{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,7]],"date-time":"2025-10-07T14:40:23Z","timestamp":1759848023576},"reference-count":17,"publisher":"World Scientific Pub Co Pte Lt","issue":"03","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Asia Pac. J. Oper. Res."],"published-print":{"date-parts":[[2009,6]]},"abstract":"<jats:p> We consider the problem of minimizing the total late work [Formula: see text] on an unbounded batch processing machine, where V<jats:sub>j<\/jats:sub> = min {T<jats:sub>j<\/jats:sub>, p<jats:sub>j<\/jats:sub>} and T<jats:sub>j<\/jats:sub> = max {C<jats:sub>j<\/jats:sub> - d<jats:sub>j<\/jats:sub>, 0}. The batch processing machine can process up to B (B \u2265 n) jobs simultaneously. The jobs that are processed together form a batch, and all jobs in a batch start and complete at the same time, respectively. For a batch of jobs, the processing time of the batch is equal to the largest processing time among the jobs in this batch. In this paper, we prove that the problem [Formula: see text] is NP-hard. <\/jats:p>","DOI":"10.1142\/s0217595909002249","type":"journal-article","created":{"date-parts":[[2009,7,22]],"date-time":"2009-07-22T12:29:32Z","timestamp":1248265772000},"page":"351-363","source":"Crossref","is-referenced-by-count":20,"title":["THE NP-HARDNESS OF MINIMIZING THE TOTAL LATE WORK ON AN UNBOUNDED BATCH MACHINE"],"prefix":"10.1142","volume":"26","author":[{"given":"JIANFENG","family":"REN","sequence":"first","affiliation":[{"name":"College of Operations Research and Management Sciences, Qufu Normal University, Shangdong, 276826, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"YUZHONG","family":"ZHANG","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Qufu Normal University, Shangdong, 276826, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"GUO","family":"SUN","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Qufu Normal University, Shangdong, 276826, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2011,11,20]]},"reference":[{"key":"rf2","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1099-1425(199806)1:1<31::AID-JOS4>3.0.CO;2-R"},{"key":"rf3","doi-asserted-by":"publisher","DOI":"10.1016\/0167-6377(93)90030-K"},{"key":"rf4","doi-asserted-by":"publisher","DOI":"10.1080\/00207549308956847"},{"key":"rf5","doi-asserted-by":"publisher","DOI":"10.1023\/A:1010991401683"},{"key":"rf6","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"crossref","first-page":"231","DOI":"10.1007\/3-540-48686-0_23","volume":"1627","author":"Deng X.","year":"1999"},{"key":"rf7","doi-asserted-by":"publisher","DOI":"10.1023\/A:1027316504440"},{"key":"rf8","first-page":"31","volume":"5","author":"Graham R. L.","journal-title":"Annals of Discrete Mathematics"},{"key":"rf9","doi-asserted-by":"publisher","DOI":"10.1287\/moor.19.1.86"},{"key":"rf10","doi-asserted-by":"publisher","DOI":"10.1080\/002075499192020"},{"key":"rf11","doi-asserted-by":"publisher","DOI":"10.1287\/opre.40.4.764"},{"key":"rf12","doi-asserted-by":"publisher","DOI":"10.1016\/S0166-218X(00)00181-5"},{"key":"rf13","doi-asserted-by":"publisher","DOI":"10.1016\/S0167-6377(02)00186-4"},{"key":"rf14","volume-title":"Combinatorial optimization algorithms and complexity","author":"Papadimitriou C. H.","year":"1982"},{"key":"rf15","doi-asserted-by":"publisher","DOI":"10.1287\/opre.40.3.586"},{"key":"rf16","doi-asserted-by":"publisher","DOI":"10.1016\/0167-6377(92)90001-J"},{"key":"rf17","first-page":"46","volume":"30","author":"Ren J.","journal-title":"Acta Mathmaticae Applicate Sinica"},{"key":"rf18","first-page":"13","volume":"25","author":"Zhang Y.","journal-title":"Journal of System Science and Mathmatic Science"}],"container-title":["Asia-Pacific Journal of Operational Research"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0217595909002249","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,6]],"date-time":"2019-08-06T22:43:14Z","timestamp":1565131394000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0217595909002249"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,6]]},"references-count":17,"journal-issue":{"issue":"03","published-online":{"date-parts":[[2011,11,20]]},"published-print":{"date-parts":[[2009,6]]}},"alternative-id":["10.1142\/S0217595909002249"],"URL":"https:\/\/doi.org\/10.1142\/s0217595909002249","relation":{},"ISSN":["0217-5959","1793-7019"],"issn-type":[{"value":"0217-5959","type":"print"},{"value":"1793-7019","type":"electronic"}],"subject":[],"published":{"date-parts":[[2009,6]]}}}