{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,29]],"date-time":"2026-05-29T20:59:04Z","timestamp":1780088344121,"version":"3.54.0"},"reference-count":32,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2025,6,8]],"date-time":"2025-06-08T00:00:00Z","timestamp":1749340800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi","award":["2022L415"],"award-info":[{"award-number":["2022L415"]}]},{"name":"Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi","award":["2024Y059"],"award-info":[{"award-number":["2024Y059"]}]},{"name":"Statistical Science Research Project of Shanxi Province","award":["2022L415"],"award-info":[{"award-number":["2022L415"]}]},{"name":"Statistical Science Research Project of Shanxi Province","award":["2024Y059"],"award-info":[{"award-number":["2024Y059"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This paper proposes a two-stage weighted k-out-of-n (W-k\/n) system reliability model based on uncertainty theory to address the problem of traditional W-k\/n system theory being difficult to apply to the reliability evaluation of complex systems with two-layer structures. This model effectively solves the evaluation problem in the case of insufficient data by decomposing the reliability assessment of complex systems into two stages: system structure reliability and subsystem reliability. In the model, the system state and weights are described by uncertain variables that follow a certain distribution. Under the framework of uncertainty theory, two-stage reliability measures and importance measures are defined, and corresponding calculation formulas are derived. In order to reduce computational complexity, a corresponding numerical solution method including traverse search algorithm and binary search algorithm is proposed. The effectiveness of the method is also demonstrated through a project management case of electronic consumer goods. This study provides new theoretical ideas and methodological support for reliability modeling of complex systems under data loss conditions.<\/jats:p>","DOI":"10.3390\/sym17060908","type":"journal-article","created":{"date-parts":[[2025,6,9]],"date-time":"2025-06-09T09:13:01Z","timestamp":1749460381000},"page":"908","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Importance Measures Analysis of Two-Stage Uncertain Weighted k-out-of-n Systems"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4228-2555","authenticated-orcid":false,"given":"Chun","family":"Wei","sequence":"first","affiliation":[{"name":"College of Physics and Electronics Science, Shanxi Datong University, Datong 037009, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5719-8893","authenticated-orcid":false,"given":"Haiyan","family":"Shi","sequence":"additional","affiliation":[{"name":"College of Mathematics and Statistics, Shanxi Datong University, Datong 037009, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zhiqiang","family":"Zhang","sequence":"additional","affiliation":[{"name":"College of Mathematics and Statistics, Shanxi Datong University, Datong 037009, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2025,6,8]]},"reference":[{"key":"ref_1","unstructured":"Barlow, R., and Proschan, F. 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