Eigen spherical fuzzy set and its application to decision-making problem

Document Type : Article


Department of Mathematics, Jaypee University of Information Technology, Waknaghat, Solan, Pin-173 234, Himachal Pradesh, India


Eigen fuzzy set of a fuzzy relation often occurs to be invariant under different computational aspects. The present communication introduces the novel concept of eigen spherical fuzzy set of spherical fuzzy relation along with various composition operators for the first time. We have proposed two algorithms to determine the greatest eigen spherical fuzzy sets and least eigen spherical fuzzy sets using the $max-min$ and $min-max$ composition operators respectively and illustrated the steps with the help of flow charts. Further, two numerical examples related to different fields of decision-making problems have been taken into account for illustrating the proposed methodology. The scope of future work in the field of image information retrieval, genetic algorithm for image reconstruction and notion of eigen spherical fuzzy soft sets/matrices has been duly outlined. The comparative remarks and advantages of the proposed eigen spherical fuzzy sets have also been included for a better readability.


Main Subjects

1. Zadeh, L.A. Fuzzy sets", Information and Control, 8,
pp. 338{353 (1965).
2. Atanassov, K.T. Intuitionistic fuzzy sets", Fuzzy Sets
and Systems, 20, pp. 87{96 (1986).
3. Yager, R.R. Pythagorean fuzzy subsets", In: Proceedings
of Joint IFSA World Congress and NAFIPS
Annual Meeting, Edmonton, Canada, pp. 57{61
4. Peng, X.D. New operations for interval-valued
pythagorean fuzzy set", Scientia Iranica, 26(2), pp.
1049{1076 (2019). DOI: 10.24200/sci.2018.5142.1119
5. Cuong, B.C. Picture fuzzy sets- rst results Part 1",
Seminar on neuro{fuzzy systems with applications,
Institute of Mathematics, Hanoi (2013).
6. Cuong, B.C. Picture fuzzy sets", Journal of Computer
Science and Cybernetics, 30(4), pp. 409{420 (2014).
7. Mahmood, T., Kifayat, U., Khan, Q., and Naeem,
J. An approach toward decision-making and medical
diagnosis problems using the concept of spherical fuzzy
sets", Neural Computing and Applications (2018).
DOI: https : ==doi:org=10:1007=s00521􀀀018􀀀3521􀀀2
8. Kifayat, U., Mahmood, T., and Naeem, J. Similarity
measures for T-spherical fuzzy sets with applications
in pattern recognition", Symmetry, 10, p. 193 (2018).
9. Zadeh, L.A. Similarity relations and fuzzy orderings",
Information Science, 3, pp. 177{200 (1971).
10. Sanchez, E. Resolution of eigen fuzzy sets equations",
Fuzzy Set and Systems, 1, pp. 69{74 (1978).
11. Sanchez, E. Eigen fuzzy sets and fuzzy relations",
Journal of Mathematical Analysis and Applications,
81, pp. 399{421 (1981).
12. Nobuhara, H., Bede, B., and Hirota, K. On various
eigen fuzzy sets and their application to Image reconstruction",
Information Sciences, 176(20), pp. 2988{
3010 (2006).
13. Di Martino, F., Sessa, S., and Nobuhara, H. Eigen
fuzzy sets and image information retrieval", Proceedings
of the International Conference on Fuzzy Information
Systems, 3, pp. 1385{1390, Budapest (2004).
14. Nobuhara, H. and Hirota, K. A solution for eigen
fuzzy sets of adjoint max-min composition and its
application to image analysis", IEEE International
Symposium on Intelligent Signal Processing, Budapest,
Hungary, pp. 27{30 (2003).
15. Di Martino, F. and Sessa, S., A Genetic Algorithm
Based on Eigen Fuzzy Sets for Image Reconstruction,
F. Masulli, S. Mitra, and G. Pasi (Eds.): LNAI 4578,
Springer, pp. 342{348 (2007).
16. Andersson, E.R., The Greatest and the Least Eigen
Fuzzy Sets in Evaluation of the Drug E ectiveness
Levels, L. Rutkowski et al. (Eds.): ICAISC 2006, LNAI
4029, Springer, pp. 982{987 (2006).
17. Avrachenkov, K.E. and Sanchez, E. Fuzzy Markov
chains and decision-making", Fuzzy Optimization and
Decision Making, 1, pp. 143{159 (2002).
18. Tang, J. and Meng, F. An approach to interval-valued
intuitionistic fuzzy decision making based on induced
generalized symmetrical Choquet Shapley operator",
Scientia Iranica, 25(3), pp. 1456{1470 (2018).
19. Sanchez, E. Resolution of composite fuzzy relation
equation", Information and Control, 30, pp. 38{48
20. Sanchez, E. Solution of fuzzy equations with extended
operations", Fuzzy Sets and Systems, 12, pp. 237{248
21. Di Nola, A., Pedricz, W., Sessa, S., and Sanchez, E.
Fuzzy relation equations theory as a basis of fuzzy
modelling: An overview", Fuzzy Sets and Systems, 40,
pp. 415{429 (1991).
22. Goetschel, R. and Voxman, W. Eigen fuzzy number
sets", Fuzzy Sets and Systems, 16, pp. 75{85 (1985).
23. Dubois, D. and Prade, H. Operation on fuzzy numbers",
International Journal of Systems Science, 9(9),
pp. 613{626 (1978).
24. Fernandez, M.J., Suarez, F. and Gil, P., T-eigen fuzzy
sets", Information Sciences, 75, pp. 63{80 (1993).
25. Tazaki, E. and Amagasa, M. Heuristic structure
synthesis in a class of systems using a fuzzy automaton",
In IEEE Transactions on Systems, Man, and
Cybernetics, 9(2), pp. 73{79 (1979).
26. Cao, Z.Q. The eigen fuzzy sets of a fuzzy matrix" in
Approximate Reasoning in Decision Analysis, M. M.
Gupta and E. Sanchez, Eds., North-Holland, Amsterdam,
pp. 61{63 (1982).
530 A. Guleria and R.K. Bajaj/Scientia Iranica, Transactions E: Industrial Engineering 28 (2021) 516{531
27. Gerstenkorn, T. and Rakus, E. An application of
fuzzy set theory to di erentiating the e ectiveness of
drugs in treatment of in
ammation of genital organs",
Fuzzy Sets and Systems, 68, pp. 327{333 (1994).
28. Atanassov, K.T. Geometrical interpretation of the
elements of the intuitionistic fuzzy objects", Preprint
IMMFAIS1{89, So a, 1989. Reprinted: Int. J. Bioautomation,
20(1), pp. S27{S42 (2016).
29. Klement, E.P., Mesiar, R., and Pap, E., Triangular
Norms, Kluwer Academic Publishers, Dordrecht
30. Klement, E.P., Mesiar, R., and Pap, E. Triangular
norms. Position paper I: basic analytical and algebraic
properties", Fuzzy Sets and Systems, 143, pp. 5{26
31. Sandhya, S. and Garg, R. Implementation of multicriteria
decision-making approach for the team leader
selection in IT sector", Journal of Project Management,
1(2), pp. 67{75 (2016).
32. Sandhya, S., Garg, R., and Kumar, R. Computational
MADM evaluation and ranking of cloud service
providers using distance-based approach", International
Journal of Information and Decision Sciences,
10(3), pp. 222{234 (2018).
33. Garg, R., Sharma, R., and Sharma, K. Ranking
and selection of commercial o -the-shelf using fuzzy
distance-based approach", Decision Science Letters,
5(2), pp. 201{210 (2016).
34. Garg, R., Kumar, R., and Garg, S. MADM-based
parametric selection and ranking of E-learning websites
using fuzzy COPRAS", IEEE Transactions on
Education, 62(1), pp. 11{18 (2019).
35. Jain, D., Garg, R., Bansal, A., and Saini, K.K.
Selection and ranking of E-learning websites using
weighted distance-based approximation", Journal of
Computers in Education, 3(2), pp. 193{207 (2016).
36. Garg, R. Optimal selection of E-learning websites using
multiattribute decision-making approaches", Journal
of Multi-Criteria Decision Analysis, 24(3{4), pp.
187{196 (2017).
37. Garg, R. and Arora, S. Performance evaluation and
selection of nancial fraud detection models using
MCDM approach", International Journal of Recent
Research Aspects, 4(2), pp. 172{178 (2017).
38. Molodstov, D.A. Soft set theory- rst result", Computers
and Mathematics with Application, 27, pp. 19{
31 (1999).
39. Kumar, T. and Bajaj, R.K. On complex intuitionistic
fuzzy soft sets with distance measures and entropies",
Journal of Mathematics, 2014, Article ID-972198
40. Peng, X.D., Yang, Y., Song, J., and Jiang, Y.
Pythagorean fuzzy soft set and its application", Computer
Engineering, 41, pp. 224{229 (2015).
41. Thirunavukarasu, P., Suresh, R., and Ashokkumarm,
V. Theory of complex fuzzy soft set and its applications",
International Journal for Innovative Research
in Science & Technology, 3(10), pp. 13{18 (2017).
42. Naim, C. and Serdar, E. Soft matrix theory and its
decision making", Computers and Mathematics with
Applications, 59, pp. 3308{3314 (2010).
43. Tas, N., Ozgur, N.Y., and Demir, P. An application
of soft set and fuzzy soft set theories to stock management",
Journal of Natural and Applied Sciences, 21(3),
pp. 791{796 (2017).
44. Alcantud, J.C.R., Rambaud S.C., and Torrecillas
M.J.M. Valuation fuzzy soft sets: A
exible fuzzy soft
set based decision making procedure for the valuation
of assets", Symmetry, 9(11), p. 253 (2017).
45. Guleria, A. and Bajaj, R.K. On pythagorean fuzzy
soft matrices, operations and their applications in decision
making and medical diagnosis", Soft Computing,
23(17), pp. 7889{7900 (2018). DOI: 10:1007=s00500􀀀 018 􀀀 3419 􀀀 z
46. Garg, R.K., Sharma, K., Nagpal, C.K., Garg, R.,
Kumar, R., and Sandhya, Ranking of software engineering
metrics by fuzzy-based matrix methodology",
Software: Testing, Veri cation and Reliability, 23(2),
pp. 149{168 (2013).
47. Garg, R., Sharma, R., and Sharma, K. MCDM based
evaluation and ranking of commercial o -the-shelf
using fuzzy based matrix method", Decision Science
Letters, 6(2), pp. 117{136 (2017).
48. Bansal, A., Kumar, B., and Garg, R. Multi-criteria
decision-making approach for the selection of software
e ort estimation model", Management Science Letters,
7(6), pp. 285{296 (2017).
49. Garg, R. Performance evaluation and selection of software
e ort estimation models based on multi-criteria
decision-making method", International Journal of
Recent Research Aspects, 4(3), pp. 252{257 (2017).
50. Bajaj, R.K. and Guleria, A. Dimensionality reduction
technique in decision making using pythagorean
fuzzy soft matrices", Recent Patents on Computer
Science, 13(3), pp. 406{413 (2020). DOI: