Minimum tiling of a rectangle by squares

Detalhes bibliográficos
Autor(a) principal: Santos, André Gustavo dos
Data de Publicação: 2018
Outros Autores: Monaci, Michele
Tipo de documento: Artigo
Idioma: eng
Título da fonte: LOCUS Repositório Institucional da UFV
Texto Completo: https://doi.org/10.1007/s10479-017-2746-2
http://www.locus.ufv.br/handle/123456789/22903
Resumo: We consider a two-dimensional problem in which one is required to split a given rectangular bin into the smallest number of items. The resulting items must be squares to be packed, without overlapping, into the bin so as to cover all the given rectangle. We present a mathematical model and a heuristic algorithm that is proved to find the optimal solution in some special cases. Then, we introduce a relaxation of the problem and present different exact approaches based on this relaxation. Finally, we report computational experiments on the performances of the algorithms on a large set of randomly generated instances.
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spelling Santos, André Gustavo dosMonaci, Michele2019-01-03T15:56:05Z2019-01-03T15:56:05Z2018-121572-9338https://doi.org/10.1007/s10479-017-2746-2http://www.locus.ufv.br/handle/123456789/22903We consider a two-dimensional problem in which one is required to split a given rectangular bin into the smallest number of items. The resulting items must be squares to be packed, without overlapping, into the bin so as to cover all the given rectangle. We present a mathematical model and a heuristic algorithm that is proved to find the optimal solution in some special cases. Then, we introduce a relaxation of the problem and present different exact approaches based on this relaxation. Finally, we report computational experiments on the performances of the algorithms on a large set of randomly generated instances.engAnnals of Operations ResearchVolume 271, Issue 2, Pages 831– 851, December 2018Springer Science+Business Media, LLC, part of Springer Nature 2018.info:eu-repo/semantics/openAccessTwo-dimensional packingMathematical modelsExact algorithmsComputational experimentsMinimum tiling of a rectangle by squaresinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articleapplication/pdfreponame:LOCUS Repositório Institucional da UFVinstname:Universidade Federal de Viçosa (UFV)instacron:UFVORIGINALartigo.pdfartigo.pdfTexto completoapplication/pdf726419https://locus.ufv.br//bitstream/123456789/22903/1/artigo.pdf845dbdfe09ecb01d40361851e2b759fcMD51LICENSElicense.txtlicense.txttext/plain; charset=utf-81748https://locus.ufv.br//bitstream/123456789/22903/2/license.txt8a4605be74aa9ea9d79846c1fba20a33MD52123456789/229032019-01-03 14:08:34.186oai:locus.ufv.br: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Repositório InstitucionalPUBhttps://www.locus.ufv.br/oai/requestfabiojreis@ufv.bropendoar:21452019-01-03T17:08:34LOCUS Repositório Institucional da UFV - Universidade Federal de Viçosa (UFV)false
dc.title.en.fl_str_mv Minimum tiling of a rectangle by squares
title Minimum tiling of a rectangle by squares
spellingShingle Minimum tiling of a rectangle by squares
Santos, André Gustavo dos
Two-dimensional packing
Mathematical models
Exact algorithms
Computational experiments
title_short Minimum tiling of a rectangle by squares
title_full Minimum tiling of a rectangle by squares
title_fullStr Minimum tiling of a rectangle by squares
title_full_unstemmed Minimum tiling of a rectangle by squares
title_sort Minimum tiling of a rectangle by squares
author Santos, André Gustavo dos
author_facet Santos, André Gustavo dos
Monaci, Michele
author_role author
author2 Monaci, Michele
author2_role author
dc.contributor.author.fl_str_mv Santos, André Gustavo dos
Monaci, Michele
dc.subject.pt-BR.fl_str_mv Two-dimensional packing
Mathematical models
Exact algorithms
Computational experiments
topic Two-dimensional packing
Mathematical models
Exact algorithms
Computational experiments
description We consider a two-dimensional problem in which one is required to split a given rectangular bin into the smallest number of items. The resulting items must be squares to be packed, without overlapping, into the bin so as to cover all the given rectangle. We present a mathematical model and a heuristic algorithm that is proved to find the optimal solution in some special cases. Then, we introduce a relaxation of the problem and present different exact approaches based on this relaxation. Finally, we report computational experiments on the performances of the algorithms on a large set of randomly generated instances.
publishDate 2018
dc.date.issued.fl_str_mv 2018-12
dc.date.accessioned.fl_str_mv 2019-01-03T15:56:05Z
dc.date.available.fl_str_mv 2019-01-03T15:56:05Z
dc.type.status.fl_str_mv info:eu-repo/semantics/publishedVersion
dc.type.driver.fl_str_mv info:eu-repo/semantics/article
format article
status_str publishedVersion
dc.identifier.uri.fl_str_mv https://doi.org/10.1007/s10479-017-2746-2
http://www.locus.ufv.br/handle/123456789/22903
dc.identifier.issn.none.fl_str_mv 1572-9338
identifier_str_mv 1572-9338
url https://doi.org/10.1007/s10479-017-2746-2
http://www.locus.ufv.br/handle/123456789/22903
dc.language.iso.fl_str_mv eng
language eng
dc.relation.ispartofseries.pt-BR.fl_str_mv Volume 271, Issue 2, Pages 831– 851, December 2018
dc.rights.driver.fl_str_mv Springer Science+Business Media, LLC, part of Springer Nature 2018.
info:eu-repo/semantics/openAccess
rights_invalid_str_mv Springer Science+Business Media, LLC, part of Springer Nature 2018.
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv Annals of Operations Research
publisher.none.fl_str_mv Annals of Operations Research
dc.source.none.fl_str_mv reponame:LOCUS Repositório Institucional da UFV
instname:Universidade Federal de Viçosa (UFV)
instacron:UFV
instname_str Universidade Federal de Viçosa (UFV)
instacron_str UFV
institution UFV
reponame_str LOCUS Repositório Institucional da UFV
collection LOCUS Repositório Institucional da UFV
bitstream.url.fl_str_mv https://locus.ufv.br//bitstream/123456789/22903/1/artigo.pdf
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repository.name.fl_str_mv LOCUS Repositório Institucional da UFV - Universidade Federal de Viçosa (UFV)
repository.mail.fl_str_mv fabiojreis@ufv.br
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