Abstract
In many practical situations, we need to optimize under fuzzy constraints. There is a known Bellman-Zadeh approach for solving such problems, but the resulting solution, in general, depends on the choice of a not well-defined constant M. We show that this dependence disappears if we use an algebraic t-norm (and-operation) \( f_ \& (a,b)=a\cdot b\), and we also prove that the algebraic product is the only t-norm for which the corresponding solution is independent on M.
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References
Bellman, R.E., Zadeh, L.A.: Decision making in a fuzzy environment. Manag. Sci. 47(4), B141–B145 (1970)
Bouchon-Meunier, B., Kreinovich, V., Nguyen, H.T.: “Non-Associative Operations”. In: Proceedings of the Second International Conference on Intelligent Technologies InTech’2001, Bangkok, Thailand, November 27–29, pp. 39–46 (2001)
Goodman, I.R., Trejo, R.A., Kreinovich, V., Martinez, J., Gonzalez, R.: “An even more realistic (non-associative) interval logic and its relation to psychology of human reasoning”, In: Proceedings of the Joint 9th World Congress of the International Fuzzy Systems Association and 20th International Conference of the North American Fuzzy Information Processing Society IFSA/NAFIPS’2001, Vancouver, Canada, July 25–28, pp. 1586–1591 (2001)
Klir, G.J., Yuan, B.: Fuzzy Sets and Fuzzy Logic. Prentice Hall, Upper Saddle River, New Jersey (1995)
Goodman, I.R., Trejo, R.A., Kreinovich, V., Martinez, J., Gonzalez, R.: “An even more realistic (non-associative) interval logic and its relation to psychology of human reasoning”, In: Proceedings of the Joint 9th World Congress of the International Fuzzy Systems Association and 20th International Conference of the North American Fuzzy Information Processing Society IFSA/NAFIPS’2001, Vancouver, Canada, July 25–28, pp. 1586–1591 (2001)
Nguyen, H.T., Walker, E.A.: First Course In Fuzzy Logic. CRC Press, Boca Raton, Florida (2006)
Trejo, R., Kreinovich, V., Goodman, I.R., Martinez, J., Gonzalez, R.: “A realistic (Non-associative) logic and a possible explanations of 7\(\pm \)2 Law”. Int. J. Approx. Reason. 29, 235–266 (2002)
Zadeh, L.A.: Fuzzy sets. Inf. Control. 8, 338–353 (1965)
Zimmerman, H.H., Zysno, P.: Latent connectives in human decision making. Fuzzy Sets Syst. 4, 37–51 (1980)
Acknowledgements
This work was supported in part by the National Science Foundation grants 0953339, HRD-0734825 and HRD-1242122 (Cyber-ShARE Center of Excellence) and DUE-0926721, by Grants 1 T36 GM078000-01 and 1R43TR000173-01 from the National Institutes of Health, and by a grant N62909-12-1-7039 from the Office of Naval Research.
This work was performed when Juan Carlos Figueroa-García was a visiting researcher at the University of Texas at El Paso.
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Figueroa-García, J.C., Ceberio, M., Kreinovich, V. (2018). Algebraic Product is the only T-norm for Which Optimization Under Fuzzy Constraints is Scale-Invariant. In: Ceberio, M., Kreinovich, V. (eds) Constraint Programming and Decision Making: Theory and Applications. Studies in Systems, Decision and Control, vol 100. Springer, Cham. https://doi.org/10.1007/978-3-319-61753-4_8
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DOI: https://doi.org/10.1007/978-3-319-61753-4_8
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