{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,17]],"date-time":"2026-07-17T15:49:10Z","timestamp":1784303350756,"version":"3.55.0"},"reference-count":26,"publisher":"MIT Press - Journals","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Evolutionary Computation"],"published-print":{"date-parts":[[2021,3]]},"abstract":"<jats:p> An objective normalization strategy is essential in any evolutionary multiobjective or many-objective optimization (EMO or EMaO) algorithm, due to the distance calculations between objective vectors required to compute diversity and convergence of population members. For the decomposition-based EMO\/EMaO algorithms involving the Penalty Boundary Intersection (PBI) metric, normalization is an important matter due to the computation of two distance metrics. In this article, we make a theoretical analysis of the effect of instabilities in the normalization process on the performance of PBI-based MOEA\/D and a proposed PBI-based NSGA-III procedure. Although the effect is well recognized in the literature, few theoretical studies have been done so far to understand its true nature and the choice of a suitable penalty parameter value for an arbitrary problem. The developed theoretical results have been corroborated with extensive experimental results on three to 15-objective convex and non-convex instances of DTLZ and WFG problems. The article, makes important theoretical conclusions on PBI-based decomposition algorithms derived from the study. <\/jats:p>","DOI":"10.1162\/evco_a_00276","type":"journal-article","created":{"date-parts":[[2020,6,22]],"date-time":"2020-06-22T13:56:29Z","timestamp":1592834189000},"page":"157-186","source":"Crossref","is-referenced-by-count":18,"title":["Effect of Objective Normalization and Penalty Parameter on Penalty Boundary Intersection Decomposition-Based Evolutionary Many-Objective Optimization Algorithms"],"prefix":"10.1162","volume":"29","author":[{"given":"Lei","family":"Chen","sequence":"first","affiliation":[{"name":"School of Applied Mathematics, Guangdong University of Technology, Guangzhou, 51000, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Kalyanmoy","family":"Deb","sequence":"additional","affiliation":[{"name":"Department of Electrical and Computer Engineering and BEACON Center for the Study of Evolution in Action, Michigan State University, East Lansing, MI 48824, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hai-Lin","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Applied Mathematics, Guangdong University of Technology, Guangzhou, 51000, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Qingfu","family":"Zhang","sequence":"additional","affiliation":[{"name":"Department of Computer Science, City University of Hong Kong, Hong Kong and the Shenzhen Research Institute, City University of Hong Kong, Shenzhen, 518057, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"281","reference":[{"key":"B1","doi-asserted-by":"publisher","DOI":"10.1145\/1774088.1774325"},{"key":"B2","doi-asserted-by":"publisher","DOI":"10.1109\/TEVC.2003.810761"},{"key":"B3","doi-asserted-by":"publisher","DOI":"10.1109\/TEVC.2016.2519378"},{"key":"B4","doi-asserted-by":"publisher","DOI":"10.1137\/S1052623496307510"},{"key":"B5","doi-asserted-by":"publisher","DOI":"10.1109\/4235.996017"},{"key":"B6","doi-asserted-by":"publisher","DOI":"10.1109\/TEVC.2013.2281535"},{"key":"B7","doi-asserted-by":"publisher","DOI":"10.1109\/TEVC.2010.2041667"},{"key":"B8","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-70928-2_9"},{"key":"B9","doi-asserted-by":"publisher","DOI":"10.1007\/1-84628-137-7_6"},{"key":"B10","doi-asserted-by":"publisher","DOI":"10.1109\/TEVC.2005.861417"},{"key":"B11","doi-asserted-by":"publisher","DOI":"10.1016\/0377-2217(88)90257-3"},{"key":"B12","doi-asserted-by":"publisher","DOI":"10.1109\/TEVC.2014.2315442"},{"key":"B13","doi-asserted-by":"publisher","DOI":"10.1109\/CEC.2016.7743913"},{"key":"B14","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-45823-6_47"},{"key":"B15","doi-asserted-by":"publisher","DOI":"10.1007\/s40747-017-0061-9"},{"key":"B16","doi-asserted-by":"publisher","DOI":"10.1109\/5326.704576"},{"key":"B17","doi-asserted-by":"publisher","DOI":"10.1007\/3-540-36970-8_27"},{"key":"B18","doi-asserted-by":"publisher","DOI":"10.1109\/TEVC.2008.925798"},{"key":"B19","doi-asserted-by":"publisher","DOI":"10.1109\/TEVC.2014.2373386"},{"key":"B20","doi-asserted-by":"publisher","DOI":"10.1109\/TEVC.2017.2725902"},{"issue":"18","key":"B21","first-page":"773","volume":"18","author":"Liu H.-L.","year":"2013","journal-title":"IEEE Transactions on Evolutionary Computation"},{"key":"B22","doi-asserted-by":"publisher","DOI":"10.1162\/evco.2010.18.1.18103"},{"issue":"23","key":"B23","first-page":"503","volume":"23","author":"Seada H.","year":"2018","journal-title":"IEEE Transactions on Evolutionary Computation"},{"key":"B24","doi-asserted-by":"publisher","DOI":"10.1109\/TEVC.2015.2420112"},{"key":"B25","doi-asserted-by":"publisher","DOI":"10.1109\/TEVC.2007.892759"},{"key":"B26","first-page":"1","author":"Zhang Q.","year":"2010","journal-title":"IEEE Congress on Evolutionary Computation"}],"container-title":["Evolutionary Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mitpressjournals.org\/doi\/pdf\/10.1162\/evco_a_00276","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,3,12]],"date-time":"2021-03-12T21:59:18Z","timestamp":1615586358000},"score":1,"resource":{"primary":{"URL":"https:\/\/direct.mit.edu\/evco\/article\/29\/1\/157-186\/97345"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,3]]},"references-count":26,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2021,3]]}},"alternative-id":["10.1162\/evco_a_00276"],"URL":"https:\/\/doi.org\/10.1162\/evco_a_00276","relation":{},"ISSN":["1063-6560","1530-9304"],"issn-type":[{"value":"1063-6560","type":"print"},{"value":"1530-9304","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,3]]}}}