Document Type : Research Paper
Authors
Department of Mathematics, University of Management and Technology, Lahore, Pakistan.
Abstract
Zadeh conceptualized the theory of fuzzy set to provide a tool for the basis of the theory of possibility. Atanassov extended this theory with the introduction of intuitionistic fuzzy set. Smarandache introduced the concept of refined intuitionistic fuzzy set by further subdivision of membership and non-membership value. The meagerness regarding the allocation of a single membership and non-membership value to any object under consideration is addressed with this novel refinement. In this study, this novel idea is utilized to characterize the essential elements e.g. subset, equal set, null set, and complement set, for refined intuitionistic fuzzy set. Moreover, their basic set theoretic operations like union, intersection, extended intersection, restricted union, restricted intersection, and restricted difference, are conceptualized. Furthermore, some basic laws are also discussed with the help of an illustrative example in each case for vivid understanding.
Keywords
Main Subjects
[1] Zadeth, L. A. (1965). Fuzzy sets. Information and control, 8, 338-353.
[2] Zadeh, L. A. (1999). Fuzzy sets as a basis for a theory of possibility. Fuzzy sets and systems, 100, 9-34.
[3] Dubois, D., & Prade, H. (1986). Fuzzy sets and statistical data. European journal of operational research, 25(3), 345-356.
[4] Dubois, D., & Prade, H. (1991). Random sets and fuzzy interval analysis. Fuzzy sets and systems, 42(1), 87-101.
[5] Dubois, D., & Prade, H. (1983). Ranking fuzzy numbers in the setting of possibility theory. Information sciences, 30(3), 183-224.
[6] Beg, I., & Ashraf, S. (2009). Similarity measures for fuzzy sets. Appl. comput. math, 8(2), 192-202.
[7] Beg, I., & Ashraf, S. (2008). Fuzzy similarity and measure of similarity with Lukasiewicz implicator. New mathematics and natural computation, 4(02), 191-206.
[8] Beg, I., & Ashraf, S. (2009). Fuzzy inclusion and fuzzy similarity with gödel fuzzy implicator. New mathematics and natural computation, 5(03), 617-633.
[9] Neog, T. J., & Sut, D. K. (2011). Complement of an extended fuzzy set. International journal of computer applications, 29(3), 39-45.
[10] Uygun, O., Yalcin, S., Kiraz, A., & Furkan Erkan, E. (2020). A novel assessment approach to EFQM driven institutionalization using integrated fuzzy multi-criteria decision-making methods. Scientia ironical, 27(2), 880-892.
[11] Damirchi-Darasi, S. R., Zarandi, M. F., Turksen, I. B., & Izadi, M. (2019). Type-2 fuzzy rule-based expert system for diagnosis of spinal cord disorders. Scientia Iranica. Transaction E, industrial engineering, 26(1), 455-471.
[12] Brown, J. G. (1971). A note on fuzzy sets. Information and control, 18(1), 32-39.
[13] Dubois, D., & Prade, H. (1989). Fuzzy sets, probability and measurement. European journal of operational research, 40(2), 135-154.
[14] Dubois, D., & Prade, H. (1997). The three semantics of fuzzy sets. Fuzzy sets and systems, 90(2), 141-150.
[15] Isah, A. I. (2020). Some algebraic structures of multi-fuzzy set. Science world journal, 15(1), 21-25.
[16] Meng, F. Y., Tang, J., & Fujita, H. (2019). Consistency-based algorithms for decision-making with interval fuzzy preference relations. IEEE transactions on fuzzy systems, 27(10), 2052-2066.
[17] Atanassov, K. (1986). Intuitionistic fuzzy sets. Fuzzy sets and systems 20 (1), 87-96.
[18] Atanassov, K. T. (1999). Intuitionistic fuzzy sets. In Intuitionistic fuzzy sets (pp. 1-137). Physica, Heidelberg.
[19] Ejegwa, P. A., Akowe, S. O., Otene, P. M., & Ikyule, J. M. (2014). An overview on intuitionistic fuzzy sets. International journal of scientific and technology research, 3(3), 142-145.
[20] Szmidt, E., & Kacprzyk, J. (2001). Entropy for intuitionistic fuzzy sets. Fuzzy sets and systems, 118(3), 467-477.
[21] Szmidt, E., & Kacprzyk, J. (2000). Distances between intuitionistic fuzzy sets. Fuzzy sets and systems, 114(3), 505-518.
[22] Ersoy, B. A., & Davvaz, B. (2013, September). Structure of intuitionistic fuzzy sets in-semihyperrings. In Abstract and applied analysis (Vol. 2013). Hindawi.
[23] Bustince, H., & Burillo, P. (1996). Vague sets are intuitionistic fuzzy sets. Fuzzy sets and systems, 79(3), 403-405.
[24] Garg, H., & Kaur, G. (2020). Extended TOPSIS method for multi-criteria group decision-making problems under cubic intuitionistic fuzzy environment. Scientia iranica, 27(1), 396-410.
[25] Garg, H. (2016). A novel correlation coefficients between pythagorean fuzzy sets and its applications to decision‐making processes. International journal of intelligent systems, 31(12), 1234-1252.
[26] Ejegwa, P. A., & Onyeke, I. C. (2020). Medical diagnostic analysis on some selected patients based on modified Thao et al.’s correlation coefficient of intuitionistic fuzzy sets via an algorithmic approach. Journal of fuzzy extension and applications, 1(2), 130-141.
[27] Edalatpanah, S. A. (2019). A data envelopment analysis model with triangular intuitionistic fuzzy numbers. International journal of data envelopment analysis, 7(4), 47-58.
[28] Edalatpanah, S. A. (2020). Neutrosophic structured element. Expert systems. https://doi.org/10.1111/exsy.12542
[29] Kumar, R., Edalatpanah, S. A., Jha, S., Gayen, S., & Singh, R. (2019). Shortest path problems using fuzzy weighted arc length. International journal of innovative technology and exploring engineering, 8(6), 724-731.
[30] Gulzar, M., Alghazzawi, D., Mateen, M. H., & Kausar, N. (2020). A certain class of t-intuitionistic fuzzy subgroups. IEEE access, 8, 163260-163268.
[31] Zhang, L., Zhan, J., & Yao, Y. (2020). Intuitionistic fuzzy TOPSIS method based on CVPIFRS models: an application to biomedical problems. Information sciences, 517, 315-339.
[32] Mohan, S., Kannusamy, A. P., & Samiappan, V. (2020). A new approach for ranking of intuitionistic fuzzy numbers. Journal of fuzzy extension and applications, 1(1), 15-26.
[33] Imeni, M. (2020). Fuzzy logic in accounting and auditing. Journal of fuzzy extension and applications, 1(1), 69-75.
[34] Smarandache, F. (2019). Neutrosophic Set is a Generalization of Intuitionistic Fuzzy Set, Inconsistent Intuitionistic Fuzzy Set (Picture Fuzzy Set, Ternary Fuzzy Set), Pythagorean Fuzzy Set (Atanassov’s Intuitionistic Fuzzy Set of second type), q-Rung Orthopair Fuzzy Set, Spherical Fuzzy Set, and n-HyperSpherical Fuzzy Set, while Neutrosophication is a Generalization of Regret Theory, Grey System Theory, and Three-Ways Decision (revisited). Infinite Study.
)