{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,24]],"date-time":"2026-04-24T20:23:21Z","timestamp":1777062201318,"version":"3.51.4"},"reference-count":42,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2020,3,2]],"date-time":"2020-03-02T00:00:00Z","timestamp":1583107200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>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\u2019s 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.<\/jats:p>","DOI":"10.3390\/sym12030365","type":"journal-article","created":{"date-parts":[[2020,3,2]],"date-time":"2020-03-02T07:50:53Z","timestamp":1583135453000},"page":"365","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":109,"title":["T-Spherical Fuzzy Einstein Hybrid Aggregation Operators and Their Applications in Multi-Attribute Decision Making Problems"],"prefix":"10.3390","volume":"12","author":[{"given":"Muhammad","family":"Munir","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, International Islamic University, Islamabad 44000, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5835-3349","authenticated-orcid":false,"given":"Humaira","family":"Kalsoom","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1438-6413","authenticated-orcid":false,"given":"Kifayat","family":"Ullah","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, International Islamic University, Islamabad 44000, Pakistan"},{"name":"Department of Data analysis and Mathematical Modeling, Ghent University, 9000 Ghent, Belgium"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3871-3845","authenticated-orcid":false,"given":"Tahir","family":"Mahmood","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, International Islamic University, Islamabad 44000, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0944-2134","authenticated-orcid":false,"given":"Yu-Ming","family":"Chu","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Huzhou University, Huzhou 313000, China"}]}],"member":"1968","published-online":{"date-parts":[[2020,3,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"338","DOI":"10.1016\/S0019-9958(65)90241-X","article-title":"Fuzzy sets","volume":"8","author":"Zadeh","year":"1965","journal-title":"Inf. Control."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1016\/S0165-0114(86)80034-3","article-title":"Intuitionistic fuzzy sets","volume":"20","author":"Atanassov","year":"1986","journal-title":"Fuzzy Sets Syst."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Yager, R.R. (2013, January 24\u201328). Pythagorean Fuzzy Subsets. Proceedings of the IFSA World Congress and NAFIPS Annual Meeting (IFSA\/NAFIPS), Edmonton, AB, Canada.","DOI":"10.1109\/IFSA-NAFIPS.2013.6608375"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1222","DOI":"10.1109\/TFUZZ.2016.2604005","article-title":"Generalized orthopair fuzzy sets","volume":"25","author":"Yager","year":"2017","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_5","first-page":"409","article-title":"Picture fuzzy sets","volume":"30","author":"Cuong","year":"2014","journal-title":"J. Comput. Sci. Cybern."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"7041","DOI":"10.1007\/s00521-018-3521-2","article-title":"An Approach towards Decision Making and Medical Diagnosis Problems Using the Concept of Spherical Fuzzy Sets","volume":"31","author":"Mahmood","year":"2018","journal-title":"Neural Comput. Appl."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"B-141","DOI":"10.1287\/mnsc.17.4.B141","article-title":"Decision-making in a fuzzy environment","volume":"17","author":"Bellman","year":"1970","journal-title":"Manag. Sci."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1179","DOI":"10.1109\/TFUZZ.2006.890678","article-title":"Intuitionistic fuzzy aggregation operators","volume":"15","author":"Xu","year":"2007","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1016\/j.fss.2003.06.007","article-title":"Triangular norms. Position I: Basic analytic and algebraic properties","volume":"143","author":"Klement","year":"2004","journal-title":"Fuzzy Sets Syst."},{"key":"ref_10","first-page":"190","article-title":"Pythagorean fuzzy weighted averaging aggregation operator and its application to decision making theory","volume":"54","author":"Rahman","year":"2017","journal-title":"Nucleus"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"436","DOI":"10.1002\/int.21584","article-title":"Pythagorean membership grades, complex numbers, and decision making","volume":"28","author":"Yager","year":"2013","journal-title":"Int. J. Intell. Syst."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"713","DOI":"10.3233\/JIFS-161798","article-title":"Picture fuzzy aggregation operators and their application to multiple attribute decision making","volume":"33","author":"Wei","year":"2017","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"5275","DOI":"10.1007\/s13369-017-2625-9","article-title":"Some picture fuzzy aggregation operators and their applications to multicriteria decision-making","volume":"42","author":"Garg","year":"2017","journal-title":"Arab. J. Sci. Eng."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Ullah, K., Hassan, N., Mahmood, T., Jan, N., and Hassan, M. (2019). Evaluation of investment policy based on multi-attribute decision making using interval-valued T-spherical fuzzy aggregation operators. Symmetry, 11.","DOI":"10.3390\/sym11030357"},{"key":"ref_15","unstructured":"Ullah, K., Mahmood, T., and Jan, N. (2019, January 26\u201328). Some Averaging Aggregation Operators for T-Spherical Fuzzy Sets and Their Applications in Multi-Attribute Decision Making. Proceedings of the International Conference on Soft Computing and Machine Learning (ICSCML), Wuhan, China."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"22613","DOI":"10.1109\/ACCESS.2019.2896107","article-title":"T-spherical fuzzy power Muirhead mean operator based on novel operational laws and their applications in multi-attribute group decision making","volume":"7","author":"Liu","year":"2019","journal-title":"IEEE Access"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Garg, H., Munir, M., Ullah, K., Mahmood, T., and Jan, N. (2018). Algorithm for T-Spherical Fuzzy Multi-Attribute Decision Making Based on Improved Interactive Aggregation Operators. Symmetry, 10.","DOI":"10.3390\/sym10120670"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Quek, S.G., Selvachandran, G., Munir, M., Mahmood, T., Ullah, K., Son, L.H., Thong, P.H., Kumar, R., and Priyadarshini, I. (2019). Multi-Attribute Multi-Perception Decision-Making Based on Generalized T-Spherical Fuzzy Weighted Aggregation Operators on Neutrosophic Sets. Mathematics, 7.","DOI":"10.3390\/math7090780"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Liu, P., Munir, M., Mahmood, T., and Ullah, K. (2019). Some Similarity Measures for Interval-Valued Picture Fuzzy Sets and Their Applications in Decision Making. Information, 10.","DOI":"10.3390\/info10120369"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Zeng, S., Hussain, A., Mahmood, T., Ali, M.A., Ashraf, S., and Munir, M. (2019). Covering-Based Spherical Fuzzy Rough Set Model Hybrid with TOPSIS for Multi-Attribute Decision-Making. Symmetry, 11.","DOI":"10.3390\/sym11040547"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Zeng, S., Garg, H., Munir, M., Mahmood, T., and Hussain, A. (2019). A Multi-Attribute Decision Making Process with Immediate Probabilistic Interactive Averaging Aggregation Operators of T-Spherical Fuzzy Sets and Its Application in the Selection of Solar Cells. Energies, 12.","DOI":"10.3390\/en12234436"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1080\/16583655.2019.1708541","article-title":"Pythagorean fuzzy soft rough sets and their applications in decision-making","volume":"14","author":"Hussain","year":"2020","journal-title":"J. Taibah Univ. Sci."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"571","DOI":"10.1002\/int.22217","article-title":"q-Rung orthopair fuzzy soft average aggregation operators and their application in multi-criteria decision making","volume":"35","author":"Hussain","year":"2020","journal-title":"Int. J. Intell. Syst."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"472","DOI":"10.1016\/j.knosys.2012.09.006","article-title":"Some intuitionistic fuzzy Einstein hybrid aggregation operators and their application to multiple attribute decision making","volume":"37","author":"Zhao","year":"2013","journal-title":"Knowl.-Based Syst."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1016\/j.cie.2016.08.017","article-title":"Generalized intuitionistic fuzzy interactive geometric interaction operators using Einstein t-norm and t-conorm and their application to decision making","volume":"101","author":"Garg","year":"2016","journal-title":"Comput. Ind. Eng."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"886","DOI":"10.1002\/int.21809","article-title":"A new generalized pythagorean fuzzy information aggregation using einstein operations and its application to decision making","volume":"31","author":"Garg","year":"2016","journal-title":"Int. J. Intell. Syst."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"763","DOI":"10.1080\/0952813X.2018.1467497","article-title":"Generalised Pythagorean fuzzy geometric interactive aggregation operators using Einstein operations and their application to decision making","volume":"30","author":"Garg","year":"2018","journal-title":"J. Exp. Theor. Artif. Intell."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"331","DOI":"10.3233\/IFS-131000","article-title":"Some induced Einstein aggregation operators based on the data mining with interval-valued intuitionistic fuzzy information and their application to multiple attribute decision making","volume":"27","author":"Cai","year":"2014","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_29","first-page":"2","article-title":"The induced intuitionistic fuzzy Einstein aggregation and its application in group decision-making","volume":"30","author":"Xu","year":"2013","journal-title":"J. Ind. Prod. Eng."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1049","DOI":"10.1002\/int.20498","article-title":"Intuitionistic fuzzy geometric aggregation operators based on Einstein operations","volume":"26","author":"Wang","year":"2011","journal-title":"Int. J. Intell. Syst."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"25","DOI":"10.3389\/fphy.2020.00025","article-title":"New Investigation on the Generalized K-Fractional Integral Operators","volume":"8","author":"Rashid","year":"2020","journal-title":"Front. Phys."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Chu, H.H., Kalsoom, H., Rashid, S., Idrees, M., Safdar, F., Chu, Y.M., and Baleanu, D. (2020). Quantum Analogs of Ostrowski-Type Inequalities for Raina\u2019s Function correlated with Coordinated Generalized Phi-Convex Functions. Symmetry, 12.","DOI":"10.3390\/sym12020308"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Rashid, S., Kalsoom, H., Hammouch, Z., Ashraf, R., Baleanu, D., and Chu, Y.M. (2020). New Multi-Parametrized Estimates Having pth-Order Differentiability in Fractional Calculus for Predominating h-Convex Functions in Hilbert Space. Symmetry, 12.","DOI":"10.3390\/sym12020222"},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Kalsoom, H., Rashid, S., Idrees, M., Chu, Y.M., and Baleanu, D. (2020). Two-Variable Quantum Integral Inequalities of Simpson-Type Based on Higher-Order Generalized Strongly Preinvex and Quasi-Preinvex Functions. Symmetry, 12.","DOI":"10.3390\/sym12010051"},{"key":"ref_35","first-page":"1","article-title":"Delay dynamic double integral inequalities on time scales with applications","volume":"1","author":"Rafeeq","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_36","first-page":"63","article-title":"Hermite-Hadamard Type Integral Inequalities for Functions Whose Mixed Partial Derivatives Are Co-ordinated Preinvex","volume":"52","author":"Kalsoom","year":"2020","journal-title":"Punjab Univ. J. Math."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Rashid, S., Jarad, F., Noor, M.A., Kalsoom, H., and Chu, Y.M. (2019). Inequalities by means of generalized proportional fractional integral operators with respect to another function. Mathematics, 7.","DOI":"10.3390\/math7121225"},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Deng, y., Kalsoom, H., and Wu, S. (2019). Some New Quantum Hermite-Hadamard Type Estimates Within a Class of Generalized (s,m)-Preinvex Functions. Symmetry, 11.","DOI":"10.3390\/sym11101283"},{"key":"ref_39","first-page":"65","article-title":"Some Hermite-Hadamard type integral inequalities whose $ n $-times differentiable functions are s-logarithmically convex functions","volume":"2019","author":"Kalsoom","year":"2019","journal-title":"Punjab Univ. J. Math."},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Kalsoom, H., Latif, M.A., Junjua, M.-U.-D., Hussain, S., and Shahzadi, G. (2019). Some (p, q)-Estimates of Hermite--Hadamard-Type Inequalities For Co-ordinated Convex and Quasi-Convex Functions. Mathematics, 8.","DOI":"10.3390\/math7080683"},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Kalsoom, H., Wu, J., Hussain, S., and Latif, M.A. (2019). Simpson\u2019s type inequalities for co-ordinated convex functions on quantum calculus. Symmetry, 11.","DOI":"10.3390\/sym11060768"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"201","DOI":"10.22436\/jnsa.008.03.04","article-title":"Some inequalities of Hermite-Hadamard type for n-times differentiable (\u03c1, m)-geometrically convex functions","volume":"8","author":"Zafar","year":"2015","journal-title":"J. Nonlinear Sci. Appl."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/3\/365\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:03:14Z","timestamp":1760173394000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/3\/365"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,3,2]]},"references-count":42,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2020,3]]}},"alternative-id":["sym12030365"],"URL":"https:\/\/doi.org\/10.3390\/sym12030365","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,3,2]]}}}