Exact algorithms in bar nesting: How to cut general items from linear stocks so that wastage is minimised
This paper proposes exact, polynomial-time algorithms that solve the problem of cutting items with angled sides from a single linear stock so that wastage is minimised. In industry, this problem is called “bar nesting”. Here we give an algorithmic framework that solves several important variants of...
Saved in:
| Published in | Computers & industrial engineering Vol. 200; p. 110838 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Elsevier Ltd
01.02.2025
Elsevier |
| Subjects | |
| Online Access | Get full text |
| ISSN | 0360-8352 1879-0550 |
| DOI | 10.1016/j.cie.2024.110838 |
Cover
| Abstract | This paper proposes exact, polynomial-time algorithms that solve the problem of cutting items with angled sides from a single linear stock so that wastage is minimised. In industry, this problem is called “bar nesting”. Here we give an algorithmic framework that solves several important variants of the problem, including cutting items from stocks with asymmetric cross-sections, cutting items whose sides occur on different planes, and the minimum score separation problem.
•Exact, polynomial-time algorithms are given for single-stock bar nesting problems.•Several industry-related problem variants are considered.•Correctness proofs and source code are provided. |
|---|---|
| AbstractList | This paper proposes exact, polynomial-time algorithms that solve the problem of cutting items with angled sides from a single linear stock so that wastage is minimised. In industry, this problem is called ''bar nesting''. Here we give an algorithmic framework that solves several important variants of the problem, including cutting items from stocks with asymmetric cross-sections, cutting items whose sides occur on different planes, and the minimum score separation problem. This paper proposes exact, polynomial-time algorithms that solve the problem of cutting items with angled sides from a single linear stock so that wastage is minimised. In industry, this problem is called “bar nesting”. Here we give an algorithmic framework that solves several important variants of the problem, including cutting items from stocks with asymmetric cross-sections, cutting items whose sides occur on different planes, and the minimum score separation problem. •Exact, polynomial-time algorithms are given for single-stock bar nesting problems.•Several industry-related problem variants are considered.•Correctness proofs and source code are provided. |
| ArticleNumber | 110838 |
| Author | Lewis, Rhyd Bonnet, Louis |
| Author_xml | – sequence: 1 givenname: Rhyd orcidid: 0000-0003-1046-811X surname: Lewis fullname: Lewis, Rhyd email: lewisr9@cardiff.ac.uk organization: School of Mathematics, Cardiff University, CF24 4AX, Wales, United Kingdom – sequence: 2 givenname: Louis orcidid: 0009-0005-3805-2740 surname: Bonnet fullname: Bonnet, Louis email: lbonnet@laas.fr organization: Laboratory for Analysis and Architecture of Systems, CNRS, Toulouse 31031, France |
| BackLink | https://hal.science/hal-05069687$$DView record in HAL |
| BookMark | eNqNkE1LAzEURbOoYKv-AHfZumh9mXS-dCWiVii40XV4k7xpU2cSSaLVf--UEdyJqweXcy68O2MT5x0xdi5gIUAUl7uFtrTIIFsuhIBKVhM2BVnAvJJ5dsxmMe4AYJnXYsrs3SfqxLHb-GDTto_cOt5g4I5ism5zxVd-z5Pn-j3xDTkK2HGbaADb4HveWUcDHZPXr5FHz9MWE99jTLghbiPvrbO9jWRO2VGLXaSzn3vCXu7vnm9X8_XTw-PtzXquZVakOcq2JWjbEkyeZ6IEnWeFwZIKWQvIGmnksmohr3UJjWmwlmiGcGmaRkupQZ6wbOx9d2_4tceuU2_B9hi-lAB1GEjt1DCQOgykxoEG6WKUtviLe7RqdbNWhwxyKOqiKj_EwIqR1cHHGKj9V__16NDw-YeloOKAOE3GBtJJGW__sL8B-3WREg |
| Cites_doi | 10.1016/j.ejor.2005.12.047 10.1016/j.cor.2023.106527 10.1023/A:1009871302966 10.1007/s12532-009-0002-8 10.1002/jos.86 10.1007/978-1-4419-7997-1_35 10.1137/S0036144596297514 10.1287/opre.12.5.655 10.1057/palgrave.jors.2601835 10.1016/j.ejor.2022.01.028 10.1145/2529989 10.1016/j.ejor.2011.03.016 10.1111/itor.12095 10.1007/978-3-7091-2748-3_8 |
| ContentType | Journal Article |
| Copyright | 2025 The Authors Distributed under a Creative Commons Attribution 4.0 International License |
| Copyright_xml | – notice: 2025 The Authors – notice: Distributed under a Creative Commons Attribution 4.0 International License |
| DBID | 6I. AAFTH AAYXX CITATION 1XC VOOES ADTOC UNPAY |
| DOI | 10.1016/j.cie.2024.110838 |
| DatabaseName | ScienceDirect Open Access Titles Elsevier:ScienceDirect:Open Access CrossRef Hyper Article en Ligne (HAL) Hyper Article en Ligne (HAL) (Open Access) Unpaywall for CDI: Periodical Content Unpaywall |
| DatabaseTitle | CrossRef |
| DatabaseTitleList | |
| Database_xml | – sequence: 1 dbid: UNPAY name: Unpaywall url: https://proxy.k.utb.cz/login?url=https://unpaywall.org/ sourceTypes: Open Access Repository |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Applied Sciences Engineering Physics Computer Science |
| ExternalDocumentID | 10.1016/j.cie.2024.110838 oai:HAL:hal-05069687v1 10_1016_j_cie_2024_110838 S0360835224009604 |
| GroupedDBID | --K --M -~X .DC .~1 0R~ 1B1 1RT 1~. 1~5 29F 4.4 457 4G. 5GY 5VS 6I. 7-5 71M 8P~ 9JN 9JO AAAKG AABNK AACTN AAEDT AAEDW AAFTH AAFWJ AAIKC AAIKJ AAKOC AALRI AAMNW AAOAW AAQFI AAQXK AARIN AAXKI AAXUO ABAOU ABDPE ABJNI ABMAC ABUCO ABWVN ABXDB ACDAQ ACGFO ACGFS ACNCT ACNNM ACRLP ACRPL ADBBV ADEZE ADGUI ADMUD ADNMO ADRHT ADTZH AEBSH AECPX AEIPS AEKER AENEX AFJKZ AFKWA AFTJW AGHFR AGUBO AGYEJ AHHHB AHJVU AIEXJ AIGVJ AIKHN AITUG AJOXV AKRWK ALMA_UNASSIGNED_HOLDINGS AMFUW AMRAJ ANKPU APLSM ARUGR ASPBG AVWKF AXJTR AZFZN BJAXD BKOJK BKOMP BLXMC CS3 DU5 EBS EFJIC EJD EO8 EO9 EP2 EP3 FDB FEDTE FGOYB FIRID FNPLU FYGXN G-2 G-Q GBLVA HAMUX HLZ HVGLF HZ~ H~9 IHE J1W JJJVA KOM LX9 LY1 LY7 M41 MHUIS MO0 MS~ N9A O-L O9- OAUVE OZT P-8 P-9 P2P PC. PQQKQ Q38 R2- RIG RNS ROL RPZ RXW SBC SDF SDG SDP SDS SES SET SEW SPC SPCBC SSB SSD SST SSW SSZ T5K TAE TN5 WUQ XPP ZMT ~G- AATTM AAYWO AAYXX ACLOT ACVFH ADCNI AEUPX AFPUW AGQPQ AIGII AIIUN AKBMS AKYEP APXCP CITATION EFKBS EFLBG ~HD 1XC VOOES ADTOC UNPAY |
| ID | FETCH-LOGICAL-c326t-a3ffe0ff70d552170c526da7e639102b3d348f059c70bdba93adb3d4dbbc33c03 |
| IEDL.DBID | UNPAY |
| ISSN | 0360-8352 1879-0550 |
| IngestDate | Sun Oct 26 04:11:08 EDT 2025 Wed Oct 29 08:31:40 EDT 2025 Wed Oct 01 03:39:00 EDT 2025 Sat Feb 08 15:52:31 EST 2025 |
| IsDoiOpenAccess | true |
| IsOpenAccess | true |
| IsPeerReviewed | true |
| IsScholarly | true |
| Keywords | Bin packing Graph theory Stock cutting Manufacturing and logistics Bar nesting |
| Language | English |
| License | This is an open access article under the CC BY license. Distributed under a Creative Commons Attribution 4.0 International License: http://creativecommons.org/licenses/by/4.0 cc-by |
| LinkModel | DirectLink |
| MergedId | FETCHMERGED-LOGICAL-c326t-a3ffe0ff70d552170c526da7e639102b3d348f059c70bdba93adb3d4dbbc33c03 |
| ORCID | 0000-0003-1046-811X 0009-0005-3805-2740 |
| OpenAccessLink | https://proxy.k.utb.cz/login?url=https://doi.org/10.1016/j.cie.2024.110838 |
| ParticipantIDs | unpaywall_primary_10_1016_j_cie_2024_110838 hal_primary_oai_HAL_hal_05069687v1 crossref_primary_10_1016_j_cie_2024_110838 elsevier_sciencedirect_doi_10_1016_j_cie_2024_110838 |
| ProviderPackageCode | CITATION AAYXX |
| PublicationCentury | 2000 |
| PublicationDate | 2025-02-01 |
| PublicationDateYYYYMMDD | 2025-02-01 |
| PublicationDate_xml | – month: 02 year: 2025 text: 2025-02-01 day: 01 |
| PublicationDecade | 2020 |
| PublicationTitle | Computers & industrial engineering |
| PublicationYear | 2025 |
| Publisher | Elsevier Ltd Elsevier |
| Publisher_xml | – name: Elsevier Ltd – name: Elsevier |
| References | Lewis, Holborn (b19) 2017; 21 Kolmogorov (b16) 2009; 1 Garey, Johnson (b9) 1981 Gilmore, Gomory (b11) 1964; 12 Khemani (b15) 2008 Shachnai, Tamir (b21) 2001; 4 Lewis (b17) 2021 Jansen (b14) 1999; 3 Hawa, Lewis, Thompson (b13) 2022; 302 Wäscher, Haußner, Schumann (b22) 2007; 183 Borges, Schouerya, Miyazawa (b2) 2024; 164 Garey, Johnson (b8) 1979 Goulimis (b12) 2004; 55 Lewis, Song, Dowsland, Thompson (b20) 2011; 213 Becker (b1) 2010 Burkard, Deineko, van Dal, van der Veen, Woeginger (b3) 1998; 40 Coffman, Csirik, Galambos, Martello, Vigo (b4) 2013 Cormen, Leiserson, Rivest, Stein (b5) 2009 Duan, Pettie (b6) 2014; 6 Lewis (b18) 2024 Falkenauer (b7) 1998 Garraffa, Vancroonenburg, Salassa, Vanden Berghe, Wauters (b10) 2016; 23 Lewis (10.1016/j.cie.2024.110838_b20) 2011; 213 Kolmogorov (10.1016/j.cie.2024.110838_b16) 2009; 1 Falkenauer (10.1016/j.cie.2024.110838_b7) 1998 Garey (10.1016/j.cie.2024.110838_b9) 1981 Lewis (10.1016/j.cie.2024.110838_b17) 2021 Gilmore (10.1016/j.cie.2024.110838_b11) 1964; 12 Shachnai (10.1016/j.cie.2024.110838_b21) 2001; 4 Duan (10.1016/j.cie.2024.110838_b6) 2014; 6 Jansen (10.1016/j.cie.2024.110838_b14) 1999; 3 Burkard (10.1016/j.cie.2024.110838_b3) 1998; 40 Hawa (10.1016/j.cie.2024.110838_b13) 2022; 302 Becker (10.1016/j.cie.2024.110838_b1) 2010 Garraffa (10.1016/j.cie.2024.110838_b10) 2016; 23 Goulimis (10.1016/j.cie.2024.110838_b12) 2004; 55 Khemani (10.1016/j.cie.2024.110838_b15) 2008 Garey (10.1016/j.cie.2024.110838_b8) 1979 Borges (10.1016/j.cie.2024.110838_b2) 2024; 164 Cormen (10.1016/j.cie.2024.110838_b5) 2009 Wäscher (10.1016/j.cie.2024.110838_b22) 2007; 183 Coffman (10.1016/j.cie.2024.110838_b4) 2013 Lewis (10.1016/j.cie.2024.110838_b18) 2024 Lewis (10.1016/j.cie.2024.110838_b19) 2017; 21 |
| References_xml | – volume: 40 start-page: 496 year: 1998 end-page: 546 ident: b3 article-title: Well-solvable special cases of the traveling salesman problem: A survey publication-title: SIAM Review – volume: 302 start-page: 847 year: 2022 end-page: 859 ident: b13 article-title: Exact and approximate methods for the score-constrained packing problem publication-title: European Journal of Operational Research – volume: 1 start-page: 43 year: 2009 end-page: 67 ident: b16 article-title: Blossom V: A new implementation of a minimum cost perfect matching algorithm publication-title: Mathematical Programming Computation – volume: 21 start-page: 463 year: 2017 end-page: 476 ident: b19 article-title: How to pack trapezoids: Exact and evolutionary algorithms publication-title: IEEE Transactions on Evolutionary Computation – start-page: 455 year: 2013 end-page: 531 ident: b4 article-title: Bin packing approximation algorithms: survey and classification – year: 1998 ident: b7 article-title: Genetic algorithms and grouping problems – start-page: 147 year: 1981 end-page: 172 ident: b9 article-title: Approximation algorithms for bin packing problems: a survey – year: 2021 ident: b17 article-title: Guide to graph colouring: algorithms and applications – volume: 213 start-page: 52 year: 2011 end-page: 65 ident: b20 article-title: An investigation into two bin packing problems with ordering and orientation implications publication-title: European Journal of Operational Research – volume: 164 year: 2024 ident: b2 article-title: Mathematical models and exact algorithms for the colored bin packing problem publication-title: Computers & Operations Research – volume: 183 start-page: 1109 year: 2007 end-page: 1130 ident: b22 article-title: An improved typology of cutting and packing problems publication-title: European Journal of Operational Research – volume: 3 start-page: 363 year: 1999 end-page: 377 ident: b14 article-title: An approximation scheme for bin packing with conflicts publication-title: Journal of Combinatorial Optimization – volume: 4 start-page: 313 year: 2001 end-page: 338 ident: b21 article-title: Polynomial time approximation schemes for class-constrained packing problems publication-title: Journal of Scheduling – year: 2024 ident: b18 article-title: Code and data – volume: 55 start-page: 1367 year: 2004 end-page: 1368 ident: b12 article-title: Minimum score separation—an open combinatorial problem associated with the cutting stock problem publication-title: Journal of the Operational Research Society – volume: 6 start-page: 1 year: 2014 end-page: 23 ident: b6 article-title: Linear-time approximation for maximum weight matching publication-title: Journal of the ACM – year: 2009 ident: b5 article-title: Introduction to algorithms – volume: 23 start-page: 5 year: 2016 end-page: 24 ident: b10 article-title: The one-dimensional cutting stock problem with sequence dependent cut losses publication-title: International Transactions in Operational Research – year: 1979 ident: b8 article-title: Computers and intractability—a guide to NP-completeness – year: 2010 ident: b1 article-title: Twin-constrained Hamiltonian paths on threshold graphs—an approach to the minimum score separation problem – year: 2008 ident: b15 article-title: A first course in artificial intelligence – volume: 12 start-page: 655 year: 1964 end-page: 679 ident: b11 article-title: Sequencing a one state-variable machine: A solvable case of the traveling salesman problem publication-title: Operations Research – volume: 183 start-page: 1109 year: 2007 ident: 10.1016/j.cie.2024.110838_b22 article-title: An improved typology of cutting and packing problems publication-title: European Journal of Operational Research doi: 10.1016/j.ejor.2005.12.047 – year: 2021 ident: 10.1016/j.cie.2024.110838_b17 – year: 2010 ident: 10.1016/j.cie.2024.110838_b1 – volume: 164 year: 2024 ident: 10.1016/j.cie.2024.110838_b2 article-title: Mathematical models and exact algorithms for the colored bin packing problem publication-title: Computers & Operations Research doi: 10.1016/j.cor.2023.106527 – volume: 3 start-page: 363 issue: 4 year: 1999 ident: 10.1016/j.cie.2024.110838_b14 article-title: An approximation scheme for bin packing with conflicts publication-title: Journal of Combinatorial Optimization doi: 10.1023/A:1009871302966 – volume: 1 start-page: 43 issue: 1 year: 2009 ident: 10.1016/j.cie.2024.110838_b16 article-title: Blossom V: A new implementation of a minimum cost perfect matching algorithm publication-title: Mathematical Programming Computation doi: 10.1007/s12532-009-0002-8 – year: 1998 ident: 10.1016/j.cie.2024.110838_b7 – volume: 4 start-page: 313 issue: 6 year: 2001 ident: 10.1016/j.cie.2024.110838_b21 article-title: Polynomial time approximation schemes for class-constrained packing problems publication-title: Journal of Scheduling doi: 10.1002/jos.86 – year: 2008 ident: 10.1016/j.cie.2024.110838_b15 – volume: 21 start-page: 463 year: 2017 ident: 10.1016/j.cie.2024.110838_b19 article-title: How to pack trapezoids: Exact and evolutionary algorithms publication-title: IEEE Transactions on Evolutionary Computation – start-page: 455 year: 2013 ident: 10.1016/j.cie.2024.110838_b4 doi: 10.1007/978-1-4419-7997-1_35 – volume: 40 start-page: 496 issue: 3 year: 1998 ident: 10.1016/j.cie.2024.110838_b3 article-title: Well-solvable special cases of the traveling salesman problem: A survey publication-title: SIAM Review doi: 10.1137/S0036144596297514 – volume: 12 start-page: 655 issue: 5 year: 1964 ident: 10.1016/j.cie.2024.110838_b11 article-title: Sequencing a one state-variable machine: A solvable case of the traveling salesman problem publication-title: Operations Research doi: 10.1287/opre.12.5.655 – volume: 55 start-page: 1367 year: 2004 ident: 10.1016/j.cie.2024.110838_b12 article-title: Minimum score separation—an open combinatorial problem associated with the cutting stock problem publication-title: Journal of the Operational Research Society doi: 10.1057/palgrave.jors.2601835 – volume: 302 start-page: 847 issue: 3 year: 2022 ident: 10.1016/j.cie.2024.110838_b13 article-title: Exact and approximate methods for the score-constrained packing problem publication-title: European Journal of Operational Research doi: 10.1016/j.ejor.2022.01.028 – year: 2009 ident: 10.1016/j.cie.2024.110838_b5 – volume: 6 start-page: 1 issue: 1 year: 2014 ident: 10.1016/j.cie.2024.110838_b6 article-title: Linear-time approximation for maximum weight matching publication-title: Journal of the ACM doi: 10.1145/2529989 – volume: 213 start-page: 52 year: 2011 ident: 10.1016/j.cie.2024.110838_b20 article-title: An investigation into two bin packing problems with ordering and orientation implications publication-title: European Journal of Operational Research doi: 10.1016/j.ejor.2011.03.016 – volume: 23 start-page: 5 issue: 1 year: 2016 ident: 10.1016/j.cie.2024.110838_b10 article-title: The one-dimensional cutting stock problem with sequence dependent cut losses publication-title: International Transactions in Operational Research doi: 10.1111/itor.12095 – start-page: 147 year: 1981 ident: 10.1016/j.cie.2024.110838_b9 doi: 10.1007/978-3-7091-2748-3_8 – year: 1979 ident: 10.1016/j.cie.2024.110838_b8 – year: 2024 ident: 10.1016/j.cie.2024.110838_b18 |
| SSID | ssj0004591 |
| Score | 2.438983 |
| Snippet | This paper proposes exact, polynomial-time algorithms that solve the problem of cutting items with angled sides from a single linear stock so that wastage is... |
| SourceID | unpaywall hal crossref elsevier |
| SourceType | Open Access Repository Index Database Publisher |
| StartPage | 110838 |
| SubjectTerms | Bar nesting Bin packing Computer Science Graph theory Manufacturing and logistics Mechanical engineering Mechanics Operations Research Physics Stock cutting |
| SummonAdditionalLinks | – databaseName: Science Direct dbid: .~1 link: http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwnV1Lb9RADB6VXgoHoAXUpS2yECdQ2DQzabrcqqrVqmq5QKXeIs9rN7BNVptZll747dh5VIuEQOrVmhmNbMf-ktifhXiHlEfRmSwaHTOptnEY6VinUeKd1hYp4SN_Grj6fDS-Vhc36c2GOO17Ybissov9bUxvonUnGXbaHM6LYviFYm-LH1SDw5kTVKmMpxh8_HW4xhjeTs2jxRGv7v9sNjVedCy9IiaqKYbnFpW_56ZHUy6S3FqWc7xb4Wy2loHOn4unHXSEk_Z222LDlTviWQcjoXtI6x3xZI1j8IUozn6iCYCzSbUowvS2hqIEjQsomV-jnHyCcbWCUIFZBpi0JNTQzGcDbj0BhqG0mjCi-V5DXUGYYoAVEqqcOChqYHISchZnX4rr87Ovp-Oom68QGQJtIULpvYu9z2KbUhbPYpMmRxYzR6iFcIeWVqpjT_jLZLG2GkcSLQmV1dpIaWL5SmyWVel2BUg5SpVOKIpTulPeobNeqjRF72mntQPxvtdsPm9pNPK-vuxbTgrK2Qx5a4aBUL3u8z98Iacw_69tb8lO98czb_b45DJnWZzGTAKU_TgciA_3Zvz_TV4_7CZ74nHCY4Kb4u59sRkWS3dA2CXoN41z_gYp4-xR priority: 102 providerName: Elsevier |
| Title | Exact algorithms in bar nesting: How to cut general items from linear stocks so that wastage is minimised |
| URI | https://dx.doi.org/10.1016/j.cie.2024.110838 https://hal.science/hal-05069687 https://doi.org/10.1016/j.cie.2024.110838 |
| UnpaywallVersion | publishedVersion |
| Volume | 200 |
| hasFullText | 1 |
| inHoldings | 1 |
| isFullTextHit | |
| isPrint | |
| journalDatabaseRights | – providerCode: PRVESC databaseName: Baden-Württemberg Complete Freedom Collection (Elsevier) issn: 1879-0550 databaseCode: GBLVA dateStart: 20110101 customDbUrl: isFulltext: true dateEnd: 99991231 titleUrlDefault: https://www.sciencedirect.com omitProxy: true ssIdentifier: ssj0004591 providerName: Elsevier – providerCode: PRVESC databaseName: Elsevier SD Complete Freedom Collection [SCCMFC] issn: 1879-0550 databaseCode: ACRLP dateStart: 19950101 customDbUrl: isFulltext: true dateEnd: 99991231 titleUrlDefault: https://www.sciencedirect.com omitProxy: true ssIdentifier: ssj0004591 providerName: Elsevier – providerCode: PRVESC databaseName: Elsevier SD Freedom Collection Journals [SCFCJ] issn: 1879-0550 databaseCode: AIKHN dateStart: 19950101 customDbUrl: isFulltext: true dateEnd: 99991231 titleUrlDefault: https://www.sciencedirect.com omitProxy: true ssIdentifier: ssj0004591 providerName: Elsevier – providerCode: PRVESC databaseName: Science Direct issn: 1879-0550 databaseCode: .~1 dateStart: 19950101 customDbUrl: isFulltext: true dateEnd: 99991231 titleUrlDefault: https://www.sciencedirect.com omitProxy: true ssIdentifier: ssj0004591 providerName: Elsevier – providerCode: PRVLSH databaseName: Elsevier Journals issn: 1879-0550 databaseCode: AKRWK dateStart: 19770101 customDbUrl: isFulltext: true mediaType: online dateEnd: 99991231 omitProxy: true ssIdentifier: ssj0004591 providerName: Library Specific Holdings |
| link | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwlZ3Pb9MwFMefaHsADgwGEx2jekKcQKm8OGlabtW0KeNHxYFK4xT5Z1tWkqlxV-DA385zkk5j4ufVsq3E7znv4_j5a4DnguKoMCoJRkMvqq2MCCSTcRBaI6UWFPCF_zXwbjJIp9Hrs_isEYv2Z2F-2r-v8rDIxWkZF0ZVwjoftqAziAm729CZTt6PP9Z7kSzwJOEXV8NkFDDC7u0O5q_6-F0Mas19MuTtdX4hvm7Ecnkt0pzs1DlaZSVQ6BNMzvtrJ_vq2w35xn96iftwr-FNHNcO8gBumXwXdhr2xGZml7tw95ow4UNYHH8RyqFYzorVws0_l7jIUYoV5l6UI5-9wrTYoCtQrR3OauVqrC51Q39eBT27Um0CS3VeYlmgmwuHG0EoOjO4KNErmpCHGf0IpifHH47SoLmUIVBEei4Q3FrDrE2Yjin0J0zF4UCLxBDqEKxIrnk0tARtKmFSSzHiQlNhpKVUnCvG96CdF7l5DMj5KI5kSJ9-ipGRNcJoy6M4FtZSS6278GJrpuyi1t7ItklpnzIaoMwPaVYPaReirSGzBh5qKMjIGn9q9oyMftW9F9tOx28zX8Zi5pWDksvDLry88om_P8n-f9V-AndCf6VwlQh-AG23WpunxDlO9qDV_37Yg8749E066TX-_gMwUvjn |
| linkProvider | Unpaywall |
| linkToHtml | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwnV1Lb9QwEB615VA48CggFgqMECdQ2DS2m25vVdUqwLYXWqm3yM_dwJKsNl4WLv3tHedRLRICiatlW6MZe-ZzMvMNwFtJcVRanUajg0Cqra2MVKxElDirlJEU8GX4NHB2vp9d8k9X4moDjvtamJBW2fn-1qc33robGXbaHM6LYviFfG-LH3iDw_km3OEiScML7MP13hpleNs2j2ZHYXr_a7NJ8qJ96Y2Y8CYbPtSo_Dk4bU5DluT2spzLXys5m62FoNOHcL_DjnjUivcINmy5Aw86HIndLa134N4ayeBjKE5-Su1RzibVovDT7zUWJSq5wDIQbJSTQ8yqFfoK9dLjpGWhxqZBG4baEww4lGYTSNTfaqwr9FPpcSUJVk4sFjUGdhI6LdY8gcvTk4vjLOoaLESaUJuPJHPOxs6lsREUxtNYi2TfyNQSbCHgoZhh_MARANNprIySIyYNDXKjlGZMx-wpbJVVaZ8BMjYSXCXkxinecWelNY5xIaRztNKYAbzrNZvPWx6NvE8w-5qTgvJghrw1wwB4r_v8t8OQk5__27I3ZKfb7QNxdnY0zsNYLOLAApT-2BvA-1sz_luS5_8nyWvYzi7Oxvn44_nnF3A3CT2Dm0zvXdjyi6V9SUDGq1fNQb0BHTHvdA |
| linkToUnpaywall | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwlZ3fb9MwEMdPrHsAHhgMEOWXTognUCYTx03LW4U2VQgmHqg0nqLzr7asJFPjroO_nnOSTgPx89WyrcR3zn0cn78GeE4cR8mZPBkNo6i2cZRooVWSeqe1JQ74FH8NvD8eTKbZ2xN10olFx7MwP-zfN3lY7OK8jEuzJmFdDndgd6AYu3uwOz3-MP7U7kWKJJJEXFwN81EiGLu3O5i_6uN3MWhnHpMhr6_LM_q6oeXySqQ52mtztOpGoDAmmJwerIM-MN9-km_8p5e4Dbc63sRx6yB34Jor92GvY0_sZna9DzevCBPehcXhBZmAtJxVq0WYf6lxUaKmFZZRlKOcvcZJtcFQoVkHnLXK1dhc6obxvApGduXaDJbmtMa6wjCngBtiFJ05XNQYFU3Yw5y9B9Ojw49vJkl3KUNimPRCQtJ7J7zPhVUc-nNhVDqwlDtGHYYVLa3Mhp6hzeRCW00jSZYLM6u1kdIIeR96ZVW6B4BSjlSmU_70c4zMvCNnvcyUIu-5pbV9eLE1U3HWam8U26S0zwUPUBGHtGiHtA_Z1pBFBw8tFBRsjT81e8ZGv-w-im1Pxu-KWCaUiMpB-fmrPry89Im_P8nD_6r9CG6k8UrhJhH8MfTCau2eMOcE_bTz8O-qKfZb |
| openUrl | ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Exact+algorithms+in+bar+nesting%3A+How+to+cut+general+items+from+linear+stocks+so+that+wastage+is+minimised&rft.jtitle=Computers+%26+industrial+engineering&rft.au=Lewis%2C+Rhyd&rft.au=Bonnet%2C+Louis&rft.date=2025-02-01&rft.issn=0360-8352&rft.volume=200&rft.spage=110838&rft_id=info:doi/10.1016%2Fj.cie.2024.110838&rft.externalDBID=n%2Fa&rft.externalDocID=10_1016_j_cie_2024_110838 |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=0360-8352&client=summon |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=0360-8352&client=summon |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=0360-8352&client=summon |