{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,12]],"date-time":"2026-02-12T06:19:08Z","timestamp":1770877148955,"version":"3.50.1"},"reference-count":14,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2026,2,10]],"date-time":"2026-02-10T00:00:00Z","timestamp":1770681600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100003725","name":"National Research Foundation of the Republic of Korea","doi-asserted-by":"crossref","award":["RS-2025-23524280"],"award-info":[{"award-number":["RS-2025-23524280"]}],"id":[{"id":"10.13039\/501100003725","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In industries, particularly in quality optimization, the trade-off between model bias and variance is inevitable, reflecting the tension between accuracy and uncertainty. Traditional methods often address these aspects separately, potentially leading to suboptimal decisions. This study proposes a Pareto-front optimization framework for a variance-added expected loss function within the context of interrelated quality characteristics. By integrating multivariate quadratic loss with a variance term, our approach simultaneously captures deviation from targets (bias) and system uncertainty (variance). Unlike sequential approaches that first minimize bias and then variance\u2014often increasing total risk\u2014our weighted formulation flexibly adjusts for their trade-offs. This enables a more balanced and efficient optimization process that identifies solutions with lower overall risk. Through Pareto-front analysis, we reveal trade-offs between expected loss and variance, allowing users to select optimal quality designs based on their preferred bias\u2013variance balance. Representative examples and a case study adopted from the literature validate the effectiveness and practical applicability of the proposed method.<\/jats:p>","DOI":"10.3390\/e28020199","type":"journal-article","created":{"date-parts":[[2026,2,10]],"date-time":"2026-02-10T11:16:38Z","timestamp":1770722198000},"page":"199","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Pareto-Front Optimization of Variance-Added Expected Loss with Interrelated Qualities"],"prefix":"10.3390","volume":"28","author":[{"ORCID":"https:\/\/orcid.org\/0009-0002-1372-7957","authenticated-orcid":false,"given":"Sangwon","family":"Kim","sequence":"first","affiliation":[{"name":"Department of Industrial Engineering, Hanyang University, Seoul 04763, Republic of Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5184-7151","authenticated-orcid":false,"given":"Kichun","family":"Lee","sequence":"additional","affiliation":[{"name":"Department of Industrial Engineering, Hanyang University, Seoul 04763, Republic of Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2026,2,10]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1080\/07408179308964286","article-title":"Strategies for robust multiresponse quality engineering","volume":"25","author":"Pignatiello","year":"1993","journal-title":"IIE Trans."},{"key":"ref_2","first-page":"253","article-title":"A study of the sintering properties of iron ores using the Taguchi\u2019s parameter design","volume":"28","author":"Shiau","year":"1990","journal-title":"J. Chin. Stat. Assoc."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"367","DOI":"10.1108\/02656719710170639","article-title":"The optimization of multi-response problems in the Taguchi method","volume":"14","author":"Tong","year":"1997","journal-title":"Int. J. Qual. Reliab. Manag."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2685","DOI":"10.1002\/qre.2038","article-title":"A multivariate loss function approach to robust design of systems with multiple performance characteristics","volume":"32","author":"Soh","year":"2016","journal-title":"Qual. Reliab. Eng. Int."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Costa, N., and Louren\u00e7o, J. (2022). Bi-Objective Optimization Problems\u2014A Game Theory Perspective to Improve Process and Product. 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Optim."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1084","DOI":"10.2514\/2.1071","article-title":"Ability of objective functions to generate points on non-convex Pareto frontiers","volume":"38","author":"Messac","year":"2000","journal-title":"AIAA J."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"421","DOI":"10.1016\/j.ejor.2007.05.030","article-title":"A general framework for multiresponse optimization problems based on goal programming","volume":"189","author":"Kazemzadeh","year":"2008","journal-title":"Eur. J. Oper. Res."},{"key":"ref_13","first-page":"857","article-title":"Product excellence through design of experiments","volume":"32","author":"Joglekar","year":"1987","journal-title":"Cereal Foods World"},{"key":"ref_14","unstructured":"Montgomery, D.C. (2017). Design and Analysis of Experiments, John Wiley & Sons. [9th ed.]."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/2\/199\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,2,12]],"date-time":"2026-02-12T05:29:06Z","timestamp":1770874146000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/2\/199"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,2,10]]},"references-count":14,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2026,2]]}},"alternative-id":["e28020199"],"URL":"https:\/\/doi.org\/10.3390\/e28020199","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,2,10]]}}}