IDEAS home Printed from https://ideas.repec.org/a/wly/jnljam/v2014y2014i1n959453.html

Mathematical Modeling and Optimal Blank Generation in Glass Manufacturing

Author

Listed:
  • Raymond Phillips
  • Matthew Woolway
  • Dario Fanucchi
  • M. Montaz Ali

Abstract

This paper discusses the stock size selection problem (Chambers and Dyson, 1976), which is of relevance in the float glass industry. Given a fixed integer N, generally between 2 and 6 (but potentially larger), we find the N best sizes for intermediate stock from which to cut a roster of orders. An objective function is formulated with the purpose of minimizing wastage, and the problem is phrased as a combinatorial optimization problem involving the selection of columns of a cost matrix. Some bounds and heuristics are developed, and two exact algorithms (depth‐first search and branch‐and‐bound) are applied to the problem, as well as one approximate algorithm (NOMAD). It is found that wastage reduces dramatically as N increases, but this trend becomes less pronounced for larger values of N (beyond 6 or 7). For typical values of N, branch‐and‐bound is able to find the exact solution within a reasonable amount of time.

Suggested Citation

  • Raymond Phillips & Matthew Woolway & Dario Fanucchi & M. Montaz Ali, 2014. "Mathematical Modeling and Optimal Blank Generation in Glass Manufacturing," Journal of Applied Mathematics, John Wiley & Sons, vol. 2014(1).
    • Handle: RePEc:wly:jnljam:v:2014:y:2014:i:1:n:959453
    DOI: 10.1155/2014/959453
    as

    Download full text from publisher

    File URL: https://doi.org/10.1155/2014/959453
    Download Restriction: no
    File URL: https://libkey.io/10.1155/2014/959453?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    References listed on IDEAS

    as
    1. P. C. Gilmore & R. E. Gomory, 1961. "A Linear Programming Approach to the Cutting-Stock Problem," Operations Research, INFORMS, vol. 9(6), pages 849-859, December.
    2. P. C. Gilmore & R. E. Gomory, 1965. "Multistage Cutting Stock Problems of Two and More Dimensions," Operations Research, INFORMS, vol. 13(1), pages 94-120, February.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Amanda O. C. Ayres & Betania S. C. Campello & Washington A. Oliveira & Carla T. L. S. Ghidini, 2021. "A Bi-Integrated Model for coupling lot-sizing and cutting-stock problems," OR Spectrum: Quantitative Approaches in Management, Springer;Gesellschaft für Operations Research e.V., vol. 43(4), pages 1047-1076, December.
    2. Melega, Gislaine Mara & de Araujo, Silvio Alexandre & Jans, Raf, 2018. "Classification and literature review of integrated lot-sizing and cutting stock problems," European Journal of Operational Research, Elsevier, vol. 271(1), pages 1-19.
    3. W. D. D. Madhavee & N. Saldin & U. C. Vaidyarathna & C. J. Jayawardene, 2018. "A Practical Application of the Generalized Cutting Stock Algorithm," Academic Journal of Applied Mathematical Sciences, Academic Research Publishing Group, vol. 4(3), pages 15-21, 03-2018.
    4. Mhand Hifi & Rym M'Hallah, 2005. "An Exact Algorithm for Constrained Two-Dimensional Two-Staged Cutting Problems," Operations Research, INFORMS, vol. 53(1), pages 140-150, February.
    5. Gomory, Ralph, 2016. "Origin and early evolution of corner polyhedra," European Journal of Operational Research, Elsevier, vol. 253(3), pages 543-556.
    6. Sierra-Paradinas, María & Soto-Sánchez, Óscar & Alonso-Ayuso, Antonio & Martín-Campo, F. Javier & Gallego, Micael, 2021. "An exact model for a slitting problem in the steel industry," European Journal of Operational Research, Elsevier, vol. 295(1), pages 336-347.
    7. Morabito, Reinaldo & Belluzzo, Luciano, 2007. "Optimising the cutting of wood fibre plates in the hardboard industry," European Journal of Operational Research, Elsevier, vol. 183(3), pages 1405-1420, December.
    8. Wuttke, David A. & Heese, H. Sebastian, 2018. "Two-dimensional cutting stock problem with sequence dependent setup times," European Journal of Operational Research, Elsevier, vol. 265(1), pages 303-315.
    9. Furini, Fabio & Malaguti, Enrico & Medina Durán, Rosa & Persiani, Alfredo & Toth, Paolo, 2012. "A column generation heuristic for the two-dimensional two-staged guillotine cutting stock problem with multiple stock size," European Journal of Operational Research, Elsevier, vol. 218(1), pages 251-260.
    10. Lodi, Andrea & Martello, Silvano & Monaci, Michele, 2002. "Two-dimensional packing problems: A survey," European Journal of Operational Research, Elsevier, vol. 141(2), pages 241-252, September.
    11. Yuen, Boon J., 1995. "Improved heuristics for sequencing cutting patterns," European Journal of Operational Research, Elsevier, vol. 87(1), pages 57-64, November.
    12. Mhand Hifi & Toufik Saadi, 2012. "A parallel algorithm for two-staged two-dimensional fixed-orientation cutting problems," Computational Optimization and Applications, Springer, vol. 51(2), pages 783-807, March.
    13. Gonçalves, José Fernando & Wäscher, Gerhard, 2020. "A MIP model and a biased random-key genetic algorithm based approach for a two-dimensional cutting problem with defects," European Journal of Operational Research, Elsevier, vol. 286(3), pages 867-882.
    14. de Gelder, E.R. & Wagelmans, A.P.M., 2007. "The two-dimensional cutting stock problem within the roller blind production process," Econometric Institute Research Papers EI 2007-47, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    15. Cui, Yaodong & Huang, Baixiong, 2012. "Reducing the number of cuts in generating three-staged cutting patterns," European Journal of Operational Research, Elsevier, vol. 218(2), pages 358-365.
    16. Belov, G. & Scheithauer, G., 2006. "A branch-and-cut-and-price algorithm for one-dimensional stock cutting and two-dimensional two-stage cutting," European Journal of Operational Research, Elsevier, vol. 171(1), pages 85-106, May.
    17. Kelly Poldi & Silvio Araujo, 2016. "Mathematical models and a heuristic method for the multiperiod one-dimensional cutting stock problem," Annals of Operations Research, Springer, vol. 238(1), pages 497-520, March.
    18. Morgan, Leslie O. & Morton, Alysse R. & Daniels, Richard L., 2006. "Simultaneously determining the mix of space launch vehicles and the assignment of satellites to rockets," European Journal of Operational Research, Elsevier, vol. 172(3), pages 747-760, August.
    19. Mhand Hifi & Rym M'Hallah & Toufik Saadi, 2008. "Algorithms for the Constrained Two-Staged Two-Dimensional Cutting Problem," INFORMS Journal on Computing, INFORMS, vol. 20(2), pages 212-221, May.
    20. Ben Messaoud, Said & Chu, Chengbin & Espinouse, Marie-Laure, 2008. "Characterization and modelling of guillotine constraints," European Journal of Operational Research, Elsevier, vol. 191(1), pages 112-126, November.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:wly:jnljam:v:2014:y:2014:i:1:n:959453. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Wiley Content Delivery (email available below). General contact details of provider: https://onlinelibrary.wiley.com/journal/4185 .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.