{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,25]],"date-time":"2026-02-25T10:10:18Z","timestamp":1772014218880,"version":"3.50.1"},"reference-count":27,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2026,2,18]],"date-time":"2026-02-18T00:00:00Z","timestamp":1771372800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University","award":["IMSIU-DDRSP2602"],"award-info":[{"award-number":["IMSIU-DDRSP2602"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Parallel\u2013series systems are fundamental in many industrial and engineering applications, yet their reliability assessment and improvement remain challenging, particularly when components exhibit non-constant failure rates. This study addresses this challenge by modeling a hybrid parallel\u2013series system whose components follow the Akshaya lifetime distribution, a flexible model that can capture various hazard-rate shapes. For this system, we derive closed-form analytical expressions for key reliability indices, including the system reliability function, mean time to failure (MTTF), reliability equivalence factors (REFs), and \u03b4-fractiles. To enhance system performance, four improvement strategies are formulated and analytically compared: failure-rate reduction, hot duplication, cold duplication with a perfect switch, and cold duplication with an imperfect switch. A comprehensive numerical case study validates the theoretical derivations and demonstrates the effectiveness of each strategy. The results show that cold duplication with a perfect switch yields the highest reliability gain, and the computed REFs provide a quantitative tool for balancing redundancy against component-level improvements. This work provides reliability engineers with a comprehensive analytical framework for the design and enhancement of complex parallel-series systems.<\/jats:p>","DOI":"10.3390\/axioms15020149","type":"journal-article","created":{"date-parts":[[2026,2,18]],"date-time":"2026-02-18T11:45:58Z","timestamp":1771415158000},"page":"149","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Reliability Improvement of a Parallel\u2013Series System via Duplication and Reduction Strategies Under the Akshaya Distribution"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2485-6287","authenticated-orcid":false,"given":"Ahmed T.","family":"Ramadan","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Basic Science, Galala University, Suez 43713, Egypt"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8303-1782","authenticated-orcid":false,"given":"Ahmed R.","family":"El-Saeed","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Faculty of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11432, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0009-0003-9992-6398","authenticated-orcid":false,"given":"Norah D.","family":"Alshahrani","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Sciences, University of Bisha, Bisha 61922, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8938-9187","authenticated-orcid":false,"given":"Ahlam H.","family":"Tolba","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 33516, Egypt"}]}],"member":"1968","published-online":{"date-parts":[[2026,2,18]]},"reference":[{"key":"ref_1","first-page":"339","article-title":"Reliability equivalence factors for a series\u2013parallel system of components with exponentiated Weibull lifetimes","volume":"28","author":"Alghamdi","year":"2017","journal-title":"IMA J. 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