A discussion on aggregation operators

Gómez, Daniel   ; Montero, Javier

Kybernetika, Tome 40 (2004) no. 1, pp. 107-120

Abstract (VO)

It has been lately made very clear that aggregation processes can not be based upon a unique binary operator. Global aggregation operators have been therefore introduced as families of aggregation operators $\lbrace T_n\rbrace _n$, being each one of these $T_n$ the $n$-ary operator actually amalgamating information whenever the number of items to be aggregated is $n$. Of course, some mathematical restrictions can be introduced, in order to assure an appropriate meaning, consistency and key mathematical capabilities. In this paper we shall discuss these standard conditions, pointing out their respective relevance.

Détail

Classification : 03E72, 68T30, 68T37
Keywords: aggregation rules; logical connectives; fuzzy sets

Bibliographie

[1] Aczél J.: Lectures on Functional Equations and Their Applications. Academic Press, New York 1966 | MR

[2] Amo A., Montero J., Biging, G., Cutello V.: Fuzzy classification systems. European J. Oper. Res. (to appear) | MR | Zbl

[3] Amo A. Del, Montero, J., Molina E.: Additive recursive rules. In: Preferences and Decisions Under Incomplete Knowledge (J. Fodor et al, eds.), Physica–Verlag, Heidelberg 2000 | MR

[4] Amo A. Del, Montero, J., Molina E.: Representation of consistent recursive rules. European J. Oper. Res. 30 (2001), 29–53 | DOI | MR

[5] Amo A. del, Montero, J., Biging G.: Classifying pixels by means of fuzzy relations. Internat. J. Gen. Systems 29 (2000), 605–621 | DOI

[6] Amo A., Montero J., Fernández A., López M., Tordesillas, J., Biging G.: Spectral fuzzy classification: an application. IEEE Trans. Systems, Man and Cybernetics 32 (2002), 42–48 | DOI

[7] Barlow R. E., Proschan F.: Statistical Theory of Reliability and Life Testing. To Begin With, Silver Spring 1981 | Zbl

[8] Cai K. Y.: Introduction to Fuzzy Reliability. Kluwer, Boston 1996 | Zbl

[9] Calvo T., Mayor, G., (eds.) R. Mesiar: Aggregation Operators. Physica–Verlag, Heidelberg 2002 | MR | Zbl

[10] Calvo T., Kolesárová A., Komorníková, M., Mesiar R.: Aggregation operators: properties, classes and construction methods. In: Aggregation Operators (T. Calvo et al, eds.), Physica–Verlag, Heidelberg 2002, pp. 3–104 | Zbl

[11] Cutello V., Montero J.: Recursive families of OWA operators. In: Proc. FUZZ-IEEE Conference (P. P. Bonisone, ed.), IEEE Press, Piscataway 1994, pp. 1137–1141

[12] Cutello V., Montero J.: Hierarchical aggregation of OWA operators: basic measures and related computational problems. Internat. J. Uncertainty, Fuzziness and Knowledge-based Systems 3 (1995), 17–26 | DOI | MR | Zbl

[13] Cutello V., Montero J.: Recursive connective rules. Intelligent Systems 14 (1999), 3–20 | Zbl

[14] Baets B. De: Idempotent uninorms. European J. Oper. Res. 118 (1999), 631–642 | DOI | Zbl

[15] Dombi J.: Basic concepts for a theory of evaluation: the aggregative operator. European J. Oper. Res. 10 (1982), 282–293 | DOI | MR | Zbl

[16] Fodor J. C., Roubens M.: Fuzzy Preference Modelling and Multicriteria Decision Support. Kluwer, Dordrecht 1994 | Zbl

[17] Fung L. W., Fu K. S.: An axiomatic approach to rational decision making in a fuzzy environment. In: Fuzzy Sets and Their Applications to Cognitive and Decision Processes (L. Zadeh et al, eds.), Academic Press, New York 1975, pp. 227–256 | MR | Zbl

[18] Golumbic M. C.: Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York 1980 | MR | Zbl

[19] Gómez D., Montero J., Yáñez J., González-Pachón, J., Cutello V.: Crisp dimension theory and valued preference relations. Internat. J. Gen. Systems (to appear) | Zbl

[20] Klement E. P., Mesiar, R., Pap E.: Triangular Norms. Kluwer, Dordrecht 2000 | MR | Zbl

[21] Klement E. P., Mesiar, R., Pap E.: On the relationship of associative compensatory operators to triangular norms and conorms. Internat. J. Uncertainty, Fuzziness and Knowledge-based Systems 4 (1996), 129–144 | DOI | MR | Zbl

[22] Klir G. J., Folger T. A.: Fuzzy Sets, Uncertainty and Information. Prentice Hall, London 1988 | MR | Zbl

[23] Kolesárová A., Komorníková M.: Triangular norm-based iterative compensatory operators. Fuzzy Sets and Systems 104 (1999), 109–120 | DOI | Zbl

[24] Mak K. T.: Coherent continuous systems and the generalized functional equation of associativity. Mathem. Oper. Res. 12 (1987), 597–625 | DOI | MR | Zbl

[25] Marichal J. L., Mathonet, P., Tousset E.: Characterization of some aggregation functions stable for positive linear transformations. Fuzzy Sets and Systems 102 (1999), 293–314 | MR | Zbl

[26] Mas M., Mayor G., Suñer, J., Torrens J.: Generation of multi-dimensional aggregation functions. Mathware and Soft Computing 5 (1998), 233–242 | MR | Zbl

[27] Mas M., Mayor, G., Torrens J.: T-operators. Internat. J. Uncertainty, Fuzziness and Knowledge-based Systems 7 (1999), 31–50 | DOI | MR | Zbl

[28] Mayor G., Calvo T.: On extended aggregation functions. In: Proc. IFSA Conference, Volume 1, Academia, Prague 1997, pp. 281–285

[29] Menger K.: Statistical methods. Proc. Nat. Acad. Sci. U.S.A. 8 (1942), 535–537 | DOI | MR

[30] Mesiar R.: Compensatory operators based on triangular norms and conorms. In: Proc. EUFIT Conference (H.-J. Zimmermann, ed.), Elite Foundation, Aachen 1995, pp. 131–135

[31] Mesiar R., Komorníková M.: Aggregation operators. In: Proc. PRIM Conference, Institute of Mathematics, Novi Sad 1997, pp. 193–211 | MR | Zbl

[32] Mesiar R., Komorníková M.: Triangular norm-based aggregation of evidence under fuzziness. In: Aggregation and Fussion of Imperfect Information (B. Bouchon–Meunier, ed.), Physica–Verlag, Heidelberg 1998, pp. 11–35 | MR

[33] Montero J.: A note on Fung–Fu’s theorem. Fuzzy Sets and Systems 17 (1985), 259–269 | MR | Zbl

[34] Montero J.: Aggregation of fuzzy opinions in a fuzzy group. Fuzzy Sets and Systems 25 (1988), 15–20 | DOI | MR

[35] Montero J., Amo A., Molina, E., Cutello V.: On the relevance of OWA rules. In: Proc. IPMU Conference, Volume 2, Universidad Politécnica de Madrid, Madrid 2000, pp. 992–996

[36] Montero J., Tejada, J., Yáñez J.: Structural properties of continuous systems. European J. Oper. Res. 45 (1990), 231–240 | DOI | MR

[37] Pattanaik P. : Voting and Collective Choice. Cambridge Univ. Press, Cambridge 1970

[38] Schweizer B., Sklar A.: Probabilistic Metric Spaces. North Holland, Amsterdam 1983 | MR | Zbl

[39] Shirland L. E., Jesse R. R., Thompson R. L., Iacovou C. L.: Determining attribute weights using mathematical programming. Omega 31 (2003), 423–437 | DOI

[40] Trillas E.: Sobre funciones de negación en la teoría de los subconjuntos difusos. Stochastica III-1:47–59, 1983; English version in : Advances of Fuzzy Logic Universidad de Santiago de Compostela (S. Barro et al, eds.), Santiago de Compostela 1998, pp. 31–43

[41] Walther G.: Granulometric smoothing. Ann. Statist. 25 (1999), 2273–2299 | MR

[42] Yager R. R.: On ordered averaging aggregation operators in multicriteria decision making. IEEE Trans. Systems, Man and Cybernetics 18 (1988), 183–190 | DOI | MR

[43] Yager R. R., Rybalov A.: Uninorm aggregation operators. Fuzzy Sets and Systems 80 (1996), 111–120 | DOI | MR | Zbl

[44] Zadeh L. A.: Fuzzy Sets. Inform. and Control 8 (1965), 338–353 | DOI | MR | Zbl

[45] Zimmermann H. J., Zysno P.: Latent connectives in human decision making. Fuzzy Sets and Systems 4 (1980), 37–51 | Zbl