{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,1]],"date-time":"2026-06-01T13:37:28Z","timestamp":1780321048891,"version":"3.54.1"},"reference-count":30,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2025,11,26]],"date-time":"2025-11-26T00:00:00Z","timestamp":1764115200000},"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":"crossref","award":["12361058"],"award-info":[{"award-number":["12361058"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"name":"Basic Research Funds for Universities Directly Under the Inner Mongolia Autonomous Region","award":["JY20220083"],"award-info":[{"award-number":["JY20220083"]}]},{"DOI":"10.13039\/501100004763","name":"Natural Science Foundation of Inner Mongolia","doi-asserted-by":"crossref","award":["2024LHMS01015"],"award-info":[{"award-number":["2024LHMS01015"]}],"id":[{"id":"10.13039\/501100004763","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>This paper aims to improve the accuracy of reliability estimates and the failure time prediction for products exhibiting nonlinear degradation behavior under constant-stress accelerated degradation test (CSADT). To achieve this, a novel degradation model and a life prediction method are proposed, which are based on a generalized Wiener process. Some models assume that the drift coefficients are related to accelerated stress. However, in certain applications, the diffusion coefficients are also affected by accelerated stress. The relationship between the drift parameter and accelerated stress variables can be derived by the accelerated model, and so is the relationship between the diffusion parameter and stress variables based on the principle of invariance of the acceleration factor. To account for individual variability among products, random effects are introduced. Model parameters are estimated using a combination of maximum likelihood estimation (MLE) and the expectation-maximization (EM) algorithm. Furthermore, the probability density function (PDF) of the remaining useful life under normal stress conditions is derived using the law of total probability. The effectiveness and applicability of the proposed approach are validated using simulated constant stress accelerated degradation data and stress relaxation data. The results demonstrate that the model not only fits the degradation process well but also modestly improves the accuracy of the failure time prediction, providing valuable guidance for engineering maintenance and reliability management.<\/jats:p>","DOI":"10.3390\/e27121197","type":"journal-article","created":{"date-parts":[[2025,11,26]],"date-time":"2025-11-26T10:04:57Z","timestamp":1764151497000},"page":"1197","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Reliability Modeling Method for Constant Stress Accelerated Degradation Based on the Generalized Wiener Process"],"prefix":"10.3390","volume":"27","author":[{"given":"Shanshan","family":"Li","sequence":"first","affiliation":[{"name":"College of Science, Inner Mongolia University of Technology, Hohhot 010051, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5889-1326","authenticated-orcid":false,"given":"Zaizai","family":"Yan","sequence":"additional","affiliation":[{"name":"College of Science, Inner Mongolia University of Technology, Hohhot 010051, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Junmei","family":"Jia","sequence":"additional","affiliation":[{"name":"College of Science, Inner Mongolia University of Technology, Hohhot 010051, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2025,11,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"108087","DOI":"10.1016\/j.ress.2021.108087","article-title":"Optimal design of step-stress accelerated degradation test oriented by nonlinear and distributed degradation process","volume":"217","author":"Zheng","year":"2022","journal-title":"Reliab. Eng. Syst. Safe"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"108815","DOI":"10.1016\/j.ress.2022.108815","article-title":"A new class of multi-stress acceleration models with interaction effects and its extension to accelerated degradation modelling","volume":"228","author":"Ye","year":"2022","journal-title":"Reliab. Eng. Syst. Safe"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"573","DOI":"10.1016\/j.apm.2017.03.053","article-title":"Novel reliability-based optimization method for thermal structure with hybrid random, interval and fuzzy parameters","volume":"47","author":"Wang","year":"2017","journal-title":"Appl. Math. Model."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Wang, C., Qiang, X., Xu, M.H., and Wu, T. (2022). Recent advances in surrogate modeling methods for uncertainty quantification and propagation. Symmetry, 14.","DOI":"10.3390\/sym14061219"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1016\/j.ress.2012.11.022","article-title":"Remaining useful life estimation based on stochastic deterioration models: A comparative study","volume":"112","author":"Fouladirad","year":"2013","journal-title":"Reliab. Eng. Syst. Safe"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"444","DOI":"10.1109\/TR.2009.2026784","article-title":"Mis-specification analysis of linear degradation models","volume":"58","author":"Peng","year":"2009","journal-title":"IEEE Trans. Reliab."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1023\/A:1009664101413","article-title":"Modelling accelerated degradation data using Wiener diffusion with a time scale transformation","volume":"3","author":"Whitmore","year":"1997","journal-title":"Lifetime Data Anal."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1109\/TR.2011.2182221","article-title":"Remaining useful life estimation based on a nonlinear diffusion degradation process","volume":"61","author":"Si","year":"2012","journal-title":"IEEE Trans. Reliab."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1016\/j.ress.2013.11.011","article-title":"Residual life estimation based on a generalized Wiener degradation process","volume":"124","author":"Wang","year":"2014","journal-title":"Reliab. Eng. Syst. Safe"},{"key":"ref_10","first-page":"496","article-title":"Multiple stresses optimization design of constant-stress accelerated degradation test based on Wiener process","volume":"239","author":"Li","year":"2025","journal-title":"Proc. Inst. Mech. Eng. Part O J. Risk Reliab."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"107509","DOI":"10.1016\/j.ress.2021.107509","article-title":"Optimal design of hybrid accelerated test based on the Inverse Gaussian process model","volume":"210","author":"Ma","year":"2021","journal-title":"Reliab. Eng. Syst. Safe"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1334","DOI":"10.1002\/qre.3296","article-title":"Remaining useful life evaluation for accelerated Wiener degradation process model with mixed random effects and measurement errors","volume":"39","author":"Duan","year":"2023","journal-title":"Qual. Reliab. Eng. Int."},{"key":"ref_13","first-page":"615","article-title":"Bayesian information fusion for degradation analysis of deteriorating products with individual heterogeneity","volume":"233","author":"Guo","year":"2019","journal-title":"Proc. Inst. Mech. Eng. Part O J. Risk Reliab."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Wang, X.N., Su, X.B., and Wang, J.J. (2021). Nonlinear Doubly Wiener Constant-Stress Accelerated Degradation Model Based on Uncertainties and Acceleration Factor Constant Principle. Appl. Sci., 11.","DOI":"10.3390\/app11198968"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"560726","DOI":"10.1155\/2014\/560726","article-title":"Accelerated degradation tests modeling based on the nonlinear wiener process with random effects","volume":"2014","author":"Tang","year":"2014","journal-title":"Math. Probl. Eng."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Liu, L., Li, X.Y., Sun, F.Q., and Wang, N. (2016). A general accelerated degradation model based on the Wiener process. Materials, 9.","DOI":"10.3390\/ma9120981"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"110101","DOI":"10.1016\/j.ress.2024.110101","article-title":"Accelerated degradation testing for lifetime analysis considering random effects and the influence of stress and measurement errors","volume":"247","author":"Li","year":"2024","journal-title":"Reliab. Eng. Syst. Safe"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"189262","DOI":"10.1109\/ACCESS.2020.3030632","article-title":"Remaining useful life prediction under imperfect prior degradation information","volume":"8","author":"Wan","year":"2020","journal-title":"IEEE Access"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"750","DOI":"10.1109\/TR.2014.2315773","article-title":"Accelerated degradation test planning using the inverse Gaussian process","volume":"63","author":"Ye","year":"2014","journal-title":"IEEE Trans. Reliab."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"753","DOI":"10.1002\/qre.1788","article-title":"Remaining life prediction based on Wiener processes with ADT prior information","volume":"32","author":"Wang","year":"2016","journal-title":"Qual. Reliab. Eng. Int."},{"key":"ref_21","unstructured":"Sun, L. (2019). Research on the Theory and Method of Degradation Failure Modeling Based on Stochastic Process. [Ph.D. Thesis, Nanjing University of Science and Technology]."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"8","DOI":"10.1016\/j.microrel.2016.08.004","article-title":"A Wiener process model for accelerated degradation analysis considering measurement errors","volume":"65","author":"Li","year":"2016","journal-title":"Microelectron. Reliab."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1969","DOI":"10.1002\/qre.2668","article-title":"Reliability analysis for accelerated degradation data based on the Wiener process with random effects","volume":"36","author":"Wang","year":"2020","journal-title":"Qual. Reliab. Eng. Int."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"632","DOI":"10.1109\/TASE.2013.2260740","article-title":"A generalized result for degradation model-based reliability estimation","volume":"11","author":"Si","year":"2013","journal-title":"IEEE Trans. Autom. Sci. Eng."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"2","DOI":"10.1016\/j.ress.2007.03.019","article-title":"A survey of the application of gamma processes in maintenance","volume":"94","year":"2009","journal-title":"Reliab. Eng. Syst. Safe"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1016\/0029-5493(93)90199-J","article-title":"Probabilistic methods for condition assessment and life prediction of concrete structures in nuclear power plants","volume":"142","author":"Ellingwood","year":"1993","journal-title":"Nucl. Eng. Des."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1316","DOI":"10.1080\/00949655.2012.658805","article-title":"Bivariate degradation analysis of products based on Wiener processes and copulas","volume":"83","author":"Pan","year":"2013","journal-title":"J. Stat. Comput. Sim."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"374","DOI":"10.1002\/qre.3417","article-title":"A random-effect gamma process model with random initial degradation for accelerated destructive degradation testing data","volume":"40","author":"Ling","year":"2024","journal-title":"Qual. Reliab. Eng. Int."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1111\/j.2517-6161.1977.tb01600.x","article-title":"Maximum likelihood estimation from incomplete data via the EM algorithm","volume":"39","author":"Dempster","year":"1977","journal-title":"J. R. Stat. Soc."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Yang, G. (2007). Life Cycle Reliability Engineering, Wiley.","DOI":"10.1002\/9780470117880"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/12\/1197\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,11,28]],"date-time":"2025-11-28T05:17:38Z","timestamp":1764307058000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/12\/1197"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,11,26]]},"references-count":30,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2025,12]]}},"alternative-id":["e27121197"],"URL":"https:\/\/doi.org\/10.3390\/e27121197","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,11,26]]}}}