{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,29]],"date-time":"2026-06-29T14:58:37Z","timestamp":1782745117212,"version":"3.54.5"},"reference-count":16,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2019,2,16]],"date-time":"2019-02-16T00:00:00Z","timestamp":1550275200000},"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>Neutrosophic cubic sets (NCs) are amore generalized version of neutrosophic sets(Ns) and interval neutrosophic sets (INs). Neutrosophic cubic setsare better placed to express consistent, indeterminate and inconsistent information, which provides a better platform to deal with incomplete, inconsistent and vague data. Aggregation operators play a key role in daily life, and in relation to science and engineering problems. In this paper we defined the algebraic and Einstein sum, multiplication and scalar multiplication, score and accuracy functions. Using these operations we defined geometric aggregation operators and Einstein geometric aggregation operators. First, we defined the algebraic and Einstein operators of addition, multiplication and scalar multiplication. We defined score and accuracy function to compare neutrosophic cubic values. Then we definedthe neutrosophic cubic weighted geometric operator (NCWG), neutrosophic cubic ordered weighted geometric operator (NCOWG), neutrosophic cubic Einstein weighted geometric operator (NCEWG), and neutrosophic cubic Einstein ordered weighted geometric operator (NCEOWG) over neutrosophic cubic sets. A multi-criteria decision making method is developed as an application to these operators. This method is then applied to a daily life problem.<\/jats:p>","DOI":"10.3390\/sym11020247","type":"journal-article","created":{"date-parts":[[2019,2,17]],"date-time":"2019-02-17T22:11:50Z","timestamp":1550441510000},"page":"247","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":16,"title":["Neutrosophic Cubic Einstein Geometric Aggregation Operators with Application to Multi-Criteria Decision Making Method"],"prefix":"10.3390","volume":"11","author":[{"given":"Majid","family":"Khan","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, Hazara University, Mansehra 21130, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6438-1047","authenticated-orcid":false,"given":"Muhammad","family":"Gulistan","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Hazara University, Mansehra 21130, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Naveed","family":"Yaqoob","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science Al-Zulfi, Majmaah University, Al-Zulfi 11932, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Madad","family":"Khan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, COMSATS University Iaslamabad, Abbottabad Campus, Abbottabad 22060, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5560-5926","authenticated-orcid":false,"given":"Florentin","family":"Smarandache","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of New Mexico, Albuquerque, NM 87301, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2019,2,16]]},"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":"183","DOI":"10.1016\/0165-0114(95)00167-0","article-title":"Interval valued strict preferences with Zadeh triplet","volume":"78","author":"Turksen","year":"1996","journal-title":"Fuzzy Sets Syst."},{"key":"ref_3","first-page":"28","article-title":"Outlines of new approach to the analysis of complex system and dicision procosses interval valued fuzzy sets","volume":"1","author":"Zadeh","year":"1968","journal-title":"IEEE Trans. Syst. Man Cybernet."},{"key":"ref_4","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_5","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1016\/0165-0114(89)90205-4","article-title":"Interval intuitionistic fuzzy sets","volume":"31","author":"Atanassov","year":"1989","journal-title":"Fuzzy Sets Syst."},{"key":"ref_6","first-page":"83","article-title":"Cubic sets","volume":"1","author":"Jun","year":"2012","journal-title":"Ann. Fuzzy Math. Inform."},{"key":"ref_7","unstructured":"Smarandache, F. (1999). A Unifying Field in Logics, Neutrosophic Logic, Neutrosophy, Neutrosophic Set and Neutrosophic Probabilty, American Research Press. [4th ed.]."},{"key":"ref_8","unstructured":"Wang, H., Smarandache, F., Zhang, Y.Q., and Sunderraman, R. (2005). Interval neutrosophic sets and loics. Theory and Application in Computing, Hexis."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1142\/S1793005717500041","article-title":"Neutrosophic cubic sets","volume":"13","author":"Jun","year":"2015","journal-title":"New. Math. Nat. Comput."},{"key":"ref_10","first-page":"99","article-title":"P-union and P-intersection of neutrosophic cubic sets","volume":"25","author":"Jun","year":"2017","journal-title":"An. St. Univ. Ovidius Constanta"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"377","DOI":"10.1615\/Int.J.UncertaintyQuantification.2017020446","article-title":"Applications of neutrosophic cubic sets in multi-criteria decision making","volume":"7","author":"Zhan","year":"2017","journal-title":"Int. J. Uncertain. Quabtif."},{"key":"ref_12","first-page":"64","article-title":"GRA for multi attribute decision making in neutrosophic cubic set environment","volume":"15","author":"Banerjee","year":"2017","journal-title":"Neutrosophic Sets Syst."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"121","DOI":"10.3390\/sym9070121","article-title":"Cosine measure for neutrosophic cubic sets for multiple attribte decision making","volume":"9","author":"Lu","year":"2017","journal-title":"Symmetry"},{"key":"ref_14","first-page":"44","article-title":"Neutrosophic cubic MCGDM method based on similarity measure","volume":"16","author":"Pramanik","year":"2017","journal-title":"Neutrosophic Sets Syst."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Shi, L., and Ye, J. (2018). Dombi Aggregation Operators of Neutrosophic Cubic Set for Multiple Attribute Deicision Making. Algorithms, 11.","DOI":"10.3390\/a11030029"},{"key":"ref_16","first-page":"67","article-title":"A Novel Generalized Simplified Neutrosophic Number Einstein Aggregation Operator","volume":"48","author":"Li","year":"2018","journal-title":"Int. J. Appl. Math."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/2\/247\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:32:40Z","timestamp":1760185960000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/2\/247"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,2,16]]},"references-count":16,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2019,2]]}},"alternative-id":["sym11020247"],"URL":"https:\/\/doi.org\/10.3390\/sym11020247","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,2,16]]}}}