{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,5]],"date-time":"2026-08-05T21:48:14Z","timestamp":1785966494966,"version":"3.56.0"},"reference-count":46,"publisher":"MDPI AG","issue":"15","license":[{"start":{"date-parts":[[2022,7,31]],"date-time":"2022-07-31T00:00:00Z","timestamp":1659225600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004281","name":"National Science Centre","doi-asserted-by":"publisher","award":["2018\/29\/B\/HS4\/02725"],"award-info":[{"award-number":["2018\/29\/B\/HS4\/02725"]}],"id":[{"id":"10.13039\/501100004281","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004281","name":"National Science Centre","doi-asserted-by":"publisher","award":["2021\/41\/B\/HS4\/01296"],"award-info":[{"award-number":["2021\/41\/B\/HS4\/01296"]}],"id":[{"id":"10.13039\/501100004281","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>A fuzzy set extension known as the hesitant fuzzy set (HFS) has increased in popularity for decision making in recent years, especially when experts have had trouble evaluating several alternatives by employing a single value for assessment when working in a fuzzy environment. However, it has a significant problem in its uses, i.e., considerable data loss. The probabilistic hesitant fuzzy set (PHFS) has been proposed to improve the HFS. It provides probability values to the HFS and has the ability to retain more information than the HFS. Previously, fuzzy regression models such as the fuzzy linear regression model (FLRM) and hesitant fuzzy linear regression model were used for decision making; however, these models do not provide information about the distribution. To address this issue, we proposed a probabilistic hesitant fuzzy linear regression model (PHFLRM) that incorporates distribution information to account for multi-criteria decision-making (MCDM) problems. The PHFLRM observes the input\u2013output (IPOP) variables as probabilistic hesitant fuzzy elements (PHFEs) and uses a linear programming model (LPM) to estimate the parameters. A case study is used to illustrate the proposed methodology. Additionally, an MCDM technique called the technique for order preference by similarity to ideal solution (TOPSIS) is employed to compare the PHFLRM findings with those obtained using TOPSIS. Lastly, Spearman\u2019s rank correlation test assesses the statistical significance of two rankings sets.<\/jats:p>","DOI":"10.3390\/s22155736","type":"journal-article","created":{"date-parts":[[2022,8,1]],"date-time":"2022-08-01T23:49:27Z","timestamp":1659397767000},"page":"5736","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Making Group Decisions within the Framework of a Probabilistic Hesitant Fuzzy Linear Regression Model"],"prefix":"10.3390","volume":"22","author":[{"given":"Ayesha","family":"Sultan","sequence":"first","affiliation":[{"name":"Department of Statistics, Lahore Campus, COMSATS University Islamabad, Islamabad 45550, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7076-2519","authenticated-orcid":false,"given":"Wojciech","family":"Sa\u0142abun","sequence":"additional","affiliation":[{"name":"Research Team on Intelligent Decision Support Systems, Department of Artificial Intelligence and Applied Mathematics, Faculty of Computer Science and Information Technology, West Pomeranian University of Technology in Szczecin, ul. Zo\u0142nierska 49, 71-210 Szczecin, Poland"},{"name":"National Institute of Telecommunications, Szachowa 1, 04-894 Warsaw, Poland"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2045-5323","authenticated-orcid":false,"given":"Shahzad","family":"Faizi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Virtual University of Pakistan, Lahore 54000, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Muhammad","family":"Ismail","sequence":"additional","affiliation":[{"name":"Department of Statistics, Lahore Campus, COMSATS University Islamabad, Islamabad 45550, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0834-2019","authenticated-orcid":false,"given":"Andrii","family":"Shekhovtsov","sequence":"additional","affiliation":[{"name":"Research Team on Intelligent Decision Support Systems, Department of Artificial Intelligence and Applied Mathematics, Faculty of Computer Science and Information Technology, West Pomeranian University of Technology in Szczecin, ul. Zo\u0142nierska 49, 71-210 Szczecin, Poland"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2022,7,31]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"903","DOI":"10.1109\/TSMC.1982.4308925","article-title":"Linear regression analysis with fuzzy model","volume":"12","author":"Asai","year":"1982","journal-title":"IEEE Trans. Syst. 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