A Purview and the inter-relations of “Fuzzy Sets, Intuitionistic Fuzzy Sets, Rough Sets, and their Extensions”

Babita Mishra¹

1. Department of Mathematics, Mahatma Gandhi Central University, Motihari, India.

Section:Research Paper, Product Type: Journal-Paper
Vol.9 , Issue.6 , pp.43-49, Dec-2022

Online published on Dec 31, 2022

Copyright © Babita Mishra . This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
 

View this paper at   Google Scholar | DPI Digital Library

XML View     PDF Download

How to Cite this Paper

94 Views    137 Downloads    35 Downloads

Abstract :
The handling of uncertainty and vagueness is a very tough task for any decision-making body. It is not possible to mathematically formulate a realistic problem by ignoring the terms like uncertain and vague. Therefore taking into consideration such types of linguistic variables, it is an essential tool for any decision-maker to handle such types of situations by using “fuzzy sets, intuitionistic fuzzy sets, rough sets, etc.” In this paper, I would like to define fuzzy sets and how gradually all the extensions or purview as a “rough set, type-2 fuzzy set, and intuitionistic fuzzy sets” are developed have been discussed. The outline of this paper includes historical development, definitions, and the relevance of the topic through real-world examples. The most important feature of this paper is to incorporate the same example in different extensions to understand the difference and needs of the extensions. Also, the inter-relations of all the sets (cited above) have been discussed.

Key-Words / Index Term :
Fuzzy sets, Rough set, Type-2 fuzzy set, Intuitionistic fuzzy sets, Uncertainty, Vagueness

References :
[1] L.A. Zadeh, “Fuzzy Sets,” Information and Control, Vol. 8, No.3, pp.338-353, 1965.
[2] J.A. Goguen, “L-Fuzzy Sets,” Journal of Mathematical Analysis and Applications, Vol. 18, No. 1, pp. 145-174, 1967.
[3] L.A. Zadeh, “The Concept of a Linguistic Variable and its Application to Approximate Reasoning-I,” Information Sciences, Vol. 8, No. 3, pp. 199-249, 1975.
[4] R.E., Bellman, L.A. Zadeh, “Decision-Making in a Fuzzy Environment,” Management Science, Vol. 17, No. 4, pp. B141-B164, 1970.
[5] H. Tanaka, T. Okuda, K. Asai, “Fuzzy Mathematical Programming,” Transactions of the Society of Instrument and Control Engineers, Vol. 9, No. 5, pp. 607-613, 1973.
[6] H. Tanaka, K. Asai, “Fuzzy Linear Programming Problems with Fuzzy Numbers,” Fuzzy Sets and Systems, Vol. 13, No. 1, pp. 1-10, 1984.
[7] S. Bodjanova, “Approximation of Fuzzy Concepts in Decision Making,” Fuzzy Sets and Systems, Vol. 85, No. 1, pp. 23-29, 1997.
[8] K.T. Atanassov, “Intuitionistic Fuzzy Sets,” Fuzzy Sets and Systems, Vol. 20, pp. 87-96, 1986.
[9] K.T. Atanassov, G. Gargov, “Interval-Valued Intuitionistic Fuzzy Sets,” Fuzzy Sets and Systems, Vol. 31, pp. 343-349, 1989.
[10] K.T. Atanassov, “New Operations Defined Over the Intuitionistic Fuzzy Sets,” Fuzzy Sets and Systems, Vol. 61, No. 2, pp. 137-142, 1994.
[11] K.T. Atanassov, “Intuitionistic Fuzzy Sets Theory and Applications,” Springer-Verlag Berlin Heidelberg GmbH, 1999.
[12] K.T. Atanassov, “Interval Valued Intuitionistic Fuzzy Sets,” In Intuitionistic Fuzzy Sets, pp. 139-177 Physica, Heidelberg, 1999.
[13] Zeshui Xu, Ronald R. Yager, “Some Geometric Aggregation Operators Based on Intuitionistic Fuzzy Sets,” International Journal of General Systems, Vol.35, No.4, pp.417-433, 2006.
[14] Hua Zhao, Zeshui Xu, Mingfang Ni, Shousheng Liu, “Generalized Aggregation Operators for Intuitionistic Fuzzy Sets,” International Journal of Intelligent Systems, Vol.25, No.1, pp.1-30, 2009.
[15] P.P. Angelov, “Optimization in an Intuitionistic Fuzzy Environment,” Fuzzy Sets and Systems, Vol. 86, No. 3, pp. 299-306, 1997.
[16] Z. Pawlak, “Rough Sets,” International Journal of Computer & Information Sciences, Vol. 11, No. 5, pp. 341-356, 1982.
[17] D. Dubois, H. Prade, “Two Fold Fuzzy Sets and Rough Sets—Some Issues in Knowledge Representation,” Fuzzy Sets and Systems, Vol. 23, No. 1, pp. 3-18, 1987.
[18] D. Dubois, H. Prade, “Rough Fuzzy Sets and Fuzzy Rough Sets,” International Journal of General System, Vol. 17, No. 2-3, pp. 191-209, 1990.
[19] D. Dubois, H. Prade, “The Three Semantics of Fuzzy Sets,” Fuzzy Sets and Systems, Vol. 90, No. 2, pp. 141-150, 1997.
[20] S. Nanda, S. Majumdar, “Fuzzy Rough Sets,” Fuzzy Sets and Systems, Vol. 45, No. 2, pp. 157-160, 1992.
[21] J. N. Mordeson, “Rough Set Theory Applied to (Fuzzy) Ideal Theory, ” Fuzzy Sets and Systems, Vol. 121, No. 2, pp. 315-324, 2001.
[22] N. N. Morsi, M. M. Yakout, “Axiomatics for Fuzzy Rough Sets,” Fuzzy Sets and Systems, Vol. 100, No. 1-3, pp. 327-342, 1998.
[23] L.I. Kuncheva, “Fuzzy Rough Sets: Application to Feature Selection,” Fuzzy Sets and Systems, Vol.51, No.2, pp.147-153, 1992.
[24] M. Kryszkiewicz, “Rough Set Approach to Incomplete Information Systems,” Information Sciences, Vol.112, No.1-4, pp.39-49, 1998.
[25] K. Chakrabarty, R. Biswas, S. Nanda, “Fuzziness in rough sets,” Fuzzy sets and systems, Vol.110, No.2, pp. 247-251, 2000.
[26] W.Z. Wu, J.S. Mi, W.X. Zhang, “Generalized Fuzzy Rough Sets,” Information Sciences, Vol.151, pp. 263-282, 2003.
[27] A. SOSTAKS, “Mathematics in the Context of Fuzzy Sets: Basic Ideas, Concepts, and Some Remarks on the History and Recent Trends of Development,” Mathematical Modelling and Analysis, Vol.16, No.2, pp.173-198, 2011.
[28] Dexue Zhang, “A Natural Topology for Fuzzy Numbers,” Journal of Mathematical Analysis and Applications, Vol.264, pp.344–353, 2001.
[29] Jerry M. Mendel, “Advances in type-2 fuzzy sets and systems,” Information Sciences, Vol.177, pp.84–110, 2007.
[30] Robert John, “Type 2 Fuzzy Sets: An Appraisal of Theory And Applications,” International Journal Of Uncertainty, Fuzziness And Knowledge-Based Systems, Vol.6, No.6, pp.563-576, 1998.
[31] Turkay Dereli, Adil Baykasoglu, Koray Altun, Alptekin Durmusoglu, I. Burhan Turksen, “Industrial Applications of Type-2 Fuzzy Sets and Systems: A Concise Review,” Computers in Industry, Vol.62, pp.125-137, 2011.
[32] Alireza Sadeghian, Jerry M. Mendel, Hooman Tahayori, “Advances in Type-2 Fuzzy Sets and Systems, Theory and Applications,” Springer Science+Business Media New York, 2013.
[33] Rajkumar Verma and Bhu Dev Sharma, “Some New Results on Intuitionistic Fuzzy Sets,” Proceedings of the Jangjeon Mathematical Society, Vol.16, No.1, pp.101-114, 2013.
[34] K.T. Atanassov, “Review and New Results on Intuitionistic Fuzzy Sets,” Int. J. Bioautomation, Vol. 20, No. S1, pp. S7-S16, 2016.
[35] Cengiz Kahraman, Basar Oztaysi, Cevik Onar, “A Comprehensive Literature Review of 50 Years of Fuzzy Set Theory,” International Journal of Computational Intelligent Systems, Vol.9, No.1, pp.3-24, 2016.
[36] H. Bustince et al., "A Historical Account of Types of Fuzzy Sets and Their Relationships," in IEEE Transactions on Fuzzy Systems, vol. 24, no.1, pp.179-194, 2016.
[37] Abbas Mardani, Mehrbakhsh Nilashi, Jurgita Antucheviciene, Madjid Tavana, Romualdas, Bausys, Othman Ibrahim, “Review Article Recent Fuzzy Generalisations of Rough Sets Theory: A Systematic Review and Methodological Critique of the Literature,” Hindawi Complexity, Vol.2017, pp.1-33, 2017.
[38] R. Karthika, A. Sahaya Sudha, “A New Operation on Sinusoidal Fuzzy Number,” International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.6, No.4, pp.104-107, 2019.
[39] Cengiz Kahraman, Fatma Kutlu Gündo?du, “Decision Making with Spherical Fuzzy Sets, Theory and Applications,” Springer Nature Switzerland AG, 2021.
[40] G. Prabakaran, D. Vaithiyanathan, M. Ganesan, “Fuzzy Decision Support System for Improving the Crop Productivity and Efficient Use of Fertilizers,” Computers and Electronics in Agriculture, Vol.150, pp.88-97, 2018.
[41] R. Vaghela, K. Solanki, “Fuzzy Logic Based Light Control Systems for Heterogeneous Traffic and Prospectus in Ahmedabad (India),” In Data Science and Intelligent Applications, Springer, Singapore, pp. 239-246, 2021.
[42] G. Chattopadhyay, S. Chattopadhyay, S.K. Midya, “Fuzzy Binary Relation Based Elucidation of Air Quality Over a Highly Polluted Urban Region of India,” Earth Science Informatics, Vol.14, No.3, pp.1625-1631, 2021.
[43] A. Chakraborty, M. Jindal, M.R. Khosravi, P. Singh, A. Shankar, M. Diwakar, “A Secure Iot-Based Cloud Platform Selection Using Entropy Distance Approach and Fuzzy Set Theory,” Wireless Communications and Mobile Computing, Vol.2021, pp.1-13, 2021.
[44] Chia-Wen Chang, Chin-Wang Tao, “Applications of Type-2 Fuzzy Sets with Set Approximation Approaches: A Summary,” In Statistical and Fuzzy Approaches to Data Processing, with Applications to Econometrics and Other Areas, Studies in Computational Intelligence, vol 892, pp.67-92, Springer, 2020.
[45] J. Downey, “Comparative Communication Research: Why We Really Need Some More Fuzzy Thinking,” International Journal of Communication, Vol.14, pp.5787-5798, 2020.
[46] D.E. Ighravwe, S.A. Oke, “An Integrated Approach of SWARA and Fuzzy COPRAS for Maintenance Technicians, Selection Factors Ranking,” International Journal of System Assurance Engineering and Management, Vol.10, No.6, pp.1615-1626, 2019.
[47] J.C.R. Alcantud, “Ranked Hesitant Fuzzy Sets for Multi-Criteria Multi-Agent Decisions,” Expert Systems with Applications, Vol.209, pp.118-276, 2022.
[48] He Xiaorong, Wu Yingyu, "Global Research Trends of Intuitionistic Fuzzy Set: A Bibliometric Analysis," Journal of Intelligent Systems, Vol.28, No.4, pp.621-631, 2019.
[49] Valery Tarassov, “A Panorama of Fuzzy Sets: Dedicated to the 100th Anniversary of the Birth of Lotfi Zadeh,” In Proceedings of Russian Advances in Fuzzy Systems and Soft Computing, pp.1-11, 2020
[50] Jaydip Bhattacharya, “Some Special Operations and Related Results on Intuitionistic Fuzzy Sets,” International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.8, No.4, pp.10-13, 2021.
[51] O. Castillo, P.K. Muhuri, H.Mo, et al., “ Editorial on Special Issue: “Trends and Developments on Type-2 Fuzzy Sets and Systems,” International Journal of Fuzzy Systems, Vol. 23, pp.1055–1056, 2021.
[52] Akbar Rezaie, Tahsin Oner, Tugce Katican, Florentine Smarandache, Neeraj Gandotra, “A Short History of Fuzzy, Intuitionistic Fuzzy, Neutrosophic and Plithogenic Sets,” International Journal of Neutrosophic Science, Vol.18, No.1, pp.99-116, 2022.