Skip to main content
SpringerLink
Log in

Synthesis of Judgements

  • Chapter
Modeling Decisions

Part of the book series: Cognitive Technologies ((COGTECH))

  • 756 Accesses

Abstract

In this chapter we study some aggregation operators for numerical information. The description is focused on results based on functional equations. Therefore, not only are the operators given, but also, at least for some of them, their characterization. We refer to these results as syntheses of judgements. Although the term could be used for any aggregation operator, we restrict its use to the case of characterizations using functional equations.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

4.5 Bibliographical Notes

  1. Aczél, J. (1948) Sur les opérations définies pour nombres réels, Bulletin de la Société Mathématique de France 76 59–64.

    MATH  Google Scholar 

  2. Aczél, J. (1966) Lectures on Functional Equations and their Applications, Academic Press.

    Google Scholar 

  3. Aczél, J. (1984) On weighted synthesis of judgements, Aequationes Mathematicae 27 288–307.

    Article  MATH  MathSciNet  Google Scholar 

  4. Aczél, J., Alsina, C. (1984) Classification of some classes of quasilinear functions with applications to triangular norms and to synthesizing judgements, Meth. Opl. Res. 48 3–22.

    MATH  Google Scholar 

  5. Aczél, J., Alsina, C. (1986) On synthesis of judgements, Socio-Econ. Plann. Sci 20:6 333–339.

    Article  Google Scholar 

  6. Aczél, J., Alsina, C. (1987) Synthesizing judgements: A functional equations approach, Mathematical Modelling 9:3/5 311–320.

    Article  MATH  MathSciNet  Google Scholar 

  7. Aczél, J., Dhombres, J. (1989) Functional Equations in Several Variables: with Applications to Mathematics, Information Theory and to the Natural and Social Sciences, Cambridge University Press.

    Google Scholar 

  8. Aczél, J., Roberts, F. S. (1989) On the possible merging functions, Math. Social Sci 17 205–243.

    Article  MATH  MathSciNet  Google Scholar 

  9. Aczél, J., Saaty, T. L. (1983) Procedures for synthesizing ratio judgements, Journal of Mathematical Psychology 27 93–102.

    Article  MATH  MathSciNet  Google Scholar 

  10. Alsina, C., Frank, M. J., Schweizer, B. (2006) Associative Functions: Triangular Norms and Copulas, World Scientific.

    Google Scholar 

  11. Anonymous (1821) Dissertation sur la recherche du milieu le plus probable, entre les résultats de plusieurs observations ou expériences, Annales de Mathématiques Pures et Appliquées, 12 181–204. According to Harter [123] (see p. 157), the author of this paper is Svanberg.

    Google Scholar 

  12. Bajraktarević, M. (1958) Sur une équation fonctionnelle aux valeurs moyennes, Glasnik. Mat. Fiz. I Astr 13:4 243–248.

    Google Scholar 

  13. Bajraktarević, M. (1963) Sur une généralisation des moyennes quasilineaire, Publ. Inst. Math. Beograd. 3:17 69–76.

    MathSciNet  Google Scholar 

  14. Bajraktarević, M. (1969) Über die Vergleichbarkeit der mit Gewichtsfunktionen gebildeten Mittelwerte, Stud. Sci. Math. Hungar. 4 3–8.

    MATH  Google Scholar 

  15. Barral Souto, J. (1938) El modo y otras medias, casos particulares de una misma expression matemática, Boletín Matemático 11:3 29–41.

    MATH  Google Scholar 

  16. Bemporad, G. (1926) Sul principio della media aritmetica, Atti Accad. Naz. Lincei 6:3 87–91.

    Google Scholar 

  17. Brecht, T., Arjomandi, E., Li, C., Pham, H. (2001) Controlling garbage collection and heap growth to reduce the execution time of Java applications, Proc. of the ACM Conference on Object-Oriented Programming, Systems, Languages, and Applications (OOPSLA), 353–366.

    Google Scholar 

  18. Buchanan, B., Shortliffe, E. (1984) Rule-based expert systems, The MYCIN experiments of the Stanford heuristic programming project, Addison-Wesley, Reading, MA.

    Google Scholar 

  19. Bull, J. M., Smith, L. A., Ball, C., Pottage, L., Freeman, R. (2003) Benchmarking Java against C and Fortran for scientific applications, Concurrency and Computation: Practice and Experience 15 417–430.

    Article  MATH  Google Scholar 

  20. Bullen, P. S., Mitrinović, D. S., Vasić, P. M. (1988) Means and their Inequalities, D. Reidel Publishing Company.

    Google Scholar 

  21. Bullen, P. S. (2003) Handbook of Means and Their Inequalities, Kluwer Academic Publishers.

    Google Scholar 

  22. Calvo, T., Kolesárová, A., Komorníková, M., Mesiar, R. (2002) Aggregation operators: properties, classes and construction methods, in T. Calvo, G. Mayor, R. Mesiar (eds.) Aggregation Operators, Physica-Verlag, 3–104.

    Google Scholar 

  23. Cauchy, A. L. (1821) Cours d’Analyse de l’Ecole Royale Polytechnique, Ire Partie, Analyse Algébraique, Paris.

    Google Scholar 

  24. Cheng, Y., Kashyap, R. L. (1992) Recursive fusion operators: Desirable properties and illustrations, in M. Al Abidi, R. C. Gonzalez (eds.) Data Fusion in Robotics and Machine Intelligence, Academic Press, 245–265.

    Google Scholar 

  25. Chisini, O. (1929) Sul concetto di media, Periodico di Matematiche, Serie 4 Vol. 9:2 106–116.

    Google Scholar 

  26. De Baets, B., Fodor, J. (1999) Residual operators of uninorms, Soft Computing 3 89–100.

    Google Scholar 

  27. De Baets, B., Fodor, J. (1999) Van Melle’s combining function in MYCIN is a representable uninorm: An alternative proof, Fuzzy Sets and Systems 104 133–136.

    Article  MATH  MathSciNet  Google Scholar 

  28. De Finetti, B. (1931) Sul concetto di media, Giorn. Ist. Ital. Attuari 3:2 369–396.

    Google Scholar 

  29. Dombi, J. (1982) Basic concepts for the theory of evaluation: The aggregative operator, European Journal of Operations Research 10 282–293.

    Article  MATH  MathSciNet  Google Scholar 

  30. Duda, R., Hart, P., Nilsson, N. (1976) Subjective bayesian methods for rule-based inference systems, AFIPS Conference Proceedings of the 1976 Nat. Computer Conference, 45 1075–1082.

    Google Scholar 

  31. Einstein, A. (1922) The Meaning of Relativity, Princeton University Press (Spanish translation, El Significado de la Relatividad, Espasa-Calpe, 1984).

    MATH  Google Scholar 

  32. Hardy, G. H., Littlewood, J. E., Pólya, G. (1934) Inequalities, Cambridge University Press, 2nd Edition, 1952.

    Google Scholar 

  33. Harter, H. L. (1974) The method of least squares and some alternatives-Part I, Int. Stat. Rev. 42:2 147–174.

    Article  MathSciNet  Google Scholar 

  34. Fechner, G. Th. (1878) Ueber den Ausgangswerth der kleinsten Abweichungssumme, dessen Bestimmung, Verwendung und Verallgemeinerung, Abhandlungen der Königliche Sächsische Gesellschaft der Wissenshaft zu Leipzig, XVIII 1–76.

    Google Scholar 

  35. Fodor, J. C., Marichal, J.-L. (1997) On nonstrict means, Aequationes Mathematicae 54 308–327.

    MATH  MathSciNet  Google Scholar 

  36. Fodor, J., Roubens, M. (1994) Fuzzy Preference Modelling and Multicriteria Decision Support, Kluwer Academic Publishers.

    Google Scholar 

  37. Fodor, J. C., Yager, R. R., Rybalov, A. (1997) Structure of uninorms, Int. J. of Unc., Fuzz. and Knowledge Based Systems 5 411–427.

    Article  MathSciNet  Google Scholar 

  38. Gini, C. (1958) Le Medie, Unione Tipografico-Editrice Torinese.

    Google Scholar 

  39. Hájek, P. (1985) Combining function for certainty degrees in consulting systems, Int. J. of Man-Machine Studies 22 59–67.

    Article  MATH  Google Scholar 

  40. Hájek, P., Valdes, J. (1990) Algebraic foundations of uncertainty processing in rule-based expert systems (group theoretic approach), Computers and Artificial Intelligence 9:4 325–344.

    MATH  MathSciNet  Google Scholar 

  41. Hamacher, H. (1975) Über logische Verknüpfungen unscharfer Aussagen und deren zugehörige Bewertungsfunktionen, Working Paper 75/14, RWTH, Aachen.

    Google Scholar 

  42. Heckerman, D. (1986) Probabilistic interpretations for MYCIN’s certainty factors, in L. N. Kanal, J. F. Lemmer (eds.) Uncertainty in Artificial Intelligence, Elsevier Science Publishers, 167–196.

    Google Scholar 

  43. Jessen, B. (1930) Über die Verallgeneinerung des arithmetischen Mittels, Acta Scientiarum Mathematicarum (Szeged), 5 108–116.

    Google Scholar 

  44. Klement, E.-P., Mesiar, R., Pap, E. (1996) On the relationship of associative compensatory operators to triangular norms and conorms, Int. J. of Unc., Fuzz. and Knowledge Based Systems 4 129–144.

    Article  MathSciNet  Google Scholar 

  45. Klement, E. P., Mesiar, R., Pap, E. (2000) Triangular Norms, Kluwer Academic Publisher.

    Google Scholar 

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

    Google Scholar 

  47. López de Mántaras, R. (1990) Approximate Reasoning Models, Ellis Horwood Limited, England.

    Google Scholar 

  48. Losonczi, L. (1971) Über eine neue Klasse von Mittelwerte, Acta Sci. Math (Acta Univ. Szeged) 32 71–78.

    MATH  MathSciNet  Google Scholar 

  49. Marichal, J.-L. (1998) Aggregation Operators for Multicriteria Decision Aid, Ph. D. Dissertation, Institute of Mathematics, University of Liège, Liége, Belgium.

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  51. Marichal, J.-L. (2000) On comparison meaningfulness of aggregation functions, Journal of Mathematical Psychology 45 213–223.

    Article  MathSciNet  Google Scholar 

  52. Marichal, J.-L. (2002) On order invariant synthesizing functions, Journal of Mathematical Psychology 46 661–676.

    Article  MATH  MathSciNet  Google Scholar 

  53. Marichal, J.-L., Mathonet, P., Tousset, E. (1999) Characterization of some aggregation functions stable for positive linear transformations, Fuzzy Sets and Systems 102 293–314.

    Article  MATH  MathSciNet  Google Scholar 

  54. Marichal, J.-L., Roubens, M. (1993) Characterization of some stable aggregation functions, Proc. Int. Conf. on Industrial Engineering and Production Management, Mons, Belgium, 187–196.

    Google Scholar 

  55. Nagumo, M. (1930) Über eine Klasse der Mittelwerte, Japan. J. of Math 6 71–79.

    Google Scholar 

  56. Nelsen, R. B. (1993) Proofs Without Words: Exercises in Visual Thinking, Mathematical Association of America.

    Google Scholar 

  57. Orlov, A. (1981) The connection between mean quantities and admissible transformations, Mathematical Notes 30 774–778.

    MathSciNet  Google Scholar 

  58. Ovchinnikov, S. (1996) Means on ordered sets, Mathematical Social Sciences 32 39–56.

    Article  MATH  MathSciNet  Google Scholar 

  59. Ovchinnikov, S. (1998) Invariant functions on simple orders, Order 14 365–371.

    MATH  MathSciNet  Google Scholar 

  60. Pappus (1982) La Collection Mathématique, Librairie Scientifique et Technique Albert Blanchard. Translation and Notes by Paul Ver Eecke.

    Google Scholar 

  61. Pedersen, O. (1974) A Survey of the Almagest, Odense University Press.

    Google Scholar 

  62. Plackett, R. L. (1958) Studies in the history of probability and statistics: VII: The principle of the arithmetic mean, Biometrika 45:1/2 130–135.

    Article  MATH  Google Scholar 

  63. Sander, W. (2002) Associative aggregation operators, in T. Calvo, G. Mayor, R. Mesiar (eds.) Aggregation Operators, Physica-Verlag, 124–158.

    Google Scholar 

  64. Shortliffe, E. H., Buchanan, B. G. (1975) A model of inexact reasoning in medicine, Mathematical Biosciences 23 351–379.

    Article  MathSciNet  Google Scholar 

  65. Shudo, K. (2004) Performance comparison of Java/.NET Runtimes, http://www.shudo.net/jit/perf/

    Google Scholar 

  66. Simpson, T. (1756) A letter to the Right Honourable George Earl of Macclesfield, President of the Royal Society, on the advantage of taking the mean of a number of observations, in practical astronomy, Philosophical Transactions of the Royal Society of London for 1755, 49:1 82–93.

    Google Scholar 

  67. Taguchi, T. (1974) On Fechner’s thesis and statistics with norm p, Annals of the Institute of Statistical Mathematics 26 175–193.

    MATH  MathSciNet  Google Scholar 

  68. Tsadiras, A. K., Margaritis, K. G. (1998) The MYCIN certainty factor handling function as uninorm operator and its use as a threshold function in artificial neurons, Fuzzy Sets and Systems 93 263–274.

    Article  MathSciNet  Google Scholar 

  69. Weisstein, E. W., et al. (2006) Pythagorean means, from MathWorld-A Wolfram Web Resource. http://mathworld.wolfram.com/PythagoreanMeans.html

    Google Scholar 

  70. Yager, R. R. (1994) Misrepresentations and challenges: A response to Elkan, IEEE Expert August 41–42.

    Google Scholar 

  71. Yager, R. R. (2002) Using importances in group preference aggregation to block strategic manipulation, in T. Calvo, G. Mayor, R. Mesiar (eds.) Aggregation Operators, Physica-Verlag, 177–191.

    Google Scholar 

  72. Yager, R. R., Rybalov, A. (1996) Uninorm aggregation operators, Fuzzy Sets and Systems 80 111–120. Also as Tech. Report #MII-1407, Machine Intelligence Institute, Iona College, New Rochelle, N. Y., 1994.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Rights and permissions

Reprints and permissions

Copyright information

© 2007 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

(2007). Synthesis of Judgements. In: Modeling Decisions. Cognitive Technologies. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-68791-7_4

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-68791-7_4

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-68789-4

  • Online ISBN: 978-3-540-68791-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation