Abstract
In this paper, the problem of measuring the degree of inclusion and equivalence measure for Atanassov intuitionistic fuzzy setting is considered. We propose inclusion and equivalence measure by using the partial or linear order on Atanassov intuitionistic fuzzy setting. Moreover, some properties of inclusion and equivalence measures and some correlation between them and aggregation operators are examined.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Aczél, J.: Lectures on Functional Equations and Their Applications. Academic Press, New York (1966)
Asiain, M.J., Bustince, H., Bedregal, B., Takáč, Z., Baczyński, M., Paternain, D., Dimuro, G.: About the use of admissible order for defining implication operators. In: Carvalho, J.P., et al. (eds.) IPMU 2016, Part I. CCIS 610, pp. 353–362. Springer (2016)
Atanassov, K.T.: Intuitionistic fuzzy sets. Fuzzy Sets Syst. 20, 87–96 (1986)
Beliakov, G., Bustince, H., Calvo, T.: A Practical Guide to Averaging Functions. Studies in Fuzziness and Soft Computing, vol. 329. Springer, Cham (2016)
Bodenhofer, U., De Baets, B., Fodor, J.: A compendium of fuzzy weak orders: representations and constructions. Fuzzy Sets Syst. 158, 811–829 (2007)
Bosteels, K., Kerre, E.E.: On a reflexivity-preserving family of cardinality-based fuzzy comparison measures. Inf. Sci. 179, 2342–2352 (2009)
Bustince, H.: Indicator of inclusion grade for interval-valued fuzzy sets. Application to approximate reasoning based on interval-valued fuzzy sets. Int. J. Approximate Reasoning 23(3), 137–209 (2000)
Bustince, H., Barrenechea, E., Pagola, M., Fernández, J., Xu, Z., Bedregal, B., Montero, J., Hagras, H., Herrera, F., De Baets, B.: A historical account of types of fuzzy sets and their relationships. IEEE Trans. Fuzzy Syst. 24(1), 174–194 (2016)
Bustince, H., Barrenechea, E., Pagola, M.: Image thresholding using restricted equivalence functions and maximizing the measures of similarity. Fuzzy Sets Syst. 158, 496–516 (2007)
Bustince, H., Fernandez, J., Kolesárová, A., Mesiar, R.: Generation of linear orders for intervals by means of aggregation functions. Fuzzy Sets Syst. 220, 69–77 (2013)
De Baets, B., De Meyer, H., Naessens, H.: A class of rational cardinality-based similarity measures. J. Comput. Appl. Math. 132, 51–69 (2001)
De Baets, B., De Meyer, H.: Transitivity frameworks for reciprocal relations: cycle-transitivity versus FG-transitivity. Fuzzy Sets Syst. 152, 249–270 (2005)
De Baets, B., Janssens, S., De Meyer, H.: On the transitivity of a parametric family of cardinality-based similarity measures. Int. J. Approximate Reasoning 50, 104–116 (2009)
De Cock, M., Kerre, E.E.: Why fuzzy T-equivalence relations do not resolve the Poincar’e paradox, and related issues. Fuzzy Sets Syst. 133, 181–192 (2003)
De Miguel, L., Bustince, H., Fernandéz, J., Induráin, E., Kolesárová, A., Mesiar, R.: Construction of admissible linear orders for interval-valued Atanassov intuitionistic fuzzy sets with an application to decision making. Inf. Fusion 27, 189–197 (2016)
De Miguel, L., Bustince, H., Barrenechea, E., Pagola, M., Jurio, A., Sanz, J., Elkano, M.: Unbalanced interval-valued OWA operators. Progress Artif. Intell. 5(3), 207–214 (2016)
Deschrijver, G., Cornelis, C., Kerre, E.E.: On the representation of intuitonistic fuzzy t-norms and t-conorms. IEEE Trans. Fuzzy Syst. 12, 45–61 (2004)
Deschrijver, G., Kerre, E.E.: Implicators based on binary aggregation operators in interval-valued fuzzy set theory. Fuzzy Sets Syst. 153(2), 229–248 (2005)
Drygaś, P.: Preservation of intuitionistic fuzzy preference relations. In: Advances in Intelligent Systems Research, EUSFLAT/LFA, 18–22 July 2011, pp. 554–558 (2011)
Freson, S., De Baets, B., De Meyer, H.: Closing reciprocal relations w.r.t. stochastic transitivity. Fuzzy Sets Syst. 241, 2–26 (2014)
Jayaram, B., Mesiar, R.: I-Fuzzy equivalence relations and I-fuzzy partitions. Inf. Sci. 179, 1278–1297 (2009)
Komorníková, M., Mesiar, R.: Aggregation functions on bounded partially ordered sets and their classification. Fuzzy Sets Syst. 175, 48–56 (2011)
Lin, L., Yuan, X.-H., Xia, Z.-Q.: Multicriteria fuzzy decision-making methods based on intuitionistic fuzzy sets. J. Comput. Syst. Sci. 73, 84–88 (2007)
Ovchinnikov, S.: Numerical representation of transitive fuzzy relations. Fuzzy Sets Syst. 126, 225–232 (2002)
Palmeira, E.S., Bedregal, B., Bustince, H., Paternain, D., De Miguel, L.: Application of two different methods for extending lattice-valued restricted equivalence functions used for constructing similarity measures on L-fuzzy sets. Inf. Sci. 441, 95–112 (2018)
Sambuc, R.: Fonctions \(\phi \)-floues: application á l’aide au diagnostic en pathologie thyroidienne. Ph.D. thesis, Universit\(\acute{e}\) de Marseille, France (1975). (in French)
Świtalski, Z.: General transitivity conditions for fuzzy reciprocal preference matrices. Fuzzy Sets Syst. 137, 85–100 (2003)
Takáč, Z.: Inclusion and subsethood measure for interval-valued fuzzy sets and for continuous type-2 fuzzy sets. Fuzzy Sets Syst. 224, 106–120 (2013)
Takáč, Z., Minárová, M., Montero, J., Barrenechea, E., Fernandez, J., Bustince, H.: Interval-valued fuzzy strong S-subsethood measures, interval-entropy and P-interval-entropy. Inf. Sci. 432, 97–115 (2018)
Zadeh, L.A.: Fuzzy sets. Inf. Control 8, 338–353 (1965)
Zadeh, L.A.: The concept of a linguistic variable and its application to approximate reasoning. Part I Inf. Sci. 8, 199–251 (1975)
Zadeh, L.A.: The concept of a linguistic variable and its application to approximate reasoning. Part II Inf. Sci. 8, 301–357 (1975)
Zadeh, L.A.: The concept of a linguistic variable and its application to approximate reasoning. Part III Inf. Sci. 9, 43–80 (1975)
Zapata, H., Bustince, H., Montes, S., Bedregal, B., Dimuro, G.P., Takáč, Z., Baczyński, M., Fernandez, J.: Interval-valued implications and interval-valued strong equality index with admissible orders. Int. J. Approximate Reasoning 88, 91–109 (2017)
Zeng, W.Y., Guo, P.: Normalized distance, similarity measure, inclusion measure and entropy of interval-valued fuzzy sets and their relationship. Inf. Sci. 178, 1334–1342 (2008)
Acknowledgements
This work was partially supported by the Centre for Innovation and Transfer of Natural Sciences and Engineering Knowledge of University of Rzeszów, Poland, the project RPPK.01.03.00-18-001/10. Moreover, Urszula Bentkowska acknowledges the support of the Polish National Science Centre grant number 2018/02/X/ST6/00214. Humberto Bustince and Javier Fernandez were partially supported by Research project TIN2016-77356-P(AEI/UE/FEDER) of the Spanish Government.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this paper
Cite this paper
Pȩkala, B., Bentkowska, U., Bustince, H., Fernandez, J., Lafuente, J. (2021). New Type of Equivalence Measure for Atanassov Intuitionistic Fuzzy Setting. In: Atanassov, K., et al. Uncertainty and Imprecision in Decision Making and Decision Support: New Challenges, Solutions and Perspectives. IWIFSGN 2018. Advances in Intelligent Systems and Computing, vol 1081. Springer, Cham. https://doi.org/10.1007/978-3-030-47024-1_2
Download citation
DOI: https://doi.org/10.1007/978-3-030-47024-1_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-47023-4
Online ISBN: 978-3-030-47024-1
eBook Packages: Intelligent Technologies and RoboticsIntelligent Technologies and Robotics (R0)