Modeling the Combined Problem of Determining the Lot Sizing and Cutting Stock of Raw Materials and Solving With Deep Reinforcement Learning and Exact Method

Document Type : Original Article

Authors

1 Department of Industrial Engineering, Faculty of Technology and Engineering, University of Qom, Qom, Iran.

2 Department of Management, Faculty of Management, University of Tehran, Tehran, Iran. Email: amohaghar@ut.ac.ir

3 Department of Industrial Engineering, Faculty of Technology and Engineering, University of Qom, Qom, Iran. Email: mokhtariy@yahoo.com

4 Department of Management, Faculty of Management, University of Qom, Qom, Iran. Email: bozorgmehr.maleki1363@gmail.com

Abstract

Determining production lot sizes and the raw material cutting problem are key issues in many manufacturing industries. These two problems play a significant role in raw material management, production planning, and cost control in industries such as metals, paper, furniture, and aluminum. Determining the lot size relates to decisions about the quantity of each product to produce in different periods, while the raw material cutting problem deals with determining cutting patterns with minimal waste and cost. Given the interdependence of these two problems, integrated decision-making concerning them can significantly impact the reduction of overall system costs and the improvement of operational productivity. This research presents an integrated mathematical model that combines these two problems. In addition to the multi-level and multi-period production structure, the model also considers costs imposed on the system, including order delivery delays, overtime, and raw material procurement. The main innovation of this research lies in its problem-solving approach; instead of using exact methods or common metaheuristic algorithms, a method based on deep reinforcement learning is used to enable decision-making within the large and complex problem space. In this approach, the problem is designed so that the learning method can make the best decisions regarding production quantity, inventory levels, and the selection of cutting patterns according to different system states. This method can adapt to various operational conditions and functions in an integrated manner without needing to decompose the problem into separate parts

Keywords

Main Subjects

References

  • Chen, Q., & Chen, Y. (2024). Heuristics for the two-dimensional cutting stock problem with usable leftover. Intelligent data analysis, (Preprint), 1–21. https://arxiv.org/abs/2108.08810v2
  • Cemil Dikili, A., Can Takinacı, A., & Akman Pek, N. (2008). A new heuristic approach to one-dimensional stock-cutting problems with multiple stock lengths in ship production. Ocean engineering, 35(7), 637–645. https://doi/science/article/pii/S0029801808000097
  • Cui, Y. (2012). A CAM system for one-dimensional stock cutting. Advances in engineering software, 47(1), 7–16.
    https://doi/article/pii/S0965997811003206
  • Cherri, A. C., Cherri, L. H., Oliveira, B. B., Oliveira, J. F., & Carravilla, M. A. (2023). A stochastic programming approach to the cutting stock problem with usable leftovers. European journal of operational research, 308(1), 38–53. https://arxiv.org/abs/2108.08810v2
  • Hindi, K. S. (1995). Computationally efficient solution of the multi-item, capacitated lot-sizing problem. Computers & industrial engineering, 28(4), 709–719. https://doi/article/pii/036083529500021R
  • Delorme, M., & Iori, M. (2020). Enhanced pseudo-polynomial formulations for bin packing and cutting stock problems. INFORMS journal on computing, 32(1), 101–119. https://doi.org/10.48550/arXiv.1509.08971
  • Kroth, K. M. (2024). Cutting and Packaging Optimization. In Operations research and management: quantitative methods for planning and decision-making in business and economics (pp. 45–67). Springer. https://doi.org/10.1007/s12559-020-09734-4
  • Kuik, R., & Salomon, M. (1990). Multi-level lot-sizing problem: evaluation of a simulated-annealing heuristic. European journal of operational research, 45(1), 25–37. https://doi/pii/0377221790901533
  • Wu, T., Shi, L., Geunes, J., & Akartunalı, K. (2011). An optimization framework for solving capacitated multi-level lot-sizing problems with backlogging. European journal of operational research, 214(2), 428–441. https://doi
    /article/pii/S0377221711003730
  • Roll, Y., & Karni, R. (1991). Multi-item, multi-level lot sizing with an aggregate capacity constraint. European journal of operational research, 51(1), 73–87. https://doi.org/10.1007/s12559-020-09734-4
  • Zhao, Q., Xie, C., & Xiao, Y. (2012). A variable neighborhood decomposition search algorithm for multilevel capacitated lot-sizing problems. Electronic notes in discrete mathematics, 39, 129–135. https://doi/S1571065312000194
  • Bertolini, M., Mezzogori, D., & Zammori, F. (2024). Hybrid heuristic for the one-dimensional cutting stock problem with usable leftovers and additional operating constraints. International journal of industrial engineering computations, 15(1), 149–170. https://doi.org/10.48550/arXiv.1509.08971
  • Melega, G. M., de Araujo, S. A., & Jans, R. (2018). Classification and literature review of integrated lot-sizing and cutting stock problems. European journal of operational research, 271(1), 1–19. https://arxiv.org/abs/2108.08810v2
  • Luo, Q., Du, B., Rao, Y., & Guo, X. (2022). Metaheuristic algorithms for a special cutting stock problem with multiple stocks in the transformer manufacturing industry. Expert systems with applications, 210, 118578.
    https://doi/article/pii/S0957417422016396
  • Fang, J., Rao, Y., Luo, Q., & Xu, J. (2023). Solving one-dimensional cutting stock problems with the deep reinforcement learning. Mathematics, 11(4), 1028. https://doi./227-7390/11/4/1028
  • Cucuzza, R., Domaneschi, M., Rosso, M. M., Martinelli, L., & Marano, G. C. (2023). Cutting stock problem (CSP) applied to structural optimization for the minimum waste cost. Cepapers, 6(5), 1066–1073. https://doi/abs/10.1002/cepa.2208
  • do Nascimento, D. N., de Araujo, S. A., & Cherri, A. C. (2022). Integrated lot-sizing and one-dimensional cutting stock problem with usable leftovers. Annals of operations research, 316, 1–19. https://doi.org/10.1007/s12559-020-
    09734-4
  • Pierini, L. M., & Poldi, K. C. (2021). Lot Sizing and cutting stock problems in a paper production process. Pesquisa
    operacional, 41(SPE), e235094. https://doi.org/10.48550/arXiv.1509.08971
  • Andrade, P. R. de L. (2021). Optimization of the production process in an automotive spring industry. Universidade Estadual Paulista (Unesp). https://doi.org/10.48550/arXiv.1509.08971
  • - Viegas, J. L., Vieira, S. M., Henriques, E. M. P., & Sousa, J. M. C. (2016). Heuristics for three-dimensional steel cutting with usable leftovers considering large time periods. European journal of industrial engineering, 10(4), 431–454.
    https://doi/abs/10.1504/EJIE.2016.078141
  • Suliman, S. M. A. (2012). An algorithm for solving lot sizing and cutting stock problem within aluminum fabrication industry [presentation]. Proceedings of the 2012 international conference on industrial engineering and operations management (pp. 783–793). https://arxiv.org/abs/2108.08810v2
  • Vanzela, M., Melega, G. M., Rangel, S., & de Araujo, S. A. (2017). The integrated lot sizing and cutting stock problem with saw cycle constraints applied to furniture production. Computers & operations research, 79, 148–160.
    https://doi.org/10.1007/s12559-020-09734-4
  • Wu, T., Akartunalı, K., Jans, R., & Liang, Z. (2017). Progressive selection method for the coupled lot-sizing and cuttingstock problem. INFORMS journal on computing, 29(3), 523–543. https://pubsonline.informs.org/doi/abs/10.1287/ijoc.2017.0746
  • Gramani, M. C. N., França, P. M., & Arenales, M. N. (2009). A lagrangian relaxation approach to a coupled lot-sizing and cutting stock problem. International journal of production economics, 119(2), 219–227.
    https://doi.org/10.48550/arXiv.1509.08971
  • Gramani, M. C. N., França, P. M., & Arenales, M. N. (2011). A linear optimization approach to the combined production planning model. Journal of the franklin institute, 348(7), 1523–1536. https://doi/article/pii/S0016003210001560
  • José Alem, D., & Morabito, R. (2012). Production planning in furniture settings via robust optimization. Computers & operations research, 39(2), 139–150. https://doi.org/10.48550/arXiv.1509.08971
  • Alem, D., & Morabito, R. (2013). Risk-averse two-stage stochastic programs in furniture plants. OR spectrum, 35(4), 773–806. https://doi/10.1007/s00291-012-0312-5
  • Christofoletti, M. M., de Araujo, S. A., & Cherri, A. C. (2021). Integrated lot-sizing and cutting stock problem applied to the mattress industry. Journal of the operational research society, 72(6), 1279–1293.
    https://doi/abs/10.1080/01605682.2020.1718013
  • Ayres, A. O. C., Campello, B. S. C., Oliveira, W. A., & Ghidini, C. T. L. S. (2021). A Bi-integrated model for coupling lotsizing and cutting-stock problems. OR spectrum, 43(4), 1047–1076. https://doi.org/10.1007/s12559-020-09734-4
  • Melega, G. M., de Araujo, S. A., & Morabito, R. (2020). Mathematical model and solution approaches for integrated lotsizing, scheduling and cutting stock problems. Annals of operations research, 295(2), 695–736.
    https://arxiv.org/abs/2304.03589v1
  • Melega, G. M., de Araujo, S. A., Jans, R., & Morabito, R. (2023). Formulations and exact solution approaches for a coupled bin-packing and lot-sizing problem with sequence-dependent setups. Flexible services and manufacturing journal, 35(4), 1276–1312. https://doi.org/10.48550/arXiv.2009.06732
  • Silva, E. M., Melega, G. M., Akartunalı, K., & de Araujo, S. A. (2023). Formulations and theoretical analysis of the onedimensional multi-period cutting stock problem with setup cost. European journal of operational research, 304(2), 443–460. https://doi.org/10.48550/arXiv.1709.01686
  • de Lara Andrade, P. R., de Araujo, S. A., Cherri, A. C., & Lemos, F. K. (2021). The integrated lot sizing and cutting stock problem in an automotive spring factory. Applied mathematical modelling, 91, 1023–1036.
    https://doi.org/10.1145/3581783.3611762
  • Melega, G. M., Araujo, S. A. de, & Jans, R. (2016). Comparison of mip models for the integrated lot-sizing and onedimensional cutting stock problem. Pesquisa operacional, 36(1), 167–196. https://doi.org/10.48550/arXiv.2012.15840
  • Campello, B. S. C., Ghidini, C. T. L. S., Ayres, Ao., & Oliveira, W. A. (2022). A residual recombination heuristic for onedimensional cutting stock problems. Top, 30(1), 194–220. https://doi.org/10.48550/arXiv.1804.00819
  • Pierini, L. M., & Poldi, K. C. (2023). Optimization of the cutting process integrated to the lot sizing in multi-plant paper production industries. Computers & operations research, 153, 106157. https://doi.org/10.48550/arXiv.2010.04159
Send comment about this article
CAPTCHA Image
Engineering Management and Soft Computing
Volume 11, Issue 2 - Serial Number 21
December 2025
Pages 326-347

Files

Share

How to cite

Statistics