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(p, q)-fuzzy aggregation operators and their applications to decision-making

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Abstract

A (p, q)-fuzzy set is the resulting structure when the fuzzy membership and non-membership values are bounded by a general nonlinear relation. This set enhances the range of depicting uncertain information by making the feasible region larger, and thereby widening the purview of decision-making. Data aggregation is crucial in optimal decision-making. This article attempts to formulate aggregation operators for (p, q)-fuzzy sets using additive generators of strict t-norms and strict t-conorms. The utility of these operators is showcased by examining a decision-making problem, where the best decision is obtained by ranking the alternatives based on their score values. A comparative study is also carried out using some existing aggregation operators to test the validity and effectiveness of the introduced operators.

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Abbreviations

AO:

Aggregation operator

FFS:

Fermatean fuzzy set

IFS:

Intuitionistic fuzzy set

MADM:

Multi-attribute decision-making

n,m-ROFS:

n,m-Rung orthopair fuzzy set

n,m-ROFWPA:

n,m-Rung orthopair fuzzy weighted power average

PFS:

Pythagorean fuzzy set

(p, q)-FS:

(p, q)-Fuzzy set

(p, q)-FWA:

(p, q)-Fuzzy weighted averaging

(p, q)-FWG:

(p, q)-Fuzzy weighted geometric

(p, q)-FEWA:

(p, q)-Fuzzy Einstein weighted averaging

(p, q)-FEWG:

(p, q)-Fuzzy Einstein weighted geometric

(p, q)-FHWA:

(p, q)-Fuzzy Hamacher weighted averaging

(p, q)-FHWG:

(p, q)-Fuzzy Hamacher weighted geometric

q-ROFS:

q-Rung orthopair fuzzy set

References

  1. Al-shami, T.M., and A. Mhemdi. 2023. Generalized frame for orthopair fuzzy sets:(m, n)-Fuzzy sets and their applications to multi-criteria decision-making methods, Information, 14(1): 56.

    Article  Google Scholar 

  2. Atanassov, K.T. 1999. Intuitionistic fuzzy Sets: theory and applications. Physica-Verlag HD.

  3. Beliakov, G., H. Bustince, D.P. Goswami, U.K. Mukherjee, and N.R. Pal. 2011. On averaging operators for Atanassov’s intuitionistic fuzzy sets. Information Sciences 181 (6): 1116–24.

    Article  MathSciNet  Google Scholar 

  4. Beliakov, G., Pradera, A., Calvo, T. 2007. Aggregation functions: a guide for practitioners, Vol. 221. Springer.

  5. Calvo, T., Mayor, G., Mesiar, R. 2002. Aggregation operators: new trends and applications, Vol. 97. Springer Science & Business Media.

  6. Darko, A.P., and D. Liang. 2020. Some q-rung orthopair fuzzy Hamacher aggregation operators and their application to multiple attribute group decision making with modified EDAS method. Engineering Applications of Artificial Intelligence 87: 103259.

    Article  Google Scholar 

  7. Garg, H. 2016. A new generalized pythagorean fuzzy information aggregation using einstein operations and its application to decision making. International Journal of Intelligent Systems 31 (9): 886–920.

    Article  Google Scholar 

  8. Grabisch, M., Marichal, J.L., Mesiar, R., Pap, E. 2009. Aggregation functions, Vol. 127. Cambridge University Press.

  9. Hadi, A., W. Khan, and A. Khan. 2021. A novel approach to MADM problems using Fermatean fuzzy Hamacher aggregation operators. International Journal of Intelligent Systems 36 (7): 3464–3499.

    Article  Google Scholar 

  10. Huang, J.Y. 2014. Intuitionistic fuzzy Hamacher aggregation operators and their application to multiple attribute decision making. Journal of Intelligent & Fuzzy Systems 27 (1): 505–513.

    Article  MathSciNet  Google Scholar 

  11. Ibrahim, H.Z., and I. Alshammari. 2022. n, m-rung orthopair fuzzy sets with applications to multicriteria decision making. IEEE Access 10: 99562–99572.

    Article  Google Scholar 

  12. Ibrahim, H.Z., T.M. Al-Shami, and O.G. Elbarbary. 2021. \((3, 2)\)-fuzzy sets and their applications to topology and optimal choices. Computational Intelligence and Neuroscience 2021: 1272266.

    Article  Google Scholar 

  13. Klement, E. P., Mesiar, R., Pap. E. 2013. Triangular norms, Vol. 8. Springer Science & Business Media.

  14. Liu, P., and P. Wang. 2018. Multiple-attribute decision-making based on Archimedean Bonferroni operators of q-rung orthopair fuzzy numbers. IEEE Transactions on Fuzzy systems 27 (5): 834–848.

    Article  Google Scholar 

  15. Maes, K.C., S. Saminger, and B. De Baets. 2007. Representation and construction of self-dual aggregation operators. European Journal of Operational Research 177 (1): 472–487.

    Article  MathSciNet  Google Scholar 

  16. Rani, P., and A.R. Mishra. 2021. Fermatean fuzzy Einstein aggregation operators-based MULTIMOORA method for electric vehicle charging station selection. Expert Systems with Applications 182: 115267.

    Article  Google Scholar 

  17. Senapati, T., and R.R. Yager. 2020. Fermatean fuzzy sets. Journal of Ambient Intelligence and Humanized Computing 11 (2): 663–674.

    Article  Google Scholar 

  18. Wang, W., and X. Liu. 2012. Intuitionistic fuzzy information aggregation using Einstein operations. IEEE Transactions on Fuzzy Systems 20 (5): 923–938.

    Article  Google Scholar 

  19. Wu, S.J., and G.W. Wei. 2017. Pythagorean fuzzy Hamacher aggregation operators and their application to multiple attribute decision making. International Journal of Knowledge-Based and Intelligent Engineering Systems 21(3): 189–201.

    Article  Google Scholar 

  20. Xia, M., Z. Xu, and B. Zhu. 2012. Some issues on intuitionistic fuzzy aggregation operators based on archimedean t-conorm and t-norm. Knowledge-Based Systems 31: 78–88.

    Article  Google Scholar 

  21. Xu, Z., and Q.L. Da. 2003. An overview of operators for aggregating information. International Journal of intelligent systems 18 (9): 953–969.

    Article  Google Scholar 

  22. Yager, R.R. 2013. Pythagorean fuzzy subsets. In 2013 joint IFSA World congress and NAFIPS annual meeting (IFSA/NAFIPS), pp. 57–61.

  23. Yager, R.R. 2013. Pythagorean membership grades in multicriteria decision making. IEEE Transactions on Fuzzy Systems 22 (4): 958–965.

    Article  Google Scholar 

  24. Yager, R.R. 2016. Generalized orthopair fuzzy sets. IEEE Transactions on Fuzzy Systems 25 (5): 1222–1230.

    Article  Google Scholar 

  25. Yang, Y., Kwai-Sang Chin, Heng Ding, Hong-Xia Lv, and Yan-Lai Li (2019.U) Pythagorean fuzzy bonferroni means based on t-norm and its dual t-conorm. International Journal of Intelligent Systems 34 (6): 1303–1336.

  26. Zadeh, L.A. 1965. Fuzzy sets. Information and Control 8 (3): 338–353.

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The first author gratefully acknowledges the financial assistance from the Ministry of Education of the Government of India and the National Institute of Technology Calicut during the preparation of this paper.

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Author notes

  1. Sunil Jacob John and T. M. Athira contributed equally to this work.

Authors and Affiliations

  1. Department of Mathematics, National Institute of Technology Calicut, NIT Campus P.O, Kozhikode, 673601, Kerala, India

    Aparna Sivadas, Sunil Jacob John & T. M Athira

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  1. Aparna Sivadas
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  2. Sunil Jacob John
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Correspondence to Aparna Sivadas.

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Sivadas, A., John, S.J. & Athira, T.M. (p, q)-fuzzy aggregation operators and their applications to decision-making. J Anal (2024). https://doi.org/10.1007/s41478-023-00693-1

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