{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,1]],"date-time":"2026-02-01T11:33:07Z","timestamp":1769945587167,"version":"3.49.0"},"reference-count":56,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2020,7,24]],"date-time":"2020-07-24T00:00:00Z","timestamp":1595548800000},"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>The concept of linear Diophantine fuzzy sets (LDFSs) is a new approach for modeling uncertainties in decision analysis. Due to the addition of reference or control parameters with membership and non-membership grades, LDFS is more flexible and reliable than existing concepts of intuitionistic fuzzy sets (IFSs), Pythagorean fuzzy sets (PFSs), and q-rung orthopair fuzzy sets (q-ROFSs). In this paper, the notions of linear Diophantine fuzzy soft rough sets (LDFSRSs) and soft rough linear Diophantine fuzzy sets (SRLDFSs) are proposed as new hybrid models of soft sets, rough sets, and LDFS. The suggested models of LDFSRSs and SRLDFSs are more flexible to discuss fuzziness and roughness in terms of upper and lower approximation operators. Certain operations on LDFSRSs and SRLDFSs have been established to discuss robust multi-criteria decision making (MCDM) for the selection of sustainable material handling equipment. For these objectives, some algorithms are developed for the ranking of feasible alternatives and deriving an optimal decision. Meanwhile, the ideas of the upper reduct, lower reduct, and core set are defined as key factors in the proposed MCDM technique. An application of MCDM is illustrated by a numerical example, and the final ranking in the selection of sustainable material handling equipment is computed by the proposed algorithms. Finally, a comparison analysis is given to justify the feasibility, reliability, and superiority of the proposed models.<\/jats:p>","DOI":"10.3390\/sym12081215","type":"journal-article","created":{"date-parts":[[2020,7,24]],"date-time":"2020-07-24T11:16:35Z","timestamp":1595589395000},"page":"1215","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":81,"title":["Linear Diophantine Fuzzy Soft Rough Sets for the Selection of Sustainable Material Handling Equipment"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8115-9168","authenticated-orcid":false,"given":"Muhammad","family":"Riaz","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of the Punjab, Lahore 54590, Pakistan"}]},{"given":"Masooma Raza","family":"Hashmi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of the Punjab, Lahore 54590, Pakistan"},{"name":"Department of Mathematics, Khawaja Fareed University of Engineering and Information Technology (KFUEIT), Rahim Yar Khan 64200, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5835-3349","authenticated-orcid":false,"given":"Humaira","family":"Kalsoom","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8522-1942","authenticated-orcid":false,"given":"Dragan","family":"Pamucar","sequence":"additional","affiliation":[{"name":"Department of Logistics, University of Defence, 11000 Belgrade, Serbia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0944-2134","authenticated-orcid":false,"given":"Yu-Ming","family":"Chu","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Huzhou University, Huzhou 313000, China"},{"name":"Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, Changsha University of Science &amp; Technology, Changsha 410114, China"}]}],"member":"1968","published-online":{"date-parts":[[2020,7,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1287\/mnsc.17.4.B141","article-title":"Decision-making in a fuzzy environment","volume":"4","author":"Bellman","year":"1970","journal-title":"Manag. 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