×

zbMATH — the first resource for mathematics

The mean value of a fuzzy number. (English) Zbl 0634.94026
The concept of a mean value (expectation) of a fuzzy number is introduced being consistent with its counterpart in probability theory. Fuzzy numbers (intervals) are viewed as consonant random sets, and the fact that a possibility measure is a particular case of Dempster’s upper probability is employed. Upper and lower expectations of a fuzzy number are proposed. Finally, issues related to employing measure-theoretic concepts in possibility theory are mentioned.
Reviewer: J.Kacprzyk

MSC:
94D05 Fuzzy sets and logic (in connection with information, communication, or circuits theory)
60D05 Geometric probability and stochastic geometry
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Dempster, A.P., Upper and lower probabilities induced by a multivalued mapping, Ann. math. statist., 38, 325-339, (1967) · Zbl 0168.17501
[2] Dubois, D., Modèles mathématiques de l’imprécis et de l’incertain, en vue d’applications aux techniques d’aide à la Décision, () · Zbl 0546.94036
[3] Dubois, D.; Prade, H., Fuzzy sets and systems: theory and applications, (1980), Academic Press New York · Zbl 0444.94049
[4] Dubois, D.; Prade, H., Upper and lower possibilitistic expectations and some applications, (), Unpublished paper · Zbl 0562.94023
[5] Dubois, D.; Prade, H., On several representations of an uncertain body of evidence, (), 167-181
[6] Dubois, D.; Prade, H., Fuzzy numbers: an overview, (), (Also in: Tech. Rep. No. 219, L.S.I., Univ. P. Sabatier, Toulouse) · Zbl 0571.04001
[7] Dubois, D.; Prade, H., Fuzzy cardinality and the modelling of imprecise quantification, Fuzzy sets and systems, 16, 199-230, (1985) · Zbl 0601.03006
[8] Dubois, D.; Prade, H., Evidence measures based on fuzzy information, Automatica, 21, 547-562, (1985) · Zbl 0596.62007
[9] Goodman, I.R., Fuzzy sets as equivalence classes of random sets, (), 327-343
[10] Höhle, U., Representation theorems for L-fuzzy quantities, Fuzzy sets and systems, 5, 83-108, (1981) · Zbl 0448.03041
[11] Klement, E.P., Operations on fuzzy sets and fuzzy numbers related to triangular norms, (), 218-225 · Zbl 0547.04003
[12] Loève, M., Probability theory, (1962), Van Nostrand Princeton, NJ · Zbl 0108.14202
[13] Matheron, G., Random sets and integral geometry, (1975), Wiley New York · Zbl 0321.60009
[14] Moore, R., Interval analysis, (1966), Prentice Hall Englewood Cliffs, NJ · Zbl 0176.13301
[15] Nguyen, H., On random sets and belief functions, J. math. anal. appl., 64, 531-542, (1978) · Zbl 0409.60016
[16] Nguyen, H., A note on the extension principle for fuzzy sets, J. math. anal. appl., 64, 369-380, (1978) · Zbl 0377.04004
[17] Nguyen, H.T., On modeling of linguistic information using random sets, Inform. sci., 34, 265-274, (1984) · Zbl 0557.68066
[18] Ralescu, D., A survey of the representation of fuzzy concepts and its applications, (), 77-91
[19] Rodabaugh, S., Fuzzy addition in the L-fuzzy real line, Fuzzy sets and systems, 8, 39-52, (1982) · Zbl 0508.54002
[20] Shafer, G., Allocation of probability, (), Available from
[21] Shafer, G., A mathematical theory of evidence, (1976), Princeton Univ. Press Princeton, NJ · Zbl 0359.62002
[22] Shafer, G., Belief functions and possibility measures, () · Zbl 0655.94025
[23] Strat, T.M., Continuous belief functions for evidential reasoning, (), 308-313
[24] Wang, P.Z.; Sanchez, E., Treating a fuzzy subset as a projectable random set, (), 213-219
[25] Zadeh, L.A., Similarity relations and fuzzy orderings, Inform. sci., 3, 177-200, (1971) · Zbl 0218.02058
[26] Zadeh, L.A., Fuzzy sets as a basis for a theory of possibility, Fuzzy sets and systems, 1, 3-28, (1978) · Zbl 0377.04002
[27] Artstein, Z.; Vitale, R.A., A strong law of large numbers for random compact sets, Ann. probab., 3, 879-882, (1975) · Zbl 0313.60012
[28] Kwakernaak, H.; Kwakernaak, H., Part II: algorithms and examples for the discrete case, Inform. sci., Inform. sci., 17, 253-278, (1979) · Zbl 0438.60005
[29] Puri, M.L.; Ralescu, D.A., The concept of normality for fuzzy random variables, Ann. probability, 13, 1373-1379, (1985) · Zbl 0583.60011
[30] Kruse, R., The strong law of large numbers for fuzzy random variables, Inform. sci., 28, 233-241, (1982) · Zbl 0571.60039
[31] Miyakoshi, M.; Shimbo, M., A strong law of large numbers for fuzzy random variables, Fuzzy sets and systems, 12, 133-142, (1984) · Zbl 0551.60035
[32] Féron, R., Ensembles aléatoires flous, C.R. acad. sci. Paris ser. A, 282, 903-906, (1976) · Zbl 0327.60004
[33] Dempster, A.P., Upper and lower probabilities generated by a random closed interval, Amer. math. statist., 39, 957-966, (1967) · Zbl 0251.62010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.