{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,7]],"date-time":"2026-07-07T20:33:55Z","timestamp":1783456435634,"version":"3.55.0"},"reference-count":40,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2019,4,16]],"date-time":"2019-04-16T00:00:00Z","timestamp":1555372800000},"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>In real life, human opinion cannot be limited to yes or no situations as shown in an ordinary fuzzy sets and intuitionistic fuzzy sets but it may be yes, abstain, no, and refusal as treated in Picture fuzzy sets or in Spherical fuzzy (SF) sets. In this article, we developed a comprehensive model to tackle decision-making problems, where strong points of view are in the favour; neutral; and against some projects, entities, or plans. Therefore, a new approach of covering-based spherical fuzzy rough set (CSFRS) models by means of spherical fuzzy    \u03b2   -neighborhoods (SF    \u03b2   -neighborhoods) is adopted to hybrid spherical fuzzy sets with notions of covering the rough set. Then, by using the principle of TOPSIS (Technique for Order Preference by Similarity to an Ideal Solution) to present the spherical fuzzy, the TOPSIS approach is presented through CSFRS models by means of SF    \u03b2   -neighborhoods. Via the SF-TOPSIS methodology, a multi-attribute decision-making problem is developed in an SF environment. This model has stronger capabilities than intuitionistic fuzzy sets and picture fuzzy sets to manage the vague and uncertainty. Finally, the proposed method is demonstrated through an example of how the proposed method helps us in decision-making problems.<\/jats:p>","DOI":"10.3390\/sym11040547","type":"journal-article","created":{"date-parts":[[2019,4,17]],"date-time":"2019-04-17T03:02:01Z","timestamp":1555470121000},"page":"547","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":84,"title":["Covering-Based Spherical Fuzzy Rough Set Model Hybrid with TOPSIS for Multi-Attribute Decision-Making"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3604-0843","authenticated-orcid":false,"given":"Shouzhen","family":"Zeng","sequence":"first","affiliation":[{"name":"School of Managment, Fudan University, Shanghai 200433, China"},{"name":"School of Business, Ningbo University, Ningbo 315211, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Azmat","family":"Hussain","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Faculty of Basic and Applied Sciences, International Islamic University, Islamabad 44000, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3871-3845","authenticated-orcid":false,"given":"Tahir","family":"Mahmood","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Faculty of Basic and Applied Sciences, International Islamic University, Islamabad 44000, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Muhammad","family":"Irfan Ali","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Islamabad Model College for Girls F-6\/2, Islamabad 44000, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8616-8829","authenticated-orcid":false,"given":"Shahzaib","family":"Ashraf","sequence":"additional","affiliation":[{"name":"Department of Mathmatics, Abdul Wali Khan University, Mardan 23200, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Muhammad","family":"Munir","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Faculty of Basic and Applied Sciences, International Islamic University, Islamabad 44000, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2019,4,16]]},"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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