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Measurement of Membership Functions: Theoretical and Empirical Work

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Fundamentals of Fuzzy Sets

Part of the book series: The Handbooks of Fuzzy Sets Series ((FSHS,volume 7))

Abstract

This chapter presents a review of various interpretations of the fuzzy membership function together with ways of obtaining a membership function. We emphasize that different interpretations of the membership function call for different elicitation methods. We try to make this distinction clear using techniques from measurement theory.

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References

  • Aczél, J. (1966). Lectures on Functional Equations and Applications, Academic Press, New York.

    MATH  Google Scholar 

  • Alsina, C. (1985). On a family of connectives for fuzzy sets, Fuzzy Sets and Systems 16:231–235.

    MathSciNet  Google Scholar 

  • Handler, W. & Kohout, L. J. (1986). The use of checklist paradigm in inference systems, in C. V. Negoita & H. Prade (eds), Fuzzy Logic in Knowledge Engineering, Verlag TÜV, Rheinland, Köln, chapter 7, pp. 95–111.

    Google Scholar 

  • Bellman, R. & Giertz, M. (1973). On analytic formalism of the theory of fuzzy sets, Information Sciences 5: 149–156.

    MathSciNet  MATH  Google Scholar 

  • Bezdek, J. C. & Harris, J. D. (1979). Convex decompostions of fuzzy partitions, Journal of Mathematical Analysis and Applications 6: 490–512.

    MathSciNet  Google Scholar 

  • Bilgiç, T. (1995). Measurement-Theoretic Frameworks for Fuzzy Set Theory with Applications to Preference Modelling, PhD thesis, University of Toronto, Dept. of Industrial Engineering Toronto Ontario M5S 1 A4 Canada.

    Google Scholar 

  • Bilgiç, T. (1996). The archimedean assumption in fuzzy set theory, in ML H. Smith, M. A. Lee, J. Keller & J. Yen (eds), Proceedings of NAFIPS’ 96, IEEE Press, pp. 299–303.

    Google Scholar 

  • Bilgiç, T. & Turkmen, L. (1995a). Measurement-theoretic justification of fuzzy set connectives, Fuzzy Sets and Systems 76(3): 289–308.

    MathSciNet  MATH  Google Scholar 

  • Bilgiç, T. & Turkmen, I. B. (1995b). Measurement-theoretic frameworks for fuzzy set theory, Working notes of the IJCAI-95 workshop on “Fuzzy Logic in Artificial Intelligence”, pp. 55–65. 14th International Joint Conference on Artificial Intelligence Montréal, Canada.

    Google Scholar 

  • Bilgiç, T. & Türk§en, I. B. (1997). Elicitation of membership functions: How far can theory take us?, Proceedings of FUZZIEEE’97, IEEE Press, Barcelona.

    Google Scholar 

  • Bolinger, D. L. M. (1972). Degree words, Vol. 53 of Janua linguarum. Series maior, The Hague Mouton.

    Google Scholar 

  • Bollmann-Sdorra, P., Wong, S. K. M. & Yao, Y. Y. (1993). A measurement-theoretic axiomatization of fuzzy sets, Fuzzy Sets and Systems 60(3): 295–307.

    MathSciNet  MATH  Google Scholar 

  • Chameau, J. L. & Santamarina, J. C. (1987a). Membership functions part I: Comparing method of measurement, International Journal of Approximate Reasoning 1:287–301.

    Google Scholar 

  • Chameau, J. L. & Santamarina, J. C. (1987b). Membership functions part II: Trends in fuzziness and implications, International Journal of Approximate Reasoning 1:303–317.

    Google Scholar 

  • Chan, C. C. (1959). A new proof of the completeness of the lukesiewicz axioms, Transactions of Maerican Mathematical Society 93: 74–80.

    Google Scholar 

  • Chang, C. C. (1958). Algebraic analysis of many-valued logics, Transactions of American Mathematical Society 88: 467–490.

    MATH  Google Scholar 

  • Chellas, B. F. (1980). Modal logic: an introduction, Cambridge University Press, Cambridge, UK.

    MATH  Google Scholar 

  • Chen, J. E. & Otto, K. N. (1995). Constructing membership functions using interpolation and measurement theory, Fuzzy Sets and Systems 73: 313–327.

    MathSciNet  MATH  Google Scholar 

  • Civanîar, M. R. & Trussel, H. J. (1986). Constructing membership functions using statistical data, Fuzzy Sets and Systems 18: 1–14.

    MathSciNet  Google Scholar 

  • de Cooman, G. ( 1991 ). Evaluation sets and mappings-the order-theoretic aspect of the meaning of properties, in E. E. Kerre (ed.), Introduction to the Basic Principles of Fuzzy Set Theory and Some of its Applications, Universteit Gent, Communication and Cognition, Gent, Belgium, pp. 159–213.

    Google Scholar 

  • de Cooman, G. & Kerre, E. E. (1994). Order norms on bounded partially ordered sets, The Journal of Fuzzy Mathematics 2: 281–310.

    MATH  Google Scholar 

  • Dombi, J. (1990). Membership function as an evaluation, Fuzzy Sets and Systems 35: 1–22.

    MathSciNet  MATH  Google Scholar 

  • Dubois, D. (1986). Belief structures, possibility theory and decomposable confidence measures on finite sets, Computers and Artificial Intelligence 5(5): 403–416.

    MATH  Google Scholar 

  • Dubois, D. (1988). Possibility theory: Searching for normative foundations, in B. R. Munier (ed.), Risk, Decision and Rationality, D. Reidel Publishing Company, pp. 601–614.

    Google Scholar 

  • Dubois, D. & Prade, H. (1980). Fuzzy sets and systems: theory and applications, Academic Press, New York.

    MATH  Google Scholar 

  • Dubois, D. & Prade, H. (1982). A class of fuzzy measures based on triangular norms: a general framework for the combination of information, International Journal of General Systems 8: 43–61.

    MathSciNet  MATH  Google Scholar 

  • Dubois, D. & Prade, H. (1986). Fuzzy sets and statistical data, European Journal of Operational Research 25: 345–356.

    MathSciNet  MATH  Google Scholar 

  • Dubois, D. & Prade, H. (1989). Fuzzy sets, probability and measurement, European Journal of Operational Research 40: 135–154.

    MathSciNet  MATH  Google Scholar 

  • Dubois, D. & Prade, H. (1991). Random sets and fuzzy interval analysis, Fuzzy Sets and Systems 42: 87–101.

    MathSciNet  MATH  Google Scholar 

  • Dubois, D. & Prade, H. (1993). Fuzzy sets and probability: Misunderstandings, bridges and gaps, Second IEEE International Conference on Fuzzy Systems, IEEE, pp. 1059–1068. San Fransisco, California March 28-April 1, 1993.

    Google Scholar 

  • Erickson, R. P., Lorenzo, P. M. D. & Woodbury, M. A. (1994). Classification of taste responses in brain stem: Membership in fuzzy sets, Journal of Neurophysiology 71(6): 2139–2150.

    Google Scholar 

  • Fishburn, P. C. (1970). Utility theory for decision making, Operations Research Society of America. Publications in operations research, no. 18, Wiley, New York.

    MATH  Google Scholar 

  • Fodor, J. C. (1993). A new look at fuzzy connectives, Fuzzy Sets and Systems 57: 141–148.

    MathSciNet  MATH  Google Scholar 

  • Fodor, J. C. & Keresztfalvi, T. (1995). Nonstandard conjunctions and implications in fuzzy logic, International Journal of Approximate Reasoning 12: 69–84.

    MathSciNet  MATH  Google Scholar 

  • French, S. (1984). Fuzzy decision analysis: Some criticisms, in H. Zimmermann, L. Zadeh & B. Gaines (eds), Fuzzy Sets and Decision Analysis, North Holland, pp. 29–44.

    Google Scholar 

  • Fuchs, L. (1963). Partially Ordered Algebraic Systems, Pergamon Press, London.

    MATH  Google Scholar 

  • Fung, L. W. & Fu, S. K. (1975). An axiomatic approach to rational decision making in a fuzzy environment, in H. J. Zimmermann, L. A. Zadeh & B. R. Gaines (eds), Fuzzy Sets and Their Application to Cognitive and Decision Processes, Academic Press, New York, pp. 227–256.

    Google Scholar 

  • Furukawa, M. & Yamakawa, T. (1995). The design algorithms of membership functions for a fuzzy neuron, Fuzzy Sets and Systems 71(3): 329–343.

    Google Scholar 

  • Gaines, B. R. (1978). Fuzzy and probabilistic uncertainty logics, Information and Control 38(2): 154–169.

    MathSciNet  MATH  Google Scholar 

  • Giles, R. (1976), Lukasiewicz logic and fuzzy set theory, International Journal of Man-Machine Studies 8(3): 312–327.

    MathSciNet  Google Scholar 

  • Giles, R. (1988).The concept of grade of membership, Fuzzy Sets and Systems 25:297–323.

    MathSciNet  MATH  Google Scholar 

  • Goguen, J. (1969). The logic of inexact concepts, Synthese 19: 325–373.

    MATH  Google Scholar 

  • Henry E. Kyburg, J. (1992). Measuring errors of measurement, in C. W. Savage & P. Ehrlich (eds), Philosophical and Foundational Issues in Measurement Theory, Lawrence Erlbaum Associates Inc. Publishers, New Jersey, USA, chapter 5, pp. 75–92.

    Google Scholar 

  • Hersh, H. & Carmazza, A. (1976). A fuzzy set approach to modifiers and vagueness in natural language, Journal of Experimental Psychology: General 105(3): 254–276.

    Google Scholar 

  • Hersh, H., Carmazza, A. & Brownell, H. H. (1979). Effects of context on fuzzy membership functions, in M. M. Gupta, R. M. Ragade & R. R. Yager (eds), Advances in Fuzzy Set Theory, North-Holland, Amsterdam, pp, 389–408.

    Google Scholar 

  • Hisdal, E. (1985). Reconciliation of the Yes-No versus grade of membership dualism in human thinking, in M. M. Gupta, A. Kandei, W. Bandîer & J. B. Kiszka (eds), Approximate Reasoning in Expert Systems, North-Holland, Amsterdam, pp. 33–46. Also in Dubois, D,, H. Prade, and R. Yager eds., Readings in Fuzzy Sets for Intelligent Systems, Morgan Kaufmann, pp. 854–860, 1993.

    Google Scholar 

  • Hisdal, E. (1988). Are grades of membership probabilities?, Fuzzy Sets and Systems 25: 325–348.

    MathSciNet  MATH  Google Scholar 

  • Kamp, J. A. W. (1975). Two theories about adjectives, in E. L. Keenan (ed.), Formal Semantics of Natural Language, Cambridge University Press, London, pp. 123–155.

    Google Scholar 

  • Kempton, W. (1981). The folk classification of ceramics: A study of cognitive prototypes, Academic Press, New York.

    Google Scholar 

  • Klir, G. J. (1994). Multivalued logics versus modal logics: alternative frameworks for uncertainty modeling, in P. P. Wang (ed.), Advances in Fuzzy Theory and Technology, Duke University, Durham, North Carolina U.S.A., pp. 3–48. Vol. 2.

    Google Scholar 

  • Kneale, W. C. (1962). The development of logic, Oxford, Clarendon Press, England.

    MATH  Google Scholar 

  • Kochen, M. (1975). Applications of fuzzy sets in psychology, in L. A. Zadeh, K. S. Fu, K. Tanaka & M. Shimura (eds), Fuzzy Sets and thei applications to cognitive and decision processes, Academic Press, pp. 395–408.

    Google Scholar 

  • Kochen, M. & Badre, A. N. (1974). On the precision of adjectives which denote fuzzy sets, Journal of Cybernetics 4: 49–59.

    Google Scholar 

  • Kohout, L. J. & Bandler, W. (1993). Interval-valued systems for approximate reasoning based on checklist paradigm semantics, in P. P. Wang (ed.), Advances of Fuzzy Set Theory, Bookwrigths Press, Durham, North Carolina USA, pp. 167–194. Volume 1.

    Google Scholar 

  • Kosko, B. (1991). Neural Networks and Fuzzy Systems, Prentice-Hall, Engiewood Cliffs, NJ.

    Google Scholar 

  • Krantz, D. H., Luce, R. D., Suppes, P. & Tversky, A. (1971). Foundations of Measurement, Vol. 1, Academic Press, San Diego.

    MATH  Google Scholar 

  • Kruse, R., Gebhardt, J. & Klawonn, F. (1994), Foundations of Fuzzy Systems, John Wiley and Sons, New York.

    Google Scholar 

  • Kruse, R. & Meyer, K. D. (1987). Statistics with Vague data, Theory and Decision Library: Series B Mathematical and Statistical Methods, D. Reidell publishing company, Dordrech Holland.

    Google Scholar 

  • Kulka, J. & Novak, V. (1984). Have fuzzy operators a psychologocal correspondence?, Studia Psychological 26: 131–140.

    Google Scholar 

  • Labov, W. (1973). The boundaries of words and their meanings, in C. J. Bailey & R. W. Shuy (eds), New Ways of Analyzing Variation in English, Georgetown University Press, Washington.

    Google Scholar 

  • Lakoff, G. (1973). Hedges: A study in meaning criteria and the logic of fuzzy concepts, Journal of Philosophical Logic 2: 458–508.

    MathSciNet  MATH  Google Scholar 

  • Lakoff, G. (1987). Women, Fire, and Dangerous Things: What categories reveal about the mind, The University of Chicago Press, Chicago.

    Google Scholar 

  • Lambert, J. (1992). The fuzzy set membership problem using the hierarchy decision method, Fuzzy Sets and Systems 48(3): 323–330.

    MathSciNet  Google Scholar 

  • Laviolette, M. & J. W. Seaman, J. (1994). The efficacy of fuzzy representations of uncertainty, IEEE Transactions on Fuzzy Systems 2(1): 4–15. With comments from D. Dubois and H. Prade, E. Hisdal, G.J. Klir, B. Kosko, N. Wilson, D. V. Lindley.

    Google Scholar 

  • Li, H. X. & Yen, V. C. (1995). Fuzzy Sets and Fuzzy Decision-Making, CRC Press, Boca Raton.

    MATH  Google Scholar 

  • Ling, C. H. (1965). Representation of associative functions, Publicationes Mathemat-icae Debrecen 12: 189–212.

    Google Scholar 

  • Luce, R. D. (1996). The ongoing dialog between empirical science and measurement theory, Journal of Mathematical Psychology 40: 78–98

    MathSciNet  MATH  Google Scholar 

  • Luce, R. D. & Raiffa, H. (1957). Games and Decisions, Wiley, New York.

    MATH  Google Scholar 

  • Luce, R., Krantz, D., Suppes, P. & Tversky, A. (1990). Foundations of Measurement, Vol. 3, Academic Press, San Diego, USA.

    MATH  Google Scholar 

  • Mabuchi, S. (1992). An interpretation of membership functions and the properties of general probabilistic operators as fuzzy set operators. 1. case of type-1 fuzzy-sets, Fuzzy Sets and Systems 49(3): 271–283.

    MathSciNet  MATH  Google Scholar 

  • McCall, S. & Ajdukiewicz, K. (eds) (1967). Polish logic, 1920-1939, Oxford, Clarendon P. papers by Ajdukiewicz [and others]; with, an introduction by Tadeusz Kotarbinski, edited by Storrs McCall, translated by B. Gruchman [and others].

    MATH  Google Scholar 

  • McCloskey, M. E. & Glucksberg, S. (1978). Natural categories: Well defined or fuzzy sets?, Memory and cognition 6: 462–472.

    Google Scholar 

  • Michell, J. (1990). An introduction to the logic of psychological measurement, L. Erlbaum Associates, Hillsdale, N.J.

    Google Scholar 

  • Murphy, C. K. (1993). Limits on the analytic hierarchy process from its consistency index, European Journal of Operational Research 65: 138–139.

    MATH  Google Scholar 

  • Nakanishi, H., Turksen, I. B. & Sugeno, M. (1993). A review and comparison of six reasoning methods, Fuzzy Sets and Systems 57: 257–294.

    MathSciNet  MATH  Google Scholar 

  • Marens, L., (1985). Abstract Measurement Theory, MIT Press, Cambridge, Mass.

    Google Scholar 

  • Nola, A. D. (1993). MV algebras in the treatment of uncertainty, in R. Lowen & M. Roubens (eds), Fuzzy logic: state of the art, Kluwer Academic Publishers, Dordrecht; Boston, pp. 123–131.

    Google Scholar 

  • Nola, A. D., & Gerla, G. (1986). Nonstandard fuzzy sets, Fuzzy Sets and Systems 18: 173–181.

    MathSciNet  MATH  Google Scholar 

  • Norwich, A. M. & Türkmen, I. B. (1982a). The construction of membership functions, in R. R. Yager (ed.), Fuzzy Sets and Possibility Theory: Recent Developments, Pergamon Press, New York, pp. 61–67.

    Google Scholar 

  • Norwich, A. M. & Türkmen, I. B. (1982b). The fundamental measurement of fuzziness, in R. R. Yager (ed.), Fuzzy Sets and Possibility Theory: Recent Developments, Pergamon Press, New York, pp. 49–60.

    Google Scholar 

  • Norwich, A. M. & Türk§en, L B. (1982c). Stochastic fuzziness, in M. M. Gupta & E. Sanchez (eds), Approximate Reasoning in Decision Analysis, North-Holland, pp. 13–22.

    Google Scholar 

  • Norwich, A. M. & Türkmen, I. B. (1984). A model for the measurement of membership and the consequences of its empirical implementation, Fuzzy Sets and Systems 12: 1–25.

    MathSciNet  MATH  Google Scholar 

  • Nowakowska, M. ( 1977). Methodological problems of measurement of fuzzy concepts in the social sciences, Behavioral Science 22: 107–115.

    Google Scholar 

  • Oden, G. C. (1979). Fuzzy propositional approach to psyholinguistic problems: an application of fuzzy set theory in cognitive science, in M. M. Gupta, R. K. Ragade & R. R. Yager (eds), Advances in Fuzzy Set Theory and Applications, North-Holland, Amsterdam, pp. 409–420.

    Google Scholar 

  • Osherson, D. N. & Smith, E. E. (1981). On the adequacy of prototype theory as a theory of concepts, Cognition 9: 35–58.

    Google Scholar 

  • Osherson, D. N. & Smith, E. E. (1982). Gradedness and conceptual combination, Cognition 12: 299–318.

    Google Scholar 

  • Palmer, F. (1981). Semantics, 2 edn, Cambridge University Press, New York.

    Google Scholar 

  • Pedrycz, W. (1994). Why triangular membership functions?, Fuzzy Sets and Systems 64(1): 21–30.

    MathSciNet  Google Scholar 

  • Pipino, L. L., van Gich, J. P. & Tom, G. (1981). Experiments in the representation and manipulation of labels of fuzzy sets, Behavioral Science 26: 216–228.

    Google Scholar 

  • Roberts, F. (1979). Measurement Theory, Addison Wesley Pub. Co.

    Google Scholar 

  • Robinson, A., (1966). Non-standard Analysis, North-Holland Pub. Co., Amsterdam.

    MATH  Google Scholar 

  • Rocha, A. F. (1982). Basic properties of neural circuits, Fuzzy Sets and Systems 7:109–121.

    MathSciNet  MATH  Google Scholar 

  • Rosch, E. & Mervis, C. B. (1975). Family resemblances: Studies in the internal structure of categories, Cognitive Pschology 7: 573–605.

    Google Scholar 

  • Rosser, J. B. & Turquette, A. R. (1977). Many-Valued Logics, Greenwood Press, Westport Connecticut.

    Google Scholar 

  • Ruspini, E. H. (1969). A new approach to clustering, Information Control 15(1): 22–32.

    MATH  Google Scholar 

  • Ruspini, E. H. (1991), On the semantics of fuzzy logic, International Journal of Approximate Reasoning 5(1): 45–88.

    MathSciNet  MATH  Google Scholar 

  • Saaty, T. L. (1974). Measuring the fuzziness of sets, Journal of Cybernetics 4: 53–61.

    Google Scholar 

  • Saaty, T. L. (1977). A scaling method for priorities in hierarchical structures, Journal of Mathematical Psychology 15: 234–281.

    MathSciNet  MATH  Google Scholar 

  • Saaty, T. L. (1980). The Analytic Hierarcy Process, McGraw-Hill, New York.

    Google Scholar 

  • Saaty, T. L. (1986). Scaling the membership function, European Journal of Operational Research 25: 320–329.

    MathSciNet  MATH  Google Scholar 

  • Saaty, T. L. (1994). Homogeneity and clustering in AHP ensures the validity of the scale, European Journal of Operational Research 72: 598–601.

    MATH  Google Scholar 

  • Sapir, E. (1944). Grading: a study in semantics, Philosophy of Science 11: 93–116.

    Google Scholar 

  • Schweizer, B. & Sklar, A. (1983). Probabilistic Metric Spaces, North-Holland, Amsterdam.

    MATH  Google Scholar 

  • Scott, D. (1976). Does many-valued logic have any use?, in S. Körner (ed.), Philosophy of logic. Camelot Press, Southampton, Great Britain, chapter 2, pp. 64–95. with comments by T. J. Smiley, J.P. Cleave and R. Giles. Bristol Conference on Critical Philosophy, 3d, 1974.

    Google Scholar 

  • Smithson, M. (1987). Fuzzy Set Analysis for Behavioral and Social Sciences, Recent Research in Psychology, Springer-Verlag, Berlin, Germany.

    Google Scholar 

  • Sugeno, M. & Yasukawa, T. (1993). A fuzzy logic based approach to qualitative modelling, IEEE Transactions on Fuzzy Systems 1(1): 1–24.

    Google Scholar 

  • Suppes, P. (1974). The measurement of belief, Journal of Royal Statistical Society Series B 36(2): 160–175.

    MathSciNet  MATH  Google Scholar 

  • Suppes, R., Krantz, D., Luce, R. & Tversky, A. (1989). Foundations of Measurement, Vol. 2, Academic Press, San Diego.

    MATH  Google Scholar 

  • Takagi, H. (1990). Fusion technology of fuzzy theory and neural network: Survey and future directions, Proc. of International Conference on Fuzzy Logic and Neural Networks, lizuka, Japan, pp. 13—26.

    Google Scholar 

  • Takagi, H. & Hayashi, I. (1991). Nn-driven fuzzy reasoning, International Journal of Approximate Reasoning 5(3): 191–213. (Special Issue of IIZUKA’88).

    MATH  Google Scholar 

  • Thole, U., Zimmermann, H. J. & Zysno, P. (1979). On the suitability of minimum and product operators for the interpretation of fuzzy sets, Fuzzy Sets and Systems 2: 167–180.

    MATH  Google Scholar 

  • Thomas, S. F. (1979). A theory of semantics and possible inference, with application to decision analysis, PhD thesis, University of Toronto.

    Google Scholar 

  • Thomas, S. R (1995). Fuzziness and Probability, ACG Press, Wichita Kansas USA.

    Google Scholar 

  • Triantaphyllou, E. & Mann, S. H. (1990). An evaluation of the eigenvalue approach for determining the membership values in fuzzy sets, Fuzzy Sets and Systems 35:295–301.

    MathSciNet  Google Scholar 

  • Trillas, E. (1979). Sobre funciones de negaciön en la teoria de conjuntos difusos, Stochastica 3: 47–59.

    MathSciNet  MATH  Google Scholar 

  • Türk§en, I. B. (1986). Interval valued fuzzy sets based on normal forms, Fuzzy Sets and Systems 20: 191–210.

    MathSciNet  Google Scholar 

  • Türk§en, I. B. (1988), Stochasic fuzzy sets: a survey, Combining Fuzzy Imprecision with Probabilistic Uncertainty in Decision Making, Springer-Verlag, New York, pp. 168–183.

    Google Scholar 

  • Türk§en, I. B. (1991). Measurement of membership functions and their assessment, Fuzzy Sets and Systems 40: 5–38.

    MathSciNet  Google Scholar 

  • Türk§en, I. B. (1992). Interval valued fuzzy sets and * Compensatory AND*, Fuzzy Sets and Systems 51: 295–307.

    MathSciNet  Google Scholar 

  • Tversky, A. (1977). Features of similarity, Psychological Review 84(4): 327–352.

    Google Scholar 

  • Weber, S. (1983). A general concept of fuzzy connectives, negations and implications based on t-norms and t-conorms, Fuzzy Sets and Systems 11: 115–134.

    MathSciNet  MATH  Google Scholar 

  • Yager, R. R. (1979). A measurement-informational discussion of fuzzy union and intersection, Internationaljournal of Man-Machine Studies 11: 189–200.

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  • Yamakawa, T. & Furukawa, M. (1992). A design algorithm of membership functions for a fuzzy neuron using example-based learning, Proceedings of the First IEEE Conference on Fuzzy Systems, San Diego, pp. 75–82.

    Google Scholar 

  • Zadeh, L. A. (1965). Fuzzy sets, Information Control 8: 338–353.

    MathSciNet  MATH  Google Scholar 

  • Zadeh, L. A. (1971). Similarity relations and fuzzy orderings, Information Sciences 3: 177–200.

    MathSciNet  MATH  Google Scholar 

  • Zadeh, L. A. (1975). Fuzzy logic and approximate reasoning, Synthese 30: 407–428.

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  • Zimmermann, H., L & Zysno, P. (1985). Quantifying vagueness in decision models, European Journal of Operational Research 22: 148–154.

    MathSciNet  MATH  Google Scholar 

  • Zwick, R. (1987). A note on random sets and the Thurstonian scaling methods, Fuzzy Sets and Systems 21: 351–356.

    MathSciNet  MATH  Google Scholar 

  • Zwick, R., Carlstein, E. & Budescu, D. V. (1987). Measures of similarity among fuzzy concepts: A comparative analysis, International Journal of Approximate Reasoning 1(2): 221–242.

    MathSciNet  Google Scholar 

  • Zysno, P. (1981). Modeling membership functions, in B.B. Rieger (ed.), Empirical Semantics I, Vol. 1 of Quantitative Semantics, Vol. 12, Studienverlag Brockmeyer, Bochum, pp. 350–375.

    Google Scholar 

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  1. Taner Bilgiç
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  2. I. Burhan Türkşen
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  1. Institut de Recherche en Informatique de Toulouse, Université Paul Sabatier, 118 route de Narbonne, 31062, Toulouse Cedex 4, France

    Didier Dubois  & Henri Prade  & 

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Bilgiç, T., Türkşen, I.B. (2000). Measurement of Membership Functions: Theoretical and Empirical Work. In: Dubois, D., Prade, H. (eds) Fundamentals of Fuzzy Sets. The Handbooks of Fuzzy Sets Series, vol 7. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-4429-6_4

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