{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,5]],"date-time":"2026-03-05T10:25:35Z","timestamp":1772706335085,"version":"3.50.1"},"reference-count":52,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2022,11,30]],"date-time":"2022-11-30T00:00:00Z","timestamp":1669766400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"German Research Foundation and the TU Berlin"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Rough set (RS) and fuzzy set (FS) theories were developed to account for ambiguity in the data processing. The most persuasive and modernist abstraction of an FS is the linear Diophantine FS (LD-FS). This paper introduces a resilient hybrid linear Diophantine fuzzy RS model (LDF-RS) on paired universes based on a linear Diophantine fuzzy relation (LDF-R). This is a typical method of fuzzy RS (F-RS) and bipolar FRS (BF-RS) on two universes that are more appropriate and customizable. By using an LDF-level cut relation, the notions of lower approximation (L-A) and upper approximation (U-A) are defined. While this is going on, certain fundamental structural aspects of LD-FAs are thoroughly investigated, with some instances to back them up. This cutting-edge LDF-RS technique is crucial from both a theoretical and practical perspective in the field of medical assessment.<\/jats:p>","DOI":"10.3390\/axioms11120686","type":"journal-article","created":{"date-parts":[[2022,11,30]],"date-time":"2022-11-30T05:45:22Z","timestamp":1669787122000},"page":"686","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["Linear Diophantine Fuzzy Rough Sets on Paired Universes with Multi Stage Decision Analysis"],"prefix":"10.3390","volume":"11","author":[{"given":"Saba","family":"Ayub","sequence":"first","affiliation":[{"name":"Department of Mathematics, Quaid-i-Azam University, Islamabad 45320, Pakistan"}]},{"given":"Muhammad","family":"Shabir","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Quaid-i-Azam University, Islamabad 45320, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8115-9168","authenticated-orcid":false,"given":"Muhammad","family":"Riaz","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of the Punjab, Lahore 54590, Pakistan"}]},{"given":"Faruk","family":"Karaaslan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, \u00c7ank\u0131r\u0131 Karatekin University, \u00c7ank\u0131r\u0131 18100, Turkey"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3583-9434","authenticated-orcid":false,"given":"Dragan","family":"Marinkovic","sequence":"additional","affiliation":[{"name":"Department of Structural Analysis, Technical University of Berlin, 10623 Berlin, Germany"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4553-8827","authenticated-orcid":false,"given":"Djordje","family":"Vranjes","sequence":"additional","affiliation":[{"name":"Academy of Technical and Art Applied Studies Belgrade, University of Belgrade, 11000 Belgrade, Serbia"}]}],"member":"1968","published-online":{"date-parts":[[2022,11,30]]},"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":"Intuintionistic Fuzzy Sets","volume":"20","author":"Atanssov","year":"1986","journal-title":"Fuzzy Sets Syst."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1016\/0165-0114(89)90215-7","article-title":"More on Intuintionistic Fuzzy Sets","volume":"33","author":"Atanssov","year":"1989","journal-title":"Fuzzy Sets Syst."},{"key":"ref_4","unstructured":"Zhang, W.R. (1994, January 18\u201321). Bipolar fuzzy sets and relations: A computational framework for cognitive modeling and multiagent decision analysis. Proceedings of the Industrial Fuzzy Control and Intelligent Systems Conference, and The NASA Joint Technology Workshop on Neural Networks and Fuzzy Logic and Fuzzy Information Processing Society Biannual Conference, San Antonio, TX, USA."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Yager, R.P. (2013, January 24\u201328). Pythagorean fuzzy subsets. Proceedings of the IFSA World Congress and NAFIPS Anual Meeting, Edmonton, AB, Canada.","DOI":"10.1109\/IFSA-NAFIPS.2013.6608375"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"958","DOI":"10.1109\/TFUZZ.2013.2278989","article-title":"Pythagorean membership grades in multi-criteria decision maiking","volume":"22","author":"Yager","year":"2014","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_7","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_8","doi-asserted-by":"crossref","first-page":"5417","DOI":"10.3233\/JIFS-190550","article-title":"Linear Diophantine fuzzy set and its applications towards multi-attribute decision-making problems","volume":"37","author":"Riaz","year":"2019","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1687","DOI":"10.1007\/s12652-021-03130-y","article-title":"A new approach to q-linear Diophantine fuzzy emergency decision support system for COVID19","volume":"13","author":"Almagrabi","year":"2022","journal-title":"J. Ambient. Intell. Humaniz. Comput."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"10353","DOI":"10.1007\/s12652-020-02826-x","article-title":"Linear Diophantine fuzzy algebraic structures","volume":"12","author":"Kamaci","year":"2021","journal-title":"J. Ambient. Intell. Humaniz. Comput."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Riaz, M., Hashmi, M.R., Kulsoom, H., Pamucar, D., and Chu, Y.M. (2020). Linear Diophantine fuzzy soft rough sets for the selection of sustainable material handling equipment. Symmetry, 12.","DOI":"10.3390\/sym12081215"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Ayub, S., Shabir, M., Riaz, M., Aslam, M., and Chinram, R. (2021). Linear Diophantine Fuzzy relations and their algebraic properties with decision making. Symmetry, 13.","DOI":"10.3390\/sym13060945"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1016\/S0020-0255(71)80005-1","article-title":"Similarity Relations and Fuzzy Orderings","volume":"3","author":"Zadeh","year":"1971","journal-title":"Inf. Sci."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/978-3-540-78311-4_1","article-title":"Mathematics of fuzziness-Basic Issues","volume":"Volume 245","author":"Wang","year":"2009","journal-title":"Studies in Fuzziness and Soft Computing"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1007\/978-3-642-29127-2_8","article-title":"Intuitionistic Fuzzy Relations (IFRs)","volume":"Volume 283","author":"Atanassov","year":"2012","journal-title":"On Intuitionistic Fuzzy Sets Theory. Studies in Fuzziness and Soft Computing"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1016\/S0898-1221(99)00056-5","article-title":"Soft set theory-first results","volume":"37","author":"Molodtsov","year":"1999","journal-title":"Comput. Math. Appl."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"273","DOI":"10.1007\/s40815-019-00763-2","article-title":"m-polar neutrosophic topology with applications to multi-criteria decision-making in medical diagnosis and clustering analysis","volume":"22","author":"Hashmi","year":"2020","journal-title":"Int. J. Fuzzy Syst."},{"key":"ref_18","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. Inf. Comp. Sci."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Pawlak, Z. (1991). Rough Sets-Theoretical Aspects of Reasoning Anout Data, Kluwer Academic Publishing.","DOI":"10.1007\/978-94-011-3534-4"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1016\/S0020-0255(98)00012-7","article-title":"Constructive and algebraic methods of the theory of rough sets","volume":"109","author":"Yao","year":"1998","journal-title":"Inf. Sci."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"4997","DOI":"10.1016\/j.ins.2007.05.037","article-title":"Generalized rough sets based on relations","volume":"177","author":"Zhu","year":"2007","journal-title":"Inf. Sci."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"3247","DOI":"10.1016\/j.ins.2008.03.014","article-title":"A short note on algebraic T-rough sets","volume":"178","author":"Davvaz","year":"2008","journal-title":"Inf. Sci."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"245","DOI":"10.3233\/FI-1996-272311","article-title":"Tolerance approximation spaces","volume":"27","author":"Skowron","year":"1996","journal-title":"Fundam. Inform."},{"key":"ref_24","first-page":"249","article-title":"Similarity relation as a basis for rough approximations","volume":"53","author":"Slowinski","year":"1995","journal-title":"ICS Res. Rep."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"4138","DOI":"10.1016\/j.ins.2008.07.002","article-title":"Generalized rough sets baased on reflexive and transitive relations","volume":"178","author":"Qin","year":"2008","journal-title":"Inf. Sci."},{"key":"ref_26","first-page":"1","article-title":"Roughness in semigroups by soft binary relations","volume":"35","author":"Kanwal","year":"2018","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"839","DOI":"10.1007\/s00500-016-2077-2","article-title":"Rough approximations based on soft binary relations and knowledge bases","volume":"21","author":"Li","year":"2017","journal-title":"Soft Comput."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"3329","DOI":"10.1016\/j.asoc.2011.01.003","article-title":"A note on soft sets, rough soft sets, fuzzy soft sets","volume":"11","author":"Ali","year":"2011","journal-title":"Appl. Soft. Comput."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"949","DOI":"10.1016\/j.ins.2009.11.023","article-title":"MGRS: A Multigranulation rough set","volume":"180","author":"Qian","year":"2010","journal-title":"Inf. Sci."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1007\/s40314-020-1098-8","article-title":"New types of soft rough sets in groups by normal soft groups","volume":"39","author":"Ayub","year":"2020","journal-title":"Comput. Appl. Math."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"2381","DOI":"10.1007\/s10462-018-9617-3","article-title":"A novel type of soft rough covering and its application to multicriteria group decision making","volume":"52","author":"Zhan","year":"2019","journal-title":"Artif. Intell. Rev."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"281","DOI":"10.1016\/j.knosys.2014.07.010","article-title":"Attribute analysis of information systems based on elementary soft implications","volume":"70","author":"Feng","year":"2014","journal-title":"Knowl.-Based Syst."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"318","DOI":"10.31181\/rme20023062022r","article-title":"Picture fuzzy aggregation approach with application to third-party logistic provider selection process","volume":"3","author":"Riaz","year":"2022","journal-title":"Rep. Mech. Eng."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"301","DOI":"10.31181\/rme20020042022a","article-title":"Interval-Valued Picture Fuzzy Maclaurin Symmetric Mean Operator with application in Multiple Attribute Decision-Making","volume":"3","author":"Ashraf","year":"2022","journal-title":"Rep. Mech. Eng."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"105","DOI":"10.31181\/rme2001020105t","article-title":"Einstein Geometric Aggregation Operators using a Novel Complex Interval-valued Pythagorean Fuzzy Setting with Application in Green Supplier Chain Management","volume":"2","author":"Ali","year":"2021","journal-title":"Rep. Mech. Eng."},{"key":"ref_36","first-page":"399","article-title":"Risk priority evaluation of power transformer parts based on hybrid FMEA framework under hesitant fuzzy environment","volume":"20","author":"Zhou","year":"2022","journal-title":"Facta Univ. Ser. Mech. Eng."},{"key":"ref_37","first-page":"1","article-title":"Applications of roughness in soft-intersection groups","volume":"8","author":"Ayub","year":"2019","journal-title":"Comput. Appl. Math."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"93088","DOI":"10.1109\/ACCESS.2019.2927317","article-title":"A study of roughness in modules of fractions","volume":"7","author":"Chen","year":"2019","journal-title":"IEEE Access"},{"key":"ref_39","first-page":"555","article-title":"An extended single-valued neutrosophic AHP and MULTIMOORA method to evaluate the optimal training aircraft for flight training organizations","volume":"19","author":"Karabasevic","year":"2021","journal-title":"Facta Univ. Ser. Mech. Eng."},{"key":"ref_40","first-page":"473","article-title":"Evaluation of logistic flows in green supply chains based on the combined DEMATEL-ANP method","volume":"19","author":"Osintsev","year":"2021","journal-title":"Facta Univ. Ser. Mech. Eng."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"2585","DOI":"10.3233\/JIFS-162177","article-title":"The lower and upper approximations and homomorphisms between lower and upper approximations in quotient groups","volume":"33","author":"Mahmood","year":"2017","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_42","first-page":"1","article-title":"A comparision between lower and upper approximations in groups with respect to group homomorphisms","volume":"35","author":"Mahmood","year":"2018","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_43","first-page":"1","article-title":"A new methodolgy for fuzzification of rough sets based on \u03b1-indiscernibility","volume":"16","author":"Shabir","year":"2016","journal-title":"Fuzzy Sets Syst."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"1130","DOI":"10.1109\/TFUZZ.2006.889960","article-title":"Attributes reduction using fuzzy rough sets","volume":"16","author":"Tsang","year":"2008","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1080\/03081079008935107","article-title":"Fuzzy rough sets and rough fuzzy sets","volume":"17","author":"Dubois","year":"1990","journal-title":"Int. J. Gen. Syst."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s40314-020-01174-y","article-title":"Roughness of a set by (\u03b1,\u03b2)-indiscernibility of Bipolar fuzzy relation","volume":"39","author":"Gul","year":"2020","journal-title":"Comput. Appl. Math."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"110","DOI":"10.1016\/j.knosys.2009.06.011","article-title":"Rough set theory based on two universal sets and its applications","volume":"23","author":"Liu","year":"2010","journal-title":"Knowl. Based Syst."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"1798","DOI":"10.1016\/j.apm.2010.10.010","article-title":"Fuzzy rough set model on two different universes and its applications","volume":"35","author":"Sun","year":"2011","journal-title":"Appl. Math. Model."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"892","DOI":"10.1016\/j.ins.2007.09.006","article-title":"Rough fuzzy approximations on two universes of discourse","volume":"178","author":"Li","year":"2008","journal-title":"Inf. Sci."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"465","DOI":"10.1016\/j.knosys.2010.11.004","article-title":"A note on Rough set theory based on two universal sets and its applications","volume":"24","author":"Yang","year":"2010","journal-title":"Knowl.-Based Syst."},{"key":"ref_51","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 it applications","volume":"35","author":"Yang","year":"2012","journal-title":"Knowl.-Based Syst."},{"key":"ref_52","doi-asserted-by":"crossref","unstructured":"Ayub, S., Shabir, M., Riaz, M., Mahmood, W., Bozanic, D., and Marinkovic, D. (2022). Linear Diophantine Fuzzy Rough Sets: A New Rough Set Approach with Decision Making. 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