Abstract
The goal of this study is to provide a framework for a novel concept, which is highly effective at expressing ambiguous information in two dimensions. As a result, of the absence of a neutral degree in interval-valued complex Pythagorean fuzzy sets, we developed the theory of interval-valued complex spherical fuzzy sets. Moreover, a network security system prevents a variety of potential threats from entering or spreading within a network, thereby protecting a company's infrastructure from damage. This is a broad and comprehensive term that refers to processes, regulations, and settings pertaining to network use, accessibility, and overall threat prevention, as well as hardware and software solutions. It is the field of cybersecurity that focuses on protecting computer networks against cyber threats. To represent and solve the given problem, we establish the new mathematical concept known as interval-valued complex spherical fuzzy relations, defined as the cartesian product of two interval-valued complex spherical fuzzy sets. Furthermore, a range of interval-valued complex spherical fuzzy relations has been developed, characterized by interesting properties and outcomes. The interval-valued complex spherical fuzzy set and interval-valued complex spherical fuzzy relations supersede all pre-existing frameworks and methods for dealing with fuzziness. Additionally, this research investigates the application of interval-valued complex spherical fuzzy relations to assess the impact of various communication network security measures and the threats encountered by these networks. We further examine the positive, neutral, and negative effects of one factor on the other factors in the proposed applications, denoted by the membership grade, abstinence grade, and non-membership grade, respectively. Finally, we examine the proposed method to reveal its advantages and superiority to the current approaches.
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References
Zadeh, L.A.: Information and control. Fuzzy Sets 8(3), 338–353 (1965)
Mendel, J.M.: Fuzzy logic systems for engineering: a tutorial. Proc. IEEE 83, 345–377 (1995)
Zadeh, L.A.: The COIIC of a linguistic variable and its application to approximate reasoning (I), (II), (III). Inform. Sci. 8, 199–249 (1975)
Bustince, H., Burillo, P.: Mathematical analysis of interval-valued fuzzy relations: application to approximate reasoning. Fuzzy Sets Syst. 113(2), 205–219 (2000)
Ouannou, A., Brouri, A., Kadi, L., Oubouaddi, H.: Identification of switched reluctance machine using fuzzy model. Int. J. Syst. Assur. Eng. Manag. 2, 1–14 (2022)
Deschrijver, G., Kerre, E.E.: On the relationship between some extensions of fuzzy set theory. Fuzzy Sets Syst. 133(2), 227–235 (2003)
Goguen, J.A., Jr.: Concept representation in natural and artificial languages: axioms, extensions and applications for fuzzy sets. Int. J. Man Mach. Stud. 6(5), 513–561 (1974)
Román Flores, H., Barros, L.C., Bassanezi, R.C.: A note on Zadeh’s extensions. Fuzzy Sets Syst. 117(3), 327–331 (2001)
Gehrke, M., Walker, C., Walker, E.: Some comments on interval valued fuzzy sets! Structure 1, 2 (1996)
Bustince, H.: Indicator of inclusion degree for interval-valued fuzzy set: application to approximate reasoning based on interval-valued fuzzy sets. Int. J. Approx. Reason. 23(3), 137–209 (2000)
Ramot, D., Milo, R., Friedman, M., Kandel, A.: Complex fuzzy sets. IEEE Trans. Fuzzy Syst. 10(2), 171–186 (2002)
Ramot, D., Friedman, M., Langholz, G., Kandel, A.: Complex fuzzy logic. IEEE Trans. Fuzzy Syst. 11, 450–461 (2003)
Greenfield, S., Chiclana, F., & Dick, S.: Interval-valued complex fuzzy logic. In: 2016 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE) (pp. 2014–2019).
Yazdanbakhsh, O., Dick, S.: A systematic review of complex fuzzy sets and logic. Fuzzy Sets Syst. 338, 1–22 (2018)
Nasir, A., Jan, N., Gumaei, A., Khan, S.U.: Medical diagnosis and life span of sufferer using interval valued complex fuzzy relations. IEEE Access 9, 93764–93780 (2021)
Chen, Z., Aghakhani, S., Man, J., Dick, S.: ANCFIS: A neurofuzzy architecture employing complex fuzzy sets. IEEE Trans. Fuzzy Syst. 19(2), 305–322 (2010)
Tamir, D.E., Rishe, N.D., Kandel, A.: Complex fuzzy sets and complex fuzzy logic an overview of theory and applications. Fifty Years Fuzzy Logic Its Appl. 15, 661–681 (2015)
Dai, S., Bi, L., Hu, B.: Distance measures between the interval-valued complex fuzzy sets. Mathematics 7(6), 549 (2019)
Greenfield, S., Chiclana, F., & Dick, S.: Join and meet operations for interval-valued complex fuzzy logic. In: 2016 Annual Conference of the North American Fuzzy Information Processing Society (NAFIPS) (pp. 1–5), (2016).
Atanassov, K.T.: Intuitionistic fuzzy sets. Fuzzy Sets Syst 20(1), 87–96 (1986)
Kumar, P.S.: Finding the solution of balanced and unbalanced intuitionistic fuzzy transportation problems by using different methods with some software packages. In: Handbook of Research on Applied AI for International Business and Marketing Applications (pp. 278–320). (2021).
Kumar, P.S.: Algorithms for solving the optimization problems using fuzzy and intuitionistic fuzzy set. Int. J. Syst. Assurance Eng. Manag. 11(1), 189–222 (2020)
Kumar, P.S.: Integer programming approach for solving solid and intuitionistic fuzzy solid assignment problems. Int. J. Logist. Syst. Manag. 10, 661–675 (2019)
Kumar, P.S.: PSK method for solving type-1 and type-3 fuzzy transportation problems. Int. J. Fuzzy Sys. Appl. (IJFSA) 5(4), 121–146 (2016)
Kumar, P.S.: Developing a new approach to solve solid assignment problems under intuitionistic fuzzy environment. Int. J. Fuzzy Sys. Appl. (IJFSA) 9(1), 1–34 (2020)
Burillo, P., Bustince, H.: Intuitionistic fuzzy relations (Part I). Mathware and soft computing 2(1), 5–38 (1995)
Li, D.F.: Multiattribute decision making models and methods using intuitionistic fuzzy sets. J. Comput. Syst. Sci. 70(1), 73–85 (2005)
Atanassov, K.T.: Interval valued intuitionistic fuzzy sets. In: Intuitionistic Fuzzy Sets (pp. 139–177). (1999).
Alkouri, A.M.D.J.S., & Salleh, A.R.: Complex intuitionistic fuzzy sets. In: AIP conference proceedings (vol. 1482, no. 1, pp. 464–470). (2012).
Garg, H., Rani, D.: Complex interval-valued intuitionistic fuzzy sets and their aggregation operators. Fund. Inform. 164(1), 61–101 (2019)
Xiao, F.: A distance measure for intuitionistic fuzzy sets and its application to pattern classification problems. IEEE Trans. Syst. Man Cybern. Syst. 51(6), 3980–3992 (2019)
Vlachos, I.K., Sergiadis, G.D.: Intuitionistic fuzzy information–applications to pattern recognition. Pattern Recogn. Lett. 28(2), 197–206 (2007)
Lee, K.M., Lee, K.M., & Cios, K.J.: Comparison of interval-valued fuzzy sets, intuitionistic fuzzy sets, and bipolar-valued fuzzy sets. In: Computing and information technologies: exploring emerging technologies (pp. 433–439), (2001).
Grzegorzewski, P.: Distances between intuitionistic fuzzy sets and/or interval-valued fuzzy sets based on the Hausdorff metric. Fuzzy Sets Syst. 148, 319–328 (2004)
Nasir, A., Jan, N., Gumaei, A., Khan, S.U., Albogamy, F.R.: Cybersecurity against the loopholes in industrial control systems using interval-valued complex intuitionistic fuzzy relations. Appl. Sci. 11(16), 7668 (2021)
Jan, N., Nasir, A., Alhilal, M.S., Khan, S.U., Pamucar, D., Alothaim, A.: Investigation of cyber-security and cyber-crimes in oil and gas sectors using the innovative structures of complex intuitionistic fuzzy relations. Entropy 23(9), 1112 (2021)
Ali, M., Tamir, D.E., Rishe, N.D., & Kandel, A.: Complex intuitionistic fuzzy classes. In: 2016 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE) (pp. 2027–2034). (2016).
Liu, Y., Jiang, W.: A new distance measure of interval-valued intuitionistic fuzzy sets and its application in decision making. Soft. Comput. 24(9), 6987–7003 (2020)
Park, D.G., Kwun, Y.C., Park, J.H., Park, I.Y.: Correlation coefficient of interval-valued intuitionistic fuzzy sets and its application to multiple attribute group decision making problems. Math. Comput. Model. 50(9–10), 1279–1293 (2009)
Cuong, B.C., Kreinovich, V.: Picture fuzzy sets. J. Comput. Sci. Cybernet. 30(4), 409–420 (2014)
Singh, P.: Correlation coefficients for picture fuzzy sets. J. Intell. Fuzzy Syst. 28(2), 591–604 (2015)
Bo, C., Zhang, X.: New operations of picture fuzzy relations and fuzzy comprehensive evaluation. Symmetry 9(11), 268 (2017)
Khalil, A.M., Li, S.G., Garg, H., Li, H., Ma, S.: New operations on interval-valued picture fuzzy set, interval-valued picture fuzzy soft set and their applications. IEEE Access 7, 51236–51253 (2019)
Akram, M., Bashir, A., Garg, H.: Decision-making model under complex picture fuzzy Hamacher aggregation operators. Comput. Appl. Math. 39(3), 1–38 (2020)
Shit, C., Ghorai, G., Xin, Q., Gulzar, M.: Harmonic aggregation operator with trapezoidal picture fuzzy numbers and its application in a multiple-attribute decision-making problem. Symmetry 14(1), 135 (2022)
Shit, C., Ghorai, G.: Multiple attribute decision-making based on different types of Dombi aggregation operators under Fermatean fuzzy information. Soft. Comput. 25(22), 13869–13880 (2021)
Ali, Z., Mahmood, T., Yang, M.S.: TOPSIS method based on complex spherical fuzzy sets with Bonferroni mean operators. Mathematics 8(10), 1739 (2020)
Akram, M., Kahraman, C., Zahid, K.: Group decision-making based on complex spherical fuzzy VIKOR approach. Knowl.-Based Syst. 216, 106793 (2021)
Acknowledgements
This work was supported in part by the “Regional Innovation Strategy (RIS)” through the National Research Foundation of Korea(NRF) funded by the Ministry of Education(MOE) (2021RIS-001 (1345341783)) and the Brain Pool program funded by the Ministry of Science and ICT through the National Research Foundation of Korea (2022H1D3A2A02060097).
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Jan, N., Gwak, J., Hussain, S. et al. Mathematical Investigation of Communication and Network Securities Under Interval-Valued Complex Spherical Fuzzy Information. Int. J. Fuzzy Syst. 26, 87–104 (2024). https://doi.org/10.1007/s40815-023-01578-y
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DOI: https://doi.org/10.1007/s40815-023-01578-y