{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,29]],"date-time":"2026-05-29T20:57:19Z","timestamp":1780088239100,"version":"3.54.0"},"reference-count":25,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2025,6,9]],"date-time":"2025-06-09T00:00:00Z","timestamp":1749427200000},"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>A class of complex systems can be structurally decomposed into several typical multi-state weighted k\/n systems. In view of this, this paper proposes a reliability model to evaluate the reliability of such systems. The system reliability evaluation process is divided into two critical stages: (1) Primary stage: Structural reliability analysis (system-level) and (2) Secondary stage: Subsystem reliability verification (component-level). In practical applications, when there is a lack of operational data, expert experience is needed to evaluate the states and availability weights of components, which is a typical application scenario for uncertainty theory. Under the framework of uncertainty theory, two-stage reliability measures and importance measures are defined, and corresponding calculation formulas are derived. To improve computational efficiency, a binary search algorithm is proposed. Finally, an application case of SDN was proposed to demonstrate the effectiveness of the theory and method. This study provides an idea for the reliability modeling of complex systems (multiple pipeline transmissions, power systems, distributed computing, etc.) in the absence of operational data.<\/jats:p>","DOI":"10.3390\/sym17060912","type":"journal-article","created":{"date-parts":[[2025,6,9]],"date-time":"2025-06-09T09:13:01Z","timestamp":1749460381000},"page":"912","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Reliability Evaluation of Two-Stage Uncertain Multi-State 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":[[2025,6,9]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1109\/24.295016","article-title":"An algorithm for computing the reliability of weighted- k-out-of-n systems","volume":"43","author":"Wu","year":"1994","journal-title":"IEEE Trans. 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