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T-Spherical Fuzzy Einstein Hybrid Aggregation Operators and Their Applications in Multi-Attribute Decision Making Problems

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Department of Mathematics and Statistics, International Islamic University, Islamabad 44000, Pakistan
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School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China
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Department of Mathematics, Huzhou University, Huzhou 313000, China
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Department of Data analysis and Mathematical Modeling, Ghent University, 9000 Ghent, Belgium
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Author to whom correspondence should be addressed.
Symmetry 2020, 12(3), 365; https://doi.org/10.3390/sym12030365
Received: 28 January 2020 / Revised: 24 February 2020 / Accepted: 27 February 2020 / Published: 2 March 2020
T-spherical fuzzy set is a recently developed model that copes with imprecise and uncertain events of real-life with the help of four functions having no restrictions. This article’s aim is to define some improved algebraic operations for T-SFSs known as Einstein sum, Einstein product and Einstein scalar multiplication based on Einstein t-norms and t-conorms. Then some geometric and averaging aggregation operators have been established based on defined Einstein operations. The validity of the defined aggregation operators has been investigated thoroughly. The multi-attribute decision-making method is described in the environment of T-SFSs and is supported by a comprehensive numerical example using the proposed Einstein aggregation tools. As consequences of the defined aggregation operators, the same concept of Einstein aggregation operators has been proposed for q-rung orthopair fuzzy sets, spherical fuzzy sets, Pythagorean fuzzy sets, picture fuzzy sets, and intuitionistic fuzzy sets. To signify the importance of proposed operators, a comparative analysis of proposed and existing studies is developed, and the results are analyzed numerically. The advantages of the proposed study are demonstrated numerically over the existing literature with the help of examples. View Full-Text
Keywords: picture fuzzy set; spherical fuzzy set; T-spherical fuzzy sets; multi-attribute decision making; Einstein aggregation operators picture fuzzy set; spherical fuzzy set; T-spherical fuzzy sets; multi-attribute decision making; Einstein aggregation operators
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MDPI and ACS Style

Munir, M.; Kalsoom, H.; Ullah, K.; Mahmood, T.; Chu, Y.-M. T-Spherical Fuzzy Einstein Hybrid Aggregation Operators and Their Applications in Multi-Attribute Decision Making Problems. Symmetry 2020, 12, 365. https://doi.org/10.3390/sym12030365

AMA Style

Munir M, Kalsoom H, Ullah K, Mahmood T, Chu Y-M. T-Spherical Fuzzy Einstein Hybrid Aggregation Operators and Their Applications in Multi-Attribute Decision Making Problems. Symmetry. 2020; 12(3):365. https://doi.org/10.3390/sym12030365

Chicago/Turabian Style

Munir, Muhammad; Kalsoom, Humaira; Ullah, Kifayat; Mahmood, Tahir; Chu, Yu-Ming. 2020. "T-Spherical Fuzzy Einstein Hybrid Aggregation Operators and Their Applications in Multi-Attribute Decision Making Problems" Symmetry 12, no. 3: 365. https://doi.org/10.3390/sym12030365

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