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The subgradient optimization technique can be then applied to improve this bound by updating the Lagrange multipliers. The second part of the article presents a new, simple, and efficient heuristic to find a feasible and approximate solution that is also an upper bound to the original problem.<\/jats:p>","DOI":"10.1002\/net.3230210606","type":"journal-article","created":{"date-parts":[[2007,5,12]],"date-time":"2007-05-12T11:21:23Z","timestamp":1178968883000},"page":"689-704","source":"Crossref","is-referenced-by-count":26,"title":["Approximate solutions of capacitated fixed\u2010charge minimum cost network flow problems"],"prefix":"10.1002","volume":"21","author":[{"given":"Do Ba","family":"Khang","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Okitsugu","family":"Fujiwara","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":"10.1002\/net.3230190202"},{"key":"e_1_2_1_3_2","volume-title":"Mathematical Theory of Production Planning","author":"Bensoussan A.","year":"1983"},{"key":"e_1_2_1_4_2","volume-title":"Programming, Games, and Transportation Networks","author":"Berge C.","year":"1965"},{"key":"e_1_2_1_5_2","volume-title":"Dynamic Programming","author":"Denardo E. 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E.Burkard Numerical experiments on an efficient method for the min concave cost flow problem over circuitless single\u2010source uncapacitated networks. Preprint N. 150 Institute of Mathematics Gratz University Austria (November1989)."},{"key":"e_1_2_1_27_2","doi-asserted-by":"publisher","DOI":"10.1002\/net.3230010205"},{"key":"e_1_2_1_28_2","doi-asserted-by":"publisher","DOI":"10.1287\/mnsc.14.7.429"},{"key":"e_1_2_1_29_2","doi-asserted-by":"publisher","DOI":"10.1287\/mnsc.15.9.506"}],"container-title":["Networks"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fnet.3230210606","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/net.3230210606","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,23]],"date-time":"2023-10-23T03:43:40Z","timestamp":1698032620000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/net.3230210606"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1991,10]]},"references-count":28,"journal-issue":{"issue":"6","published-print":{"date-parts":[[1991,10]]}},"alternative-id":["10.1002\/net.3230210606"],"URL":"https:\/\/doi.org\/10.1002\/net.3230210606","archive":["Portico"],"relation":{},"ISSN":["0028-3045","1097-0037"],"issn-type":[{"value":"0028-3045","type":"print"},{"value":"1097-0037","type":"electronic"}],"subject":[],"published":{"date-parts":[[1991,10]]}}}