Document Type : Review Paper

Authors

Department of Mathematics, Statistics and Computer Science, University of Agriculture, P.M.B. 2327, Makurdi, Nigeria.

Abstract

Fermatean Fuzzy Sets (FFSs) provide an effective way to handle uncertainty and vagueness by expanding the scope of membership and Non-Membership Degrees (NMDs) of Intuitionistic Fuzzy Set (IFS) and Pythagorean Fuzzy Set (PFS), respectively. FFS handles uncertain information more easily in the process of decision making. The concept of composite relation is an operational information measure for decision making. This study establishes Fermatean fuzzy composite relation based on max-average rule to enhance the viability of FFSs in machine learning via soft computing approach. Some numerical illustrations are provided to show the merit of the proposed max-average approach over existing the max-min-max computational process. To demonstrate the application of the approach, we discuss some pattern recognition problems of building materials and mineral fields with the aid of the Fermatean fuzzy modified composite relation and Fermatean fuzzy max-min-max approach to underscore comparative analyses. In recap, the objectives of the paper include: 1) discussion of FFS and its composite relations, 2) numerical demonstration of Fermatean fuzzy composite relations, 3) establishment of a decision application framework under FFS in pattern recognition cases, and 4) comparative analyses to showcase the merit of the new approach of Fermatean fuzzy composite relation. In future, this Fermatean fuzzy modified composite relation could be studied in different environments like picture fuzzy sets, spherical fuzzy sets, and so on.

Keywords

Main Subjects

  1. Zadeh, L. A. (1965). Fuzzy sets. Information control, 8(3), 338–353. https://doi.org/10.1016/S0019-9958(65)90241-X
  2. Atanassov, K.T. (1986). Intuitionistic fuzzy sets. Fuzzy sets and systems, 20(1), 87–96.
  3. Ejegwa, P. A. (2015). Intuitionistic fuzzy sets approach in appointment of positions in an organization via max-min-max rule. Global journal of science frontier research: f mathematics and decision sciences15(6), 1-6.
  4. Ejegwa, P. A., & Onasanya, B. O. (2019). Improved intuitionistic fuzzy composite relation and its application to medical diagnostic process. Note IFS25(1), 43-58.
  5. Dengfeng, L., & Chuntian, C. (2002). New similarity measures of intuitionistic fuzzy sets and application to pattern recognitions. Pattern recognition letters23(1-3), 221-225.
  6. Yager, R. R. (2013, June). Pythagorean fuzzy subsets. 2013 joint IFSA world congress and NAFIPS annual meeting (IFSA/NAFIPS)(pp. 57-61). IEEE.
  7. Peng, X., & Yang, Y. (2015). Some results for Pythagorean fuzzy sets. International journal of intelligent systems30(11), 1133-1160.
  8. Zhang, X., & Xu, Z. (2014). Extension of TOPSIS to multiple criteria decision making with Pythagorean fuzzy sets. International journal of intelligent systems29(12), 1061-1078.
  9. Ejegwa, P. A. (2019). Pythagorean fuzzy set and its application in career placements based on academic performance using max–min–max composition. Complex & intelligent systems5(2), 165-175.
  10. Senapati, T., & Yager, R. R. (2019). Fermatean fuzzy weighted averaging/geometric operators and its application in multi-criteria decision-making methods. Engineering applications of artificial intelligence85, 112-121.
  11. Wang, H., Wang, X., & Wang, L. (2019). Multicriteria decision making based on Archimedean Bonferroni mean operators of hesitant Fermatean 2-tuple linguistic terms. Complexity2019. https://doi.org/10.1155/2019/5705907
  12. Liu, D., Liu, Y., & Wang, L. (2019). Distance measure for Fermatean fuzzy linguistic term sets based on linguistic scale function: an illustration of the TODIM and TOPSIS methods. International journal of intelligent systems34(11), 2807-2834.
  13. Liu, D., Liu, Y., & Chen, X. (2019). Fermatean fuzzy linguistic set and its application in multicriteria decision making. International journal of intelligent systems34(5), 878-894.
  14. Senapati, T., & Yager, R. R. (2019). Some new operations over Fermatean fuzzy numbers and application of Fermatean fuzzy WPM in multiple criteria decision making. Informatica30(2), 391-412.
  15. Sahoo, L. (2022). Similarity measures for Fermatean fuzzy sets and its applications in group decision-making. Decision science letters11(2), 167-180.
  16. Sahoo, L. (2021). A new score function based Fermatean fuzzy transportation problem. Results in control and optimization4, 100040. https://doi.org/10.1016/j.rico.2021.100040
  17. Sahoo, L. (2021). Some score functions on Fermatean fuzzy sets and its application to bride selection based on TOPSIS method. International Journal of fuzzy system applications (IJFSA)10(3), 18-29.
  18. Sahoo, L. (2019). Solving matrix games with linguistic payoffs. International journal of system assurance engineering and management10(4), 484-490.
  19. Sahoo, L., & Ghosh, S. K. (2017). Solving assignment problem with linguistic costs. Journal of new theory, (17), 26-37.
  20. Sahoo, L. (2015). Effect of defuzzification methods in solving fuzzy matrix games. Journal of new theory, (8), 51-64.
  21. Ejegwa, P. A., Nwankwo, K. N., Ahmad, M., Ghazal, T. M., & Khan, M. A. (2021). Composite relation under Fermatean fuzzy context and its application in disease diagnosis. Informatica32(10), 87-101.
  22. De, S. K., Biswas, R., & Roy, A. R. (2001). An application of intuitionistic fuzzy sets in medical diagnosis. Fuzzy sets and systems117(2), 209-213.
  23. Ejegwa, P. A. (2020). Improved composite relation for Pythagorean fuzzy sets and its application to medical diagnosis. Granular computing5(2), 277-286.
  24. Ejegwa, P. A., Jana, C., & Pal, M. (2022). Medical diagnostic process based on modified composite relation on Pythagorean fuzzy multi-sets. Granular computing7(1), 15-23.
  25. Wang, W., & Xin, X. (2005). Distance measure between intuitionistic fuzzy sets. Pattern recognition letters26(13), 2063-2069.