{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:00:25Z","timestamp":1760241625526,"version":"build-2065373602"},"reference-count":56,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2018,6,8]],"date-time":"2018-06-08T00:00:00Z","timestamp":1528416000000},"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>A relation is viewed as a granularity from a granular computing perspective. A classic rough set contains only one granularity. A multi-granulation rough set contains multiple granularities, which promotes the applications of classical rough set. Firstly, this paper uses the incomplete interval-valued decision information system (IIVDIS) as research object and constructs two rough set models in the light of single granularity rough set model for applying the rough set theory to real life more widely, which are optimistic multi-granulation rough set (OMGRS) model and pessimistic multi-granulation rough set (PMGRS) model in the IIVDIS. Secondly, we design two algorithms to compute the roughness and the degree of dependence that are two tools for measuring uncertainty of rough set. Finally, several experiments are performed on six UCI data sets to verify the validity of the proposed theorems.<\/jats:p>","DOI":"10.3390\/sym10060208","type":"journal-article","created":{"date-parts":[[2018,6,8]],"date-time":"2018-06-08T11:19:31Z","timestamp":1528456771000},"page":"208","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Multi-Granulation Rough Set for Incomplete Interval-Valued Decision Information Systems Based on Multi-Threshold Tolerance Relation"],"prefix":"10.3390","volume":"10","author":[{"given":"Bingyan","family":"Lin","sequence":"first","affiliation":[{"name":"School of Sciences, Chongqing University of Technology, Chongqing 400054, China"}]},{"given":"Weihua","family":"Xu","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Southwest University, Chongqing 400715, China"}]}],"member":"1968","published-online":{"date-parts":[[2018,6,8]]},"reference":[{"key":"ref_1","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. Inf. Sci."},{"key":"ref_2","first-page":"661","article-title":"Rough set theory and its application to analysis","volume":"29","author":"Pawlak","year":"1998","journal-title":"J. Cybern."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1016\/S0957-4174(99)00053-6","article-title":"The integrated methodology of rough set theory and artificial neural network for business failure prediction","volume":"18","author":"Ahn","year":"2000","journal-title":"Expert Syst. Appl."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2264","DOI":"10.1109\/TPWRD.2010.2055587","article-title":"Knowledge discovery in distance relay event report: A comparative data-mining strategy of rough set theory with decision tree","volume":"25","author":"Othman","year":"2010","journal-title":"IEEE Trans. Power Deliv."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"567","DOI":"10.1002\/int.20019","article-title":"Rough set spatial data modeling for data mining","volume":"19","author":"Beaubouef","year":"2004","journal-title":"Int. J. Intell. Syst."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1007\/s13042-011-0019-y","article-title":"Extreme learning machines: A survey","volume":"2","author":"Huang","year":"2011","journal-title":"Int. J. Mach. Learn. Cybern."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"725","DOI":"10.1016\/j.ins.2004.12.010","article-title":"Rough prime ideals and rough fuzzy prime ideals in semigroups","volume":"176","author":"Xiao","year":"2006","journal-title":"Inf. Sci."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"3658","DOI":"10.1016\/j.ins.2006.02.014","article-title":"Roughness in modules","volume":"176","author":"Davvaz","year":"2006","journal-title":"Inf. Sci."},{"key":"ref_9","first-page":"40","article-title":"A hybrid clustering method based on improved artificial bee colony and fuzzy C-means algorithm","volume":"15","author":"Kumar","year":"2017","journal-title":"Int. J. Artif. Intell."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1016\/j.knosys.2016.08.009","article-title":"Feature subset selection based on fuzzy neighborhood rough sets","volume":"111","author":"Wang","year":"2016","journal-title":"Knowl.-Based Syst."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Wang, C.Z., Hu, Q.H., Wang, X.Z., Chen, D.G., Qian, Y.H., and Dong, Z. (2017). Feature selection based on neighborhood discrimination index. IEEE Trans. Neural Netw. Learn. Syst., 1\u201314.","DOI":"10.1109\/TNNLS.2017.2710422"},{"key":"ref_12","unstructured":"Wang, C.Z., He, Q., Shao, M.W., and Hu, Q.H. (2017). Feature selection based on maximal neighborhood discernibility. Int. J. Mach. Learn. Cybern., 1\u201312."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1016\/S0020-0255(03)00056-2","article-title":"Reduction and axiomization of covering generalized rough sets","volume":"152","author":"Zhu","year":"2003","journal-title":"Inf. Sci."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"323","DOI":"10.1016\/j.trit.2016.11.001","article-title":"A survey on rough set theory and its applications","volume":"1","author":"Zhang","year":"2016","journal-title":"CAAI Trans. Intell. Technol."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1997","DOI":"10.1007\/s00500-014-1387-5","article-title":"Complement information entropy for uncertainty measure in fuzzy rough set and its applications","volume":"19","author":"Zhao","year":"2015","journal-title":"Soft Comput."},{"key":"ref_16","first-page":"415","article-title":"Non-monotonic attribute reduction in decision-theoretic rough sets","volume":"126","author":"Li","year":"2013","journal-title":"Fundam. Inf."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"18","DOI":"10.1016\/j.knosys.2017.07.017","article-title":"A unified information measure for general binary relations","volume":"135","author":"Wang","year":"2017","journal-title":"Knowl.-Based Syst."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"12249","DOI":"10.1016\/j.eswa.2011.04.003","article-title":"Rule extraction based on granulation order in interval-valued fuzzy information system","volume":"38","author":"Cheng","year":"2011","journal-title":"Expert Syst. Appl."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1016\/j.ins.2013.04.031","article-title":"Variable-precision-dominance-based rough set approach to interval-valued information systems","volume":"244","author":"Zhang","year":"2013","journal-title":"Inf. Sci."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"620","DOI":"10.1016\/j.ijar.2012.01.001","article-title":"Rough sets based matrix approaches with dynamic attribute variation in set-valued information systems","volume":"53","author":"Zhang","year":"2012","journal-title":"Int. J. Approx. Reason."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1512","DOI":"10.4236\/am.2013.411204","article-title":"Attribute reduction in interval and set-valued decision information systems","volume":"4","author":"Wang","year":"2013","journal-title":"Appl. Math."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1799","DOI":"10.3233\/IFS-130859","article-title":"Uncertainty measure of Atanassov\u2019s intuitionistic fuzzy T equivalence information systems","volume":"26","author":"Xu","year":"2014","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1007\/s13042-015-0492-9","article-title":"Attributes reduction and rules acquisition in an lattice-valued information system with fuzzy decision","volume":"8","author":"Zhang","year":"2017","journal-title":"Int. J. Mach. Learn. Cybern."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1347","DOI":"10.1016\/0895-7177(89)90373-7","article-title":"Rough classification in incomplete information systems","volume":"12","author":"Sowiski","year":"1989","journal-title":"Math. Comput. Model."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"164","DOI":"10.1016\/j.ejor.2004.03.032","article-title":"Knowledge acquisition in incomplete information systems: A rough set approach","volume":"168","author":"Leung","year":"2006","journal-title":"Eur. J. Oper. Res."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"858","DOI":"10.1016\/j.knosys.2011.03.007","article-title":"Neighborhood systems-based rough sets in incomplete information system","volume":"24","author":"Yang","year":"2011","journal-title":"Knowl.-Based Syst."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1016\/S0020-0255(98)10019-1","article-title":"Rough set approach to incomplete information systems","volume":"112","author":"Kryszkiewicz","year":"1998","journal-title":"Inf. Sci."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1331","DOI":"10.1016\/j.datak.2009.07.007","article-title":"Dominance-based rough set approach to incomplete interval-valued information system","volume":"68","author":"Yang","year":"2009","journal-title":"Data Knowl. Eng."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"691","DOI":"10.1016\/j.camwa.2004.06.028","article-title":"Incomplete information system and its optimal selections","volume":"48","author":"Zhang","year":"2004","journal-title":"Comput. Math. Appl."},{"key":"ref_30","first-page":"1881","article-title":"A new method of attribute reduction based on information quantity in an incomplete system","volume":"7","author":"Xu","year":"2012","journal-title":"J. Softw."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"255","DOI":"10.3103\/S0146411608050040","article-title":"Extensions of dominance-based rough set approach in incomplete information system","volume":"42","author":"Wei","year":"2008","journal-title":"Autom. Control Comput. Sci."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"334","DOI":"10.1016\/j.ins.2014.10.003","article-title":"\u03b1-Dominance relation and rough sets in interval-valued information systems","volume":"294","author":"Yang","year":"2015","journal-title":"Inf. Sci."},{"key":"ref_33","first-page":"469","article-title":"\u03b8-Improved limited tolerance relation model of incomplete information system for evaluation of water conservancy project management modernization","volume":"6","author":"Gao","year":"2013","journal-title":"Water Sci. Eng."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1219","DOI":"10.1016\/j.ins.2007.09.019","article-title":"Dominance-based rough set approach and knowledge reductions in incomplete ordered information System","volume":"178","author":"Yang","year":"2008","journal-title":"Inf. Sci."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1002\/int.20051","article-title":"Dominance relation and rules in an incomplete ordered information system","volume":"20","author":"Shao","year":"2005","journal-title":"Int. J. Intell. Syst."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"106","DOI":"10.1016\/j.ins.2016.01.098","article-title":"Dominance-based rough set approach to incomplete ordered information systems","volume":"346\u2013347","author":"Du","year":"2016","journal-title":"Inf. Sci."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1019","DOI":"10.1007\/s13042-015-0408-8","article-title":"Decision rules acquisition based on interval knowledge granules for incomplete ordered decision information systems","volume":"6","author":"Huang","year":"2015","journal-title":"Int. J. Mach. Learn. Cybern."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Dai, J.H., Wei, B.J., Shi, H., and Liu, W. (2017, January 21\u201323). Uncertainty measurement for incomplete interval-valued information systems by \u03b8-rough set model. Proceedings of the 3rd International Conference on Information Management (ICIM), Chengdu, China.","DOI":"10.1109\/INFOMAN.2017.7950378"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1016\/j.knosys.2017.09.009","article-title":"Uncertainty measurement for incomplete interval-valued information systems based on \u03b1-weak similarity","volume":"136","author":"Dai","year":"2017","journal-title":"Knowl.-Based Syst."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1002\/1098-111X(200101)16:1<87::AID-INT7>3.0.CO;2-S","article-title":"Information granulation and rough set approximation","volume":"16","author":"Yao","year":"2001","journal-title":"Int. J. Intell. Syst."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"420","DOI":"10.1109\/TSMCA.2009.2035436","article-title":"Incomplete multigranulation rough set","volume":"40","author":"Qian","year":"2010","journal-title":"IEEE Trans. Syst. Man Cybern. Part A"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1002\/1098-111X(200101)16:1<57::AID-INT6>3.0.CO;2-Y","article-title":"Information granules: Towards foundations of granular computing","volume":"16","author":"Skowron","year":"2001","journal-title":"Int. J. Intell. Syst."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"1244","DOI":"10.1016\/j.asoc.2009.03.007","article-title":"Knowledge granulation, knowledge entropy and knowledge uncertainty measure in ordered information systems","volume":"9","author":"Xu","year":"2009","journal-title":"Appl. Soft. Comput."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1016\/S0165-0114(97)00077-8","article-title":"Toward a theory of fuzzy information granulation and its centrality in human reasoning and fuzzy logic","volume":"90","author":"Zadeh","year":"1997","journal-title":"Fuzzy Sets Syst."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"949","DOI":"10.1016\/j.ins.2009.11.023","article-title":"MGRS: A multi-granulation rough set","volume":"180","author":"Qian","year":"2010","journal-title":"Inf. Sci."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"712","DOI":"10.1016\/j.ins.2009.11.018","article-title":"Multi-adjoint t-concept lattices","volume":"180","author":"Medina","year":"2010","journal-title":"Inf. Sci."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"86","DOI":"10.1016\/j.fss.2011.12.016","article-title":"Signatures: Definitions, operators and applications to fuzzy modelling","volume":"201","author":"Pozna","year":"2012","journal-title":"Fuzzy Sets Syst."},{"key":"ref_48","first-page":"1","article-title":"Medical image retrieval using vector quantization and fuzzy S-tree","volume":"41","year":"2017","journal-title":"J. Med. Syst."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"1159","DOI":"10.12785\/amis\/080327","article-title":"Incomplete variable multigranulation rough sets decision","volume":"8","author":"Zhang","year":"2014","journal-title":"Appl. Math. Inf. Sci."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"548","DOI":"10.3923\/itj.2013.548.550","article-title":"Interval-valued information systems rough set model based on multi-granulations","volume":"12","author":"Yang","year":"2013","journal-title":"Inf. Technol. J."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"475","DOI":"10.1080\/03081079.2012.673598","article-title":"Multiple granulation rough set approach to ordered information systems","volume":"41","author":"Xu","year":"2012","journal-title":"Int. J. Gen. Syst."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"323","DOI":"10.1007\/978-3-642-24553-4_44","article-title":"Incomplete multigranulation rough sets in incomplete ordered decision system","volume":"Volume 6840","author":"Huang","year":"2012","journal-title":"Bio-Inspired Computing and Applications"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1007\/s13042-011-0054-8","article-title":"On multigranulation rough sets in incomplete information system","volume":"3","author":"Yang","year":"2012","journal-title":"Int. J. Mach. Learn. Cybern."},{"key":"ref_54","first-page":"67","article-title":"Set pair analysis and prelimiary application","volume":"13","author":"Zhao","year":"1994","journal-title":"Explor. Nat."},{"key":"ref_55","first-page":"163","article-title":"Attribute reduction of interval-valued information system based on variable precision tolerance relation","volume":"36","author":"Chen","year":"2009","journal-title":"Comput. Sci."},{"key":"ref_56","first-page":"524","article-title":"Attribute reduction algorithm based on rough set in incomplete interval-valued information","volume":"37","author":"Zeng","year":"2013","journal-title":"J. Nanjing Univ. Sci. Technol."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/10\/6\/208\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:07:55Z","timestamp":1760195275000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/10\/6\/208"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,6,8]]},"references-count":56,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2018,6]]}},"alternative-id":["sym10060208"],"URL":"https:\/\/doi.org\/10.3390\/sym10060208","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2018,6,8]]}}}