Exploring The Boundaries of Uncertainty: Interval Valued Pythagorean Neutrosophic Set and Their Properties
DOI:
https://doi.org/10.11113/mjfas.v20n4.3482Keywords:
Interval valued pythagorean fuzzy set, interval valued neutrosophic set, interval valued pythagorean neutrosophic set, algebraic operations.Abstract
The interval-valued Pythagorean fuzzy set (IVPFS) presents a novel approach to tackling vagueness and uncertainty, while neutrosophic sets, a broader concept encompassing of fuzzy sets and intuitionistic fuzzy sets, are tailored to depict real-world data characterized by uncertainty, imprecision, inconsistency, and incompleteness. Additionally, the development of Interval Value Neutrosophic Sets (IVNS) enhances precision in handling problems involving a range of numbers within the real unit interval, rather than focusing solely on a single value. However, despite these advancements, there is a deficiency in research addressing practical implementation challenges, conducting comparative analyses with existing methods, and applying these concepts across various fields. This study aims to bridge this research gap by proposing a novel concept based on the Interval Valued Pythagorean Neutrosophic Set (IVPNS), which is a generalization of the IVPFS and INS. The development of IVPNS provides a more comprehensive framework for handling uncertainty, ambiguity, and incomplete information in various fields, leading to more robust decision-making processes, improved problem-solving capabilities, and better management of complex systems. Furthermore, this research introduces the algebraic operations for IVPNS, including addition, multiplication, scalar multiplication, and exponentiation and provides a comparative analysis with IVPFS and IVNS. The study incorporates illustrative numerical examples to demonstrate these operations in practice. Additionally, this study provides and rigorously proves the algebraic properties of IVPNS, specifically discussing their commutative and associative properties. This validation ensures compliance with established conditions for IVPNS, reinforcing their theoretical soundness and practical applicability.
References
Ajay, D., & Chellamani, P. (2022). Pythagorean neutrosophic soft sets and their application to decision-making scenario. In Intelligent and Fuzzy Techniques for Emerging Conditions and Digital Transformation: Proceedings of the INFUS 2021 Conference, held August 24-26, 2021. Volume 2 (pp. 552-560). Springer International Publishing.
Atanassov K.T. and Gargov, G. (1989). Interval-valued intuitionistic fuzzy sets. Fuzzy Sets and Systems,3(3), 343–349.
Atanassov, K. T., & Stoeva, S. (1986). Intuitionistic fuzzy sets. Fuzzy sets and Systems, 20(1), 87-96.
Atanassov, K.T. (1994). Operators over interval-valued intuitionistic fuzzy sets. Fuzzy Sets and Systems, 64(2),159–174.
Awang, N. A., Isa, N. I. M., Hashim, H., & Abdullah, L. (2023). AHP Approach using Interval Neutrosophic Weighted Averaging (INWA) Operator for Ranking Flash Floods Contributing Factors. Neutrosophic Sets and Systems, 57(1), 10.
Biswas, A., Sarkar, B. (2018). Pythagorean fuzzy multicriteria group decision-making through similarity measure based on point operators. Int J Intell Syst, 33(8),1731–1744.
Broumi, S., & Smarandache, F. (2015). Interval-valued neutrosophic soft, rough sets. International Journal of Computational Mathematics, 2015, 1-13.
Broumi, S., & Smarandache, F. (2015). New operations on interval neutrosophic sets. Journal of new theory, (1), 24-37.
Broumi, S., Prabha, S.K. (2023). Fermatean Neutrosophic Matrices and Their Basic Operations. Neutrosophic Sets and Systems, 58(35),572–595.
Broumi. S, Smarandache. F., Dhar, M. (2024). Rough neutrosophic sets. Neutrosophic Theory Appl., 32(32),493–502.
Dung, V. Thuy, L.T., Mai, P. Q., Mai's, P.Q., Nguyen V. D., and Nguyen T. M. L. (2018). TOPSIS Approach Using Interval Neutrosophic Sets for Personnel Selection. Asian Journal of Scientific Research, 1(3), 434–440.
Dutta, P., Borah, G., Gohain, B., Chutia, R. (2023). Nonlinear distance measures under the framework of Pythagorean fuzzy sets with applications in problems of pattern recognition, medical diagnosis, and COVID-19 medicine selection. Beni-Suef University Journal of Basic and Applied Sciences,12(1), 42.
Garg, H. (2017). A new improved score function of an interval-valued Pythagorean fuzzy set based TOPSIS method, Int. J. Uncertainty Quantif., 7(5), 463 – 474.
Garg, H. A. (2016). Novel accuracy function under interval-valued Pythagorean fuzzy environment for solving multicriteria decision-making problem. J. Intell. Fuzzy Syst.,31(1),529-540.
Ho, L. H., Lin, Y. L., & Chen, T. Y. (2020). A Pearson-like correlation-based TOPSIS method with interval-valued Pythagorean fuzzy uncertainty and its application to multiple criteria decision analysis of stroke rehabilitation treatments. Neural Computing and Applications, 32, 8265-8295.
İrem, O., and Cengiz, K. (2019). Analytic network process with neutrosophic sets. Studies in Fuzziness and Soft Computing, 369,525 – 542.
Jeni Seles Martina, D., & Deepa, G. (2023). Operations on Multi-Valued Neutrosophic Matrices and Its Application to Neutrosophic Simplified-TOPSIS Method. International Journal of Information Technology & Decision Making, 22(01), 37-56.
Kahraman, C., Otay, İ., Öztayşi, B., & Onar, S. Ç. (2019). An integrated AHP & DEA methodology with neutrosophic sets. Fuzzy multi-criteria decision-making using neutrosophic sets, 623-645.
Khan, M., Zeeshan, M. Iqbal, S. (2021). Neutrosophic soft metric matrices with applications in decision-making, J. Algebraic Hyperstrucres Logical Algebras, 2(4),63–81.
Khatter. K. (2021). Interval-valued trapezoidal neutrosophic set: multi-attribute decision making for prioritization of non-functional requirements. Journal of Ambient Intelligence and Humanized Computing,12(1),1039-1055.
Li D, Zeng, W. (2018). Distance measure of Pythagorean fuzzy sets. Int J Intell Syst,33(2), 348–361.
Liang, W., Zhang, X., & Liu, M. (2015). The maximizing deviation method based on interval-valued Pythagorean fuzzy weighted aggregating operator for multiple criteria group decision analysis. Discrete Dynamics in Nature and Society, 2015.
keMohd, W. W., Abdullah, L., Yusoff, B., Taib, C. M. I. C., & Merigo, J. M. (2019). An integrated MCDM model based on Pythagorean fuzzy sets for green supplier development program. Malaysian Journal of Mathematical Sciences, 13, 23-37.
Peng, X., & Yang, Y. (2016). Fundamental properties of interval‐valued Pythagorean fuzzy aggregation operators. International journal of intelligent systems, 31(5), 444-487.
Poonia, M., & Bajaj, R. K. (2021). Complex Neutrosophic Matrix with Some Algebraic Operations and Matrix Norm Convergence. Neutrosophic Sets and Systems, 47(1), 165-178.
Quynh, M. P., Thu, T. L., Huong, Q. D., Van, A. P. T., Van, H. N., & Van, D. N. (2020). Distribution center location selection using a novel multi-criteria decision-making approach under interval neutrosophic complex sets.Decision Science Letters, 9(3),501–510.
Rahman, K., Abdullah, S., Shakeel, M., Ali Khan, M. S., & Ullah, M. (2017). Interval-valued Pythagorean fuzzy geometric aggregation operators and their application to group decision-making problem. Cogent Mathematics, 4(1), 1338638.
Rahman, K., Garg, H., Ali, R., Alfalqi, S. H., & Lamoudan, T. (2023). Algorithms for decision-making process using complex Pythagorean fuzzy set and its application to hospital siting for COVID-19 patients. Engineering Applications of Artificial Intelligence, 126, 107153.
SheebaMaybell, P., & Shanmugapriya, M. M. (2022). A Significant Factor Of Fuzzy Neutrosophic Soft Matrices In Decision Making. Webology (ISSN: 1735-188X), 19(2).
Sinika, S., & Ramesh, G. (2023). To Enhance New Interval Arithmetic Operations in Solving Linear Programming Problem Using Interval-valued Trapezoidal Neutrosophic Numbers. Mathematics and Statistics, 11(4), 710-725.
Smarandache, F. (2006, May). Neutrosophic set-a generalization of the intuitionistic fuzzy set. In 2006 IEEE international conference on granular computing (pp. 38-42). IEEE.
Smarandache, F. (2010). Neutrosophic set–a generalization of the intuitionistic fuzzy set. Journal of Defense Resources Management (JoDRM), 1(1), 107-116.
Smarandache, F., Ali, M., & Khan, M. (2019). Arithmetic operations of neutrosophic sets, interval neutrosophic sets and rough neutrosophic sets. Fuzzy Multi-criteria Decision-Making Using Neutrosophic Sets, 25-42.
Stephen, S., & Helen, M. (2021). Interval-valued Neutrosophic Pythagorean Sets and their Application Decision Making using IVNP-TOPSIS. International Journal of Innovative Research in Science. Engineering and Technology, 10(1), 14571-14578.
Wei, G., & Wei, Y. (2018). Similarity measures of Pythagorean fuzzy sets based on the cosine function and their applications. International Journal of Intelligent Systems, 33(3), 634-652.
Yager, R. R. (2013). Pythagorean membership grades in multicriteria decision making. IEEE Transactions on Fuzzy Systems, 22(4), 958-965.
Yager, R. R. (2013, June). Pythagorean fuzzy subsets. In 2013 joint IFSA world congress and NAFIPS annual meeting (IFSA/NAFIPS) (pp. 57-61). IEEE.
Yager, R. R., & Abbasov, A. M. (2013). Pythagorean membership grades, complex numbers, and decision-making. International Journal of Intelligent Systems, 28(5), 436-452.
Yao, Z., Ran, H. (2023) Operational efficiency evaluation of Urban and rural residents' basic pension insurance system based on the triangular fuzzy neutrosophic GRA method. Journal of Intelligent & Fuzzy Systems, 44(6), 1-12.
Zadeh, L.A. (1965). Fuzzy sets. Journal of Information and Control,8,338 – 353.
Zeng, W., Li, D., & Yin, Q. (2018). Distance and similarity measures of Pythagorean fuzzy sets and their applications to multiple criteria group decision making. International Journal of Intelligent Systems, 33(11), 2236-2254.
Zhang, X. (2016). Multicriteria Pythagorean fuzzy decision analysis: A hierarchical QUALIFLEX approach with the closeness index-based ranking methods. Information Sciences, 330, 104-124.
Zhang, H. Y., Wang, J. Q., & Chen, X. H. (2014). Interval neutrosophic sets and their application in multicriteria decision making problems. The Scientific World Journal, 2014.
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