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Aggregation Operators: Properties, Classes and Construction Methods

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Aggregation Operators

Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 97))

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].

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References

  1. 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.

    Google Scholar 

  2. J. Aczél: On mean values. Bulletin of the American Math. Society 54 (1948) 392–400.

    Article  MATH  Google Scholar 

  3. J. Aczél: Lectures on Functional Equations and their Applications. Academic Press, New York, 1966.

    MATH  Google Scholar 

  4. 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.

    MATH  Google Scholar 

  5. 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.

    Google Scholar 

  6. P. Benvenuti and R. Mesiar Pseudo-arithmetical operations as a basis for integration with respect to a general fuzzy measure. Inform. Sc.,to appear.

    Google Scholar 

  7. B. Bouchon-Meunier, ed.: Aggregation and Fusion of Imperfect Information. Physica-Verlag, Heidelberg, 1998.

    MATH  Google Scholar 

  8. 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.

    MathSciNet  MATH  Google Scholar 

  9. T. Calvo and B. De Baets: On a generalization of the absorption equation. Int. Fuzzy. Math. Publ. 8 (2000) 141–149.

    MATH  Google Scholar 

  10. 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.

    Article  Google Scholar 

  11. T. Calvo and G. Mayor: Remarks on two types aggregation functions. Tatra Mount. Math. Publ. 16 (1999) 235–254.

    MathSciNet  MATH  Google Scholar 

  12. 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.

    MATH  Google Scholar 

  13. T. Calvo and R. Mesiar: Weighted means based on triangular conorms. Int. J. of Uncertainty, Fuzziness and Knowledge-Based Systems 9 (2001).

    Google Scholar 

  14. T. Calvo and R. Mesiar: Criteria importances in median-like aggregation. IEEE Transactions on Fuzzy Systems,to appear.

    Google Scholar 

  15. T. Calvo and R. Mesiar: Generalized medians. Fuzzy Sets and Systems, to appear.

    Google Scholar 

  16. T. Calvo and R. Mesiar: Continuous generated associative aggregation operators. Fuzzy Sets and Systems,to appear.

    Google Scholar 

  17. T. Calvo and R. Mesiar: Stability of aggregation operators Proceedings Eusflat’2001,Leicester, 2001, to appear.

    Google Scholar 

  18. F. Chiclana, F. Herrera and F. Herrera-Viedma: The ordered weighted geometric operator. Proceedings IPMU’2000, Madrid, 2000, pp. 985–991.

    Google Scholar 

  19. A.H. Clifford: Naturally totally ordered commutative semigroups. Amer. J. Math. 76 (1954) 631–646.

    Article  MathSciNet  MATH  Google Scholar 

  20. G. Choquet: Theory of capacities. Ann. Inst. Fourier 5 (1953–54) 131–295.

    Google Scholar 

  21. A.C. Climescu: Sur l’équation fonctionelle de l’associativité. Bull. École Polytechn. Iassy 1 (1946) 1–16.

    Google Scholar 

  22. B. De Baets: Idempotent uninorms. Europ. J. Oper. Research 180 (1999) 631642.

    Google Scholar 

  23. D. Denneberg: Non-additive Measure and Integral. Kluwer Academic Publishers, Dordrecht, 1994.

    MATH  Google Scholar 

  24. 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.

    Google Scholar 

  25. M. Detyniecki: Mathematical Aggregation Operators and their Applications to Video Querying. Ph.D. Thesis, University Paris V I, 2000.

    Google Scholar 

  26. J. Dombi: Basic concepts for a theory of evaluation: The aggregative operator. Europ. J. Oper. Research 10 (1982) 282–293.

    Article  MathSciNet  MATH  Google Scholar 

  27. D. Dubois and H. Prade: A review of fuzzy set aggregation connectives. Inform. Sci. 36 (1985) 85–121.

    Article  MathSciNet  MATH  Google Scholar 

  28. D. Dubois and H. Prade: Weighted minimum and maximum in fuzzy set theory. Inform. Sci. 39 (1986) 85–121.

    Article  MathSciNet  Google Scholar 

  29. J.J. Dujmovic: Weighted conjunctive and disjunctive means and their application in system evaluation. Univ. Beograd Publ. Elektrotech. Fak., 1974, pp. 147–158.

    Google Scholar 

  30. J.C. Fodor: Contrapositive symmetry of fuzzy implications. Fuzzy Sets and Systems 69 (1995) 141–156.

    Article  MathSciNet  MATH  Google Scholar 

  31. J.C. Fodor: An extension of Fung-Fu’s theorem. Int. J. of Uncertainty, Fuziness and Knowledge-Based Systems 4 (1996) 235–243.

    Article  MathSciNet  MATH  Google Scholar 

  32. 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.

    Article  Google Scholar 

  33. J.C. Fodor and J.-L. Marichal: On nonstrict means. Aequationes Mathematicae 54 (1997) 308–327.

    Article  MathSciNet  MATH  Google Scholar 

  34. J.C. Fodor and M. Roubens: Fuzzy Preference Modelling and Multicriteria Decision Support. Kluwer Academic Publishers, Dordrecht, 1994.

    MATH  Google Scholar 

  35. 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.

    Article  MathSciNet  MATH  Google Scholar 

  36. M.J Frank. (1979) On the simultaneous associativity of F(x, y) and x + y -F(x, y). Aequationes Math. 19 (1979) 194–226.

    Article  MATH  Google Scholar 

  37. 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.

    Article  MathSciNet  MATH  Google Scholar 

  38. 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.

    Google Scholar 

  39. 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.

    Google Scholar 

  40. 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.

    Google Scholar 

  41. L. Godo and V. Torra: Extending Choquet integrals for aggregation of ordinal values. Proceedings IPMU’2000, Madrid, 2000, pp. 410–417.

    Google Scholar 

  42. L. Gonzalez: A note on infinitary action of triangular norms and conorms. Fuzzy Sets and Systems 101 (1999) 177–180.

    Article  MathSciNet  MATH  Google Scholar 

  43. L. Gonzalez: Universal aggregation operators. Proceedings Eusf lat’2001,Leicester, 2001, to appear.

    Google Scholar 

  44. L. Gonzalez: What is arithmetic mean? Proceedings A GGOP’2001,Oviedo, 2001, to appear.

    Google Scholar 

  45. S. Gottwald: A Treatise on Many-Valued Logic. Research Studies Press Ltd., Baldock, Hertforshire, 2001.

    Google Scholar 

  46. M. Grabisch: Fuzzy integral in multicriteria decision making. Fuzzy Sets and Systems 69 (1995) 279–298.

    Article  MathSciNet  Google Scholar 

  47. M. Grabisch: k-order additive fuzzy measures. Proceedings IPMU’96, Granada, 1996, pp. 1345–1350.

    Google Scholar 

  48. M. Grabisch: k-order additive discrete fuzzy measures and their representation. Fuzzy Sets and Systems 92 (1997) 167–189.

    Article  MathSciNet  Google Scholar 

  49. 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.

    Google Scholar 

  50. M. Grabisch: Symmetric and asymmetric integrals: the ordinal case. Proceedings IIZUKA’2000, Iizuka, 2000, CD-rom.

    Google Scholar 

  51. M. Grabisch, J.-L. Marichal and M. Roubens: Equivalent representations of set functions. Math. Operat. Res. 25 (2000) 157–178.

    Article  MathSciNet  MATH  Google Scholar 

  52. M. Grabisch, T. Murofushi, M. Sugeno, eds.: Fuzzy Measures and Integrals. Theory and Applications. Physica-Verlag, Heidelberg, 2000.

    MATH  Google Scholar 

  53. M. Grabisch, H.T. Nguyen and E.A. Walker: Fundamentals of Uncertainty Calculi with Applications to Fuzzy Inference. Kluwer Academic Publishers, Dordercht, 1995.

    Google Scholar 

  54. P. Hajek: Metamathematics of Fuzzy Logic. Kluwer Academic Publishers, Dordrecht, 1998.

    Book  MATH  Google Scholar 

  55. P.R. Halmos: Measure Theory. Van Nostrand, New York, 1950.

    Google Scholar 

  56. H. Imaoka: On a subjective evaluation model by a generalized fuzzy integral. Int. J. of Uncertainty, Fuzziness and Knowledge-Based Systems 5 (1997) 517529.

    Google Scholar 

  57. 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.

    Google Scholar 

  58. E.P. Klement: Construction of fuzzy v-algebras using triangular norms. J. Math. Anal. Appl. 85 (1982) pp. 543–566.

    Article  MathSciNet  MATH  Google Scholar 

  59. 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.

    Article  MathSciNet  MATH  Google Scholar 

  60. 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.

    Article  MathSciNet  MATH  Google Scholar 

  61. E.P. Klement, R. Mesiar and E. Pap: Triangular Norms. Kluwer Academic Publishers, Dordrecht, 2000.

    Google Scholar 

  62. 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.

    MathSciNet  MATH  Google Scholar 

  63. E.P. Klement, R. Mesiar and E. Pap: Geometric approach to aggregation. Proceedings Eusflat’2001 Leicester, 2001, to appear.

    Google Scholar 

  64. G.J. Klir and T.A. Folger: Fuzzy Sets, Uncertainty and Information. Prentice Hall, Englewood Cliffs, 1988.

    Google Scholar 

  65. A. Kolesârovâ: On the comparison of quasi-arithmetic means. Busefal 80 (1999) 30–34.

    Google Scholar 

  66. A. Kolesârovâ: Collapsed input-based aggregation. Int. J. of Uncertainty, Fuzziness and Knowledge- Based Systems 9 (2001).

    Google Scholar 

  67. A. Kolesârovâ: Limit properties of quasi-arithmetic means. Fuzzy Sets and Systems,to appear.

    Google Scholar 

  68. A. Kolesârovâ: Parametric evaluation of aggregation operators. Preprint, submitted.

    Google Scholar 

  69. A. Kolesârovâ and M. Komorníkovâ: Triangular norm-based iterative aggregation and compensatory operators. Fuzzy Sets and Systems 104 (1999) 109–120.

    Article  MathSciNet  MATH  Google Scholar 

  70. A. Kolesârovâ and J. Mordelovâ: 1-Lipschitz and kernel aggregation operators. Proceedings of AGGOP’2001,Oviedo, 2001, to appear.

    Google Scholar 

  71. A.N. Kolmogoroff: Sur la notion de la moyenne. Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. 12 (1930) 388–391.

    MATH  Google Scholar 

  72. M. Komorníkovâ: Generated aggregation operators. Proceedings EUSFLAT’99, Palma de Mallorca, 1999, pp. 355–358.

    Google Scholar 

  73. M. Komorníkovâ: Aggregation operators and additive generators. Int. J. of Uncertainty, Fuzziness and Knowledge- Based Systems 9 (2001).

    Google Scholar 

  74. J. Lâzaro and T. Rückschlossovâ: Shift invariant binary aggregation operators. Proceedings AGGOP’2001,Oviedo, 2001, to appear.

    Google Scholar 

  75. Y.-M. Li and Z.-K. Shi: Weak uninorms aggregation operators. Inform. Sci. 124 (2000) 317–323.

    Article  MathSciNet  MATH  Google Scholar 

  76. C.M. Ling: Representation of associative functions. Publ. Math. Debrecen 12 (1965) 189–212.

    MathSciNet  Google Scholar 

  77. M.K. Luhandjula: Compensatory operators in fuzzy linear programming with multiple objectives. Fuzzy Sets and Systems 8 (1982) 245–252.

    Article  MathSciNet  MATH  Google Scholar 

  78. J.-L. Marichal: Aggregations Operators for Multi-Criteria Decision Aid. Ph.D. Thesis, University of Liége, 1998.

    Google Scholar 

  79. 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.

    Google Scholar 

  80. 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.

    Article  MathSciNet  Google Scholar 

  81. J.-L. Marichal: Aggregation of interacting criteria by means of the discrete Choquet integral. Chapter in this monograph.

    Google Scholar 

  82. J.-L. Marichal. On order invariant synthesizing functions. Preprint, submitted.

    Google Scholar 

  83. J.-L. Marichal: On an axiomatization of the quasi-arithmetic mean values without the symmetry axiom. Aequationes Mathematicae 59 (2000) 74–83.

    Article  MathSciNet  MATH  Google Scholar 

  84. 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.

    Article  MathSciNet  MATH  Google Scholar 

  85. M. Mas, G. Mayor and J. Torrens: t-operators. Int. J. Uncertainty, Fuzziness and Knowledge-Based Systems 7 (1999) 31–50

    Article  MathSciNet  MATH  Google Scholar 

  86. G. Mayor and T. Calvo: On extended aggregation functions. Proceedings IFSA ‘87, Prague, 1997, vol. I, pp. 281–285.

    Google Scholar 

  87. G. Mayor and J. Torrens: On a class of operators for expert systems. Int. J. of Intelligent Systems 8 (1988) 771–778.

    Article  Google Scholar 

  88. K. Menger: Statistical metrics. Procs. Nat. Acad. Sci. U.S.A. 37 (1942) 535537.

    Google Scholar 

  89. R. Mesiar: Compensatory operators based on triangular norms. Proceedings EUFIT’95, Aachen, 1995, pp. 131–135.

    Google Scholar 

  90. R. Mesiar: Choquet-like integrals. J. Math. Anal. Appl. 194 (1995) 477–488.

    Article  MathSciNet  MATH  Google Scholar 

  91. R. Mesiar: Generalizations of k-order additive discrete fuzzy measures. Fuzzy Sets and Systems 102 (1999) 423–428.

    Article  MathSciNet  MATH  Google Scholar 

  92. R. Mesiar: k-order additive measures. Int. J. of Uncertainty, Fuzziness and Knowledge-Based Systems 6 (1999) 561–568.

    Google Scholar 

  93. R. Mesiar and B. De Baets: New construction methods for aggregation operators. Proceedings IPMU’2000, Madrid, 2000, pp. 701–706.

    Google Scholar 

  94. R. Mesiar, T. Calvo and J. Martin: Integral based aggregation of real data. Proceedings IPMU’2000, Madrid, 2000, pp. 58–62

    Google Scholar 

  95. R. Mesiar and B. De Baets: Continuous ordinal sums of aggregation operators. Manuscript in preparation.

    Google Scholar 

  96. 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.

    Google Scholar 

  97. 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.

    Google Scholar 

  98. R. Mesiar and D. Vivona: Two-step integral with respect to fuzzy measure. Tatra Mount. Math. Publ. 16 (1999) 359–368.

    MathSciNet  MATH  Google Scholar 

  99. 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.

    Google Scholar 

  100. 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.

    Article  MathSciNet  Google Scholar 

  101. 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.

    Article  MathSciNet  MATH  Google Scholar 

  102. R. Moynihan. On TT semigroups of probability distribution functions II. Aequationes Math. 17 (1978) 19–40.

    Article  MathSciNet  MATH  Google Scholar 

  103. E. Muel and J. Mordelovâ: Kernel aggregation operators. Proceedings AGGOP’2001,Oviedo, 2001, to appear.

    Google Scholar 

  104. 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.

    Article  MathSciNet  Google Scholar 

  105. 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.

    Google Scholar 

  106. M. Nagumo: Uber eine Klasse der Mittelwerte. Japanese Journal of Mathematics 6 (1930) 71–79.

    Google Scholar 

  107. R.B. Nelsen: An Introduction to Copulas. Lecture Notes in Statistic 139, Springer, 1999.

    Google Scholar 

  108. S. Ovchinnikov and A. Dukhovny: Integral representation of invariant functionals J. Math. Anal. Appl. 244 (2000) 228–232.

    Article  MathSciNet  MATH  Google Scholar 

  109. E. Pap: Null-Additive Set Functions. Kluwer Academic Publishers, Dordrecht, 1995.

    MATH  Google Scholar 

  110. A.L. Ralescu and D.A. Ralescu: Extensions of fuzzy aggregation. Fuzzy Sets and Systems 86 (1997) 321–330.

    Article  MathSciNet  MATH  Google Scholar 

  111. T. Michâlikovâ-Rückschlossovâ: Some constructions of aggregation operators. J. Electrical Engin. 12 (2000) 29–32.

    Google Scholar 

  112. T. Rückschlossovâ: Invariant aggregation operators. Manuscript in preparation.

    Google Scholar 

  113. W. Sander: Associative aggregation operators. Chapter in this monograph.

    Google Scholar 

  114. B. Schweizer and A. Sklar: Probabilistic Metric Spaces. North Holland, New York, 1983.

    MATH  Google Scholar 

  115. C. Shannon and W. Weaver: The Mathematical Theory of Communication. University of Illinois Press, Urbana, 1949.

    MATH  Google Scholar 

  116. N. Shilkret: Maxitive measures and integration. Indag. Math. 33 (1971) 109116.

    Google Scholar 

  117. W. Silvert: Symmetric summation: A class of operations of fuzzy sets. IEEE Trans. Syst., Man Cybern. 9 (1979) 657–659.

    Article  MathSciNet  MATH  Google Scholar 

  118. D. Smutnâ: On a peculiar t-norm. Busefal 75 (1998) 60–67.

    Google Scholar 

  119. M. Sugeno: Theory of Fuzzy Integrals and Applications. Ph.D. Thesis, Tokyo Inst. of Technology, Tokyo, 1974.

    Google Scholar 

  120. M. Sugeno and T. Murofushi: Pseudo-additive measures and integrals, J. Math. Anal. Appl. 122 (1987) 197–222.

    Article  MathSciNet  MATH  Google Scholar 

  121. 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).

    Google Scholar 

  122. J. Sipos: Integral with respect to a pre-measure. Math. Slovaca 29 (1979) 141–145.

    MathSciNet  MATH  Google Scholar 

  123. V. Torra: The weighted OWA operator. Int. J. of Intelligent Systems 12 (1997) 153–166.

    Article  MATH  Google Scholar 

  124. V. Torra and L. Godo: Continuous WOWA operators with application to defuzzification. Chapter in this monograph.

    Google Scholar 

  125. I.B. Türksen: Interval-valued fuzzy sets and “compensatory AND”. Fuzzy Sets and Systems 51 (1992) 295–307.

    Article  MathSciNet  Google Scholar 

  126. P. Vicenik: A note on generators of t-norms. Busefal 75 (1998) 33–38.

    Google Scholar 

  127. P. Vicenik- Additive generators and discontinuity. Busefal 76 (1998) 25–28.

    Google Scholar 

  128. 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.

    Google Scholar 

  129. Z. Wang and G.J. Klir: Fuzzy Measure Theory, Plenum Press, 1992.

    Google Scholar 

  130. S. Weber: 1-decomposable measures and integrals for Archimedean tconorms I. J. Math. Anal. Appl. 101 (1984) 114–138.

    Article  MathSciNet  Google Scholar 

  131. S. Weber: Two integrals and some modified version—critical remarks. Fuzzy Sets and Systems 20 (1986) 97–105.

    Article  MathSciNet  MATH  Google Scholar 

  132. R.R. Yager: On a general class of fuzzy connectives. Fuzzy Sets and Systems 4 (1980) 235–242.

    Article  MathSciNet  Google Scholar 

  133. R.R. Yager: On ordered weighted averaging aggregation operators in multi-criteria decisionmaking. IEEE Trans. Syst., Man Cybern. 18 (1988) 183–190.

    Article  MathSciNet  MATH  Google Scholar 

  134. R.R. Yager: Criteria importances in OWA aggregation: An application of fuzzy modeling. Proceedings IEEE’FUZZ’97, Barcelona, 1997, pp. 1677–1682.

    Google Scholar 

  135. R.R. Yager: Fusion od ordinal information using weighted median aggregation. Int. J. Approx. Reasoning 18 (1998) 35–52.

    Article  Google Scholar 

  136. R.R. Yager: Uninorms in fuzzy modeling. Fuzzy Sets and Systems to appear.

    Google Scholar 

  137. R.R. Yager: Using importances in group preference aggregation to block strategic manipulation. Chapter in this monograph.

    Google Scholar 

  138. 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.

    MathSciNet  MATH  Google Scholar 

  139. R.R. Yager and D.P. Filev: Essentials of Fuzzy Modelling and Control. J. Wiley & Sons, New York, 1994.

    Google Scholar 

  140. R.R. Yager and J. Kacprzyk: The Ordered Weighted Averaging Operators, Theory and applications. Kluwer Academic Publishers, Boston, Dordrecht, London, 1997.

    Google Scholar 

  141. R.R. Yager and A. Rybalov: Uninorm aggregation operators. Fuzzy Sets and Systems 80 (1996) 111–120.

    Article  MathSciNet  MATH  Google Scholar 

  142. R.R.Yager and A. Rybalov: Noncommutative self—identity aggregation. Fuzzy Sets and Systems 85 (1997) 73–82.

    Article  MathSciNet  Google Scholar 

  143. L.A. Zadeh: Fuzzy sets. Inform. Control 8 (1965) 338–353.

    Article  MATH  Google Scholar 

  144. H.J Zimmermann and P. Zysno: Latent connectives in human decision making Fuzzy Sets and Systems 4 (1980) 37–51.

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. University of the Balearic Islands, E-07071, Palma de Mallorca, Spain

    Tomasa Calvo

  2. University of Alcalá, E-28871, Alcalá de Henares, Madrid, Spain

    Tomasa Calvo

  3. ChTF STU, Radlinského 9, 81237, Bratislava, Slovak Republic

    Anna Kolesárová

  4. SvF STU, Radlinského 11, 81368, Bratislava, Slovak Republic

    Magda Komorníková & Radko Mesiar

  5. ÚTIA AVČR, 18208, Prague 8, Czech Republic

    Magda Komorníková & Radko Mesiar

Authors
  1. Tomasa Calvo
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  2. Anna Kolesárová
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  3. Magda Komorníková
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  4. Radko Mesiar
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Editor information

Editors and Affiliations

  1. Department of Mathematics and Computer Science, University of the Balearic Islands, 07071, Palma de Mallorca, Spain

    Tomasa Calvo  & Gaspar Mayor  & 

  2. Department of Computer Science, University of Alcalá, 28871, Alcalá de Henares, Spain

    Tomasa Calvo

  3. Faculty of Civil Engineering, Slovak Technical University, Radlinského 11, 81368, Bratislava, Slovakia

    Radko Mesiar

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© 2002 Physica-Verlag Heidelberg

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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

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  • 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

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