregation operators Proceedings Eusflat’2001,Leicester, 2001, to appear.
F. Chiclana, F. Herrera and F. Herrera-Viedma: The ordered weighted geometric operator. Proceedings IPMU’2000, Madrid, 2000, pp. 985–991.
A.H. Clifford: Naturally totally ordered commutative semigroups. Amer. J. Math. 76 (1954) 631–646.
G. Choquet: Theory of capacities. Ann. Inst. Fourier 5 (1953–54) 131–295.
A.C. Climescu: Sur l’équation fonctionelle de l’associativité. Bull. École Polytechn. Iassy 1 (1946) 1–16.
B. De Baets: Idempotent uninorms. Europ. J. Oper. Research 180 (1999) 631642.
D. Denneberg: Non-additive Measure and Integral. Kluwer Academic Publishers, Dordrecht, 1994.
D. Denneberg: Non-additive measure and integral, basic concepts and their role for applications. In: M. Grabisch, T. Murofushi and M. Sugeno, eds. Fuzzy Measures and Integrals. Theory and Applications. Physica- Verlag, Heidelberg, 2000, pp. 42–69.
M. Detyniecki: Mathematical Aggregation Operators and their Applications to Video Querying. Ph.D. Thesis, University Paris V I, 2000.
J. Dombi: Basic concepts for a theory of evaluation: The aggregative operator. Europ. J. Oper. Research 10 (1982) 282–293.
D. Dubois and H. Prade: A review of fuzzy set aggregation connectives. Inform. Sci. 36 (1985) 85–121.
D. Dubois and H. Prade: Weighted minimum and maximum in fuzzy set theory. Inform. Sci. 39 (1986) 85–121.
J.J. Dujmovic: Weighted conjunctive and disjunctive means and their application in system evaluation. Univ. Beograd Publ. Elektrotech. Fak., 1974, pp. 147–158.
J.C. Fodor: Contrapositive symmetry of fuzzy implications. Fuzzy Sets and Systems 69 (1995) 141–156.
J.C. Fodor: An extension of Fung-Fu’s theorem. Int. J. of Uncertainty, Fuziness and Knowledge-Based Systems 4 (1996) 235–243.
J.C. Fodor, J.-L. Marichal and M. Roubens: Characterization of the ordered weighted averaging operators. IEEE Transactions on Fuzzy Systems 3 (1995) 236–240.
J.C. Fodor and J.-L. Marichal: On nonstrict means. Aequationes Mathematicae 54 (1997) 308–327.
J.C. Fodor and M. Roubens: Fuzzy Preference Modelling and Multicriteria Decision Support. Kluwer Academic Publishers, Dordrecht, 1994.
J.C. Fodor, R.R. Yager and A. Rybalov: Structure of uninorms. Int. J. of Uncertainty, Fuzziness and Knowledge-Based Systems 5 (1997) 411–427.
M.J Frank. (1979) On the simultaneous associativity of F(x, y) and x + y -F(x, y). Aequationes Math. 19 (1979) 194–226.
K. Fujimoto, T. Murofushi and M. Sugeno: Canonical hierarchical decomposition of the Choquet integral over a finite set with respect to null-additive fuzzy measure. Int. J. of Uncertainty, Fuzziness and Knowledge-Based Systems 6 (1998) 345–363.
K. Fujimoto and T. Murofushi: Hierarchical decomposition of the Choquet integral. In: M. Grabisch, T. Murofushi and M. Sugeno, eds. Fuzzy Measures and Integrals. Theory and Applications. Physica- Verlag, Heidelberg, 2000, pp. 94–103.
L.W. Fung and K.S. Fu: An axiomatic approach to rational decision making in a fuzzy environment. In: L.A. Zadeh, K.S. Fu, K. Tanaka and M. Shimura, eds., Fuzzy sets and Their Applications to Cognitive and Decision Processes. Academic Press, New York, 1975, pp. 227–256.
L. Godo and C. Sierra: A new approach to connective generation in the framework of expert systems using fuzzy logic. In: Proceedings 18th International Symposium on Multiple-Valued Logic. Palma de Mallorca, IEEE Computer Society Press, 1988, pp. 157–162.
L. Godo and V. Torra: Extending Choquet integrals for aggregation of ordinal values. Proceedings IPMU’2000, Madrid, 2000, pp. 410–417.
L. Gonzalez: A note on infinitary action of triangular norms and conorms. Fuzzy Sets and Systems 101 (1999) 177–180.
L. Gonzalez: Universal aggregation operators. Proceedings Eusf lat’2001,Leicester, 2001, to appear.
L. Gonzalez: What is arithmetic mean? Proceedings A GGOP’2001,Oviedo, 2001, to appear.
S. Gottwald: A Treatise on Many-Valued Logic. Research Studies Press Ltd., Baldock, Hertforshire, 2001.
M. Grabisch: Fuzzy integral in multicriteria decision making. Fuzzy Sets and Systems 69 (1995) 279–298.
M. Grabisch: k-order additive fuzzy measures. Proceedings IPMU’96, Granada, 1996, pp. 1345–1350.
M. Grabisch: k-order additive discrete fuzzy measures and their representation. Fuzzy Sets and Systems 92 (1997) 167–189.
M. Grabisch: The interaction and Möbius representation of fuzzy measures on finite spaces, k-additive measures. In: M. Grabisch, T. Murofushi and M. Sugeno, eds. Fuzzy Measures and Integrals. Theory and Applications. PhysicaVerlag, Heidelberg, 2000, pp. 70–93.
M. Grabisch: Symmetric and asymmetric integrals: the ordinal case. Proceedings IIZUKA’2000, Iizuka, 2000, CD-rom.
M. Grabisch, J.-L. Marichal and M. Roubens: Equivalent representations of set functions. Math. Operat. Res. 25 (2000) 157–178.
M. Grabisch, T. Murofushi, M. Sugeno, eds.: Fuzzy Measures and Integrals. Theory and Applications. Physica-Verlag, Heidelberg, 2000.
M. Grabisch, H.T. Nguyen and E.A. Walker: Fundamentals of Uncertainty Calculi with Applications to Fuzzy Inference. Kluwer Academic Publishers, Dordercht, 1995.
P. Hajek: Metamathematics of Fuzzy Logic. Kluwer Academic Publishers, Dordrecht, 1998.
P.R. Halmos: Measure Theory. Van Nostrand, New York, 1950.
H. Imaoka: On a subjective evaluation model by a generalized fuzzy integral. Int. J. of Uncertainty, Fuzziness and Knowledge-Based Systems 5 (1997) 517529.
H. Imaoka: Comparison between three integrals. In: M. Grabisch, T. Murofushi and M. Sugeno, eds. Fuzzy Measures and Integrals. Theory and Applications. Physica- Verlag, Heidelberg, 2000, pp. 273–286.
E.P. Klement: Construction of fuzzy v-algebras using triangular norms. J. Math. Anal. Appl. 85 (1982) pp. 543–566.
E.P. Klement, R. Mesiar and E. Pap: On the relationship of associative compensatory operators to triangular norms and conorms. Int. J. of Uncertainty, Fuzziness and Knowledge-Based Systems 4 (1996) 129–144.
E.P. Klement, R. Mesiar and E. Pap: Quasi-and pseudo-inverses of monotone functions, and the construction of t-norms. Fuzzy Sets and Systems 104 (1999) 3–13.
E.P. Klement, R. Mesiar and E. Pap: Triangular Norms. Kluwer Academic Publishers, Dordrecht, 2000.
E.P. Klement, R. Mesiar and E. Pap: Integration with respect to decomposable measures, based on a conditionally distributive semiring on the unit interval. Int. J. of Uncertainty, Fuzziness and Knowledge-Based Systems 8 (2000) 701–717.
E.P. Klement, R. Mesiar and E. Pap: Geometric approach to aggregation. Proceedings Eusflat’2001 Leicester, 2001, to appear.
G.J. Klir and T.A. Folger: Fuzzy Sets, Uncertainty and Information. Prentice Hall, Englewood Cliffs, 1988.
A. Kolesârovâ: On the comparison of quasi-arithmetic means. Busefal 80 (1999) 30–34.
A. Kolesârovâ: Collapsed input-based aggregation. Int. J. of Uncertainty, Fuzziness and Knowledge- Based Systems 9 (2001).
A. Kolesârovâ: Limit properties of quasi-arithmetic means. Fuzzy Sets and Systems,to appear.
A. Kolesârovâ: Parametric evaluation of aggregation operators. Preprint, submitted.
A. Kolesârovâ and M. Komorníkovâ: Triangular norm-based iterative aggregation and compensatory operators. Fuzzy Sets and Systems 104 (1999) 109–120.
A. Kolesârovâ and J. Mordelovâ: 1-Lipschitz and kernel aggregation operators. Proceedings of AGGOP’2001,Oviedo, 2001, to appear.
A.N. Kolmogoroff: Sur la notion de la moyenne. Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. 12 (1930) 388–391.
M. Komorníkovâ: Generated aggregation operators. Proceedings EUSFLAT’99, Palma de Mallorca, 1999, pp. 355–358.
M. Komorníkovâ: Aggregation operators and additive generators. Int. J. of Uncertainty, Fuzziness and Knowledge- Based Systems 9 (2001).
J. Lâzaro and T. Rückschlossovâ: Shift invariant binary aggregation operators. Proceedings AGGOP’2001,Oviedo, 2001, to appear.
Y.-M. Li and Z.-K. Shi: Weak uninorms aggregation operators. Inform. Sci. 124 (2000) 317–323.
C.M. Ling: Representation of associative functions. Publ. Math. Debrecen 12 (1965) 189–212.
M.K. Luhandjula: Compensatory operators in fuzzy linear programming with multiple objectives. Fuzzy Sets and Systems 8 (1982) 245–252.
J.-L. Marichal: Aggregations Operators for Multi-Criteria Decision Aid. Ph.D. Thesis, University of Liége, 1998.
J.-L. Marichal: On Choquet and Sugeno integrals as aggregation functions. In: M. Grabisch, T. Murofushi and M. Sugeno, eds. Fuzzy Measures and Integrals. Theory and Applications. Physica- Verlag, Heidelberg, 2000, pp. 247–272.
J.-L. Marichal: An axiomatic approach of the discrete Choquet integral as a tool to aggregate interacting criteria. IEEE Transactions on Fuzzy Systems 8 (2000) 800–807.
J.-L. Marichal: Aggregation of interacting criteria by means of the discrete Choquet integral. Chapter in this monograph.
J.-L. Marichal. On order invariant synthesizing functions. Preprint, submitted.
J.-L. Marichal: On an axiomatization of the quasi-arithmetic mean values without the symmetry axiom. Aequationes Mathematicae 59 (2000) 74–83.
J.-L. Marichal, P. Mathonet and E. Thousset: Characterization of some aggregations functions stable for positive linear transformations. Fuzzy Sets and Systems 102 (1999) 293–314.
M. Mas, G. Mayor and J. Torrens: t-operators. Int. J. Uncertainty, Fuzziness and Knowledge-Based Systems 7 (1999) 31–50
G. Mayor and T. Calvo: On extended aggregation functions. Proceedings IFSA ‘87, Prague, 1997, vol. I, pp. 281–285.
G. Mayor and J. Torrens: On a class of operators for expert systems. Int. J. of Intelligent Systems 8 (1988) 771–778.
K. Menger: Statistical metrics. Procs. Nat. Acad. Sci. U.S.A. 37 (1942) 535537.
R. Mesiar: Compensatory operators based on triangular norms. Proceedings EUFIT’95, Aachen, 1995, pp. 131–135.
R. Mesiar: Choquet-like integrals. J. Math. Anal. Appl. 194 (1995) 477–488.
R. Mesiar: Generalizations of k-order additive discrete fuzzy measures. Fuzzy Sets and Systems 102 (1999) 423–428.
R. Mesiar: k-order additive measures. Int. J. of Uncertainty, Fuzziness and Knowledge-Based Systems 6 (1999) 561–568.
R. Mesiar and B. De Baets: New construction methods for aggregation operators. Proceedings IPMU’2000, Madrid, 2000, pp. 701–706.
R. Mesiar, T. Calvo and J. Martin: Integral based aggregation of real data. Proceedings IPMU’2000, Madrid, 2000, pp. 58–62
R. Mesiar and B. De Baets: Continuous ordinal sums of aggregation operators. Manuscript in preparation.
R. Mesiar and M. Komorníkovâ: Aggregation operators. In: D. Herceg and K. Surla, eds., Proceedings PRIM’96, XI. Conference on Applied Mathematics, 1996, pp. 193–211.
R. Mesiar and M. Komorníkovâ: Triangular norm-based aggregation of evidence under fuzziness. In: B. Bouchon-Meunier, ed., Aggregation and Fusion of Imperfect Information. Physica-Verlag, Heidelberg, 1998.
R. Mesiar and D. Vivona: Two-step integral with respect to fuzzy measure. Tatra Mount. Math. Publ. 16 (1999) 359–368.
R. Mesiar and H. Thiele: On T-quantifiers and S- quantifiers. In: V. Novak and I. Perfilieva, eds., Discovering the Word with Fuzzy Logic. Physica-Verlag, Heidelberg, 2000, pp. 310–326.
M. Mizumoto: Pictorial representations of fuzzy connectives, Part I.: Cases of t-norms, t-conorms and averaging operators. Fuzzy Sets and Systems 31 (1989) 217–242.
M. Mizumoto: Pictorial representations of fuzzy connectives, Part II.: Cases of compensatory operators and self-dual operators. Fuzzy Sets and Systems 32 (1989) 45–79.
R. Moynihan. On TT semigroups of probability distribution functions II. Aequationes Math. 17 (1978) 19–40.
E. Muel and J. Mordelovâ: Kernel aggregation operators. Proceedings AGGOP’2001,Oviedo, 2001, to appear.
T. Murofushi and M. Sugeno: Fuzzy t-conorm integrals with respect to fuzzy measures: generalizations of Sugeno integral and Choquet integral. Fuzzy Sets and Systems 42 (1991) 51–57.
T. Murofushi and M. Sugeno: Fuzzy measures and fuzzy integrals. In: M. Grabisch, T. Murofushi and M. Sugeno, eds. Fuzzy Measures and Integrals. Theory and Applications. Physica- Verlag, Heidelberg, 2000, pp. 3–41.
M. Nagumo: Uber eine Klasse der Mittelwerte. Japanese Journal of Mathematics 6 (1930) 71–79.
R.B. Nelsen: An Introduction to Copulas. Lecture Notes in Statistic 139, Springer, 1999.
S. Ovchinnikov and A. Dukhovny: Integral representation of invariant functionals J. Math. Anal. Appl. 244 (2000) 228–232.
E. Pap: Null-Additive Set Functions. Kluwer Academic Publishers, Dordrecht, 1995.
A.L. Ralescu and D.A. Ralescu: Extensions of fuzzy aggregation. Fuzzy Sets and Systems 86 (1997) 321–330.
T. Michâlikovâ-Rückschlossovâ: Some constructions of aggregation operators. J. Electrical Engin. 12 (2000) 29–32.
T. Rückschlossovâ: Invariant aggregation operators. Manuscript in preparation.
W. Sander: Associative aggregation operators. Chapter in this monograph.
B. Schweizer and A. Sklar: Probabilistic Metric Spaces. North Holland, New York, 1983.
C. Shannon and W. Weaver: The Mathematical Theory of Communication. University of Illinois Press, Urbana, 1949.
N. Shilkret: Maxitive measures and integration. Indag. Math. 33 (1971) 109116.
W. Silvert: Symmetric summation: A class of operations of fuzzy sets. IEEE Trans. Syst., Man Cybern. 9 (1979) 657–659.
D. Smutnâ: On a peculiar t-norm. Busefal 75 (1998) 60–67.
M. Sugeno: Theory of Fuzzy Integrals and Applications. Ph.D. Thesis, Tokyo Inst. of Technology, Tokyo, 1974.
M. Sugeno and T. Murofushi: Pseudo-additive measures and integrals, J. Math. Anal. Appl. 122 (1987) 197–222.
M. Sabo, A. Kolesârovâ and S. Varga: RET operators generated by triangular norms and copulas. Int. J. Uncertainty, Fuzziness and Knowledge-Based Systems 9 (2001).
J. Sipos: Integral with respect to a pre-measure. Math. Slovaca 29 (1979) 141–145.
V. Torra: The weighted OWA operator. Int. J. of Intelligent Systems 12 (1997) 153–166.
V. Torra and L. Godo: Continuous WOWA operators with application to defuzzification. Chapter in this monograph.
I.B. Türksen: Interval-valued fuzzy sets and “compensatory AND”. Fuzzy Sets and Systems 51 (1992) 295–307.
P. Vicenik: A note on generators of t-norms. Busefal 75 (1998) 33–38.
P. Vicenik- Additive generators and discontinuity. Busefal 76 (1998) 25–28.
P. Vicenik: Additive generators of non-continuous triangular norms. In: S. Rodabaugh and P. Klement, eds., Proceedings of Linz Seminar 1999,Kluwer Academic Publishers, to appear.
Z. Wang and G.J. Klir: Fuzzy Measure Theory, Plenum Press, 1992.
S. Weber: 1-decomposable measures and integrals for Archimedean tconorms I. J. Math. Anal. Appl. 101 (1984) 114–138.
S. Weber: Two integrals and some modified version—critical remarks. Fuzzy Sets and Systems 20 (1986) 97–105.
R.R. Yager: On a general class of fuzzy connectives. Fuzzy Sets and Systems 4 (1980) 235–242.
R.R. Yager: On ordered weighted averaging aggregation operators in multi-criteria decisionmaking. IEEE Trans. Syst., Man Cybern. 18 (1988) 183–190.
R.R. Yager: Criteria importances in OWA aggregation: An application of fuzzy modeling. Proceedings IEEE’FUZZ’97, Barcelona, 1997, pp. 1677–1682.
R.R. Yager: Fusion od ordinal information using weighted median aggregation. Int. J. Approx. Reasoning 18 (1998) 35–52.
R.R. Yager: Uninorms in fuzzy modeling. Fuzzy Sets and Systems to appear.
R.R. Yager: Using importances in group preference aggregation to block strategic manipulation. Chapter in this monograph.
R.R. Yager, M. Detyniecki and B. Bouchon—Meunier: Specifying t—norms based on the value of T(1/2,1/2). Mathware and Soft Computing 7 (2000) 77–78.
R.R. Yager and D.P. Filev: Essentials of Fuzzy Modelling and Control. J. Wiley & Sons, New York, 1994.
R.R. Yager and J. Kacprzyk: The Ordered Weighted Averaging Operators, Theory and applications. Kluwer Academic Publishers, Boston, Dordrecht, London, 1997.
R.R. Yager and A. Rybalov: Uninorm aggregation operators. Fuzzy Sets and Systems 80 (1996) 111–120.
R.R.Yager and A. Rybalov: Noncommutative self—identity aggregation. Fuzzy Sets and Systems 85 (1997) 73–82.
L.A. Zadeh: Fuzzy sets. Inform. Control 8 (1965) 338–353.
H.J Zimmermann and P. Zysno: Latent connectives in human decision making Fuzzy Sets and Systems 4 (1980) 37–51.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2002 Physica-Verlag Heidelberg
About this chapter
Cite this chapter
Calvo, T., Kolesárová, A., Komorníková, M., Mesiar, R. (2002). Aggregation Operators: Properties, Classes and Construction Methods. In: Calvo, T., Mayor, G., Mesiar, R. (eds) Aggregation Operators. Studies in Fuzziness and Soft Computing, vol 97. Physica, Heidelberg. https://doi.org/10.1007/978-3-7908-1787-4_1
Download citation
DOI: https://doi.org/10.1007/978-3-7908-1787-4_1
Publisher Name: Physica, Heidelberg
Print ISBN: 978-3-662-00319-0
Online ISBN: 978-3-7908-1787-4
eBook Packages: Springer Book Archive