{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,7]],"date-time":"2026-03-07T12:00:35Z","timestamp":1772884835286,"version":"3.50.1"},"reference-count":49,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2022,2,10]],"date-time":"2022-02-10T00:00:00Z","timestamp":1644451200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["51475086"],"award-info":[{"award-number":["51475086"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003787","name":"Natural Science Foundation of Hebei Province","doi-asserted-by":"publisher","award":["E2020501013"],"award-info":[{"award-number":["E2020501013"]}],"id":[{"id":"10.13039\/501100003787","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The degradation rates of many engineering systems are always positive in practical applications. In some cases, the degradation process is approximately linear, such as train wheel wear degradation and an increase in the operating current of laser devices. In this paper we introduce a degradation modeling and reliability estimation method based on the Wiener process with a transmuted-truncated normal distribution (TTND). The TTND is employed to characterize unit-to-unit variability due to its flexibility in capturing symmetry and the asymmetrical behavior of the drift variable of the Wiener process. A Wiener process with TTND is applied to model the degradation processes of the deteriorating systems in two cases. In the first case, we assume that the degradation process is unaffected by measurement uncertainties, whereas in the second case, the measurement uncertainties are considered. The exact and explicit expressions of PDFs and the reliability functions are derived based on the concept of first hitting time. The Gibbs sampling technique and the Metropolis\u2013Hastings algorithm are employed to obtain the Bayesian estimates of the model\u2019s parameters. The efficiency and the feasibility of the presented approach are demonstrated through numerical examples, with Monte Carlo simulation and a practical application with laser degradation data. Furthermore, deviance information criteria (DIC) are used to compare the proposed model and some existing models. The result indicated that the proposed approach provided better reliability estimation results.<\/jats:p>","DOI":"10.3390\/sym14020353","type":"journal-article","created":{"date-parts":[[2022,2,11]],"date-time":"2022-02-11T02:40:17Z","timestamp":1644547217000},"page":"353","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Reliability Analysis with Wiener-Transmuted Truncated Normal Degradation Model for Linear and Non-Negative Degradation Data"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3899-1626","authenticated-orcid":false,"given":"Isyaku","family":"Muhammad","sequence":"first","affiliation":[{"name":"School of Mechanical Engineering and Automation, Northeastern University, Shenyang 110819, China"}]},{"given":"Xingang","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Mechanical Engineering and Automation, Northeastern University, Shenyang 110819, China"},{"name":"College of Mechanical and Electrical Engineering, Guangdong University of Petrochemical Technology, Maoming 525000, China"}]},{"given":"Changyou","family":"Li","sequence":"additional","affiliation":[{"name":"School of Mechanical Engineering and Automation, Northeastern University, Shenyang 110819, China"}]},{"given":"Mingming","family":"Yan","sequence":"additional","affiliation":[{"name":"School of Mechanical Engineering and Automation, Northeastern University, Shenyang 110819, China"}]},{"given":"Mustapha","family":"Mukhtar","sequence":"additional","affiliation":[{"name":"School of Economics and Management, Guangdong University of Petrochemical Technolog, Maoming 525000, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0329-5898","authenticated-orcid":false,"given":"Mustapha","family":"Muhammad","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Hebei Normal University, Shijiazhuang 050024, China"},{"name":"Department of Mathematical Sciences, Bayero University Kano, Kano 700241, Nigeria"}]}],"member":"1968","published-online":{"date-parts":[[2022,2,10]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1016\/j.finel.2012.08.002","article-title":"Optimization of the truss-type structures using the generalized perturbation-based Stochastic Finite Element Method","volume":"63","author":"Solecka","year":"2013","journal-title":"Finite Elem. 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