Managing the Uncertainty: From Probability to Fuzziness, Neutrosophy and Soft Sets
محورهای موضوعی : Transactions on Fuzzy Sets and Systems
Michael Voskoglou
1
(
Department of Mathematics, School of Technological Applications, Patras, Greece.
)
کلید واژه: Uncertainty, Fuzzy set (FS), Interval valued FS (IVFS), Type-2 FS, Intuitionistic FS (IFS), Neutrosophic set (NS), Rough set, Soft set, Grey system (GS),
چکیده مقاله :
The present paper reviews and compares the main theories reported in the literature for managing the existing real life uncertainty by listing their advantages and disadvantages. Starting with a comparison of the bivalent logic (including probability) and fuzzy logic, proceeds to a brief description of the primary generalizations of fuzzy sets (FSs) including interval valued FSs, type-2 FSs, intuitionistic FSs, neutrosophic sets, etc. Alternative theories related to fuzziness are also examined including grey system theory, rough sets and soft sets. The conclusion obtained at the end of this discussion is that there is no ideal model for managing the uncertainty; it all depends upon the form, the available data and the existing knowledge about the problem under solution. The combination of all the existing models, however, provides a sufficient framework for efficiently tackling several types of uncertainty appearing in real life.
[1] K. T. Atanassov, Intuitionistic Fuzzy sets, Fuzzy Sets and Systems, 20(1) (1986), 87-96.
[2] Black, M., Vagueness, Phil. of Science, 4 (1937) 427-455. Reprinted in Int. J. of General Systems, 17 (1990), 107-128.
[3] S. L. Chang, Fuzzy topological spaces, Journal of Mathematical Analysis and Applications, 24(1) (1968), 182-190.
[4] B. C. Cuong, Picture Fuzzy sets, Journal of Computer Science and Cybernetics, 30(4) (2014), 409-420.
[5] F. Dernoncourt, Fuzzy logic: Between human reasoning and Artificial Intelligence, Report, Ecole Normale Supperieure, Paris, (2011). Retrieved from: https://www.researchgate.net/publication/235333084-Fuzzy-logic-between-human-reasoning-and-artificial-intelligence
[6] J. Deng, Control problems of grey systems, Systems and Control Letters, (1982), 288-294.
[7] J. Deng, Introduction to grey system theory, The Journal of Grey System, 1 (1989), 1-24.
[8] D. Dubois and H. Prade, Interval-Valued Fuzzy Sets, Possibility Theory and Imprecise Probability, Proceedings EUSFLAT-LFA, (2005), 314-319. Retrieved from: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.207.932&rep=rep1&type=pdf
[9] J. Fox, Towards a reconciliation of fuzzy logic and standard logic, Int. J. of Man-Machine Studies, 15 (1981), 213-220.
[10] S. Haack, Do we need fuzzy logic? Int. J. of Man-Machine Studies, 11 (1979), 437-445.
[11] J.-S. R. Jang, ANFIS: adaptive network-based fuzzy inference system, IEEE Transactions on Systems, Man and Cybernetics, 23(3) (1993), 665-685.
[12] E. T. Jaynes, Probability Theory: The Logic of Science, Cambridge University Press, UK, 8th Printing, (2011), (first published, 2003).
[13] A. Kaufmann and M. Gupta, Introduction to Fuzzy Arithmetic, Van Nostrand Reinhold Company, New York, (1991).
[14] A. Kharal and B. Ahmad, Mappings on Soft Classes, New Mathematics and Natural Computation , 7(3) (2011), 471-481.
[15] G. J. Klir and T. A. Folger, Fuzzy Sets, Uncertainty and Information, Prentice-Hall, London, (1988).
[16] S. Korner, Laws of Thought, In Encyclopedia of Philosophy; Mac Millan: New York, NY, USA., 4 (1967), 414-417.
[17] B. Kosko, Fuzzy Thinking: The New Science of Fuzzy Logic, Hyperion: New York, (1993).
[18] B. Kosko, Fuzziness Vs Probability, Int. J. of General Systems, 17(2-3) (1990), 211-240.
[19] S. F. Liu and Y. Lin. (Eds.), Advances in Grey System Research, Berlin-Heidelberg: Springer, (2010).
[20] P. K. Maji, R. Biswas, and A. R. Roy, Soft Set Theory, Computers and Mathematics with Applications, 45 (2003), 555-562.
[21] E. H. Mamdani and S. Assilian, An experiment in linguistic synthesis with a fuzzy logic controller, Int. J. of Man-Machine Studies, 7(1) (1975), 1-13.
[22] J. M. Mendel, Uncertain Rule-Based Fuzzy Logic Systems: Introduction and New Directions, Prentice-Hall, Upper-Saddle River, NJ, (2001).
[23] J. M. Mendel, Fuzzy Sets for Words: a New Beginning, Proc. IEEE FUZZ Conference, St. Louis, MO, May 26-28, (2003), 37-42.
[24] D. Molodtsov, Soft Set Theory-First Results, Computers and Mathematics with Applications, 37(4-5) (1999), 19-31.
[25] R. A. Moore, R. B. Kearfort and M. J. Clood, Introduction to Interval Analysis, 2nd Printing, Philadelphia, SIAM, (1995).
[26] D. Mumford, The Dawing of the Age of Stochasticity, in V. Amoid, M. Atiyah, P. Laxand & B. Mazur (Eds.), Mathematics: Frontiers and Perspectives, AMS, (2000), 197-218.
[27] A. P. Paplinski, Neuro-Fuzzy Computing, Lecture Notes, Monash University, Australia, (2005).
[28] Z. Pawlak, Rough Sets: Aspects of Reasoning about Data, Kluer Academic Publishers, Dordrecht, (1991).
[29] D. Ramot, R. Milo, M. Friedman and A. Kandel, Complex fuzzy set, IEEE Transactions on Fuzzy Systems, 10 (2002), 171-186.
[30] F. Smarandache, Neutrosophy/Neutrosophic probability, set, and logic, Proquest, Michigan, USA, (1998).
[31] F. Smarandache, Indeterminancy in Neutrosophic Theories and their Applications, International Journal of Neutrosophic Science, 15(2) (2021), 89-97.
[32] M. Sugeno, Industrial applications of fuzzy control, Elsevier Science Pub. Co., (1985).
[33] B. K. Tripathy and K. R. Arun, Soft Sets and Its Applications, J. S. Jacob (Ed.), Handbook of Research on Generalized and Hybrid Set Structures and Applications for Soft Computing, IGI Global, Hersey, PA, (2016), 65-85.
[34] V. Torra and Y. Narukawa, On hesitant fuzzy sets and decision, 18th IEEE Int. Conference on Fuzzy Systems, Jeju island, Korea, 544 (2009), 1378-1382.
[35] E. Van Broekhoven, and B. De Baets, Fast and accurate centre of gravity defuzzification of fuzzy systems outputs defined on trapezoidal fuzzy partitions, Fuzzy Sets Syst., 157 (2006), 904-918.
[36] M. Gr. Voskoglou, Finite Markov Chain and Fuzzy Logic Assessment Models: Emerging Research and Opportunities, Create Space Independent Publishing Platform, Amazon, Columbia, SC., USA., (2017).
[37] M. Gr. Voskoglou, Methods for Assessing Human-Machine Performance under Fuzzy Conditions, Mathematics, 7(3), article 230, (2019). DOI:10.3390/math7030230
[38] M. Gr. Voskoglou and E. Athanassopoulos, The Importance of Bayesian Reasoning in Everyday Life and Science, Int. J. of Education, Development, Society and Technology, 8(2) (2020), 24-33.
[39] M. Gr. Voskoglou, Fuzzy Control Systems, WSEAS Transactions on Systems, 19 (2020), 295-300. DOI:10.37394/23202.2020.19.33
[40] M. Gr. Voskoglou, Use of Soft Sets and the Bloom’s Taxonomy for Assessing Learning Skills, Transactions on Fuzzy Sets and Systems, 1(1) (2022), 106-113.
[41] M. Gr. Voskoglou, A Combined Use of Soft Sets and Grey Numbers in Decision Making, Journal of Computational and Cognitive Engineering https://doi.org/10.47852/bonviewjcce2202237, (2022).
[42] I. G. Umbers and P. J. King, An analysis of human decision-making in cement kiln control and the implications for automation, Int. J. of Man-Mach. Stud., 12 (1980), 11-23.
[43] R. R. Yager, Pythagorean fuzzy subsets, in Proceedings of Joint IFSA World Congress and NAFIPS Annual Meeting, Edmonton, Canada, (2013), 57-61.
[44] L. A. Zadeh, Fuzzy Sets, Information and Control, 8 (1965), 338-353.
[45] L. A. Zadeh, Outline of a new approach to the analysis of complex systems and decision processes, IEEE Trans. Syst. Man Cybern., 3 (1980), 28-44.
[46] L. A. Zadeh, The Concept of a Linguistic Variable and its Application to Approximate Reasoning, Information Science, 8 (1975), 199-249.
[47] L. A. Zadeh, Fuzzy logic=computing with words, IEEE Trans. on Fuzzy Systems, 4 (1996), 103-111.
[48] Z. Zhang and Z. Hu, Extension of TOPSIS to Multiple Criteria Decision Making with Pythagorean Fuzzy Sets, Int. J. of Information Systems, 29(12) (2014), 1061-1078.
