{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,6]],"date-time":"2026-05-06T18:45:08Z","timestamp":1778093108013,"version":"3.51.4"},"reference-count":38,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2021,5,26]],"date-time":"2021-05-26T00:00:00Z","timestamp":1621987200000},"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>Binary relations are most important in various fields of pure and applied sciences. The concept of linear Diophantine fuzzy sets (LDFSs) proposed by Riaz and Hashmi is a novel mathematical approach to model vagueness and uncertainty in decision-making problems. In LDFS theory, the use of reference or control parameters corresponding to membership and non-membership grades makes it most accommodating towards modeling uncertainties in real-life problems. The main purpose of this paper is to establish a robust fusion of binary relations and LDFSs, and to introduce the concept of linear Diophantine fuzzy relation (LDF-relation) by making the use of reference parameters corresponding to the membership and non-membership fuzzy relations. The novel concept of LDF-relation is more flexible to discuss the symmetry between two or more objects that is superior to the prevailing notion of intuitionistic fuzzy relation (IF-relation). Certain basic operations are defined to investigate some significant results which are very useful in solving real-life problems. Based on these operations and their related results, it is analyzed that the collection of all LDF-relations gives rise to some algebraic structures such as semi-group, semi-ring and hemi-ring. Furthermore, the notion of score function of LDF-relations is introduced to analyze the symmetry of the optimal decision and ranking of feasible alternatives. Additionally, a new algorithm for modeling uncertainty in decision-making problems is proposed based on LDFSs and LDF-relations. A practical application of proposed decision-making approach is illustrated by a numerical example. Proposed LDF-relations, their operations, and related results may serve as a foundation for computational intelligence and modeling uncertainties in decision-making problems.<\/jats:p>","DOI":"10.3390\/sym13060945","type":"journal-article","created":{"date-parts":[[2021,5,26]],"date-time":"2021-05-26T21:56:44Z","timestamp":1622066204000},"page":"945","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":70,"title":["Linear Diophantine Fuzzy Relations and Their Algebraic Properties with Decision Making"],"prefix":"10.3390","volume":"13","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"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6443-4962","authenticated-orcid":false,"given":"Muhammad","family":"Aslam","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Sciences, King Khalid University, Abha 61413, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6113-3689","authenticated-orcid":false,"given":"Ronnason","family":"Chinram","sequence":"additional","affiliation":[{"name":"Algebra and Applications Research Unit, Division of Computational Science, Faculty of Science, Prince of Songkla University, Hat Yai, Songkhla 90110, Thailand"}]}],"member":"1968","published-online":{"date-parts":[[2021,5,26]]},"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. 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