Abstract
Possibility measures are not closed under convex combinations, since the nestedness property of focal sets is not preserved under this aggregation mode. This paper intends to investigate the eventwise aggregations of possibility measures that preserve the consonance property of these set-functions. It is proved that these operations are defined as the union (via the maximum) of some monotonic pointwise transformations of the fuzzy sets that characterize the possibility measures ; especially the previously introduced weighted maximum operations are among the natural solutions. This work also suggests to view a fuzzy set as a disjunctive aggregation of possibly dissonant imprecise, individual opinions, the weights reflecting the degrees of membership of individuals in the group. The extension of these results to decomposable set-functions is considered.
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References
Aczel J. (1966) Lectures on Functional Equations and Applications. Academic Press, New York.
Arrazola I., Plainfossé A., Prade H., and Testemale C. (1989) Extrapolation of fuzzy values from incomplete data bases. Information Systems 14(6), to appear.
Choleva W. (1985) Aggregation of fuzzy opinions: an axiomatic approach. Fuzzy Sets and Systems 17, 249–258.
Dubois D. (1986) Belief structures, possibility theory and decomposable confidence measures on finite sets. Computers and Artificial Intelligence (Bratislava) 5, 403–416.
Dubois D., and Koning J.L. (1989) Social choice axioms for fuzzy set aggregation. Fuzzy Sets and Systems, Special Issue on ‘Aggregation and Best Choice with Imprecise Opinions’, to appear.
Dubois D., and Prade H. (1982) A class of fuzzy measures based on triangular norms — A general framework for the combination of uncertain information. Int. J. of General Systems 8 (1), 43–61.
Dubois D., and Prade H. (1986) Weighted minimum and maximum in fuzzy set theory. Information Sciences 39, 205–210.
Dubois D., and Prade H. (1988a) Possibility Theory: an Approach to Computerized Processing of Uncertainty. Plenum Press, New York.
Dubois D., and Prade H. (1988b) Modeling uncertainty and inductive inference. Acta Psychologica 68, 53–78.
French S. (1985) Group consensus probability distributions: a critical survey, in J.M. Bernardo, M.H. DeGroot, D.V. Lindley and A.F.M. Smith (Eds.): Bayesian Statistics 2, North- Holland, Amsterdam, pp. 183–202.
Fung L.W., and Fu K.S. (1975) An axiomatic approach to rational decision making in a fuzzy environment, in L.A. Zadeh, K.S. Fu, K. Tanaka and M. Shimura (Eds.): Fuzzy Sets and their Applications to Cognitive and Decision Processes, Academic Press, New York, pp. 227–256.
Herstein I.N., and Milnor J.W. (1953) An axiomatic approach to measurable utility. Econometrica 21, 291–297.
Lehrer K., and Wagner C.G. (1981) Rational Consensus in Science and Society. D. Reidel Publ. Co., Boston.
McConway K. (1981) Marginalization and linear opinion pools. J. Amer. Statistical Assoc. 76, 410–414.
Montero de Juan F.J. (1987) Aggregation of fuzzy opinions in a non-homogeneous group. Fuzzy Sets and Systems 25, 15–20.
Schweizer B., and Sklar A. (1983) Probabilistic Metric Spaces. North-Holland, Amsterdam.
Shackle G.L.S. (1961) Decision, Order and Time in Human Affairs. Cambridge University Press, Cambridge, U.K..
Shafer G. (1976) A Mathematical Theory of Evidence. Princeton University Press, Princeton.
Wagner C.G. (1988) Consensus for belief functions and related uncertainty measures. Report ORNL/TM-10748, Oak Ridge National Laboratory, Oak Ridge, Tenn..
Weber S. (1984) ⊥ — decomposable measures and integrals for archimedean t-conorms ⊥. J. Math. Anal. Appl. 101, 114–138.
Zadeh L.A. (1971) Similarity relations and fuzzy orderings. Information Sciences 3, 177–200.
Zadeh L.A. (1978) Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems 1, 3–28.
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© 1990 Kluwer Academic Publishers
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Dubois, D., Prade, H. (1990). Aggregation of Possibility Measures. In: Kacprzyk, J., Fedrizzi, M. (eds) Multiperson Decision Making Models Using Fuzzy Sets and Possibility Theory. Theory and Decision Library, vol 18. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-2109-2_4
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DOI: https://doi.org/10.1007/978-94-009-2109-2_4
Publisher Name: Springer, Dordrecht
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