Examples of the use of triangular norms

  • Jaime Gil-Aluja
Part of the Applied Optimization book series (APOP, volume 11)


In previous sections, the interdependence among several qualities or abilities using operators ∧ (and) as well as ∨ (and/or) has been studied. Now, we will incorporate a weakened «and» an also a strengthened «and/or», two dual triangular norms. For greater generalisation, we will make recourse to expertons.


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  1. (1).
    For a mathematician this interval would take on the name of a segment Google Scholar
  2. (2).
    Even in this case, in which b 2 = a 1, as we are dealing with a limit situation, when the candidate reaches a1, by means of the upper limit, recourse must be made to (2.22)Google Scholar
  3. (4).
    A whole range of distances can be defined, on the condition that the axiomatic of the notion of distance is respectedGoogle Scholar
  4. (5).
    We are not going to linger over this question. In our times, who is not aware of the theory of fuzzy sub-sets?Google Scholar
  5. (6).
    Kaufmann, A. and Gil Aluja, J.: Introducción de la teoría de los subconjuntos borrosos a la gestión de las empresas. Ed. Milladoiro, Santiago de Compostela, 3rd ed. 1993, pages 143–146Google Scholar
  6. (7).
    For greater ease the square brackets for the intervals in Φ1-fuzzy subset and the Φ-fuzzy sub-set will not be usedGoogle Scholar
  7. (8).
    The sense of the word dual is more general than that used in mathematics (for example in linear programming).Google Scholar
  8. (9).
    The upper index (i) indicates job E i Google Scholar
  9. (10).
    In this case the fuzzy relation has been considered without its being divided by 5, which is the number of qualitites or abilities. We have acted in a different way from (3.26) to show that both cases are correctGoogle Scholar
  10. (11).
    The notion of distance should not be confused with the metric notion. In a metric notion (X=Y)⇔(d(X, Y) = 0)Google Scholar
  11. (12).
    μ(x) is called a membership function or function of belonging of x: μ(x) {0, 1}Google Scholar
  12. (13).
    Kaufmann, A and Gil Aluja, J.: Introducción de la teoría de los conjuntos borrosos a la gestión de la empresa. Ed. Milladoiro, Santiago de Compostela, 3rd. ed. 1993, pages 148–150Google Scholar
  13. (14).
    The expected value of x is expressed by means of ε(x)Google Scholar
  14. (15).
    Recourse is made to the same experts as for the other candidatesGoogle Scholar
  15. (16).
    We calculate first\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\cdot}$}}{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{C}} ({P_1},{C_5} \bot {C_5})\) and then\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\cdot}$}}{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{C}} ({P_1},{C_4} \bot {C_5}_ \bot {C_6})\)since the T-norms and the T-co- norms are associative.Google Scholar

Copyright information

© Kluwer Academic Publishers 1998

Authors and Affiliations

  • Jaime Gil-Aluja
    • 1
  1. 1.Departament d’Economia i Organització d’EmpresesFacultat de Ciències Economiques i Empresarials de la Universität de BarcelonaBarcelonaSpain

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