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A multistate network consists of multistate components and possesses multiple different performance levels. For such a network, reliability is concerned with the probability of the network capacity level greater than or equal to a predetermined demand level <jats:italic>d<\/jats:italic>. One major method for multistate network reliability evaluation is using <jats:italic>d<\/jats:italic>\u2010minimal paths. This paper proposes an efficient algorithm to find <jats:italic>d<\/jats:italic>\u2010minimal paths. First, a new concept of qualified state vector is defined so as to fix a relatively smaller search space of <jats:italic>d<\/jats:italic>\u2010minimal paths, and a sufficient and necessary condition for a qualified state vector to be <jats:italic>d<\/jats:italic>\u2010minimal path is established. Then, the max\u2010flow algorithm and the enumeration algorithm are integrated to search for <jats:italic>d<\/jats:italic>\u2010minimal paths in the determined search space that is recursively divided into subspaces such that the searching efficiency can be increased as much as possible. Both analytical and numerical results show that the proposed algorithm is more efficient in finding all <jats:italic>d<\/jats:italic>\u2010minimal paths. In addition, a case study related to power transmission network is performed to demonstrate the implication of network reliability.<\/jats:p>","DOI":"10.1155\/2019\/4561845","type":"journal-article","created":{"date-parts":[[2019,3,21]],"date-time":"2019-03-21T23:30:54Z","timestamp":1553211054000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Efficient Enumeration of <i>d<\/i>\u2010Minimal Paths in Reliability Evaluation of Multistate Networks"],"prefix":"10.1155","volume":"2019","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8618-9566","authenticated-orcid":false,"given":"Xiu-Zhen","family":"Xu","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7246-8089","authenticated-orcid":false,"given":"Yi-Feng","family":"Niu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6202-4441","authenticated-orcid":false,"given":"Qing","family":"Li","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2019,3,21]]},"reference":[{"key":"e_1_2_11_1_2","doi-asserted-by":"publisher","DOI":"10.1007\/s11518-011-5161-2"},{"key":"e_1_2_11_2_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.ress.2016.11.012"},{"key":"e_1_2_11_3_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.ress.2011.04.001"},{"key":"e_1_2_11_4_2","doi-asserted-by":"publisher","DOI":"10.1155\/2018\/4084917"},{"key":"e_1_2_11_5_2","doi-asserted-by":"publisher","DOI":"10.1109\/TFUZZ.2016.2574910"},{"key":"e_1_2_11_6_2","doi-asserted-by":"publisher","DOI":"10.1155\/2017\/2017634"},{"key":"e_1_2_11_7_2","doi-asserted-by":"publisher","DOI":"10.1080\/00207721.2016.1157222"},{"key":"e_1_2_11_8_2","doi-asserted-by":"publisher","DOI":"10.1080\/0740817X.2013.876130"},{"key":"e_1_2_11_9_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.tre.2010.09.012"},{"key":"e_1_2_11_10_2","doi-asserted-by":"publisher","DOI":"10.1002\/nav.21507"},{"key":"e_1_2_11_11_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.tre.2017.01.008"},{"key":"e_1_2_11_12_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.tre.2014.04.009"},{"key":"e_1_2_11_13_2","doi-asserted-by":"publisher","DOI":"10.1155\/2018\/9142542"},{"key":"e_1_2_11_14_2","doi-asserted-by":"publisher","DOI":"10.1155\/2015\/621264"},{"key":"e_1_2_11_15_2","doi-asserted-by":"publisher","DOI":"10.1155\/2018\/8950794"},{"key":"e_1_2_11_16_2","doi-asserted-by":"publisher","DOI":"10.1007\/s11424-010-0205-0"},{"key":"e_1_2_11_17_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2010.09.033"},{"key":"e_1_2_11_18_2","doi-asserted-by":"publisher","DOI":"10.1002\/qre.1772"},{"key":"e_1_2_11_19_2","doi-asserted-by":"publisher","DOI":"10.1155\/2018\/5967604"},{"key":"e_1_2_11_20_2","doi-asserted-by":"publisher","DOI":"10.1155\/2018\/9259348"},{"key":"e_1_2_11_21_2","doi-asserted-by":"publisher","DOI":"10.1155\/2018\/8094292"},{"key":"e_1_2_11_22_2","doi-asserted-by":"publisher","DOI":"10.1109\/24.387394"},{"key":"e_1_2_11_23_2","doi-asserted-by":"publisher","DOI":"10.1109\/tr.2008.920792"},{"key":"e_1_2_11_24_2","doi-asserted-by":"publisher","DOI":"10.1109\/TR.2010.2046805"},{"key":"e_1_2_11_25_2","doi-asserted-by":"publisher","DOI":"10.1002\/net.3230250306"},{"key":"e_1_2_11_26_2","doi-asserted-by":"publisher","DOI":"10.1057\/palgrave.jors.2601951"},{"key":"e_1_2_11_27_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0305-0548(00)00039-3"},{"key":"e_1_2_11_28_2","doi-asserted-by":"publisher","DOI":"10.1007\/s001700200127"},{"key":"e_1_2_11_29_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.jocs.2016.05.011"},{"key":"e_1_2_11_30_2","doi-asserted-by":"publisher","DOI":"10.1007\/s11518-017-5343-7"},{"key":"e_1_2_11_31_2","doi-asserted-by":"publisher","DOI":"10.1080\/02533839.2012.730231"},{"key":"e_1_2_11_32_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.ress.2015.04.013"},{"key":"e_1_2_11_33_2","doi-asserted-by":"publisher","DOI":"10.1109\/TR.2017.2771495"},{"key":"e_1_2_11_34_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.ress.2008.06.002"},{"key":"e_1_2_11_35_2","doi-asserted-by":"publisher","DOI":"10.1515\/9781400875184"},{"key":"e_1_2_11_36_2","volume-title":"Network Flows: Theory, Algorithms, and Applications","author":"Ahuja R. 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