{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,13]],"date-time":"2026-02-13T18:29:12Z","timestamp":1771007352612,"version":"3.50.1"},"reference-count":60,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2022,6,23]],"date-time":"2022-06-23T00:00:00Z","timestamp":1655942400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004242","name":"Princess Nourah bint Abdulrahman University Researchers","doi-asserted-by":"publisher","award":["PNURSP2022R50"],"award-info":[{"award-number":["PNURSP2022R50"]}],"id":[{"id":"10.13039\/501100004242","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>It is extremely frequent for systems to fail in their demanding operating environments in many real-world contexts. When systems reach their lowest, highest, or both extreme operating conditions, they usually fail to perform their intended functions, which is something that researchers pay little attention to. The goal of this paper is to develop inference for multi-reliability using unit alpha power exponential distributions for stress\u2013strength variables based on the progressive first failure. As a result, the problem of estimating the stress\u2013strength function R, where X, Y, and Z come from three separate alpha power exponential distributions, is addressed in this paper. The conventional methods, such as maximum likelihood for point estimation, Bayesian and asymptotic confidence, boot-p, and boot-t methods for interval estimation, are also examined. Various confidence intervals have been obtained. Monte Carlo simulations and real-world application examples are used to evaluate and compare the performance of the various proposed estimators.<\/jats:p>","DOI":"10.3390\/sym14071306","type":"journal-article","created":{"date-parts":[[2022,6,23]],"date-time":"2022-06-23T22:43:00Z","timestamp":1656024180000},"page":"1306","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["Optimal Plan of Multi-Stress\u2013Strength Reliability Bayesian and Non-Bayesian Methods for the Alpha Power Exponential Model Using Progressive First Failure"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3888-1275","authenticated-orcid":false,"given":"Ehab M.","family":"Almetwally","sequence":"first","affiliation":[{"name":"Department of Statistical, Faculty of Business Administration, Delta University for Science and Technology, Gamasa 11152, Egypt"},{"name":"Department of Mathematical Statistical, Faculty of Graduate Studies for Statistical Research, Cairo University, Cairo 12613, Egypt"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9449-7489","authenticated-orcid":false,"given":"Refah","family":"Alotaibi","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, College of Science, Princess Nourah Bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia"}]},{"given":"Aned Al","family":"Mutairi","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, College of Science, Princess Nourah Bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2208-3498","authenticated-orcid":false,"given":"Chanseok","family":"Park","sequence":"additional","affiliation":[{"name":"Applied Statistics Laboratory, Department of Industrial Engineering, Pusan National University, Busan 46241, Korea"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7501-7232","authenticated-orcid":false,"given":"Hoda","family":"Rezk","sequence":"additional","affiliation":[{"name":"Department of Statistics, Al-Azhar University, Cairo 11751, Egypt"}]}],"member":"1968","published-online":{"date-parts":[[2022,6,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"83","DOI":"10.2307\/1269555","article-title":"Testing reliability in a stress-strength model when X and Y are normally distributed","volume":"34","author":"Weerahandi","year":"1992","journal-title":"Technometrics"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1023\/A:1011352923990","article-title":"Inference for reliability and stress-strength for a scaled Burr Type X distribution","volume":"7","author":"Surles","year":"2001","journal-title":"Lifetime Data Anal."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1443","DOI":"10.1080\/03610926.2011.563011","article-title":"Inferences on stress-strength reliability from Lindley distributions","volume":"42","author":"Ghitany","year":"2013","journal-title":"Commun. 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