Complex fuzzy sets with applications in decision-making

Document Type : Research Paper

Authors

1 Department of Mathematics, COMSATS University Islamabad, Islamabad Campus, Pakistan

2 Department of Mathematics, COMSATS University Islamabad, Abbottabad Campus, Pakistan

Abstract

In this paper, we discussed the conjunctive
normal form, disjunctive normal form, duality principle, equality of two sets
and a semi Boolean algebra of complex fuzzy sets (CFSs). We established some
basic results and particular examples with respect to standard complex fuzzy
intersection, standard complex fuzzy union and standard complex fuzzy
complement functions with the same function for determining the phase term. We
used CFSs in signals and systems because the behavior of CFSs is similar to
Fourier transforms in certain cases. Moreover, we developed a new algorithm
using a Cartesian product of complex fuzzy sets for applications in signals
and systems by which we identified a reference signal out of the large number
of signals detected by a digital receiver.

Keywords


[1] M. Ali, F. Smarandache, Complex neutrosophic set, Neural Computing and Applications, 28(7) (2017), 1817-1834.
[2] A. Alkouri, A. R. Salleh, Complex intuitionistic fuzzy sets, In: International Conference on Fundamental and Applied Sciences, AIP Conference Proceedings, 1482 (2012), 464-470.
[3] K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20(1) (1986), 87-96.
[4] K. T. Atanassov, Interval valued intuitionistic fuzzy sets, In Intuitionistic Fuzzy Sets, Physica, Heidelberg, (1999), 139-177.
[5] T. Aydın, S. Enginoğlu, Interval-valued intuitionistic fuzzy parameterized interval-valued intuitionistic fuzzy soft sets and their application in decision-making, Journal of Ambient Intelligence and Humanized Computing, 12(1) (2021), 1541-1558.
[6] L. Bi, Z. Zeng, B. Hu, S. Dai, Two classes of entropy measures for complex fuzzy sets, Mathematics, 7(1) (2019), 96.
[7] T. Y. Chen, The likelihood-based optimization ordering model for multiple criteria group decision making with Pythagorean fuzzy uncertainty, Neural Computing and Applications, 33(10) (2021), 4865-4900.
[8] S. Dai, A generalization of rotational invariance for complex fuzzy operations, IEEE Transactions on Fuzzy Systems, 29(5) (2020), 1152-1159.
[9] S. Dai, Complex fuzzy ordered weighted distance measures, Iranian Journal of Fuzzy Systems, 17(6) (2020), 107-114.
[10] S. Dai, L. Bi, B. Hu, Distance measures between the interval-valued complex fuzzy sets, Mathematics, 7(6) (2019), 549.
[11] S. Dick, Toward complex fuzzy logic, IEEE Transactions on Fuzzy Systems, 13(3) (2005), 405-414.
[12] H. Garg, A novel accuracy function under interval-valued Pythagorean fuzzy environment for solving multicriteria decision making problem, Journal of Intelligent and Fuzzy Systems, 31(1) (2016), 529-540.
[13] A. Garrido, Fuzzy mathematical analysis, In Conference Paper, September 2007.
[14] W. Huang, F. Zhang, S. Xu, A complete ranking method for interval-valued intuitionistic fuzzy numbers and its applications to multicriteria decision making, Soft Computing, 25(3) (2021), 2513-2520.
[15] N. N. Karnik, J. M. Mendel, Operations on type-2 fuzzy sets, Fuzzy Sets and Systems, 122(2) (2001), 327-348.
[16] Y. Liu, F. Liu, An adaptive neuro-complex-fuzzy-inferential modeling mechanism for generating higher-order TSK models, Neurocomputing, 365 (2019), 94-101.
[17] X. Ma, J. Zhan, M. Khan, M. Zeeshan, S. Anis, A. S. Awan, Complex fuzzy sets with applications in signals, Computational and Applied Mathematics, 38(4) (2019), 1-34.
[18] T. Mahmood, A novel approach towards bipolar soft sets and their applications, Journal of Mathematics, (2020). DOI: 10.1155/2020/4690808.
[19] T. T. Ngan, L. T. H. Lan, M. Ali, D. Tamir, L. H. Son, T. M. Tuan, N. Rishe, A. Kandel, Logic connectives of complex fuzzy sets, Romanian Journal of Information Science and Technology, 21(4) (2018), 344-358.
[20] L. Pan, X. Gao, Y. Deng, K. H. Cheong, The constrained Pythagorean fuzzy sets and its similarity measure, IEEE Transactions on Fuzzy Systems, 30(4) (2021), 1102-1113.
[21] D. Ramot, R. Milo, M. Friedman, A. Kandel, Complex fuzzy sets, IEEE Transactions on Fuzzy Systems, 10(2) (2002), 171-186.
[22] P. Rani, A. R. Mishra, A. Saha, D. Pamucar, Pythagorean fuzzy weighted discrimination-based approximation approach to the assessment of sustainable bioenergy technologies for agricultural residues, International Journal of Intelligent Systems, 36(6) (2021), 2964-2990.
[23] I. W. Selesnick, G. Schuller, The discrete fourier transform, 2nd chapter of the book The transform and data compression Handbook, editted by K. R. Rao and P. C. Yip, CRC Press, Boca Raton, 2001.
[24] G. Selvachandran, S. G. Quek, L. T. H Lan, N. L. Giang, W. Ding, M. Abdel-Basset, V. H. C. Albuquerque, A new design of Mamdani complex fuzzy inference system for multi-attribute decision making problems, IEEE Transactions on Fuzzy Systems, 29(4) (2019), 716-730.
[25] V. Torra, Hesitant fuzzy sets, International Journal of Intelligent Systems, 25(6) (2010), 529-539.
[26] T. M. Tuan, L. T. H. Lan, S. Y. Chou, T. T. Ngan, L. H. Son, N. L. Giang, M. Ali, M-CFIS-R: Mamdani complex fuzzy inference system with rule reduction using complex fuzzy measures in granular computing, Mathematics, 8(5) (2020), 707.
[27] Y. Xue, Y. Deng, H. Garg, Uncertain database retrieval with measure–Based belief function attribute values under intuitionistic fuzzy set, Information Sciences, 546 (2021), 436-447.
[28] R. R. Yager, Pythagorean fuzzy subsets, In 2013 Joint IFSA World Congress and NAFIPS Annual Meeting (IFSA/NAFIPS), IEEE, 57-61. DOI: 10.1109/IFSA-NAFIPS.2013.6608375.
[29] O. Yazdanbakhsh, S. Dick, FANCFIS: Fast adaptive neuro-complex fuzzy inference system, International Journal of Approximate Reasoning, 105 (2019), 417-430.
[30] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353.
[31] G. Zhang, T. S. Dillon, K. Y. Cai, J. Ma, J. Lu, Operation properties and δ-equalities of complex fuzzy sets, International Journal of Approximate Reasoning, 50(8) (2009), 1227-1249.
[32] Y. Zhou, W. Cao, L. Liu, S. Agaian, C. P. Chen, Fast fourier transform using matrix decomposition, Information Sciences, 291 (2015), 172-183.