{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,31]],"date-time":"2026-07-31T15:58:02Z","timestamp":1785513482756,"version":"3.56.0"},"reference-count":43,"publisher":"MDPI AG","issue":"15","license":[{"start":{"date-parts":[[2026,7,31]],"date-time":"2026-07-31T00:00:00Z","timestamp":1785456000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Hubei Provincial Natural Science Foundation and Xiangyang Innovation and Development Joint Fund of China","award":["2025AFD104"],"award-info":[{"award-number":["2025AFD104"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Materials"],"abstract":"<jats:p>Chemical formulation design is a challenging multi-objective optimization problem involving mixed continuous and discrete variables, feasibility constraints, and limited experimental data. Existing virtual formulation generation methods are often restricted to interpolation within historically observed regions and therefore lack sufficient exploratory capability. In this work, a problem-setup-centric multi-objective optimization framework for virtual chemical formulation generation is proposed. The framework uses a pymoo-based mixed-variable evolutionary workflow to optimize machine-learning-predicted properties under application-specific objectives and constraints. One energetic-material formulation dataset and one steel-alloy composition dataset are used to evaluate the proposed approach. The generated Pareto solutions include candidates that remain similar to historical formulations and candidates that show greater statistical deviation from the historical data distribution. The t-SNE projections provide qualitative visualizations of local structural relationships, while Mahalanobis-distance and feature-distribution analyses characterize statistical deviation in the original feature space. Overall, this study provides a flexible and practical workflow for data-driven virtual formulation design with customized mixed-variable constraints.<\/jats:p>","DOI":"10.3390\/ma19153248","type":"journal-article","created":{"date-parts":[[2026,7,31]],"date-time":"2026-07-31T15:02:01Z","timestamp":1785510121000},"page":"3248","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Problem-Setup-Centric Framework for Mixed-Variable Chemical Formulation Optimization with Customized Constraints"],"prefix":"10.3390","volume":"19","author":[{"given":"Yingjun","family":"Zhang","sequence":"first","affiliation":[{"name":"Department of Chemistry, Shanghai University, Shanghai 200444, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Shaochen","family":"Chen","sequence":"additional","affiliation":[{"name":"Hubei Institute of Aerospace Chemotechnology, Xiangyang 441003, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Suqi","family":"Gao","sequence":"additional","affiliation":[{"name":"Hubei Institute of Aerospace Chemotechnology, Xiangyang 441003, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5513-8072","authenticated-orcid":false,"given":"Pengcheng","family":"Xu","sequence":"additional","affiliation":[{"name":"Hubei Institute of Aerospace Chemotechnology, Xiangyang 441003, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5361-6122","authenticated-orcid":false,"given":"Wencong","family":"Lu","sequence":"additional","affiliation":[{"name":"Department of Chemistry, Shanghai University, Shanghai 200444, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,7,31]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2","DOI":"10.1016\/j.cherd.2016.10.014","article-title":"Computer-aided molecular design: An introduction and review of tools, applications, and solution techniques","volume":"116","author":"Austin","year":"2016","journal-title":"Chem. 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