Abstract
This paper presents a generalization of the extension principle for fuzzy numbers. The minimum is substituted by a general binary aggregation function. It is used to extend the usual metric for real numbers to fuzzy numbers, generating a new family of fuzzy-valued distances between fuzzy numbers. Then, some conditions on these aggregation functions are studied to hold the fuzzy number properties of the generated distances.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Asmus, T., Dimuro, G., Bedregal, B.: On two-player interval-valued fuzzy bayesian games: interval-valued fuzzy bayesian games. Int. J. Intell. Syst. 32, 557–596 (2016)
Bednar, J.: Fuzzy distances. Kybernetika 41, 375–388 (2005)
Beliakov, G., Pradera, A., Calvo, T.: Aggregation Functions: A Guide for Practitioners, 1st edn. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-73721-6
Costa, C., Bedregal, B., Neto, A.D.: Atanassov’s intuitionistic probability and Markov chains. Knowl. Based Syst. 43, 52–62 (2013)
Dubois, D., Prade, H.: Additions of interactive fuzzy numbers. IEEE Trans. Autom. Control 26(4), 926–936 (1981)
Dubois, D., Prade, H.: Fuzzy Sets and Systems: Theory and Applications, 1st edn. Academic Press (1980)
George, A., Veeramani, P.: On some results in fuzzy metric spaces. Fuzzy Sets Syst. 64, 395–399 (1994)
Kaleva, O., Seikkala, S.: On fuzzy metric spaces. Fuzzy Sets Syst. 12, 215–229 (1984)
Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms, 1st edn. Springer, Dordrecht (2000). https://doi.org/10.1007/978-94-015-9540-7
Klir, G.J., Yuan, B.: Fuzzy Sets and Fuzzy Logic: Theory and Applications, 1st edn. Prentice-Hall, Upper Sadle River (1995)
Kramosil, I., Michálek, J.: Fuzzy metrics and statistical metric spaces. Kybernetica 11, 326–334 (1975)
Otto, K.N., Lewis, A.D., Antonsson, E.K.: Approximating \(\alpha \)-cuts with the vertex method. Fuzzy Sets Syst. 55(1), 43–50 (1993)
Schweizer, B., Sklar, A.: Statistical metric spaces. Pac. J. Math. 10(1), 22 (1960)
Souza, E.L., Santiago, R.H.N., Canuto, A.M.P., Nunes, R.O.: Gradual complex numbers and their application for performance evaluation classifiers. IEEE Trans. Fuzzy Syst. 26(2), 1058–1065 (2018)
Yager, R.R.: A characterization of the extension principle. Fuzzy Sets Syst. 18(3), 205–217 (1986)
Zadeh, L.A.: Fuzzy sets. Inf. Control 8, 338–353 (1965)
Zadeh, L.: The concept of a linguistic variable and its application to approximate reasoning-i. Inf. Sci. 8(3), 199–249 (1975)
Zumelzu, N., Bedregal, B., Mansilla, E., Bustince, H., Diaz, R.: Admissible orders on fuzzy numbers. IEEE Trans. Fuzzy Syst. (2022). https://doi.org/10.1109/TFUZZ.2022.3160326
Acknowledgements
The authors are also grateful to the reviewers for their valuable comments. This work was supported by the Brazilian Coordination for the Improvement of Higher Education Personnel (CAPES).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2022 Springer Nature Switzerland AG
About this paper
Cite this paper
Araújo, J., Bedregal, B., Santiago, R. (2022). Fuzzy-Valued Distance Between Fuzzy Numbers Based on a Generalized Extension Principle. In: Ciucci, D., et al. Information Processing and Management of Uncertainty in Knowledge-Based Systems. IPMU 2022. Communications in Computer and Information Science, vol 1601. Springer, Cham. https://doi.org/10.1007/978-3-031-08971-8_38
Download citation
DOI: https://doi.org/10.1007/978-3-031-08971-8_38
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-08970-1
Online ISBN: 978-3-031-08971-8
eBook Packages: Computer ScienceComputer Science (R0)