Improving Decision-Making Under Uncertainty: A Comparative Study of Fuzzy Set Extensions
Abstract
Fuzzy sets have revolutionized decision-making by providing a mathematical tool for modeling
uncertainty and imprecision. However, traditional fuzzy sets may not be sufficient in certain
situations, leading to the development of extensions such as Type-2 fuzzy sets, Intuitionistic
fuzzy sets, and Type-2 intuitionistic fuzzy sets. This paper provides an overview of these sets,
comparing and contrasting them using operations of union, intersection, and distance measures.
Additionally, a new distance measure is proposed for Type-2 intuitionistic fuzzy sets, which
is demonstrated with a numerical example. By understanding the properties and applications
of these sets, informed decisions can be made in real-world situations with uncertainty and
imprecision.
References
Atanassov, K. T. (1999). Intuitionistic fuzzy sets. In Intuitionistic fuzzy sets (pp. 1-137). Physica, Heidelberg.
Atanassov, K. T. (2017). Type-1 fuzzy sets and intuitionistic fuzzy sets. Algorithms, 10(3), 106.
Bag, T., & Samanta, S. K. (2008). A comparative study of fuzzy norms on a linear space. Fuzzy sets and systems, 159(6), 670-684.
Castillo, O. (2012). Introduction to type-2 fuzzy logic control. In Type-2 fuzzy logic in intelligent control applications (pp. 3-5). Springer, Berlin, Heidelberg.
Zadeh, L. A. (1971). Quantitative fuzzy semantics. Information sciences, 3(2), 159-176.
Dubois, D., & Prade, H. (1979). Fuzzy real algebra: some results. Fuzzy sets and systems, 2(4), 327-348.
Dubois, D. J. (1980). Fuzzy sets and systems: theory and applications (Vol. 144). Academic press.
Dombi, J. (1982). A general class of fuzzy operators, the DeMorgan class of fuzzy operators and fuzziness measures induced by fuzzy operators. Fuzzy sets and systems, 8(2), 149-163.
Dubois, D., & Prade, H. (1982). A class of fuzzy measures based on triangular norms a general framework for the combination of uncertain information. International journal of general systems, 8(1), 43-61.
Dan, S., Kar, M. B., Majumder, S., Roy, B., Kar, S., & Pamucar, D. (2019). Intuitionistic type-2 fuzzy set and its properties. Symmetry, 11(6), 808.
Fodor, J., & Rudas, I. J. (2007). On continuous triangular norms that are migrative. Fuzzy Sets and Systems, 158(15), 1692-1697.
García, J. C. F. (2009, October). Solving fuzzy linear programming problems with interval type-2 RHS. In 2009 IEEE International Conference on Systems, Man and Cybernetics (pp. 262-267). IEEE.
Hidalgo, D., Melin, P., & Castillo, O. (2012). An optimization method for designing type-2 fuzzy inference systems based on the footprint of uncertainty using genetic algorithms. Expert Systems with Applications, 39(4), 4590-4598.
Kacprzyk, J. (1997). Multistage fuzzy control: a prescriptive approach.
Karnik, N. N., & Mendel, J. M. (2001). Centroid of a type-2 fuzzy set. information SCiences, 132(1-4), 195-220.
Klement, E. P., Mesiar, R., & Pap, E. (2004). Triangular norms. Position paper I: basic analytical and algebraic properties. Fuzzy sets and systems, 143(1), 5-26.
Kundu, P., Kar, S., & Maiti, M. (2014). Fixed charge transportation problem with type-2 fuzzy variables. Information sciences, 255, 170-186.
Kar, M. B., Roy, B., Kar, S., Majumder, S., & Pamucar, D. (2019). Type-2 multi-fuzzy sets and their applications in decision making. Symmetry, 11(2), 170.
Mizumoto, M., & Tanaka, K. (1976). Some properties of fuzzy sets of type 2. Information and control, 31(4), 312-340.
Mizumoto, M., & Tanaka, K. (1981). Fuzzy sets and type 2 under algebraic product and algebraic sum. Fuzzy Sets and Systems, 5(3), 277-290.
Mendel, J. M., & John, R. B. (2002). Type-2 fuzzy sets made simple. IEEE Transactions on fuzzy systems, 10(2), 117-127.
Maes, K. C., & De Baets, B. (2007). The triple rotation method for constructing t-norms. Fuzzy Sets and Systems, 158(15), 1652-1674.
Mendel, J. M. (2007). Advances in type-2 fuzzy sets and systems. Information sciences, 177(1), 84-110.
Montiel, O., Castillo, O., Melin, P., & Sepulveda, R. (2008). Mediative fuzzy logic: a new approach for contradictory knowledge management. In Forging New Frontiers: Fuzzy Pioneers II (pp. 135-149). Springer, Berlin, Heidelberg.
Mahapatra, G. S., & Roy, T. K. (2013). Intuitionistic fuzzy number and its arithmetic operation with application on system failure. Journal of uncertain systems, 7(2), 92-107.
Mo, H.,Wang, F. Y., Zhou, M., Li, R., & Xiao, Z. (2014). Footprint of uncertainty for type-2 fuzzy sets. Information Sciences, 272, 96-110.
Marasini, D., Quatto, P., & Ripamonti, E. (2016). Fuzzy analysis of students’ ratings. Evaluation Review, 40(2), 122-141.
Singh, S., & Garg, H. (2017). Distance measures between type-2 intuitionistic fuzzy sets and their application to multicriteria decision-making process. Applied Intelligence, 46(4), 788-799.
Singh, P. (2014). Some new distance measures for type-2 fuzzy sets and distance measure based ranking for group decision making problems. Frontiers of Computer Science, 8(5), 741-752.
Takáč, Z. (2014). Aggregation of fuzzy truth values. Information Sciences, 271, 1-13.
Wang, W., & Xin, X. (2005). Distance measure between intuitionistic fuzzy sets. Pattern recognition letters, 26(13), 2063-2069.
Yager, R. R., Walker, C. L., & Walker, E. A. (2005). Generalizing Leximin to t-norms and t-conorms: the LexiT and LexiS Orderings. Fuzzy sets and systems, 151(2), 327-340.
Zadeh, L. A. (1965). Fuzzy sets. Information and control, 8(3), 338-353.
Zadeh, L. A. (1973). Outline of a new approach to the analysis of complex systems and decision processes. IEEE Transactions on systems, Man, and Cybernetics, (1), 28-44.
Zadeh, L. A. (1975). The concept of a linguistic variable and its application to approximate reasoning—I. Information sciences, 8(3), 199-249.
Zadeh, L. A. (1975). The concept of a linguistic variable and its application to approximate reasoning—II. Information sciences, 8(4), 301-357.
Zadeh, L. A. (1999). Fuzzy sets as a basis for a theory of possibility. Fuzzy sets and systems, 100, 9-34.
Zadeh, L. A. (2005). From imprecise to granular probabilities. Fuzzy Sets and Systems, 154(3), 370-374.
Zimmerman, H. J. (2001). Fuzzy Set Theory and Applications. 4-th rev. ed.
Ganie, A. H. (2022). Multicriteria decision-making based on distance measures and knowledge measures of Fermatean fuzzy sets. Granular Computing, 7(4), 979-998.
Singh, S., & Ganie, A. H. (2022). Generalized hesitant fuzzy knowledge measure with its application to multi-criteria decision-making. Granular Computing, 7(2), 239-252.
Ganie, A. H. (2023). Some t-conorm-based distance measures and knowledge measures for Pythagorean fuzzy sets with their application in decision-making. Complex & Intelligent Systems, 9(1), 515-535.
Almulhim, T., & Barahona, I. (2023). An extended picture fuzzy multicriteria group decision analysis with different weights: A case study of COVID-19 vaccine allocation. Socio-Economic Planning Sciences, 85, 101435.
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Copyright (c) 2023 Suhail Ahmad Ganai, Nitin Bhardwaj, Riyaz Ahmad Padder
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