{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:10:44Z","timestamp":1760145044732,"version":"build-2065373602"},"reference-count":37,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2024,6,13]],"date-time":"2024-06-13T00:00:00Z","timestamp":1718236800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Volkswagen Foundation","award":["#97775","#100072","2024"],"award-info":[{"award-number":["#97775","#100072","2024"]}]},{"name":"British Academy","award":["#97775","#100072","2024"],"award-info":[{"award-number":["#97775","#100072","2024"]}]},{"name":"Technological Institute of Sonora (ITSON), Mexico","award":["#97775","#100072","2024"],"award-info":[{"award-number":["#97775","#100072","2024"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Sphere packing consists of placing several spheres in a container without mutual overlapping. While packing into regular-shape containers is well explored, less attention is focused on containers with nonlinear boundaries, such as ellipsoids or paraboloids. Packing n-dimensional spheres into a minimum-height container bounded by a parabolic surface is formulated. The minimum allowable distances between spheres as well as between spheres and the container boundary are considered. A normalized \u03a6-function is used for analytical description of the containment constraints. A nonlinear programming model for the packing problem is provided. A solution algorithm based on the feasible directions approach and a decomposition technique is proposed. The computational results for problem instances with various space dimensions, different numbers of spheres and their radii, the minimal allowable distances and the parameters of the parabolic container are presented to demonstrate the efficiency of the proposed approach.<\/jats:p>","DOI":"10.3390\/axioms13060396","type":"journal-article","created":{"date-parts":[[2024,6,13]],"date-time":"2024-06-13T10:41:03Z","timestamp":1718275263000},"page":"396","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Packing Spheres into a Minimum-Height Parabolic Container"],"prefix":"10.3390","volume":"13","author":[{"given":"Yuriy","family":"Stoyan","sequence":"first","affiliation":[{"name":"Pidhornyi Institute of Mechanical Engineering Problems, vul. Komunalnykiv, 2\/10, 61046 Kharkiv, Ukraine"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Georgiy","family":"Yaskov","sequence":"additional","affiliation":[{"name":"Pidhornyi Institute of Mechanical Engineering Problems, vul. Komunalnykiv, 2\/10, 61046 Kharkiv, Ukraine"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tetyana","family":"Romanova","sequence":"additional","affiliation":[{"name":"Pidhornyi Institute of Mechanical Engineering Problems, vul. Komunalnykiv, 2\/10, 61046 Kharkiv, Ukraine"},{"name":"Leeds University Business School, University of Leeds, Leeds LS2 9JT, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1850-4755","authenticated-orcid":false,"given":"Igor","family":"Litvinchev","sequence":"additional","affiliation":[{"name":"Graduate Program in Systems Engineering, Nuevo Leon State University (UANL), Av. Universidad s\/n, Col. Ciudad Universitaria, San Nicolas de los Garza 66455, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1697-8551","authenticated-orcid":false,"given":"Jos\u00e9 Manuel","family":"Velarde Cant\u00fa","sequence":"additional","affiliation":[{"name":"Technological Institute of Sonora (ITSON), Navojoa-City 85870, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3728-9576","authenticated-orcid":false,"given":"Mauricio L\u00f3pez","family":"Acosta","sequence":"additional","affiliation":[{"name":"Technological Institute of Sonora (ITSON), Navojoa-City 85870, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,6,13]]},"reference":[{"key":"ref_1","first-page":"385","article-title":"Introduction to Cutting and Packing Optimization","volume":"Volume 263","author":"Scheithauer","year":"2018","journal-title":"International Series in Operations Research & Management Science"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1109","DOI":"10.1016\/j.ejor.2005.12.047","article-title":"An improved typology of cutting and packing problems","volume":"183","author":"Schumann","year":"2007","journal-title":"Eur. 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