Mathematics and Computational Sciences

Similarity measures for complex Fermatean fuzzy sets and their applications in decision-making and clustering problems

Document Type : Original Article

Authors

Nisha Devi Gunasegaran 1
Hemavathi Perumal 1
Vinod Kumar Raja 2
Suresh Kandhasamy 3

1 Department of Mathematics, Saveetha Institute of Medical and Technical Sciences (SIMATS), Saveetha School of Engineering, Thandalam, India.

2 Department of Mathematics, Rajalakshmi Engineering College (Autonomous), Thandalam, India.

3 Department of Mathematics, St. Joseph College of Engineering, Chennai, India.

https://doi.org/10.30511/mcs.2026.2078363.1638
Abstract
The quantification of similarity between fuzzy objects is central to decision-making and clustering under uncertainty. While numerous measures have been developed within Intuitionistic, Pythagorean, and Fermatean fuzzy frameworks, these approaches cannot be directly applied to Complex Fermatean Fuzzy Sets (CFFSs) because CFFSs represent membership and nonmembership degrees through complex amplitudes and phases. This study proposes novel similarity measures specifically designed for CFFSs, establishes their fundamental mathematical properties, and demonstrates their performance through applications in decision-making and clustering. The results confirm that the proposed measures effectively capture complex-valued uncertainty and provide a reliable foundation for practical fuzzy analysis.

Keywords

Complex Fermatean fuzzy Sets
Similarity measure
Decision-making
Clustering
Subjects
[1] Akram, M., Amjad, U., Alcantud, J. C. R., & Santos-García, G. (2023). Complex Fermatean fuzzy n-soft sets: A new hybrid model with applications. Journal of Ambient Intelligence and Humanized Computing, 14, 87658798. 
[2] Anandarajan, H., & Robinson, P. J. (2025). A Neural Network-Based framework for complemented linguistic intuitionistic Fuzzy Aggregation in MAGDM problems. Mathematics and Computational Sciences, 6(4), 5063. 
[3] Alkouri, A. M. J. S., & Salleh, A. R. (2012). Complex intuitionistic fuzzy sets. AIP Conference Proceedings, 1482, 464470. 
[4] Arora, H. D., Naithani, A., & Chanian, R. (2020). Similarity measures for Q-rung orthopair fuzzy sets and applications to decision-making. Mathematical Foundations of Computing, 3(2), 112125. 
[5] Chinnadurai, V., Thayalan, S., & Bobin, A. (2023). Distance measures of complex Fermatean fuzzy number and their application to multi-criteria decision-making problems. Applied and Computational Mathematics, 18(1), 14. 
[6] Ejegwa, P. A. (2020). New similarity measures for Pythagorean fuzzy sets with applications. International Journal of Fuzzy Computation and Modelling, 4(1), 112.
[7] Garg, H., & Rani, D. (2019). Some generalized complex intuitionistic fuzzy aggregation operators and their application to multicriteria decision-making processes. Arabian Journal of Science & Engineering, 44(4), 26792698. 
[8] Grzegorzewski, P. (2004). Distances between intuitionistic fuzzy sets and interval-valued fuzzy sets based on the Hausdorff metric. Fuzzy Sets and Systems, 148(2), 319328. 
[9] Kahn, F. M., & Khan, I. (2022). A benchmark similarity measure for Fermatean fuzzy sets. Bulletin of the Section of Logic, 51(1), 4560. 
[10] Liu, Z. (2024). Hellinger distance measures on Pythagorean fuzzy environment via their applications. International Journal of Knowledge-Based and Intelligent Engineering Systems, 28(1), 211229.
[11] Liu, Z. (2024). Fermatean fuzzy similarity measures based on Tanimoto and Sørensen coefficients with applications to pattern classification, medical diagnosis, and clustering analysis. Engineering Applications of Artificial Intelligence, 132, 107878. 
[12] Liu, Z., & Yang, M. S. (2019). Distance and similarity measures of Pythagorean fuzzy sets based on the Hausdorff metric with application to fuzzy TOPSIS. International Journal of Intelligent Systems, 34(10), 26332654. 
[13] Nguyen, X. T., Nguyen, V. D., Nguyen, V. H., & Garg, H. (2019). Exponential similarity measures for Pythagorean fuzzy sets and their applications to pattern recognition and decision-making. Complex & Intelligent Systems, 5(3), 217228. 
[14] Roohollah Abbasi Shureshjani, G., Gholam Hassan Shirdel, M., Madineh Farnam, M., & Majid Darehmiraki, M. (2024). Parametric distance measure for trapezoidal intuitionistic fuzzy numbers and application in Multi-criteria group decision-making. Mathematics and Computational Sciences, 5(4), 6184. 
[15] Sathya, R., & Anusuya, V. (2022). Cosine similarity and its application by Pythagorean fuzzy sets. Global Journal of Advanced Research, 9(2), 4555.
[16] Senapati, T., & Yager, R. (2020). Fermatean fuzzy sets. Journal of Ambient Intelligence and Humanized Computing, 11(2), 663674.
[17] Sahoo, L. (2022). Similarity measures for Fermatean fuzzy sets and its applications in group decision-making. Decision Science Letters, 11(2), 167180.
[18] Wang, H., Tuo, C., & Wang, Z. (2024). Enhancing similarity and distance measurements in Fermatean fuzzy sets: Tanimoto-inspired measures and decision-making applications. Symmetry, 16(1), 345374. 
[19] Yager, R. R. (2013, June). Pythagorean fuzzy subsets. In 2013 joint IFSA world congress and NAFIPS annual meeting (IFSA/NAFIPS) (pp. 5761). IEEE. 
[20] Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8(3), 338353. 
[21] Zaman, M., Ghani, F., Khan, A., Abdullah, S., & Khan, K. (2023). Complex Fermatean fuzzy extended TOPSIS method and its applications in decision making. Heliyon, 9(2), e19170. 
[22] Zeeshan, M., Khan, M., Shafqat, R., Althobaiti, A., & Bedada, T. B. (2024). Novel similarity measures under complex Pythagorean fuzzy soft matrices and their application in decision-making problems. Scientific Reports, 14(1), 17129. 
[23] Zhao, R. R., Luo, M. X., & Li, S. G. (2023). A parametric similarity measure between picture fuzzy sets and its applications in multi-attribute decision-making. Iranian Journal of Fuzzy Systems, 20(1), 87102
Mathematics and Computational Sciences
Volume 7, Issue 2
Spring 2026
Pages 238-264

  • Receive Date 19 November 2025
  • Revise Date 13 February 2026
  • Accept Date 14 May 2026

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