The Calculus of Fuzzy Quantities

  • Didier Dubois
  • Henri Prade


This chapter gives methods of calculation for expressions containing imprecise quantities, represented by possibility distributions on the real numbers. These methods are in complete agreement with what is commonly called interval analysis, of which they constitute an extension to the case of weighted intervals. Their usefulness is illustrated by some examples at the end of the chapter. Moreover, fuzzy quantities will enter extensively in Chapters 3, 5, and 6. In essence, the calculus of fuzzy quantities constitutes a refinement of sensitivity analysis, which thereby acquires nuance, and this without great increase in the amount of calculation required. The calculus of fuzzy quantities can replace the calculus of random functions (cf. Papoulis [21]) when this proves too intractable, though of course with more or less loss of information according to the type of problem. A more detailed account of the theoretical part of this chapter may be found in Ref. 27. An introductory text is Ref. 28.


Membership Function Fuzzy Number Interval Analysis Fuzzy Relation Possibility Distribution 
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  1. 1.
    DEMPSTER, A. P.(1967). Upper and lower probabilities induced by a multivalued mapping. Ann. Math. Stat., 38, 325–339.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    DUBOIS, D. (1981). A fuzzy set-based method for the optimization of machining operations. Proc. Int. Conf. Cybernetics and Society, Atlanta, Georgia, pp. 331–334.Google Scholar
  3. 3.
    DUBOIS, D. (1987). An application of fuzzy arithmetic to the optimization of industrial machining processes. Math. Modelling 9, 461–475.MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    DUBOIS, D. (1983). Modèles Mathématiques de l’Imprécis et de l’Incertain en Vue d’Applications aux Techniques d’Aide à la Décision, Thesis, University of Grenoble.Google Scholar
  5. 5.
    DUBOIS, D., and PRADE, H. (1978). Algorithmes de plus court chemin pour traiter des données floues. RAIRO, Rech. Opérât., 12(2), 213–227.zbMATHGoogle Scholar
  6. 6.
    DUBOIS, D., and PRADE, H. (1978). Operations on fuzzy numbers. Int. J. Syst. Sci., 9, 613–626.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    DUBOIS, D., and PRADE, H. (1979). Fuzzy real algebra: Some results. Fuzzy Sets Syst., 2, 327–348.MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    DUBOIS, D., and PRADE, H. (1980). Fuzzy Sets and Systems: Theory and Applications. Academic, New York.zbMATHGoogle Scholar
  9. 9.
    DUBOIS, D., and PRADE, H. (1981). Addition of interactive fuzzy numbers. IEEE Trans. Automatic Control, 26, 926–936.MathSciNetCrossRefGoogle Scholar
  10. 10.
    DUBOIS, D., and PRADE, H. (1982). The use of fuzzy numbers in decision analysis. In Fuzzy Information and Decision Processes (M. M. Gupta and E. Sanchez, eds), North-Holland, Amsterdam, pp. 309–321.Google Scholar
  11. 11.
    DUBOIS, D., and PRADE, H. (1982). Upper and lower possibilistic expectations and applications. 4th Int. Seminar on Fuzzy Set Theory, Linz, Austria. Technical Report no. 174, LSI, Université P. Sabatier, Toulouse, France.Google Scholar
  12. 12.
    DUBOIS, D., and PRADE, H. (1983). Inverse operations for fuzzy numbers. Proc. IFAC Symposium on Fuzzy Information, Knowledge Representation and Decision Analysis (E. Sanchez, ed.), Marseille, Pergamon, Oxford, pp. 399–404.Google Scholar
  13. 13.
    DUBOIS, D., and PRADE, H. (1984). Fuzzy set-theoretic differences and inclusions and their use in the analysis of fuzzy equations. Control Cybern. (Warsaw), 13, 129–146.MathSciNetzbMATHGoogle Scholar
  14. 14.
    GONDRAN, M., and MINOUX, M. (1979). Graphes et Algorithmes. Eyrolles, Paris.zbMATHGoogle Scholar
  15. 15.
    MIZUMOTO, M., and TANAKA, K. (1979). Some properties of fuzzy numbers. In Advances in Fuzzy Set Theory and Applications (M. M. Gupta, R. K. Ragade, and R. R. Yager, eds.). North-Holland, Amsterdam, pp. 153–164.Google Scholar
  16. 16.
    MOORE, R. (1966). Interval Analysis. Prentice-Hall, Englewood Cliffs, New Jersey.zbMATHGoogle Scholar
  17. 17.
    MOORE, R. (1979). Methods and Applications of Interval Analysis. SIAM Studies on Applied Mathematics, Vol. 2, Philadelphia.Google Scholar
  18. 18.
    NAHMIAS, S. (1978). Fuzzy variables. Fuzzy Sets Syst., 1, 97–111.MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    NEGOITA, C. V. (1978). Management Applications of Systems Theory. Birkhaüser, Basel.Google Scholar
  20. 20.
    NGUYEN, H. T. (1978). A note on the extension principle for fuzzy sets. J. Math. Anal. Appl., 64(2), 369–380.MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    PAPOULIS, A. (1965). Probability, Random Variables and Stochastic Processes. McGraw-Hill, New York, Chaps. 5 and 6.Google Scholar
  22. 22.
    SANCHEZ, E. (1984). Solution of fuzzy equations with extended operations. Fuzzy Sets Syst., 12, 237–248.zbMATHCrossRefGoogle Scholar
  23. 23.
    ZADEH, L. A. (1965). Fuzzy sets. Inf. Control, 8, 338–353.MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    ZADEH, L. A. (1975). The concept of a linguistic variable and its application to approximate reasoning. Inf. Sci., Part 1: 8, 199–249; Part 2: 8, 301–357; Part 3: 9, 43–80.Google Scholar
  25. 25.
    ZADEH, L. A. (1977). Theory of fuzzy sets. In Encyclopedia of Computer Science and Technology (J. Belzer, A. Holzman, and A. Kent, eds.), Marcel Dekker, New York.Google Scholar
  26. 26.
    DUBOIS, D., and PRADE, H. (1987). The mean value of a fuzzy number. Fuzzy Sets and Syst. 24(3), 279–300.MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    DUBOIS, D., and PRADE, H. (1987). Fuzzy numbers: An overview. In The Analysis of Fuzzy Information, Vol. 1: Mathematics and Logic (J. C. Bezdek, ed.), CRC Press, Boca Raton, Florida, pp. 3–40.Google Scholar
  28. 28.
    KAUFMANN A., and GUPTA M. M. (1986).An Introduction to Fuzzy Arithmetic. Van Nostrand Rheinhold, New York.Google Scholar
  29. BANDLER W., and KOHOUT L. (1980). Semantics of implication operators and fuzzy relational products. Int. J. Man-Machine Studies, 12, 89–116.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Plenum Press, New York 1988

Authors and Affiliations

  • Didier Dubois
    • 1
  • Henri Prade
    • 1
  1. 1.CNRS, Languages and Computer Systems (LSI)University of Toulouse IIIToulouseFrance

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