{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,29]],"date-time":"2026-01-29T19:51:30Z","timestamp":1769716290627,"version":"3.49.0"},"reference-count":28,"publisher":"SAGE Publications","issue":"6","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["IFS"],"published-print":{"date-parts":[[2023,6,1]]},"abstract":"<jats:p>A system undergoes a failure process in which the internal degradation and external loads are independent and compete with each other. In the reliability model of competitive failure, the threshold of failure is often a dynamic process that changes with time. Based on this, this paper constructs a model in which the failure threshold is an uncertain Liu process and applies it to the competitive failure reliability process, where soft and hard failure thresholds are modeled by different Liu processes. In the absence of a large amount of failure data, uncertainty theory is used as a tool to analyze the belief reliability and mean time to failure of the system. Taking gas-insulated substation (GIS) as an example, the effect of parameters on belief reliability is analyzed, and the reliability under Liu process threshold and constants threshold is compared, which shows the validity of the prosed model.<\/jats:p>","DOI":"10.3233\/jifs-224057","type":"journal-article","created":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T11:47:32Z","timestamp":1680263252000},"page":"9757-9767","source":"Crossref","is-referenced-by-count":0,"title":["Reliability analysis of failure threshold value with uncertain Liu process1"],"prefix":"10.1177","volume":"44","author":[{"given":"Chun","family":"Wei","sequence":"first","affiliation":[{"name":"College of Physics and Electronics Science, Shanxi Datong University, Datong, Shanxi, China"}]},{"given":"Haiyan","family":"Shi","sequence":"additional","affiliation":[{"name":"College of Mathematics and Statistics Science, Shanxi Datong University, Datong, Shanxi, China"}]},{"given":"Baoliang","family":"Liu","sequence":"additional","affiliation":[{"name":"College of Mathematics and Statistics Science, Shanxi Datong University, Datong, Shanxi, China"}]},{"given":"Zhiqiang","family":"Zhang","sequence":"additional","affiliation":[{"name":"College of Mathematics and Statistics Science, Shanxi Datong University, Datong, Shanxi, China"}]},{"given":"Yanqing","family":"Wen","sequence":"additional","affiliation":[{"name":"College of Mathematics and Statistics Science, Shanxi Datong University, Datong, Shanxi, China"}]}],"member":"179","reference":[{"key":"10.3233\/JIFS-224057_ref1","doi-asserted-by":"crossref","first-page":"194","DOI":"10.1016\/0377-2217(88)90329-3","article-title":"Periodical replacement problem without assuming minimal repair","volume":"37","author":"Masaaki","year":"1988","journal-title":"European Journal of Operational Research"},{"key":"10.3233\/JIFS-224057_ref2","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1016\/j.ress.2016.12.007","article-title":"Sensor-based calibrations to improve reliability of systems subject to multiple dependent competing failure processes","volume":"160","author":"Kong","year":"2017","journal-title":"Reliability Engineering and System Safety"},{"key":"10.3233\/JIFS-224057_ref3","doi-asserted-by":"crossref","first-page":"176","DOI":"10.1016\/0377-2217(83)90221-7","article-title":"Analysis of a system with minimal repair and its application to replacement policy","volume":"12","author":"Nakagawa","year":"1983","journal-title":"European Journal of Operational Research"},{"key":"10.3233\/JIFS-224057_ref4","doi-asserted-by":"crossref","first-page":"106496","DOI":"10.1016\/j.ress.2019.106496","article-title":"Gamma process based optimal mission abort policy","volume":"190","author":"Qiu","year":"2019","journal-title":"Reliability Engineering and System Safety"},{"issue":"2","key":"10.3233\/JIFS-224057_ref5","doi-asserted-by":"crossref","first-page":"610","DOI":"10.1016\/j.ejor.2020.11.015","article-title":"Optimal inspection and mission abort policies for systems subject to degradation","volume":"292","author":"Zhao","year":"2021","journal-title":"European Journal of Operational Research"},{"issue":"5","key":"10.3233\/JIFS-224057_ref6","doi-asserted-by":"crossref","first-page":"483","DOI":"10.1080\/0740817X.2013.812270","article-title":"Reliability modeling for dependent competing failure processes with changing degradation rate","volume":"46","author":"Rafiee","year":"2014","journal-title":"IIE Transactions"},{"key":"10.3233\/JIFS-224057_ref8","doi-asserted-by":"crossref","first-page":"453","DOI":"10.1016\/j.apm.2021.07.038","article-title":"The belief reliability analysis of composite insulators with uncertain failure threshold","volume":"100","author":"Shi","year":"2021","journal-title":"Applied Mathematical Modelling"},{"key":"10.3233\/JIFS-224057_ref9","doi-asserted-by":"crossref","first-page":"108142","DOI":"10.1016\/j.ress.2021.108142","article-title":"Optimal warranty policy design for deteriorating products with random failure threshold","volume":"218","author":"Li","year":"2022","journal-title":"Reliability Engineering and System Safety"},{"key":"10.3233\/JIFS-224057_ref10","doi-asserted-by":"crossref","first-page":"438","DOI":"10.1016\/j.cma.2019.06.036","article-title":"Epistemic uncertainty-based reliability analysis for engineering system with hybrid evidence and fuzzy variables","volume":"355","author":"Wang","year":"2019","journal-title":"Computer Methods in Applied Mechanics and Engineering"},{"issue":"4","key":"10.3233\/JIFS-224057_ref11","doi-asserted-by":"crossref","first-page":"338","DOI":"10.1016\/S0019-9958(65)90241-X","article-title":"Fuzzy sets","volume":"38","author":"Zadeh","year":"1965","journal-title":"Information and Control"},{"key":"10.3233\/JIFS-224057_ref12","unstructured":"Liu B.D. , Uncertainty Theory, 2nd ed. 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