{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,13]],"date-time":"2026-02-13T12:06:01Z","timestamp":1770984361644,"version":"3.50.1"},"reference-count":20,"publisher":"Wiley","issue":"7","license":[{"start":{"date-parts":[[2006,10,11]],"date-time":"2006-10-11T00:00:00Z","timestamp":1160524800000},"content-version":"vor","delay-in-days":4393,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Networks"],"published-print":{"date-parts":[[1994,10]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We study a class of two\u2010stage dynamic networks with random arc capacities where the decisions in the first stage must be made before realizing the random quantities in the second stage. The expected total cost of the second stage is a function of the first\u2010stage decisions, known as the expected recourse function, which is generally intractable. This paper proposes a new method to approximate the expected recourse function as a convex, separable function of the supplies to the second stage. First, the second\u2010stage network is decomposed using Lagrangian relaxation into tree subproblems whose expected recourse functions are numerically tractable. Subgradient optimization is then used to update the Lagrange multipliers to improve the approximations. Numerical experiments show that this structural decomposition approach can produce high\u2010quality approximations of the expected recourse functions for large stochastic networks. \u00a9 1994 by John Wiley &amp; Sons, Inc.<\/jats:p>","DOI":"10.1002\/net.3230240703","type":"journal-article","created":{"date-parts":[[2007,5,12]],"date-time":"2007-05-12T18:49:21Z","timestamp":1178995761000},"page":"369-384","source":"Crossref","is-referenced-by-count":26,"title":["A network recourse decomposition method for dynamic networks with random arc capacities"],"prefix":"10.1002","volume":"24","author":[{"given":"Warren B.","family":"Powell","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Raymond K.\u2010M.","family":"Cheung","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2006,10,11]]},"reference":[{"key":"e_1_2_1_2_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01582286"},{"key":"e_1_2_1_3_2","unstructured":"R. K.CheungandW. B.Powell An algorithm for multistage dynamic networks with random arc capacitites. Technical Report SOR\u201092\u201011 Dept. of Civil Engineering and Operations Research Princeton University (1992)."},{"key":"e_1_2_1_4_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-61370-8"},{"key":"e_1_2_1_5_2","doi-asserted-by":"publisher","DOI":"10.1287\/mnsc.5.1.97"},{"key":"e_1_2_1_6_2","doi-asserted-by":"publisher","DOI":"10.1287\/trsc.24.1.40"},{"key":"e_1_2_1_7_2","doi-asserted-by":"publisher","DOI":"10.1287\/moor.16.3.650"},{"key":"e_1_2_1_8_2","first-page":"313","volume-title":"Numerical Techniques for Stochastic Optimization","author":"Kall P.","year":"1988"},{"key":"e_1_2_1_9_2","doi-asserted-by":"publisher","DOI":"10.1287\/mnsc.38.11.1642"},{"key":"e_1_2_1_10_2","first-page":"249","volume-title":"Vehicle Routing: Methods and Studies","author":"Powell W. B.","year":"1988"},{"key":"e_1_2_1_11_2","article-title":"Stochastic programs over trees with random arc capacities","author":"Powell W. B.","journal-title":"Networks"},{"key":"e_1_2_1_12_2","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0121120"},{"key":"e_1_2_1_13_2","doi-asserted-by":"publisher","DOI":"10.1287\/moor.16.1.119"},{"key":"e_1_2_1_14_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-82118-9"},{"key":"e_1_2_1_15_2","doi-asserted-by":"publisher","DOI":"10.1137\/0117061"},{"key":"e_1_2_1_16_2","doi-asserted-by":"publisher","DOI":"10.1002\/net.3230160306"},{"key":"e_1_2_1_17_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF02604638"},{"key":"e_1_2_1_18_2","doi-asserted-by":"publisher","DOI":"10.1016\/0167-6377(89)90007-2"},{"key":"e_1_2_1_19_2","doi-asserted-by":"publisher","DOI":"10.1137\/0114008"},{"key":"e_1_2_1_20_2","volume-title":"Handbooks in OR & MS: Optimization","author":"Wets R. 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