{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,13]],"date-time":"2026-05-13T23:35:43Z","timestamp":1778715343753,"version":"3.51.4"},"reference-count":45,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2017,7,17]],"date-time":"2017-07-17T00:00:00Z","timestamp":1500249600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61473181;11526163"],"award-info":[{"award-number":["61473181;11526163"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012226","name":"Fundamental Research Funds For the Central Universities","doi-asserted-by":"publisher","award":["GK201702008"],"award-info":[{"award-number":["GK201702008"]}],"id":[{"id":"10.13039\/501100012226","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Youth Academic Innovation Team in Northwest University of Political Science and Law"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we extend the rough set model on two different universes in intuitionistic fuzzy approximation spaces to a single-valued neutrosophic environment. Firstly, based on the     ( \u03b1 , \u03b2 , \u03b3 )    -cut relation      R \u02dc   { ( \u03b1 , \u03b2 , \u03b3 ) }     , we propose a rough set model in generalized single-valued neutrosophic approximation spaces. Then, some properties of the new rough set model are discussed. Furthermore, we obtain two extended models of the new rough set model\u2014the degree rough set model and the variable precision rough set model\u2014and study some of their properties. Finally, we explore an example to illustrate the validity of the new rough set model.<\/jats:p>","DOI":"10.3390\/sym9070119","type":"journal-article","created":{"date-parts":[[2017,7,18]],"date-time":"2017-07-18T03:45:16Z","timestamp":1500349516000},"page":"119","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["A Novel Rough Set Model in Generalized Single Valued Neutrosophic Approximation Spaces and Its Application"],"prefix":"10.3390","volume":"9","author":[{"given":"Zhi-Lian","family":"Guo","sequence":"first","affiliation":[{"name":"College of Economics, Northwest University of Political Science and Law, Xi\u2019an 710063, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yan-Ling","family":"Liu","sequence":"additional","affiliation":[{"name":"College of Mathematics and Information Science, Shaanxi Normal University, Xi\u2019an 710119, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hai-Long","family":"Yang","sequence":"additional","affiliation":[{"name":"College of Mathematics and Information Science, Shaanxi Normal University, Xi\u2019an 710119, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2017,7,17]]},"reference":[{"key":"ref_1","unstructured":"Smarandache, F. (1998). Neutrosophy. Neutrosophic Probability, Set, and Logic, American Research Press."},{"key":"ref_2","unstructured":"Smarandache, F. (1999). A Unifying Field in Logics. Neutrosophy: Neutrosophic Probability, Set and Logic, American Research Press."},{"key":"ref_3","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":"Inform. Control"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1016\/0165-0114(86)90077-1","article-title":"Interval valued fuzzy sets based on normal forms","volume":"20","author":"Turksen","year":"1986","journal-title":"Fuzzy Sets Syst."},{"key":"ref_5","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_6","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1016\/0165-0114(89)90205-4","article-title":"Interval valued intuitionistic fuzzy sets","volume":"31","author":"Atanassov","year":"1989","journal-title":"Fuzzy Sets Syst."},{"key":"ref_7","first-page":"410","article-title":"Single valued neutrosophic sets","volume":"4","author":"Wang","year":"2010","journal-title":"Multispace Multistruct."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"386","DOI":"10.1080\/03081079.2012.761609","article-title":"Multicriteria decision making method using the correlation coefficient under single-valued neutrosophic enviroment","volume":"42","author":"Ye","year":"2013","journal-title":"Int. J. Gen. Syst."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2453","DOI":"10.3233\/IFS-141215","article-title":"Improved correlation coefficients of single valued neutrosophic sets and interval neutrosophic sets for multiple attributedecision making","volume":"27","author":"Ye","year":"2014","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_10","first-page":"91","article-title":"Application of neutrosophic set to multi-criteria decision making by copras","volume":"49","author":"Zavadskas","year":"2015","journal-title":"J. Econ. Comput. Econ. Cybern. Stud. Res."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"15923","DOI":"10.3390\/su71215792","article-title":"Sustainable assessment of alternative sites for the construction of a waste incineration plant by applying WASPAS method with single-valued neutrosophic set","volume":"7","author":"Zavadskas","year":"2015","journal-title":"Sustainability"},{"key":"ref_12","first-page":"85","article-title":"Selection of lead-zinc flotation circuit design by applying WASPAS method with single-valued neutrosophic set","volume":"21","author":"Zavadskas","year":"2016","journal-title":"Acta Montan. Slovaca"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"74","DOI":"10.1016\/j.asoc.2017.03.040","article-title":"Sustainable market valuation of buildings by the single-valued neutrosophic MAMVA method","volume":"57","author":"Zavadskas","year":"2017","journal-title":"Appl. Soft Comput."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"85","DOI":"10.15388\/Informatica.2016.78","article-title":"Some single-valued neutrosophic number heronian mean operators and their application in multiple attribute group decision making","volume":"27","author":"Li","year":"2016","journal-title":"Informatica"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"421","DOI":"10.3846\/13923730.2016.1268645","article-title":"Garage location selection for residential house by WASPAS-SVNS method","volume":"23","year":"2017","journal-title":"J. Civ. Eng. Manag."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"165","DOI":"10.3233\/IFS-120724","article-title":"Similarity measures between interval neutrosophic sets and their applications in multi-criteria decision making","volume":"26","author":"Ye","year":"2014","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_17","first-page":"204","article-title":"Vector Similarity measures of simplified neutrosophic sets and their application in multi-criteria decision making","volume":"16","author":"Ye","year":"2014","journal-title":"Int. J. Fuzzy Syst."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1245","DOI":"10.3233\/IFS-130810","article-title":"On similarity and entropy of neutrosophic sets","volume":"26","author":"Majudar","year":"2014","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"336","DOI":"10.1016\/j.asoc.2014.08.070","article-title":"An outranking approach for multi-criteria decision making problems with simplified neutrosophic sets","volume":"25","author":"Peng","year":"2014","journal-title":"Appl. Soft Comput."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"681","DOI":"10.1007\/s11424-015-4010-7","article-title":"Interval neutrosophic prioritized OWA operator and its application to multiple attribute decision making","volume":"29","author":"Liu","year":"2016","journal-title":"J. Syst. Sci. Complex."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2829","DOI":"10.3233\/JIFS-169165","article-title":"Steam turbine fault diagnosis based on single-valued neutrosophic multi-granulation rough sets over two universes","volume":"30","author":"Zhang","year":"2016","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1177","DOI":"10.1007\/s00521-015-2131-5","article-title":"Cross-entropy measure on interval neutrosophic sets and its applications in multi-criteria decision making","volume":"8","author":"Sahin","year":"2017","journal-title":"Neural Comput. Appl."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"142","DOI":"10.1016\/j.cmpb.2015.10.002","article-title":"Multi-period medical diagnosis method using a single-valued neutrosophic similarity measure based on tangent function","volume":"123","author":"Ye","year":"2016","journal-title":"Comput. Methods Progr. Biomed."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1045","DOI":"10.3233\/IFS-151827","article-title":"On single-valued neutrosophic relations","volume":"30","author":"Yang","year":"2016","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"981","DOI":"10.1007\/s13042-015-0456-0","article-title":"Simplified neutrosophic harmonic averaging projection-based method for multiple attribute decision making problems","volume":"8","author":"Ye","year":"2017","journal-title":"Int. J. Mach. Learn. Cybern."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"181","DOI":"10.15388\/Informatica.2017.125","article-title":"A neutrosophic extension of the MULTIMOORA method","volume":"28","author":"Stanujkic","year":"2017","journal-title":"Informatica"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1007\/BF01001956","article-title":"Rough sets","volume":"11","author":"Pawlak","year":"1982","journal-title":"Int. J. Comput. Inform. Sci."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Pawlak, Z. (1991). Rough Sets: Theoretical Aspects of Reasoning about Data, Kluwer Academic Publishers.","DOI":"10.1007\/978-94-011-3534-4"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"169","DOI":"10.3233\/IFS-141287","article-title":"Rough set over dual-universes in intuitionistic fuzzy approximation space and its application","volume":"28","author":"Guo","year":"2015","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"892","DOI":"10.1016\/j.ins.2007.09.006","article-title":"Rough fuzzy approximation on two universes of discourse","volume":"178","author":"Li","year":"2008","journal-title":"Inform. Sci."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1139","DOI":"10.3233\/IFS-151837","article-title":"Rough and rough fuzzy sets on two universes via covering approach","volume":"30","author":"Mareay","year":"2016","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"557","DOI":"10.1007\/s00500-010-0562-6","article-title":"Variable precision rough set model over two universes and its properties","volume":"15","author":"Shen","year":"2011","journal-title":"Soft Comput."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1079","DOI":"10.1057\/jors.2012.75","article-title":"An approach to decision making based on intuitionistic fuzzy rough sets over two universes","volume":"64","author":"Sun","year":"2013","journal-title":"J. Oper. Res. Soc."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"817","DOI":"10.1016\/j.knosys.2010.05.006","article-title":"Research on the model of rough set over dual-universes","volume":"23","author":"Yan","year":"2010","journal-title":"Knowl. Based Syst."},{"key":"ref_35","first-page":"79","article-title":"Rough set over dual-universes in fuzzy approximation space","volume":"9","author":"Yan","year":"2012","journal-title":"Iran. J. Fuzzy Syst."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"94","DOI":"10.1016\/j.knosys.2012.01.001","article-title":"Bipolar fuzzy rough set model on two different universes and its application","volume":"35","author":"Yang","year":"2012","journal-title":"Knowl. Based Syst."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"60","DOI":"10.1016\/j.knosys.2011.07.012","article-title":"Transformation of bipolar fuzzy rough set models","volume":"27","author":"Yang","year":"2012","journal-title":"Knowl. Based Syst."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1410","DOI":"10.1016\/j.ijar.2013.05.001","article-title":"Fuzzy probabilistic rough set model on two universes and its applications","volume":"54","author":"Yang","year":"2013","journal-title":"Int. J. Approx. Reason."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1080\/03081079508908047","article-title":"A non-numeric approach to uncertain reasoning","volume":"23","author":"Yao","year":"1995","journal-title":"Int. J. Gen. Syst."},{"key":"ref_40","first-page":"26833","article-title":"Roughness of neutrosophic sets","volume":"74","author":"Salama","year":"2014","journal-title":"Elix. Appl. Math."},{"key":"ref_41","first-page":"62","article-title":"Rough neutrosophic sets","volume":"3","author":"Broumi","year":"2014","journal-title":"Neutrosophic Sets Syst."},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Broumi, S., and Smarandache, F. (2015, August 07). Single Valued Neutrosophic Information Systems Based on Rough Set Theory. Available online: https:\/\/www.researchgate.net\/publication\/280742564.","DOI":"10.1155\/2015\/232919"},{"key":"ref_43","first-page":"23","article-title":"Interval neutrosophic rough set","volume":"7","author":"Broumi","year":"2015","journal-title":"Neutrosophic Sets Syst."},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Yang, H.L., Zhang, C.L., Guo, Z.L., Yiu, Y.L., and Liao, X.W. (2016). A hybrid model of single-valued neutrosophic sets and rough sets: Single valued neutrosophic rough set model. Soft Comput.","DOI":"10.1007\/s00500-016-2356-y"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"2459","DOI":"10.3233\/IFS-130916","article-title":"A multi-criteria decsion-making method using aggregation operators for simplified neutrosophic sets","volume":"26","author":"Ye","year":"2014","journal-title":"J. Intell. Fuzzy Syst."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/9\/7\/119\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T18:42:56Z","timestamp":1760208176000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/9\/7\/119"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,7,17]]},"references-count":45,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2017,7]]}},"alternative-id":["sym9070119"],"URL":"https:\/\/doi.org\/10.3390\/sym9070119","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,7,17]]}}}