Measures of Distance and Entropy Based on the Fermatean Fuzzy-Type Soft Sets Approach

Year 2024, , 12 - 29, 18.03.2024

Murat Kirisci

https://doi.org/10.32323/ujma.1379260

Cited By: 2

Abstract

The definition of Fermatean fuzzy soft sets and some of its features are introduced in this study. A Fermatean fuzzy soft set is a parameterized family of Fermatean fuzzy sets and a generalization of intuitionistic and Pythagorean fuzzy soft sets. This paper presents a definition of the Fermatean fuzzy soft entropy. Also acquired are the formulae for standard distance measures such as Hamming and Euclidean distance. Other formulas have also been proposed for calculating the entropy and distance measurements of FFSSs. Even if the entropy and distance measures are defined for other set extensions, they cannot be applied directly to Fermatean fuzzy soft sets. It can be used to determine the uncertainty associated with a Fermatean fuzzy soft set, discover similarities between any two Fermatean fuzzy soft sets using the proposed distance measures, and compare it to other existing structures in the literature. Fermatean fuzzy soft set applications in decision-making and pattern recognition difficulties are also examined. Finally, comparison studies with other known equations are performed.

Keywords

Decision-making, Entropy, Fermatean fuzzy soft set, Missile position estimation

References

• [1] C. E. Shannon, A mathematical theory of communication, Bell Syst. Tech. J., 27 (1948), 379–423 623–656.
• [2] M. Akram, M. Ashraf, Multi-criteria group decision-making based on spherical fuzzy rough numbers, Granul. Comput. 8 (2023), 1267–1298.
• [3] M. Akram, S. Zahid, Group decision-making method with a Pythagorean fuzzy rough number for the evaluation of best design concept, Granul. Comput. 8 (2023), 1121–1148.
• [4] M. Akram, S. Zahid, M. Deveci, Enhanced CRITIC-REGIME method for decision making based on Pythagorean fuzzy rough number, Expert Systems with Applications, 238 (2023), 122014.
• [5] M. Akram, F. Ilyas, M. Deveci, Interval rough integrated SWARA-ELECTRE model: An application to machine tool remanufacturing, Expert Systems with Applications, 238 (2023), 122067.
• [6] S. Broumi, A. Bakali, M. Talea, F. Smarandache, F. Karaaslan, Interval valued neutrosophic soft graphs, New Trends in Neutrosophic Theory and Applications, 2 (2018), 218–251.
• [7] M. Palanikumar, A. Iampan, S. Broumi, L. J. Manavalan, K. Sundareswari, Multi-criteria group decision-making method in Pythagorean interval-valued neutrosophic fuzzy soft using VIKOR approach, Int. J. Neutrosophic Sci. 22 (2023), 104–113.
• [8] M. Palanikumar, A. Iampan, S. Broumi, G.Balaj, Generalization of Neutrosophic interval-valued soft sets with different aggregating operators using multi-criteria group decision-making, Int. J. Neutrosophic Sci., 22 (2023), 114–123.
• [9] S. Priyadarsini, A. V. Singh, S. Broumi, Review of Generalized Neutrosophic soft set in solving multiple expert decision making problems, Int. J. Neutrosophic Sci., 19 (2022), 48–59.
• [10] K. Zahid, M. Akram, Multi-criteria group decision-making for energy production from municipal solid waste in Iran based on spherical fuzzy sets, Granul. Comput. 8 (2023), 1299–1323.
• [11] M. Akram, H. S. Nawaz, M. Deveci, Attribute reduction, and information granulation in Pythagorean fuzzy formal contexts, Expert Syst. Appl., 222 (2023), 119794.
• [12] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets Syst. 20 (1986), 87–96.
• [13] D. Molodtsov, Soft set theory first results, Comp. Math. Appl., 7(1) (2019), 91.
• [14] L. A. Zadeh, Fuzzy sets, Inf. Comp., 8 (1965), 338–353.
• [15] R. R. Yager, Pythagorean fuzzy subsets. In: Proc Joint IFSA World Congress and NAFIPS Annual Meeting, Edmonton, Canada 57—61, (2013).
• [16] T. Senapati, R. R. Yager, Fermatean fuzzy sets, J. Ambient Intell. Humaniz. Comput., 11 (2020), 663–674.
• [17] T. Senapati, R. R. Yager, Some new operations over Fermatean fuzzy numbers and application of Fermatean fuzzy WPM in multiple criteria decision making, Informatica, 30 (2019), 391–412.
• [18] T. Senapati, R. R. Yager, Fermatean fuzzy weighted averaging/geometric operators and its application in multi-criteria DM methods, Eng. Appl. Artif. Intell., 85 (2019), 112–121.
• [19] M. Akram, G. Ali, J. C. R. Alcantud, A. Riaz, Group decision-making with Fermatean fuzzy soft expert knowledge, Artif. Intell. Rev., 55 (2022), 5349–5389.
• [20] M. Akram, U. Amjad, J. C. R. Alcantud, G. Santos-Garcis, Complex fermatean fuzzy N-soft sets: a new hybrid model with applications, J. Ambient Intell. Humaniz. Comput., 14 (2023), 8765–8798.
• [21] M. Kirişci, New cosine similarity and distance measures for Fermatean fuzzy sets and TOPSIS approach, Knowl. Inf. Syst., 65 (2023), 855–868.
• [22] M. Kirişci, I. Demir, N. Şimşek, Fermatean fuzzy ELECTRE multi-criteria group decision-making and most suitable biomedical material selection, Artif. Intell. Med., 127 (2022), 102278.
• [23] M. Kirişci, Data analysis for lung cancer: Fermatean hesitant fuzzy sets approach, Math. Models Comput. Simul., 30 (2022), 701–710.
• [24] M. Kirişci, Fermatean Hesitant Fuzzy Sets for Multiple Criteria Decision-Making with Applications, Fuzzy Information and Engineering, 15(2), (2023), 100–127. doi: 10.26599/FIE.2023.9270011
• [25] M. Kirişci, Data analysis for panoramic X-ray selection: Fermatean fuzzy type correlation coefficients approach, Engineering Applications of Artificial Intelligence, 126, (2023), 106824. doi:10.1016/j.engappai.2023.106824
• [26] G. Shahzadi, A. Akram, Hypergraphs Based on Pythagorean Fuzzy Soft Model. Math. Comput. Appl., 24, (2019), 100. doi:10.3390/mca24040100
• [27] G. Shahzadi, A. Akram, Group decision-making for the selection of an antivirus mask under fermatean fuzzy soft information. Journal of Intelligent & Fuzzy Systems, 40, (2021), 1401–1416.
• [28] P.A. Ejegwa, P. Muhiuddin, E.A. Algehyne, J.M. Agbetayo, D. Al-Kadi, An enhanced Fermatean fuzzy composition relation based on a maximumaverage approach and its application in diagnostic analysis. Journal of Mathematics, (2022), Article ID 1786221, 12 pages.
• [29] P.A. Ejegwa, I. C. Onyeke, Fermatean fuzzy similarity measure algorithm and its application in students’ admission process. International Journal of Fuzzy Computation and Modelling, 4(1), (2022), 34–50.
• [30] A. Mehmood, W. Ullah, S. Broumi, M. I. Khan, H. Qureshi, M. I. Abbas, H.Kalsoom, F. Nadeem, Neutrosophic Soft Structures, Neutrosophic Sets and Systems, 33, (2020), 23–58. [31] I. C. Onyeke, P.A. Ejegwa, Modified Senapati and Yager’s Fermatean Fuzzy Distance and Its Application in Students’ Course Placement in Tertiary Institution, In: Sahoo, L., Senapati, T., Yager, R.R. (eds) Real Life Applications of Multiple Criteria Decision Making Techniques in Fuzzy Domain; Studies in Fuzziness and Soft Computing, vol 420, 237–253, Springer.
• [32] T. M. Athira, S. J. John, H. Garg, Entropy and distance measures of Pythagorean fuzzy soft sets and their applications, J. Intell. Fuzzy Syst., 37(3) (2019), 4071–4084.
• [33] T. M. Athira, S. J. John, H. Garg, A novel entropy measure of Pythagorean fuzzy soft sets, AIMS Mathematics, 5(2) (2020), 1050–1061.
• [34] A. Guleria, R. K. Bajaj, On Pythagorean fuzzy soft matrices, operations and their applications in decision making and medical diagnosis, Soft Comput. 23 (2019), 7889–7900.
• [35] P. Maji, A. Biswas, A. Roy, Fuzzy soft sets, J. Fuzzy Math., 9(3) (2001), 589–602.
• [36] P. Maji, A. Biswas, A. Roy, Intuitionistic Fuzzy soft sets, J. Fuzzy Math., 9(3) (2001), 677–692.
• [37] P. Maji, A. Biswas, A. Roy, Soft set theory, Comp. Math Appl., 45(4-5) (2003), 555–562.
• [38] P. Majumdar, S. Samanta, Similarity measure of soft sets, New Math. Nat. Comput., 4(1) (2008), 1–12.
• [39] K. Naeem, M. Riaz, X. Peng, D. Afzal, Pythagorean fuzzy soft MCGDM methods based on TOPSIS, VIKOR, and aggregation operators, J. Intell. Fuzzy Syst., 37(5) (2019), 6937–6957.
• [40] X. Peng, Y. Yang, J. Song, Y. Jiang, Pythagorean fuzzy soft set and its application, Computer Engineering, 41(7) (2015), 224–229.
• [41] J. C. R. Alcantud, G. Varela, B. Santos-Buitrago, G. Santos-Garcia, M. F. Jimenez, Analysis of survival for lung cancer resection cases with fuzzy and soft set theory in surgical decision-making, Plos One, 14 (2019), e0218283.
• [42] M. I. Ali, F. Feng, X. Liu, W. K. Min, M. Shabir, On some new operations in soft set theory, Comput. Math. Appl. 57 (2009), 1547–1553.
• [43] N. C¸ ağman, S. Enginoğlu, F. Citak, Fuzzy soft set theory and its application, Iran. J. Fuzzy Syst., 8 (2011), 137–147.
• [44] M. Kirişci, A case study for medical decision making with the fuzzy soft sets, Afr. Mat., 31 (2020), 557–564.
• [45] A. R. Roy, P. K. Maji, A fuzzy soft set theoretic approach to decision-making problems, J. Comput. Appl. Math., 203 (2007), 412–418.
• [46] S. Saleh, R. Abu-Gdairi, T. M. Al-Shami, Mohammed S. Abdo, On categorical property of fuzzy soft topological spaces, Appl. Math. Inform. Sci., 16 (2022), 635–641.
• [47] M. T. Hamida, M. Riaz, D. Afzal, Novel MCGDM with q-rung orthopair fuzzy soft sets and TOPSIS approach under q-Rung orthopair fuzzy soft topology, J. Intell. Fuzzy Syst., 39 (2020), 3853–3871.
• [48] A. Sivadas, S. J. John, Fermatean fuzzy soft sets and its applications, In: A. Awasthi, S. J.John, S. Panda (eds), CSMCS 2020. Communications in Computer and Information Science, Springer, Singapore, 1345 (2021).
• [49] Y. J. Xu, Y. K. Sun, D. F. Li, Intuitionistic fuzzy soft set, 2nd International Workshop on Intelligent Systems and Applications, (2010), 1–4.
• [50] E. Szmidt, J. Kacprzyk, Distances between intuitionistic fuzzy sets, Fuzzy Sets Syst. 114 (2000), 505–518.
• [51] P. A. Ejegwa, E. S. Modom, Diagnosis of viral hepatitis using new distance measure of intuitionistic fuzzy sets, Int. J. Fuzzy Math. Arch. 8 (1) (2015), 1–7.
• [52] P. A. Ejegwa, Distance and similarity measures for Pythagorean fuzzy sets, Granul. Comput. 5 (2020), 225–238.
• [53] P. A. Ejegwa, Modified Zhang and Xu’s distance measure for Pythagorean fuzzy sets and its application to pattern recognition problems, Neural. Comput. Appl., 32 (2020), 10199–10208.
• [54] A. H. Ganie, S. Singh, M. M. Khalaf, M. M. A. Al-Shamiri, On some measures of similarity and entropy for Pythagorean fuzzy sets with their applications, Comp. Appl. Math., 41 (2022), 420.
• [55] X. L. Zhang, Z. S. Xu, Extension of TOPSIS to multi-criteria decision making with Pythagorean fuzzy sets, Int. J. Intell. Syst. 29 (2014), 1061—1078.
• [56] G. Wei, R. Lin, H. Wang, Distance and similarity measures for hesitant interval-valued fuzzy sets, J. Intell. Fuzzy Syst., 27(1) (2014), 19–36.
• [57] Y. Song, X. Wang, H. Zhang, A distance measure between intuitionistic fuzzy belief functions, Knowl. Based Syst., 86 (2015), 288–298.
• [58] G. Wei, Some cosine similarity measures for picture fuzzy sets and their applications to strategic decision making, Informatica, 28 (2017), 547–564.
• [59] R. A. Alahmadi, A. H. Ganie, Y. Al-Qudah, M. M. Khalaf, A. H. Ganie, Multi-attribute decision-making based on novel Fermatean fuzzy similarity measure and entropy measure. Granul. Comput., 8 (2023), 1385–1405.
• [60] G. Ali, M. N. Ansari, Multiattribute decision-making under Fermatean fuzzy bipolar soft framework, Granul. Comput., 7 (2022), 337–352.
• [61] L. A. Zadeh, Fuzzy sets and systems, In Proceedings of the Symposium on Systems Theory, (1965), 29–37.
• [62] A. De Luca, S. Termini, A definition of a nonprobabilistic entropy in the setting of fuzzy set theory, In: Readings in Fuzzy Sets for Intelligent Systems, Elsevier, (1993), 197–202.
• [63] J. Wu, J. Sun, L. Liang, Y. Zha. Determination of weights for ultimate cross efficiency using Shannon entropy, Expert Syst. Appl., 38 (2011), 5162–5165.
• [64] A. Kaufmann, Introduction to the Theory of Fuzzy Sets: Fundamental Theoretical Elements, Vol. 1, Academic Press, New York, 1975.
• [65] R. R. Yager, On the measure of fuzziness and negation, Part 1: Membership in the unit interval, Int. J. Gener. Syst., 5 (1979), 221–229.
• [66] M. Higashi, G. Klir, On measures of fuzziness and fuzzy complements, Int. J. General Syst., 8 (1982), 169–180.
• [67] V. Mohagheghi, S. M. Mousavi, B. Vahdani, Enhancing DM flexibility by introducing a new last aggregation evaluating approach based on multi-criteria group decision making and Pythagorean fuzzy sets, Appl. Soft Comput., 61 (2017), 527–535.
• [68] X. Peng, H. Yuan, Y. Yang, Pythagorean fuzzy information measures and their applications, Int. J. Intell., 32 (2017), 991–1029.
• [69] Y. Jiang, Y. Tang, H. Liu, Z. Chen, Entropy on intuitionistic fuzzy soft sets and on interval-valued fuzzy soft sets, Inf. Sci., 240 (2013), 95–114.
• [70] P. Burillo, H. Bustince, Entropy on intuitionistic fuzzy sets and interval-valued fuzzy sets, Fuzzy Sets Syst, 78(3) (1996), 305–316.
• [71] Q. Han, L. Weimin, S. Yafei, T. Zhang, R. Wang, A new method for MAGDM based on improved TOPSIS and a novel Pythagorean fuzzy soft entropy, Symmetry 11 (2019), 905.