{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,5]],"date-time":"2025-10-05T12:11:44Z","timestamp":1759666304081,"version":"3.37.3"},"reference-count":46,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2016,12,23]],"date-time":"2016-12-23T00:00:00Z","timestamp":1482451200000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/100010029","name":"University of Lancaster","doi-asserted-by":"crossref","id":[{"id":"10.13039\/100010029","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["4OR-Q J Oper Res"],"published-print":{"date-parts":[[2017,12]]},"DOI":"10.1007\/s10288-016-0336-9","type":"journal-article","created":{"date-parts":[[2016,12,24]],"date-time":"2016-12-24T13:14:15Z","timestamp":1482585255000},"page":"331-357","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Integrating column generation in a method to compute a discrete representation of the non-dominated set of multi-objective linear programmes"],"prefix":"10.1007","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7786-1026","authenticated-orcid":false,"given":"Kuan-Min","family":"Lin","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Matthias","family":"Ehrgott","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Andrea","family":"Raith","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2016,12,23]]},"reference":[{"issue":"2","key":"336_CR1","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1023\/A:1011259103627","volume":"20","author":"E Andersen","year":"2001","unstructured":"Andersen E (2001) Certificates of primal or dual infeasibility in linear programming. Comput Optim Appl 20(2):171\u2013183","journal-title":"Comput Optim Appl"},{"issue":"1\u20133","key":"336_CR2","doi-asserted-by":"crossref","first-page":"6","DOI":"10.1016\/j.dam.2005.04.008","volume":"152","author":"D Baatar","year":"2005","unstructured":"Baatar D, Hamacher H, Ehrgott M, Woeginger G (2005) Decomposition of integer matrices and multileaf collimator sequencing. Discrete Appl Math 152(1\u20133):6\u201334","journal-title":"Discrete Appl Math"},{"issue":"1","key":"336_CR3","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1002\/(SICI)1520-6750(199702)44:1<47::AID-NAV3>3.0.CO;2-M","volume":"44","author":"HP Benson","year":"1997","unstructured":"Benson HP, Say\u0131n S (1997) Towards finding global representations of the efficient set in multiple objective mathematical programming. Nav Res Logist 44(1):47\u201367","journal-title":"Nav Res Logist"},{"issue":"2","key":"336_CR4","doi-asserted-by":"crossref","first-page":"377","DOI":"10.1287\/ijoc.1120.0508","volume":"25","author":"R Bokrantz","year":"2013","unstructured":"Bokrantz R, Forsgren A (2013) An algorithm for approximating convex pareto surfaces based on dual techniques. INFORMS J Comput 25(2):377\u2013393","journal-title":"INFORMS J Comput"},{"issue":"10","key":"336_CR5","doi-asserted-by":"crossref","first-page":"5862","DOI":"10.1118\/1.4930252","volume":"42","author":"R Bokrantz","year":"2015","unstructured":"Bokrantz R, Miettinen K (2015) Projections onto the Pareto surface in multicriteria radiation therapy optimization. Med Phys 42(10):5862\u20135870","journal-title":"Med Phys"},{"issue":"23","key":"336_CR6","doi-asserted-by":"crossref","first-page":"7199","DOI":"10.1088\/0031-9155\/54\/23\/011","volume":"54","author":"S Breedveld","year":"2009","unstructured":"Breedveld S, Storchi PR, Heijmen BJ (2009) The equivalence of multi-criteria methods for radiotherapy plan optimization. Phys Med Biol 54(23):7199\u20137209","journal-title":"Phys Med Biol"},{"issue":"8","key":"336_CR7","first-page":"1","volume":"4","author":"M Broderick","year":"2009","unstructured":"Broderick M, Leech M, Coffey M (2009) Direct aperture optimization as a means of reducing the complexity of intensity modulated radiation therapy plans. Radiat Oncol 4(8):1\u20137","journal-title":"Radiat Oncol"},{"issue":"1","key":"336_CR8","doi-asserted-by":"crossref","first-page":"471","DOI":"10.1007\/s10479-013-1360-1","volume":"223","author":"F Carlsson","year":"2014","unstructured":"Carlsson F, Forsgren A (2014) On column generation approaches for approximate solutions of quadratic programs in intensity-modulated radiation therapy. Ann Oper Res 223(1):471\u2013481","journal-title":"Ann Oper Res"},{"key":"336_CR9","volume-title":"Linear programming","author":"V Chv\u00e1tal","year":"1983","unstructured":"Chv\u00e1tal V (1983) Linear programming. W.H Freeman and Company, New York"},{"issue":"24","key":"336_CR10","doi-asserted-by":"crossref","first-page":"5857","DOI":"10.1088\/0031-9155\/50\/24\/007","volume":"50","author":"D Craft","year":"2005","unstructured":"Craft D, Halabi T, Bortfeld T (2005) Exploration of tradeoffs in intensity-modulated radiotherapy. Phys Med Biol 50(24):5857\u20135868","journal-title":"Phys Med Biol"},{"issue":"9","key":"336_CR11","doi-asserted-by":"crossref","first-page":"3399","DOI":"10.1118\/1.2335486","volume":"33","author":"DL Craft","year":"2006","unstructured":"Craft DL, Halabi TF, Shih HA, Bortfeld TR (2006) Approximating convex Pareto surfaces in multiobjective radiotherapy planning. Med Phys 33(9):3399\u20133407","journal-title":"Med Phys"},{"issue":"1","key":"336_CR12","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1088\/0031-9155\/58\/1\/87","volume":"58","author":"D Craft","year":"2013","unstructured":"Craft D, Richter C (2013) Deliverable navigation for multicriteria step and shoot IMRT treatment planning. Phys Med Biol 58(1):87\u2013103","journal-title":"Phys Med Biol"},{"issue":"3","key":"336_CR13","doi-asserted-by":"crossref","first-page":"631","DOI":"10.1137\/S1052623496307510","volume":"8","author":"I Das","year":"1998","unstructured":"Das I, Dennis JE (1998) Normal-boundary intersection: a new method for generating the Pareto surface in nonlinear multicriteria optimization problems. SIAM J Optimiz 8(3):631\u2013657","journal-title":"SIAM J Optimiz"},{"key":"336_CR14","volume-title":"Multicriteria optimization","author":"M Ehrgott","year":"2005","unstructured":"Ehrgott M (2005) Multicriteria optimization, 2nd edn. Springer, Berlin, Heidelberg","edition":"2"},{"issue":"3","key":"336_CR15","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1007\/s10288-008-0083-7","volume":"6","author":"M Ehrgott","year":"2008","unstructured":"Ehrgott M, G\u00fcler \u00c7, Hamacher HW, Shao L (2008a) Mathematical optimization in intensity modulated radiation therapy. 4OR Q J Oper Res 6(3):199\u2013262","journal-title":"4OR Q J Oper Res"},{"key":"336_CR16","first-page":"27","volume-title":"Optimization in medicine","author":"M Ehrgott","year":"2008","unstructured":"Ehrgott M, Hamacher H, Nu\u00dfbaum M (2008b) Decomposition of matrices and static multileaf collimators: a survey. In: Alves C, Pardalos P, Vicente L (eds) Optimization in medicine, vol 12. Springer, Berlin, pp 27\u201348"},{"issue":"5\u20136","key":"336_CR17","doi-asserted-by":"crossref","first-page":"1272","DOI":"10.1016\/j.laa.2007.05.039","volume":"428","author":"M Ehrgott","year":"2008","unstructured":"Ehrgott M, Holder A, Reese J (2008c) Beam selection in radiotherapy design. Linear Algebra Appl 428(5\u20136):1272\u20131312","journal-title":"Linear Algebra Appl"},{"issue":"1","key":"336_CR18","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1016\/j.dam.2004.10.007","volume":"152","author":"K Engel","year":"2005","unstructured":"Engel K (2005) A new algorithm for optimal multileaf collimator field segmentation. Discrete Appl Math 152(1):35\u201351","journal-title":"Discrete Appl Math"},{"issue":"1","key":"336_CR19","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1016\/j.zemedi.2011.05.004","volume":"22","author":"M Falkinger","year":"2012","unstructured":"Falkinger M, Schell S, M\u00fcller J, Wilkens JJ (2012) Prioritized optimization in intensity modulated proton therapy. Z Med Phys 22(1):21\u201328","journal-title":"Z Med Phys"},{"issue":"3","key":"336_CR20","doi-asserted-by":"crossref","first-page":"423","DOI":"10.1007\/s11081-009-9099-x","volume":"11","author":"SL Faulkenberg","year":"2010","unstructured":"Faulkenberg SL, Wiecek MM (2010) On the quality of discrete representations in multiple objective programming. Optim Eng 11(3):423\u2013440","journal-title":"Optim Eng"},{"issue":"21","key":"336_CR21","doi-asserted-by":"crossref","first-page":"7683","DOI":"10.1088\/0031-9155\/58\/21\/7683","volume":"58","author":"A Fredriksson","year":"2013","unstructured":"Fredriksson A, Bokrantz R (2013) Deliverable navigation for multicriteria IMRT treatment planning by combining shared and individual apertures. Phys Med Biol 58(21):7683\u20137697","journal-title":"Phys Med Biol"},{"issue":"1\u20132","key":"336_CR22","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1016\/S0166-218X(01)00261-X","volume":"118","author":"HW Hamacher","year":"2002","unstructured":"Hamacher HW, K\u00fcfer KH (2002) Inverse radiation therapy planning\u2014a multiple objective optimization approach. Discrete Appl Math 118(1\u20132):145\u2013161","journal-title":"Discrete Appl Math"},{"issue":"24","key":"336_CR23","doi-asserted-by":"crossref","first-page":"6349","DOI":"10.1088\/0031-9155\/51\/24\/005","volume":"51","author":"AL Hoffmann","year":"2006","unstructured":"Hoffmann AL, Siem AYD, Den Hertog D, Kaanders JHAM, Huizenga H (2006) Derivative-free generation and interpolation of convex Pareto optimal IMRT plans. Phys Med Biol 51(24):6349\u20136369","journal-title":"Phys Med Biol"},{"issue":"1","key":"336_CR24","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1023\/A:1021970819104","volume":"6","author":"A Holder","year":"2003","unstructured":"Holder A (2003) Designing radiotherapy plans with elastic constraints and interior point methods. Health Care Manag Sci 6(1):5\u201316","journal-title":"Health Care Manag Sci"},{"issue":"7","key":"336_CR25","doi-asserted-by":"crossref","first-page":"1845","DOI":"10.1088\/0031-9155\/52\/7\/006","volume":"52","author":"KW Jee","year":"2007","unstructured":"Jee KW, McShan DL, Fraass BA (2007) Lexicographic ordering: intuitive multicriteria optimization for IMRT. Phys Med Biol 52(7):1845\u20131861","journal-title":"Phys Med Biol"},{"issue":"8","key":"336_CR26","doi-asserted-by":"crossref","first-page":"1747","DOI":"10.1088\/0031-9155\/50\/8\/009","volume":"50","author":"U Jele\u0144","year":"2005","unstructured":"Jele\u0144 U, S\u00f6hn M, Alber M (2005) A finite size pencil beam for IMRT dose optimization. Phys Med Biol 50(8):1747\u20131766","journal-title":"Phys Med Biol"},{"issue":"4","key":"336_CR27","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1118\/1.597679","volume":"23","author":"PJ Keall","year":"1996","unstructured":"Keall PJ, Hoban PW (1996) Superposition dose calculation incorporating Monte Carlo generated electron track kernels. Med Phys 23(4):479\u2013485","journal-title":"Med Phys"},{"issue":"2","key":"336_CR28","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1007\/s00291-003-0125-7","volume":"25","author":"KH K\u00fcfer","year":"2003","unstructured":"K\u00fcfer KH, Scherrer A, Monz M, Alonso F, Trinkaus H, Bortfeld T, Thieke C (2003) Intensity-modulated radiotherapy\u2014a large scale multi-criteria programming problem. OR Spectr 25(2):223\u2013249","journal-title":"OR Spectr"},{"key":"336_CR29","unstructured":"Lin KM, Ehrgott M (2016) Multiobjective navigation of external radiotherapy plans based on clinical criteria. Tech. rep., Lancaster University"},{"key":"336_CR30","first-page":"1","volume-title":"Wiley encyclopedia of operations research and management science","author":"ME L\u00fcbbecke","year":"2010","unstructured":"L\u00fcbbecke ME (2010) Column generation. In: Cochran JJ, Cox LA, Keskinocak P, Kharoufeh JP, Smith JC (eds) Wiley encyclopedia of operations research and management science. Wiley, New York, pp 1\u201314"},{"issue":"2","key":"336_CR31","doi-asserted-by":"crossref","first-page":"86","DOI":"10.1007\/s00158-002-0276-1","volume":"25","author":"A Messac","year":"2003","unstructured":"Messac A, Ismail-Yahaya A, Mattson CA (2003) The normalized normal constraint method for generating the Pareto frontier. Struct Multidiscipl Optim 25(2):86\u201398","journal-title":"Struct Multidiscipl Optim"},{"issue":"4","key":"336_CR32","doi-asserted-by":"crossref","first-page":"985","DOI":"10.1088\/0031-9155\/53\/4\/011","volume":"53","author":"M Monz","year":"2008","unstructured":"Monz M, K\u00fcfer K, Bortfeld T, Thieke C (2008) Pareto navigation\u2014algorithmic foundation of interactive multi-criteria IMRT planning. Phys Med Biol 53(4):985\u2013998","journal-title":"Phys Med Biol"},{"issue":"2","key":"336_CR33","doi-asserted-by":"crossref","first-page":"369","DOI":"10.1016\/j.ejor.2015.01.021","volume":"244","author":"S Moradi","year":"2015","unstructured":"Moradi S, Raith A, Ehrgott M (2015) A bi-objective column generation algorithm for the multi-commodity minimum cost flow problem. Eur J Oper Res 244(2):369\u2013378","journal-title":"Eur J Oper Res"},{"issue":"2","key":"336_CR34","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1007\/s10107-004-0527-6","volume":"101","author":"F Preciado-Walters","year":"2004","unstructured":"Preciado-Walters F, Rardin R, Langer M, Thai V (2004) A coupled column generation, mixed integer approach to optimal planning of intensity modulated radiation therapy for cancer. Math Program 101(2):319\u2013338","journal-title":"Math Program"},{"issue":"4","key":"336_CR35","doi-asserted-by":"crossref","first-page":"493","DOI":"10.1287\/ijoc.1100.0419","volume":"23","author":"G Rennen","year":"2011","unstructured":"Rennen G, van Dam ER, Hertog Dd (2011) Enhancement of sandwich algorithms for approximating higher-dimensional convex Pareto sets. INFORMS J Comput 23(4):493\u2013517","journal-title":"INFORMS J Comput"},{"issue":"4","key":"336_CR36","doi-asserted-by":"crossref","first-page":"643","DOI":"10.1016\/j.radphyschem.2006.05.015","volume":"76","author":"N Reynaert","year":"2007","unstructured":"Reynaert N, van der Marck SC, Schaart DR, Van der Zee W, Van Vliet-Vroegindeweij C, Tomsej M, Jansen J, Heijmen B, Coghe M, De Wagter C (2007) Monte Carlo treatment planning for photon and electron beams. Radiat Phys Chem 76(4):643\u2013686","journal-title":"Radiat Phys Chem"},{"issue":"7\u20138","key":"336_CR37","doi-asserted-by":"crossref","first-page":"1969","DOI":"10.1016\/j.mcm.2011.11.056","volume":"55","author":"H Rocha","year":"2012","unstructured":"Rocha H, Dias JM, Ferreira BC, Lopes MC (2012) Discretization of optimal beamlet intensities in IMRT: a binary integer programming approach. Math Comput Model 55(7\u20138):1969\u20131980","journal-title":"Math Comput Model"},{"issue":"3","key":"336_CR38","doi-asserted-by":"crossref","first-page":"838","DOI":"10.1137\/040606612","volume":"15","author":"HE Romeijn","year":"2005","unstructured":"Romeijn HE, Ahuja RK, Dempsey JF, Kumar A (2005) A column generation approach to radiation therapy treatment planning using aperture modulation. SIAM J Optim 15(3):838\u2013862","journal-title":"SIAM J Optim"},{"issue":"3","key":"336_CR39","doi-asserted-by":"crossref","first-page":"621","DOI":"10.1088\/0031-9155\/58\/3\/621","volume":"58","author":"E Salari","year":"2013","unstructured":"Salari E, Unkelbach J (2013) A column-generation-based method for multi-criteria direct aperture optimization. Phys Med Biol 58(3):621\u2013639","journal-title":"Phys Med Biol"},{"issue":"3","key":"336_CR40","doi-asserted-by":"crossref","first-page":"543","DOI":"10.1007\/s101070050128","volume":"87","author":"S Say\u0131n","year":"2000","unstructured":"Say\u0131n S (2000) Measuring the quality of discrete representations of efficient sets in multiple objective mathematical programming. Math Program 87(3):543\u2013560","journal-title":"Math Program"},{"key":"336_CR41","volume-title":"3D-conformal radiation therapy: a multimedia introduction to methods and techniques","author":"W Schlegel","year":"2002","unstructured":"Schlegel W, Mahr A (2002) 3D-conformal radiation therapy: a multimedia introduction to methods and techniques. Springer, Berlin"},{"key":"336_CR42","doi-asserted-by":"crossref","unstructured":"Shao L, Ehrgott M (2007) Finding representative nondominated points in multiobjective linear programming. In: 1st IEEE symposium on computational intelligence in multicriteria decision making, MCDM 2007, pp 245\u2013252","DOI":"10.1109\/MCDM.2007.369446"},{"issue":"2","key":"336_CR43","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1007\/s00186-008-0220-2","volume":"68","author":"L Shao","year":"2008","unstructured":"Shao L, Ehrgott M (2008) Approximately solving multiobjective linear programmes in objective space and an application in radiotherapy treatment planning. Math Methods Oper Res 68(2):257\u2013276","journal-title":"Math Methods Oper Res"},{"issue":"3","key":"336_CR44","doi-asserted-by":"crossref","first-page":"687","DOI":"10.1016\/j.ejor.2016.05.001","volume":"255","author":"L Shao","year":"2016","unstructured":"Shao L, Ehrgott M (2016) Discrete representation of non-dominated sets in multi-objective linear programming. Eur J Oper Res 255(3):687\u2013698","journal-title":"Eur J Oper Res"},{"key":"336_CR45","doi-asserted-by":"crossref","unstructured":"Webb S (2001) Intensity-modulated radiation therapy. Series in Med Phys, Institute of Physics Publishing, Bristol and Philadelphia","DOI":"10.1887\/0750306998"},{"issue":"6","key":"336_CR46","doi-asserted-by":"crossref","first-page":"1675","DOI":"10.1088\/0031-9155\/52\/6\/009","volume":"52","author":"JJ Wilkens","year":"2007","unstructured":"Wilkens JJ, Alaly JR, Zakarian K, Thorstad WL, Deasy JO (2007) IMRT treatment planning based on prioritizing prescription goals. Phys Med Biol 52(6):1675\u20131692","journal-title":"Phys Med Biol"}],"container-title":["4OR"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10288-016-0336-9\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10288-016-0336-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10288-016-0336-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2017,11,16]],"date-time":"2017-11-16T07:21:46Z","timestamp":1510816906000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10288-016-0336-9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,12,23]]},"references-count":46,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2017,12]]}},"alternative-id":["336"],"URL":"https:\/\/doi.org\/10.1007\/s10288-016-0336-9","relation":{},"ISSN":["1619-4500","1614-2411"],"issn-type":[{"type":"print","value":"1619-4500"},{"type":"electronic","value":"1614-2411"}],"subject":[],"published":{"date-parts":[[2016,12,23]]}}}