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AIMS Mathematics

2026, Volume 11, Issue 3: 6420-6436. doi: 10.3934/math.2026265
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Pythagorean multi-fuzzy N-soft sets: a novel hybrid model for group decision making and its applications in distance education

  • 1.
    Department of Mathematics, Faculty of Science and Technology, Universitas Terbuka, South Tangerang 15418, Indonesia
  • 2.
    Department of Management, Faculty of Economics and Business, Universitas Terbuka, South Tangerang 15418, Indonesia
  • Published: 13 March 2026

  • MSC : 03E72, 03E75

  • The rapid development of distance education has led to a pressing need for practical evaluation of learning platforms, especially in situations where decision-making involves multiple evaluators and uncertain information. To address this need, we introduce a new hybrid model called Pythagorean Multi-Fuzzy N-Soft Sets (PMFNSS). This model is designed to improve group decision-making under complex and ambiguous conditions. The proposed PMFNSS model was built by combining Pythagorean Fuzzy N-Soft Sets (PFNSS) and Multi-Fuzzy N-Soft Sets (MFNSS). First, we formally defined the concept of PMFNSS. Next, we proposed a group decision-making algorithm based on the framework. Finally, to validate its applicability, the model was used in a real-life case study on the evaluation of digital learning platforms in distance education. The results indicated that PMFNSS can serve as a multi-attribute group decision-making tool. Its application to distance education offers a systematic approach to evaluating and optimizing digital learning platforms for resource efficiency and sustainability.

    Citation: Fatia Fatimah, Elin Herlinawati, Andriyansah. Pythagorean multi-fuzzy N-soft sets: a novel hybrid model for group decision making and its applications in distance education[J]. AIMS Mathematics, 2026, 11(3): 6420-6436. doi: 10.3934/math.2026265

    Related Papers:

  • The rapid development of distance education has led to a pressing need for practical evaluation of learning platforms, especially in situations where decision-making involves multiple evaluators and uncertain information. To address this need, we introduce a new hybrid model called Pythagorean Multi-Fuzzy N-Soft Sets (PMFNSS). This model is designed to improve group decision-making under complex and ambiguous conditions. The proposed PMFNSS model was built by combining Pythagorean Fuzzy N-Soft Sets (PFNSS) and Multi-Fuzzy N-Soft Sets (MFNSS). First, we formally defined the concept of PMFNSS. Next, we proposed a group decision-making algorithm based on the framework. Finally, to validate its applicability, the model was used in a real-life case study on the evaluation of digital learning platforms in distance education. The results indicated that PMFNSS can serve as a multi-attribute group decision-making tool. Its application to distance education offers a systematic approach to evaluating and optimizing digital learning platforms for resource efficiency and sustainability.



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    [1] B. Tonbuloğlu, Determining a framework for institutional support of instructors engaged in distance education programmes using the multiple-criteria decision-making method, Int. Rev. Educ., 70 (2024), 111–141. https://doi.org/10.1007/s11159-023-10045-7 doi: 10.1007/s11159-023-10045-7
    [2] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338–353. https://doi.org/10.1016/S0019-9958(65)90241-X doi: 10.1016/S0019-9958(65)90241-X
    [3] K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Set. Syst., 20 (1986), 87–96. https://doi.org/10.1016/S0165-0114(86)80034-3 doi: 10.1016/S0165-0114(86)80034-3
    [4] R. R. Yager, Pythagorean fuzzy subsets, In: 2013 Joint IFSA world congress and NAFIPS annual meeting (IFSA/NAFIPS), Edmonton, AB, Canada, 2013, 57–61. https://doi.org/10.1109/IFSA-NAFIPS.2013.6608375
    [5] D. Molodtsov, Soft set theory—first results, Comput. Math. Appl., 37 (1999), 19–31. https://doi.org/10.1016/S0898-1221(99)00056-5 doi: 10.1016/S0898-1221(99)00056-5
    [6] P. K. Maji, R. Biswas, A. R. Roy, Soft set theory, Comput. Math. Appl., 45 (2003), 555–562. https://doi.org/10.1016/S0898-1221(03)00016-6 doi: 10.1016/S0898-1221(03)00016-6
    [7] P. K. Maji, R. Biswas, A. R. Roy, Fuzzy soft sets, The journal of fuzzy mathematics, 9 (2001), 589–602.
    [8] P. Majumdar, S. K. Samanta, Generalised fuzzy soft sets, Comput. Math. Appl., 59 (2010), 1425–1432. https://doi.org/10.1016/j.camwa.2009.12.006 doi: 10.1016/j.camwa.2009.12.006
    [9] X. Yang, T. Y. Lin, J. Yang, Y. Li, D. Yu, Combination of interval-valued fuzzy set and soft set, Comput. Math. Appl., 58 (2009), 521–527. https://doi.org/10.1016/j.camwa.2009.04.019 doi: 10.1016/j.camwa.2009.04.019
    [10] X. Peng, H. Gard, Algorithms for interval-valued fuzzy soft sets in emergency decision making based on WDBA and CODAS with new information measure, Comput. Ind. Eng., 119 (2018), 439–452. https://doi.org/10.1016/j.cie.2018.04.001 doi: 10.1016/j.cie.2018.04.001
    [11] Y. Jiang, Y. Tang, Q. Chen, H. Liu, J. Tang, Interval-valued intuitionistic fuzzy soft sets and their properties, Comput. Math. Appl., 60 (2010), 906–918. https://doi.org/10.1016/j.camwa.2010.05.036 doi: 10.1016/j.camwa.2010.05.036
    [12] S. Alkhazaleh, A. R. Salleh, N. Hassan, Possibility fuzzy softset, Advances in Decision Sciences, 2011 (2011), 479756. https://doi.org/10.1155/2011/479756 doi: 10.1155/2011/479756
    [13] X. Peng, Y. Yang, J. P. Song, Y. Jiang, Pythagorean fuzzy soft set and its application, (Chinese), Computer Engineering, 41 (2015), 224–229. https://doi.org/10.3969/j.issn.1000-3428.2015.07.043 doi: 10.3969/j.issn.1000-3428.2015.07.043
    [14] A. M. Khalil, Commentary on "generalized intuitionistic fuzzy soft sets with applications in decision-making", Appl. Soft Comput., 37 (2015), 519–520. https://doi.org/10.1016/j.asoc.2015.08.054 doi: 10.1016/j.asoc.2015.08.054
    [15] F. Wang, X. Li, X. Chen, Hesitant fuzzy soft set and its applications in multicriteria decision making, J. Appl. Math., 2014 (2014), 643785. https://doi.org/10.1155/2014/643785 doi: 10.1155/2014/643785
    [16] J. Zhan, M. I. Ali, N. Mehmood, On a novel uncertain soft set model: Z-soft fuzzy rough set model and corresponding decision making methods, Appl. Soft Comput., 56 (2017), 446–457. https://doi.org/10.1016/j.asoc.2017.03.038 doi: 10.1016/j.asoc.2017.03.038
    [17] Q. Q. Aini, I. Mukhlash, F. Fatimah, D. W. Setyawati, An algebraic composition operator on intuitionistic fuzzy soft set evaluations for multi-evaluator decision-making, Journal of Fuzzy Extension and Applications, 7 (2026), 76–88. https://doi.org/10.22105/jfea.2025.530823.1968 doi: 10.22105/jfea.2025.530823.1968
    [18] Q. Q. Aini, I. Mukhlash, F. Fatimah, D. W. Setyawati, Energy of intuitionistic fuzzy soft set for multi-criteria decision-making, In: Intelligent and fuzzy systems, Cham: Springer, 2025,399–406. https://doi.org/10.1007/978-3-031-97985-9_44
    [19] M. Asif, U. Ishtiaq, I. K. Argyros, Hamacher aggregation operators for Pythagorean fuzzy set and its application in multi-attribute decision-making problem, Spectrum of Operational Research, 2 (2025), 27–40. https://doi.org/10.31181/sor2120258 doi: 10.31181/sor2120258
    [20] M. Palanikumar, N. Kausar, Research on the redefined square root interval-valued normal Pythagorean fuzzy multi-attribute decision-making model based on aggregation operators, Spectrum of Operational Research, in press. https://doi.org/10.31181/sor58
    [21] M. Tahir, K. Kfueit, M. Rasheed, A. Hanan, M. I. Shahid, Pythagorean soft sets and hypersoft sets: a comprehensive framework for advanced uncertainty modeling in decision making, Spectrum of Decision Making and Applications, 4 (2025), 1–26. https://doi.org/10.31181/sdmap41202761 doi: 10.31181/sdmap41202761
    [22] F. Fatimah, D. Rosadi, R. B. F. Hakim, J. C. R. Alcantud, N-soft sets and their decision making algorithms, Soft Comput., 22 (2018), 3829–3842. https://doi.org/10.1007/s00500-017-2838-6 doi: 10.1007/s00500-017-2838-6
    [23] T. Mahmood, U. ur Rehman, J. Ahmmad, Complex picture fuzzy N-soft sets and their decision-making algorithm, Soft Comput., 25 (2021), 13657–13678. https://doi.org/10.1007/s00500-021-06108-2 doi: 10.1007/s00500-021-06108-2
    [24] S. Sebastian, T. V. Ramakrishnan, Multi-fuzzy sets: an extension of fuzzy sets, Fuzzy Inf. Eng., 3 (2011), 35–43. https://doi.org/10.1007/s12543-011-0064-y doi: 10.1007/s12543-011-0064-y
    [25] F. Fatimah, J. C. R. Alcantud, The multi-fuzzy N-soft set and its applications to decision-making, Neural Computing and Applications, 33 (2021), 11437–11446. https://doi.org/10.1007/s00521-020-05647-3 doi: 10.1007/s00521-020-05647-3
    [26] Y. Yang, X. Tan, C. Meng, The multi-fuzzy soft set and its application in decision making, Appl. Math. Model., 37 (2013), 4915–4923. https://doi.org/10.1016/j.apm.2012.10.015 doi: 10.1016/j.apm.2012.10.015
    [27] M. Akram, G. Ali, J. C. R. Alcantud, New decision-making hybrid model: intuitionistic fuzzy N-soft rough sets, Soft Comput., 23 (2019), 9853–9868. https://doi.org/10.1007/s00500-019-03903-w doi: 10.1007/s00500-019-03903-w
    [28] H. Zhang, D. Jia-Hua, C. Yan, Multi-attribute group decision-making methods based on pythagorean fuzzy-soft sets, IEEE Access, 8 (2020), 62298–62309. https://doi.org/10.1109/ACCESS.2020.2984583 doi: 10.1109/ACCESS.2020.2984583
    [29] A. Possolo, H. K. Iyer, Invited article: Concepts and tools for the evaluation of measurement uncertainty, Rev. Sci. Instrum., 88 (2017), 011301. https://doi.org/10.1063/1.4974274 doi: 10.1063/1.4974274
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  • © 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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