Generalized Spherical Fuzzy Einstein Aggregation Operators: Application to Multi-Criteria Group Decision-Making Problems

Abstract

The aim of this paper is to present the extension of a concept related to aggregation operators from spherical fuzzy sets to generalized spherical fuzzy sets. We first introduce Einstein sum, product and scalar multiplication for generalized spherical fuzzy sets based on Einstein triangular norm and triangular conorm. Then we give the generalized spherical fuzzy Einstein weighted averaging and generalized spherical fuzzy Einstein weighted geometric operators, namely generalized spherical fuzzy Einstein aggregation operators, constructed on these operations. After investigating some fundamental properties of these operators, we develop a model for generalized spherical fuzzy Einstein aggregation operators to solve the multiple attribute group decision-making problems. Finally, we give a numerical example to demonstrate that the developed method is suitable and effective for the decision process.

Keywords

• Generalized spherical fuzzy number
• Einstein aggregation operators
• multi-criteria group decision-making

References

1. 1) S. Ashraf, S. Abdullah, Spherical aggregation operators and their application in multiattribute group decision-making, International Journal of Intelligent Systems 34(3) (2019), 493-523.
2. 2) S. Ashraf, S. Abdullah, T. Mahmood, Spherical fuzzy Dombi aggregation operators and their application in group decision-making problems, Journal of Ambient Intelligence and Humanized Computing 11 (2020), 2731-2749.
3. 3) S. Ashraf, T. Mahmood, S. Abdullah, Q. Khan, Different approaches to multi-criteria group decision-making problems for picture fuzzy environment, Bull Braz Math Soc. 50(2) (2018), 373-397.
4. 4) S. Ashraf, T. Mahmood, S. Abdullah, Q, Khan, Picture fuzzy linguistic sets and their applications for multiple attribute group, Nucleus 55(2) (2018), 66-73.
5. 5) A. Aygünoˇglu, H. Aygün, Some notes on soft topological spaces, Neural computing and Applications 21(1) (2012), 113-119.
6. 6) K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20(1) (1986), 87-96.
7. 7) R. Bellman, L. A. Zadeh, Decision-making in a fuzzy environment, Manage Sci. 17 (1970), 141-154.
8. 8) B. Cuong, Picture fuzzy sets-first results, Seminar on neuro–fuzzy systems with applications Institute of Mathematics, Hanoi, (2013).

9. 9) V. Çetkin, A. Aygünoˇglu, H. Aygün, A new approach in handling soft decision making problems, J. Nonlinear Sci. Appl 9 (2016), 231-239.
10. 10) H. Garg, Some picture fuzzy aggregation operators and their applications to multicriteria decision-making, Arabian Journal for Science and Engineering, 42(12) (2017), 5275-5290.
11. 11) A. Guleria, R. K. Bajaj, T-spherical Fuzzy Soft Sets and its Aggregation Operators with Application in Decision Making, Scientia Iranica (2019).
12. 12) F. Kutlu Gündo˘gdu, C. Kahraman, A novel VIKOR method using spherical fuzzy sets and its application to warehouse site selection, Journal of Intelligent and Fuzzy Systems 37(1) (2019), 1197-1211.
13. 13) F. Kutlu Gundogdu, C. Kahraman, Extension of WASPAS with spherical fuzzy sets, Informatica 30(2) (2019), 269-292.
14. 14) F. Kutlu Gündo˘gdu, C. Kahraman, Spherical fuzzy sets and spherical fuzzy TOPSIS method, Journal of Intelligent and Fuzzy Systems 36(1) (2019), 337-352.
15. 15) T. S. Haque, A. Chakraborty, S. P. Mondal, S. Alam, Approach to solve multi-criteria group decision-making problems by exponential operational law in generalized spherical fuzzy environment, CAAI Transactions on Intelligence Technology 5(2) (2020), 106-114.
16. 16) J. C. Harsanyi, Cardinal welfare, individualistic ethics, and interpersonal comparisons of utility, J. Political Econ. (63) (1955), 309–321.
17. 17) Y. Jin, S. Ashraf, S. Abdullah, Spherical fuzzy logarithmic aggregation operators based on entropy and their application in decision support systems, Entropy 21(7) (2019), 628.
18. 18) Y. X. Ma, J. Q. Wang, J. Wang, X. H. Wu, An interval neutrosophic linguistic multi-criteria group decision-making method and its application in selecting medical treatment options, Neural Computing and Applications 28(9) (2017), 2745-2765.
19. 19) T. Mahmood, U. Kifayat, Q. Khan, N. Jan, An approach toward decision making and medical diagnosis problems using the concept of spherical fuzzy sets, Neural Computing and Applications 31 (2018), 7041-7053.
20. 20) P. K. Maji, A. R. Roy, An application of soft set in decision making problem, Comput. Math. Appl. 44 (2002), 1077-1083.
21. 21) P. K. Maji, R. Biswas, A. R. Roy, Soft set theory, Comput. Math. Appl. 45 (2003), 555-562.
22. 22) D. Molodtsov, Soft set theory-first results, Computers Mathematics with Appl. 37 (1999), 19-31.
23. 23) M. Munir, H. Kalsoom, K. Ullah, T. Mahmood, Y. M. Chu, T-spherical fuzzy Einstein hybrid aggregation operators and their applications in multi-attribute decision-making problems, Symmetry 12(3) (2020), 365-389.
24. 24) B. Pazar Varol, H. Aygün, Fuzzy soft topology, Hacettepe Journal of Mathematics and Statistics 41(3) (2012), 407-419.
25. 25) B. Pazar Varol, A. Sostak, H. Aygün, A new approach to soft topology, Hacettepe Journal of Mathematics and Statistics 41(5) (2012), 731-741.
26. 26) X. Peng, Y. Yang, J. Song, Y. Jiang, Pythagorean Fuzzy Soft Set and Its Application, Computer Engineering 41 (2015), 224–229.
27. 27) G. Wei, Picture fuzzy aggregation operators and their application to multiple attribute decision making, Journal of Intelligent and Fuzzy Systems 33(2) (2017), 713-724.
28. 28) G. Wei, Picture fuzzy Hamacher aggregation operators and their application to multiple attribute decision making, Fundamenta Informaticae 157(3) (2018), 271-320.
29. 29) Z. Xu, Intuitionistic fuzzy aggregation operators, IEEE Trans. Fuzzy Syst. 15 (2007), 1179-1187.
30. 30) R. R. Yager, On ordered weighted averaging aggregation operators in multi-criteria decision making, IEEE Trans. Syst., Man, Cybern. 18 (1988), 183–190.
31. 31) R. R. Yager, Pythagorean fuzzy subsets, In Proceedings of Joint IFSA World Congress and NAFIPS Annual Meeting, Edmonton, Canada (2013), 57-61.
32. 32) L. A. Zadeh, Fuzzy sets, Information and Control 8 (1965), 338-353.
33. 33) L. A. Zadeh, Similarity relations and fuzzy orderings, Inf. Sci. 3 (1971), 177-206.
34. 34) L. A. Zadeh, A fuzzy-alogorithmic approach to the definition of complex or imprecise concepts, Int. J. Man. Mach. Stud. 8 (1976), 249-291.