{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,16]],"date-time":"2025-10-16T10:05:43Z","timestamp":1760609143808,"version":"build-2065373602"},"reference-count":32,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2019,3,25]],"date-time":"2019-03-25T00:00:00Z","timestamp":1553472000000},"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>In this article, we proposed new Pythagorean trapezoidal uncertain linguistic fuzzy aggregation information\u2014namely, the Pythagorean trapezoidal uncertain linguistic fuzzy Einstein weighted averaging (PTULFEWA) operator, the Pythagorean trapezoidal uncertain linguistic fuzzy Einstein ordered weighted averaging (PTULFEOWA) operator, and the Pythagorean trapezoidal uncertain linguistic fuzzy Einstein hybrid weighted averaging (PTULFEHWA) operator\u2014using the Einstein operational laws. We studied some important properties of the suggested aggregation operators and showed that the PTULFEHWA is more general than the other proposed operators, which simplifies these aggregation operators. Furthermore, we presented a multiple attribute group decision making (MADM) process for the proposed aggregation operators under the Pythagorean trapezoidal uncertain linguistic fuzzy (PTULF) environment. A numerical example was constructed to determine the effectiveness and practicality of the proposed approach. Lastly, a comparative analysis was performed of the presented approach with existing approaches to show that the proposed method is consistent and provides more information that may be useful for complex problems in the decision-making process.<\/jats:p>","DOI":"10.3390\/sym11030440","type":"journal-article","created":{"date-parts":[[2019,3,27]],"date-time":"2019-03-27T05:03:12Z","timestamp":1553662992000},"page":"440","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["A New Ranking Methodology for Pythagorean Trapezoidal Uncertain Linguistic Fuzzy Sets Based on Einstein Operations"],"prefix":"10.3390","volume":"11","author":[{"given":"Arshad Ahmad","family":"Khan","sequence":"first","affiliation":[{"name":"College of Economic and Management, Northwest A &amp; F University Yangling, Shaanxi 712100, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7474-5115","authenticated-orcid":false,"given":"Saleem","family":"Abdullah","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Abdul WAli Khan University, Mardan 23200, Khyber Pakhtunkhwa, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Muhammad","family":"Shakeel","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Hazara Univeristy, Mansehra 21120, Khyber Pakhtunkhwa, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Faisal","family":"Khan","sequence":"additional","affiliation":[{"name":"College of Engineering and Informatics, National University Ireland, Galway H91 CF50, Ireland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Noor ul","family":"Amin","sequence":"additional","affiliation":[{"name":"Department of Information Technology, Hazara University, Mansehra 21120, Khyber Pakhtunkhwa, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jianchao","family":"Luo","sequence":"additional","affiliation":[{"name":"College of Economic and Management, Northwest A &amp; F University Yangling, Shaanxi 712100, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,3,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"338","DOI":"10.1016\/S0019-9958(65)90241-X","article-title":"A Fuzzy set","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":"Carrasco, R.A., Blasco, M.F., Garc\u00eda-Madariaga, J., and Herrera-Viedma, E. (2018). A Fuzzy Linguistic RFM Model Applied to Campaign Management. Int. J. Interact. Multimed. Artif. Intell., in press.","DOI":"10.9781\/ijimai.2018.03.003"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Revanasiddappa, M.B., and Harish, B.S. (2018). A New Feature Selection Method based on Intuitionistic Fuzzy Entropy to Categorize Text Documents. Int. J. Interact. Multimed. Artif. Intell., in press.","DOI":"10.9781\/ijimai.2018.04.002"},{"key":"ref_5","unstructured":"Morente-Molinera, J.A., Kou, G., Crespo, R.G., and Corchado, J.M. (2018, January 26\u201328). SoMeT, Managing Multi-Criteria Group Decision Making Environments with High Number of Alternatives Using Fuzzy Ontologies. Proceedings of the 17th International Conference on Intelligent Software, Granada, Spain."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1","DOI":"10.15388\/Informatica.2018.155","article-title":"Fuzzy logic models for non-programmed decision-making in personnel selection processes based on gamification","volume":"29","year":"2018","journal-title":"Informatica"},{"key":"ref_7","first-page":"19","article-title":"A generalization of operators on intuitionistic fuzzy sets using triangular norms and conorms","volume":"8","author":"Deschrijver","year":"2002","journal-title":"Notes IFS"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"68","DOI":"10.1016\/j.knosys.2016.12.001","article-title":"A social ties-based approach for group decision-making problems with incomplete additive preference relations","volume":"119","author":"Liang","year":"2017","journal-title":"Knowl. Based Syst."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"923","DOI":"10.1109\/TFUZZ.2012.2189405","article-title":"Intuitionistic fuzzy information aggregation using Einstein operations","volume":"20","author":"Wang","year":"2012","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"90","DOI":"10.1016\/j.asoc.2013.12.010","article-title":"Multiple attribute group decision making methods based on intuitionistic linguistic power generalized aggregation operators","volume":"17","author":"Liu","year":"2014","journal-title":"Appl. Soft Comput."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1491","DOI":"10.3233\/IFS-130833","article-title":"Atanassov\u2019s intuitionistic linguistic ordered weighted averaging distance operator and its application to decision making","volume":"26","author":"Su","year":"2014","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_12","first-page":"303","article-title":"A new method for financial decision making under Intuitionistic linguistic envi-ronment","volume":"50","author":"Xiao","year":"2016","journal-title":"Econ. Comput. Econ. Cybern. Stud. Res."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"58","DOI":"10.1016\/j.ins.2012.04.014","article-title":"Methods for aggregating intuitionistic uncertain linguistic variables and their application to group decision making","volume":"205","author":"Liu","year":"2012","journal-title":"Inf. Sci."},{"key":"ref_14","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_15","doi-asserted-by":"crossref","first-page":"958","DOI":"10.1109\/TFUZZ.2013.2278989","article-title":"Pythagorean membership grades in multicriteria decision making","volume":"22","author":"Yager","year":"2014","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1133","DOI":"10.1002\/int.21738","article-title":"Some results for Pythagorean fuzzy sets","volume":"30","author":"Peng","year":"2015","journal-title":"Int. J. Intell. Syst."},{"key":"ref_17","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_18","unstructured":"Shakeel, M., Abduulah, S., Shahzad, M., Mahmood, T., and Siddiqui, N. (2018). Averaging aggregation operators with Pythagorean trapezoidal fuzzy numbers and their application to group decision making. J. Intell. Fuzzy Syst., 1\u201317."},{"key":"ref_19","first-page":"147","article-title":"Averaging aggregation operators with Interval Pythagorean trapezoidal fuzzy numbers and their application to group decision making","volume":"50","author":"Shakeel","year":"2018","journal-title":"J. Math."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"633","DOI":"10.3233\/JIN-180092","article-title":"Induced interval-valued Pythagorean trapezoidal fuzzy aggregation operators based on Einstein operations and their application in group decision making","volume":"17","author":"Shakeel","year":"2018","journal-title":"J. Integr. Neurosci."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Shakeel, M., Abdullah, S., Shahzad, M., and Siddiqui, N. (2019). Geometric aggregation operators with interval-valued Pythagorean trapezoidal fuzzy numbers based on Einstein operations and their application in group decision making. Int. J. Mach. Learn. Cybern., 1\u201320.","DOI":"10.1007\/s40314-019-0858-9"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1198","DOI":"10.1002\/int.21823","article-title":"Symmetric Pythagorean fuzzy weighted geometric\/averaging operators and their application in multi-criteria decision-making problems","volume":"31","author":"Ma","year":"2016","journal-title":"Int. J. Intell. Syst."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1234","DOI":"10.1002\/int.21979","article-title":"Linguistic Pythagorean fuzzy sets and its applications in multi attribute decision-making process","volume":"33","author":"Garg","year":"2018","journal-title":"Int. J. Intell. Syst."},{"key":"ref_24","first-page":"43","article-title":"Trapezoidal\/triangular intuitionistic fuzzy numbers versus interval-valued trapezoidal\/triangular fuzzy numbers and applications to multicriteria decision making methods","volume":"20","author":"Ban","year":"2014","journal-title":"Notes Intuitionistic Fuzzy Sets"},{"key":"ref_25","first-page":"1571","article-title":"Multi-criteria decision-making method based on aggregation operators for intuitionistic linguistic fuzzy numbers","volume":"25","author":"Wang","year":"2010","journal-title":"Control Decis."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1016\/j.asoc.2017.10.017","article-title":"Partitioned Heronian means based on linguistic intuitionistic fuzzy numbers for dealing with multi-attribute group decision making","volume":"62","author":"Liu","year":"2018","journal-title":"Appl. Soft Comput."},{"key":"ref_27","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_28","doi-asserted-by":"crossref","first-page":"491","DOI":"10.3233\/IFS-130924","article-title":"Some geometric Choquet aggregation operators using Einstein operations under intuitionistic fuzzy environment","volume":"26","author":"Zhang","year":"2014","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1061","DOI":"10.1002\/int.21676","article-title":"Extension of TOPSIS to multiple criteria decision making with Pythagorean fuzzy sets","volume":"29","author":"Zhang","year":"2014","journal-title":"Int. J. Intell. Syst."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Lowen, R. (1996). Fuzzy real numbers. Fuzzy Set Theory, Springer.","DOI":"10.1007\/978-94-015-8741-9"},{"key":"ref_31","first-page":"321","article-title":"Aggregation operators on intuitionistic trapezoidal fuzzy number and its application to multi-criteria decision-making problems","volume":"20","author":"Jianqiang","year":"2009","journal-title":"J. Syst. Eng. Electron."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"54","DOI":"10.1016\/j.knosys.2017.09.010","article-title":"Solving multi-criteria group decision making problems under environments with a high number of alternatives using fuzzy ontologies and multi-granular linguistic modelling methods","volume":"137","author":"Kou","year":"2017","journal-title":"Knowl. Based Syst."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/3\/440\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:40:33Z","timestamp":1760186433000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/3\/440"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,3,25]]},"references-count":32,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2019,3]]}},"alternative-id":["sym11030440"],"URL":"https:\/\/doi.org\/10.3390\/sym11030440","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2019,3,25]]}}}