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Accurate and efficient reliability analysis methods are required for such systems in critical applications. Multiple\u2010valued decision diagram (MDD) is a common model for reliability analysis of\n                    <jats:italic>k<\/jats:italic>\n                    \u2010out\u2010of\u2010\n                    <jats:italic>n<\/jats:italic>\n                    PMSs. To construct the system MDD, each variable corresponding to a system component is assigned a different order. Different input variable orders can lead to dramatically different sizes of final system MDD, which in turn affects the reliability analysis efficiency for such systems. This paper aims to reduce the nonsink nodes during system MDD generation by investigating optimal MDD variable ordering strategies for the systems. Unlike those variable ordering strategies based on fault tree traversal or PMS\u2010BDD variable ordering, the proposed strategies take full advantage of the structural characteristics of the\n                    <jats:italic>k<\/jats:italic>\n                    \u2010out\u2010of\u2010\n                    <jats:italic>n<\/jats:italic>\n                    PMSs and incorporate all interrelationships among the number of components shared by all phases, and the system structure parameters\n                    <jats:italic>k<\/jats:italic>\n                    and\n                    <jats:italic>n<\/jats:italic>\n                    . The proposed optimal strategies are validated through both theoretical analysis and experimental evaluations.\n                  <\/jats:p>","DOI":"10.1002\/qre.70231","type":"journal-article","created":{"date-parts":[[2026,4,28]],"date-time":"2026-04-28T06:48:41Z","timestamp":1777358921000},"page":"2719-2734","update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["MDD Variables Ordering Strategies for\n                    <i>k<\/i>\n                    \u2010Out\u2010of\u2010\n                    <i>n<\/i>\n                    Phased\u2010Mission Systems"],"prefix":"10.1002","volume":"42","author":[{"ORCID":"https:\/\/orcid.org\/0009-0009-0564-9160","authenticated-orcid":false,"given":"Chaonan","family":"Wang","sequence":"first","affiliation":[{"name":"College of Cyber Security Jinan University  Guangzhou China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jingui","family":"Yu","sequence":"additional","affiliation":[{"name":"College of Cyber Security Jinan University  Guangzhou China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0002-0331-6193","authenticated-orcid":false,"given":"Yujie","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Economics &amp; Management University of Science and Technology Beijing  Beijing China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yujie","family":"Su","sequence":"additional","affiliation":[{"name":"College of Cyber Security Jinan University  Guangzhou China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"311","published-online":{"date-parts":[[2026,4,27]]},"reference":[{"key":"e_1_2_9_2_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ress.2020.106903"},{"key":"e_1_2_9_3_1","doi-asserted-by":"crossref","unstructured":"I.Mura \u201cStochastic Modeling and Analysis of Phased\u2010Mission Systems Dependability \u201d inAnnual Reliability and Maintainability Symposium (RAMS)(IEEE 2021) 1\u20136.","DOI":"10.1109\/RAMS48097.2021.9605781"},{"key":"e_1_2_9_4_1","doi-asserted-by":"publisher","DOI":"10.1109\/TR.1981.5221156"},{"key":"e_1_2_9_5_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.phycom.2023.102069"},{"key":"e_1_2_9_6_1","doi-asserted-by":"publisher","DOI":"10.1109\/JIOT.2020.2993216"},{"key":"e_1_2_9_7_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ress.2018.03.031"},{"key":"e_1_2_9_8_1","doi-asserted-by":"publisher","DOI":"10.1002\/qre.2060"},{"key":"e_1_2_9_9_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ress.2018.03.008"},{"key":"e_1_2_9_10_1","doi-asserted-by":"publisher","DOI":"10.1002\/qre.1546"},{"key":"e_1_2_9_11_1","doi-asserted-by":"publisher","DOI":"10.1109\/TSMCA.2012.2220761"},{"key":"e_1_2_9_12_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ress.2017.02.006"},{"key":"e_1_2_9_13_1","doi-asserted-by":"publisher","DOI":"10.1155\/2020\/8087242"},{"key":"e_1_2_9_14_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ress.2013.08.005"},{"key":"e_1_2_9_15_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ress.2014.11.012"},{"key":"e_1_2_9_16_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ress.2019.106786"},{"key":"e_1_2_9_17_1","doi-asserted-by":"crossref","unstructured":"S. V.Amari \u201cA Practical Method for Reliability Analysis of Phased\u2010Mission Systems \u201d inProceedings of Annual Reliability & Maintainability Symposium(IEEE 2011) https:\/\/doi.org\/10.1109\/RAMS.2011.\u20105754460.","DOI":"10.1109\/RAMS.2011.5754460"},{"key":"e_1_2_9_18_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ress.2013.08.005"},{"key":"e_1_2_9_19_1","first-page":"24","article-title":"Performance Evaluation of Non\u2010Repairable Cyclic Phased\u2010Mission Systems Using Evidential Reasoning Rule and Multi\u2010Valued Decision Diagrams","volume":"124","author":"Zhang P.","year":"2023","journal-title":"Reliability Engineering & System Safety"},{"key":"e_1_2_9_20_1","doi-asserted-by":"publisher","DOI":"10.1109\/TR.2014.2299674"},{"key":"e_1_2_9_21_1","doi-asserted-by":"crossref","unstructured":"S.Amari \u201cA New Approach to Solve Dynamic Fault Trees \u201d inProceedings of Annual Reliability & Maintainability Symposium(IEEE 2003) 374\u2013379.","DOI":"10.1109\/RAMS.2003.1182018"},{"key":"e_1_2_9_22_1","doi-asserted-by":"publisher","DOI":"10.1177\/1748006X231215830"},{"key":"e_1_2_9_23_1","doi-asserted-by":"publisher","DOI":"10.1002\/qre.3614"},{"key":"e_1_2_9_24_1","doi-asserted-by":"publisher","DOI":"10.1109\/24.765927"},{"key":"e_1_2_9_25_1","doi-asserted-by":"publisher","DOI":"10.1109\/TR.2014.2299497"},{"key":"e_1_2_9_26_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ress.2022.109085"},{"key":"e_1_2_9_27_1","doi-asserted-by":"crossref","unstructured":"M. B.Siddiqui S. N.Ahmad andM. T.Beg \u201cVariable Ordering of BDD Mapped Multi\u2010Input Multi\u2010Output Adders Using Modified Genetic Algorithm \u201d inInternational Conference on Multimedia Signal Processing and Communication Technologies (IMPACT)(2017) 218\u2013221.","DOI":"10.1109\/MSPCT.2017.8364008"},{"key":"e_1_2_9_28_1","doi-asserted-by":"crossref","unstructured":"M.Bouissou F.Bruyere andA.Rauzy \u201cBDD Based Fault\u2010Tree Processing: A Comparison of Variable Ordering Heuristics \u201d inProceedings of the European Safety and Reliability Conference (ESREL)(1997).","DOI":"10.1016\/B978-008042835-2\/50231-9"},{"key":"e_1_2_9_29_1","doi-asserted-by":"publisher","DOI":"10.1109\/TCAD.2003.819427"},{"key":"e_1_2_9_30_1","doi-asserted-by":"publisher","DOI":"10.1109\/TCAD.2005.852053"},{"key":"e_1_2_9_31_1","doi-asserted-by":"publisher","DOI":"10.1177\/1748006X19879305"},{"key":"e_1_2_9_32_1","doi-asserted-by":"publisher","DOI":"10.1145\/3716313"},{"key":"e_1_2_9_33_1","doi-asserted-by":"publisher","DOI":"10.1109\/TR.2008.2011673"},{"key":"e_1_2_9_34_1","doi-asserted-by":"crossref","unstructured":"R.Drechsler \u201cEvaluation of Static Variable Ordering Heuristics for MDD Construction \u201d inProceedings of the International Symposium on Multiple\u2010Valued Logic(IEEE 2002) 254\u2013260 https:\/\/doi.org\/10.1109\/ISMVL.2002.1011096.","DOI":"10.1109\/ISMVL.2002.1011096"},{"key":"e_1_2_9_35_1","doi-asserted-by":"publisher","DOI":"10.3969\/j.issn.1001\u20102400.2017.06.004"}],"container-title":["Quality and Reliability Engineering International"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/qre.70231","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/full-xml\/10.1002\/qre.70231","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/qre.70231","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,27]],"date-time":"2026-08-27T09:31:51Z","timestamp":1787823111000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/qre.70231"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,4,27]]},"references-count":34,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2026,10]]}},"alternative-id":["10.1002\/qre.70231"],"URL":"https:\/\/doi.org\/10.1002\/qre.70231","archive":["Portico"],"relation":{},"ISSN":["0748-8017","1099-1638"],"issn-type":[{"value":"0748-8017","type":"print"},{"value":"1099-1638","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,4,27]]},"assertion":[{"value":"2026-01-20","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2026-04-17","order":2,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2026-04-27","order":3,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}