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Grids for cutting and packing problems: a study in the 2D knapsack problem

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Abstract

Different grids of points to solve cutting and packing problems with rectangular shaped items are discussed in this work. The grids are the canonical dissections (also known as normal patterns), useful numbers, reduced raster points, regular normal patterns, and meet-in-the-middle patterns. Theoretical results involving the size and subset relations among the grids are proposed, besides practical procedures to reduce the size. Computational experiments are performed in which the two-dimensional (2D) knapsack problem is solved with an integer linear programming model. The results show the impact on the grids before and after applying the reduction procedures, concluding that the reduced raster points and meet-in-the-middle patterns are generally the grids with the smallest number of points.

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References

  • Alvarez-Valdes R, Parreño F, Tamarit JM (2005) A branch-and-cut algorithm for the pallet loading problem. Comput Oper Res 32(11):3007–3029

    Article  Google Scholar 

  • Alvarez-Valdes R, Parreño F, Tamarit JM (2009) A branch and bound algorithm for the strip packing problem. OR Spectr 31:431–459

    Article  Google Scholar 

  • Alves C, Brás P, de Carvalho JV, Pinto T (2012) New constructive algorithms for leather nesting in the automotive industry. Comput Oper Res 39(7):1487–1505

    Article  Google Scholar 

  • Arahori Y, Imamichi T, Nagamochi H (2012) An exact strip packing algorithm based on canonical forms. Comput Oper Res 39(12):2991–3011

    Article  Google Scholar 

  • Baldacci R, Boschetti MA (2007) A cutting-plane approach for the two-dimensional orthogonal non-guillotine cutting problem. Eur J Oper Res 183:1136–1149

    Article  Google Scholar 

  • Baldi MM, Perboli G, Tadei R (2012) The three-dimensional knapsack problem with balancing constraints. Appl Math Comput 218(19):9802–9818

    Google Scholar 

  • Beasley JE (1985a) Algorithms for unconstrained two-dimensional guillotine cutting. J Oper Res Soc 36(4):297–306

    Article  Google Scholar 

  • Beasley JE (1985b) An exact two-dimensional non-guillotine cutting tree search procedure. Oper Res 33:49–64

    Article  Google Scholar 

  • Birgin EG, Lobato RD, Morabito R (2010) An effective recursive partitioning approach for the packing of identical rectangles in a rectangle. J Oper Res Soc 61(2):306–320

    Article  Google Scholar 

  • Birgin EG, Lobato RD, Morabito R (2012) Generating unconstrained two-dimensional non-guillotine cutting patterns by a recursive partitioning algorithm. J Oper Res Soc 63(2):183–200

    Article  Google Scholar 

  • Boschetti MA, Montaletti L (2010) An exact algorithm for the two-dimensional strip-packing problem. Oper Res 58(6):1774–1791

    Article  Google Scholar 

  • Boschetti MA, Mingozzi A, Hadjiconstantinou E (2002) New upper bounds for the two-dimensional orthogonal non-guillotine cutting stock problem. IMA J Manag Math 13(2):95–119

    Article  Google Scholar 

  • Caprara A (2004) Knapsack problems: a book review. 4OR 2(4):317–320

    Article  Google Scholar 

  • Caprara A, Monaci M (2004) On the two-dimensional knapsack problem. Oper Res Lett 32:5–14

    Article  Google Scholar 

  • Carnieri C, Mendoza GA, Gavinho LG (1994) Solution procedures for cutting lumber into furniture parts. Eur J Oper Res 73(3):495–501

    Article  Google Scholar 

  • Christofides N, Whitlock C (1977) An algorithm for two-dimensional cutting problems. Oper Res 25(1):30–44

    Article  Google Scholar 

  • 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 191:59–83

    Article  Google Scholar 

  • Côté JF, Iori M (2018) The meet-in-the-middle principle for cutting and packing problems. INFORMS J Comput 30(4):646–661

    Article  Google Scholar 

  • Côté JF, Dell’Amico M, Iori M (2014a) Combinatorial benders’ cuts for the strip packing problem. Oper Res 62(3):643–661

    Article  Google Scholar 

  • Côté JF, Gendreau M, Potvin JY (2014b) An exact algorithm for the two-dimensional orthogonal packing problem with unloading constraints. Oper Res 62(5):1126–1141

    Article  Google Scholar 

  • Crainic TG, Perboli G, Tadei R (2008) Extreme point-based heuristics for three-dimensional bin packing. Informs J Comput 20(3):368–384

    Article  Google Scholar 

  • de Almeida A, Figueiredo MB (2010) A particular approach for the three-dimensional packing problem with additional constraints. Comput Oper Res 37(11):1968–1976

    Article  Google Scholar 

  • Fayard D, Zissimopoulos V (1995) An approximation algorithm for solving unconstrained two-dimensional knapsack problems. Eur J Oper Res 84(3):618–632

    Article  Google Scholar 

  • Garey MR, Johnson DS (1979) Comput intractability: a guide to the theory of np-completeness. Freeman, San Francisco

    Google Scholar 

  • Gonçalves RF, Queiroz TA (2014) The knapsack problem with three practical constraints. Procedia Comput Sci 29:2192–2200

    Article  Google Scholar 

  • Herz JC (1972) Recursive computational procedure for two-dimensional stock cutting. IBM J Res Dev 16(5):462–469

    Article  Google Scholar 

  • Junqueira L, Morabito R, Yamashita DS (2012) Three-dimensional container loading models with cargo stability and load bearing constraints. Comput Oper Res 39(1):74–85

    Article  Google Scholar 

  • Leung SCH, Zhang D, Zhou C, Wu T (2012) A hybrid simulated annealing metaheuristic algorithm for the two-dimensional knapsack packing problem. Comput Oper Res 39(1):64–73

    Article  Google Scholar 

  • Lins L, Lins S, Morabito R (2003) An L-approach for packing (l, w)-rectangles into rectangular and L-shaped pieces. J Oper Res Soc 54(7):777–789

    Article  Google Scholar 

  • Martello S, Toth P (1990) Knapsack problems: algorithms and computer implementations. Wiley, Chichester

    Google Scholar 

  • Martello S, Pisinger D, Vigo D (2000) The three-dimensional bin packing problem. Oper Res 48(2):256–267

    Article  Google Scholar 

  • Martello S, Monaci M, Vigo D (2003) An exact approach to the strip-packing problem. INFORMS J Compu 15(3):310–319

    Article  Google Scholar 

  • Martinovic J, Hähnel M, Scheithauer G, Dargie W, Fischer A (2018) Cutting stock problems with nondeterministic item lengths: a new approach to server consolidation. 4OR 17(2):173–200

    Article  Google Scholar 

  • Morabito R, Arenales M (1994) An and/or-graph approach to the container loading problem. Int Trans Oper Res 1(1):59–73

    Google Scholar 

  • Queiroz TA, Miyazawa FK, Wakabayashi Y, Xavier EC (2012) Algorithms for 3D guillotine cutting problems: unbounded knapsack, cutting stock and strip packing. Comput Oper Res 39(2):200–212

    Article  Google Scholar 

  • Queiroz TA, Miyazawa FK, Wakabayashi Y (2015) On the L-approach for generating unconstrained two-dimensional non-guillotine cutting patterns. 4OR 13(2):199–219

    Article  Google Scholar 

  • Queiroz TA, Hokama PHDB, Schouery RCS, Miyazawa FK (2017) Two-dimensional disjunctively constrained knapsack problem: heuristic and exact approaches. Comput Ind Eng 105:313–328

    Article  Google Scholar 

  • Scheithauer G, Terno J (1996) The G4-heuristic for the pallet loading problem. J Oper Res Soc 47(4):511–522

    Article  Google Scholar 

  • Terno J, Lindemann R, Scheithauer G (1987) Zuschnittprobleme and ihre praktische lösung. Technical report, Verlag Harry Deutsch, Thun und FrankfurtMain

  • Valle AMD, Queiroz TA, Miyazawa FK, Xavier EC (2012) Heuristics for two-dimensional knapsack and cutting stock problems with items of irregular shape. Expert Syst Appl 39(16):12,589–12,598

    Article  Google Scholar 

  • Wäscher G, Haußner H, Schumann H (2007) An improved typology of cutting and packing problems. Eur J Oper Res 183(3):1109–1130

    Article  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Counsel of Technological and Scientific Development (CNPq Grant Number 308312/2016-3) and the State of Goiás Research Foundation (FAPEG).

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  1. Institute of Mathematics and Technology, Federal University of Goiás - Campus Catalão, Catalão, GO, 75704-020, Brazil

    Jéssica Gabriela de Almeida Cunha, Vinícius Loti de Lima & Thiago Alves de Queiroz

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  1. Jéssica Gabriela de Almeida Cunha
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  2. Vinícius Loti de Lima
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  3. Thiago Alves de Queiroz
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Correspondence to Thiago Alves de Queiroz.

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de Almeida Cunha, J.G., de Lima, V.L. & de Queiroz, T.A. Grids for cutting and packing problems: a study in the 2D knapsack problem. 4OR-Q J Oper Res 18, 293–339 (2020). https://doi.org/10.1007/s10288-019-00419-9

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