Abstract
Different variations of the classic cutting stock problem (CSP) have emerged and presented increasingly complex challenges for scientists and researchers. One of these variations, which is the central subject of this work, is the two-dimensional cutting stock problem with usable leftovers (2D-CSPUL). In these problems, leftovers can be generated to reduce waste. This technique has great practical importance for many companies, with a strong economic and environmental impact. In this paper, a non-linear mathematical model and its linearization are proposed to represent the 2D-CSPUL. Due to the complexity of the model, a heuristic procedure was also proposed. Computational tests were performed with instances from the literature and randomly generated instances. The results demonstrate that the proposed model and the heuristic procedure satisfactorily solve the problem, proving to be adequate and beneficial tools when applied to real situations.
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The full description of all instances used in the computational tests can be found at http://data.mendeley.com/datasets/ddc79swng4/1.
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References
Andrade R, Birgin EG, Morabito R, Ronconi DP (2014) MIP models for two-dimensional non-guillotine cutting problems with usable leftovers. J Oper Res Soc 65(11):1649–1663. https://doi.org/10.1057/jors.2013.108
Andrade R, Birgin EG, Morabito R (2016) Two-stage two-dimensional guillotine cutting stock problems with usable leftover. Int Trans Oper Res 23:121–145. https://doi.org/10.1111/itor.12077
Arenales MN, Cherri AC, do Nascimento DN, Vianna ACG (2015) A new mathematical model for the cutting stock/leftover problem. Pesqui Oper 35(3):509–522. https://doi.org/10.1590/0101-7438.2015.035.03.0509
Birgin EG, Romão OC, Ronconi DP (2020) The multiperiod two-dimensional non-guillotine cutting stock problem with usable leftovers. Int Trans Oper Res 27(3):1392–1418. https://doi.org/10.1111/itor.12648
Bouaine A, Lebbar M, Ha MA (2018) Minimization of the wood wastes for an industry of furnishing: a two dimensional cutting stock problem. Manag Prod Eng Rev 9(2):42–51. https://doi.org/10.24425/119524
Cherri AC (2009) Algumas extensões do problema de corte de estoque com sobras de material aproveitáveis. Doctoral thesis, ICMC-USP, São Carlos
Cherri AC, Arenales MN, Yanasse HH, Poldi KC, Vianna ACG (2014) The one-dimensional cutting stock problem with usable leftovers—a survey. Eur J Oper Res 236(2):395–402
Christofides N, Hadjiconstantinou E (1995) An exact algorithm for orthogonal 2-D cutting problems using guillotine cuts. Eur J Oper Res 83(1):21–38. https://doi.org/10.1016/0377-2217(93)E0277-5
Cintra GF, Miyazawa FK, Wakabayashi Y, Xavier EC (2008) Algorithms for two-dimensional cutting stock and strip packing problems using dynamic programming and column generation. Eur J Oper Res 191:61–85. https://doi.org/10.1016/j.ejor.2007.08.007
Clautiaux F, Sadykov R, Vanderbeck F, Viaud Q (2019) Pattern-based diving heuristics for a two-dimensional guillotine cutting-stock problem with leftovers. EURO J Comput Optim 7:265–297. https://doi.org/10.1007/s13675-019-00113-9
Coelho KR, Cherri AC, Baptista EC, Jabbour CJC, Soler EM (2017) Sustainable operations: The cutting stock problem with usable leftovers from a sustainable perspective. J Clean Prod 167:545–552. https://doi.org/10.1016/j.jclepro.2017.08.153
do Nascimento DN, de Araújo SA, Cherri AC (2020) Integrated lot-sizing and one-dimensional cutting stock problem with usable leftovers. Ann Oper Res. https://doi.org/10.1007/s10479-020-03772-9
Dusberger F, Raidl GR (2015) Solving the 3-staged 2-dimensional cutting stock problem by dynamic programming and variable neighborhood search. Electron Notes Discrete Math 47:133–140. https://doi.org/10.1016/j.endm.2014.11.018
Furini F, Malaguti E (2013) Models for the two-dimensional two-stage cutting stock problem with multiple stock size. Comput Oper Res 40(8):1953–1962. https://doi.org/10.1016/j.cor.2013.02.026
Furini F, Malaguti E, Durán RM, Persiani A, Toth P (2012) A column generation heuristic for the two-dimensional two-staged guillotine cutting stock problem with multiple stock size. Eur J Oper Res 218(1):251–260. https://doi.org/10.1016/j.ejor.2011.10.018
Furini F, Malaguti E, Thomopulos D (2016) Modeling two-dimensional guillotine cutting problems via integer programming. INFORMS J Comput 28:736–751. https://doi.org/10.1287/ijoc.2016.0710
Gilmore PC, Gomory RE (1961) A linear programming approach to the cutting-stock problem. Oper Res 9(6):849–859. https://doi.org/10.1287/opre.9.6.849
Gilmore PC, Gomory RE (1963) A linear programming approach to the cutting-stock problem—part II. Oper Res 11(6):863–888. https://doi.org/10.1287/opre.11.6.863
Gilmore PC, Gomory RE (1965) Multistage cutting stock problems of two and more dimensions. Oper Res 13(1):94–120. https://doi.org/10.1287/opre.13.1.94
Kwon SJ, Joung S, Lee K (2019) Comparative analysis of pattern-based models for the two-dimensional two-stage guillotine cutting stock problem. Comput Oper Res 109:159–169. https://doi.org/10.1016/j.cor.2019.05.005
Li F, Chen Y, Hu X (2022) Manufacturing-oriented silicon steel coil lengthwise cutting stock problem with useable leftover. Eng Comput (Swansea) 39(2):477–492. https://doi.org/10.1108/EC-11-2020-0660
Liu L, Liu X, Pei J, Fan W, Pardalos PM (2017) A study on decision making of cutting stock with frustum of cone bars. Oper Res Int J 17:187–204. https://doi.org/10.1007/s12351-015-0221-x
Lodi A, Monaci M (2003) Integer linear programming models for 2-staged two-dimensional Knapsack problems. Math Program 94(2):257–278. https://doi.org/10.1007/s10107-002-0319-9
Martin M, Hokama PH, Morabito R, Munari P (2020) The constrained two-dimensional guillotine cutting problem with defects: an ILP formulation, a Benders decomposition and a CP-based algorithm. Int J Prod Res 58(9):2712–2729. https://doi.org/10.1080/00207543.2019.1630773
Morabito R (1989) Corte de estoque bidimensional. Dissertation, ICMC–USP, São Carlos
Silva E, Alvelos F, de Carvalho JV (2010) An integer programming model for two- and three-stage two-dimensional cutting stock problems. Eur J Oper Res 205(3):699–708. https://doi.org/10.1016/j.ejor.2010.01.039
Suliman S (2006) A sequential heuristic procedure for the two-dimensional cutting-stock problem. Int J Prod Econ 99(1):177–185. https://doi.org/10.1016/j.ijpe.2004.12.017
Sumetthapiwat S, Intiyot B, Jeenanunta C (2020) A column generation on two-dimensional cutting stock problem with fixed-size usable leftover and multiple stock sizes. Int J Logist Syst Manag 35(2):273–288. https://doi.org/10.1504/IJLSM.2020.104781
Tomat L, Gradisar M (2017) One-dimensional stock cutting: optimization of usable leftovers in consecutive orders. Cent Eur J Oper Res 25(2):473–489. https://doi.org/10.1007/s10100-017-0466-y
Wang PY (1983) Two algorithms for constrained two-dimensional cutting stock problems. INFORMS Oper Res 31(3):573—586. https://www.jstor.org/stable/170624
Wang D, Xiao F, Zhou L, Liang Z (2020) Two-dimensional skiving and cutting stock problem with setup cost based on column-and-row generation. Eur J Oper Res 286(2):547–563. https://doi.org/10.1016/j.ejor.2020.03.060
Wuttke DA, Heese HS (2018) Two-dimensional cutting stock problem with sequence dependent setup times. Eur J Oper Res 265(1):303–315. https://doi.org/10.1016/j.ejor.2017.07.036
Funding
This research was funded by the São Paulo Research Foundation (Fundação de Amparo à Pesquisa do Estado de São Paulo) FAPESP (Grant Numbers 2019/25041-8, 2018/16600-0, 2018/07240-0 and 2016/01860-1) and the National Council for Scientific and Technological Development (Conselho Nacional de Desenvolvimento Científico e Tecnológico) CNPq (Grant Numbers 317460/2021-8, 421130/2018-0 and 306558/2018-1). This work is partially financed by the ERDF—European Regional Development Fund through the Operational Programme for Competitiveness and Internationalisation—COMPETE 2020 Programme and by National Funds through the Portuguese funding agency, FCT—Fundação para a Ciência e a Tecnologia, I.P., within project POCI-01-0145-FEDER-029609.
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do Nascimento, D.N., Cherri, A.C. & Oliveira, J.F. The two-dimensional cutting stock problem with usable leftovers: mathematical modelling and heuristic approaches. Oper Res Int J 22, 5363–5403 (2022). https://doi.org/10.1007/s12351-022-00735-9
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DOI: https://doi.org/10.1007/s12351-022-00735-9