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Cutting and Packing

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Handbook of Heuristics

Abstract

Cutting and Packing (C&P) problems arise in many industrial and logistics applications, whenever a set of small items, with different shapes, has to be assigned to large objects with specific shapes so as to optimize some objective function. Besides some characteristics common to combinatorial optimization problems, the distinctive feature of this field is the existence of a geometric subproblem, to ensure that the items do not overlap and are completely contained in the large objects. The geometric tools required to deal with this subproblem depend on the shapes (rectangles, circles, irregular) and on the specific conditions of the problem being solved. In this chapter, after an introduction that describes and classifies Cutting and Packing problems, we review the basic strategies that have appeared in the literature for designing constructive algorithms, local search procedures, and metaheuristics for problems with regular and irregular shapes.

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References

  1. Albano A, Sapuppo G (1980) Optimal allocation of two-dimensional irregular shapes using heuristic search methods. IEEE Trans Syst Man Cybern 10(5):242–248

    Google Scholar 

  2. Alonso M, Alvarez-Valdes R, Parreño F, Tamarit J (2014) A reactive GRASP algorithm for the container loading problem with load-bearing constraints. Eur J Ind Eng 8:669–694

    Google Scholar 

  3. Alvarez-Valdes R, Parajon A, Tamarit J (2002) A computational study of LP-based heuristic algorithms for the two-dimensional guillotine cutting stock problems. OR Spectr 24:179–192

    Google Scholar 

  4. Alvarez-Valdes R, Parajon A, Tamarit J (2002) A tabu search algorithm for large-scale guillotine (un)constrained two-dimensional cutting problems. Comput Oper Res 29:925–947

    Google Scholar 

  5. Alvarez-Valdes R, Parreño F, Tamarit J (2005) A tabu search algorithm for the pallet loading problem. OR Spectr 27:43–61

    Google Scholar 

  6. Alvarez-Valdes R, Parreño F, Tamarit J (2007) A tabu search algorithm for a two-dimensional non-guillotine cutting problem. Eur J Oper Res 183:1167–1182

    Google Scholar 

  7. Alvarez-Valdes R, Marti R, Parajon A, Tamarit J (2007) GRASP and path relinking for the two-dimensional two-stage cutting-stock problem. INFORMS J Comput 19:261–272

    Google Scholar 

  8. Alvarez-Valdes R, Parreño F, Tamarit J (2008) Reactive GRASP for the strip packing problem. Comput Oper Res 35:1065–1083

    Google Scholar 

  9. Alvarez-Valdes R, Parreño F, Tamarit J (2013) A GRASP/Path relinking algorithm for two- and three-dimensional multiple bin-size bin packing problems. Comput Oper Res 40: 3081–3090

    Google Scholar 

  10. Alvarez-Valdes R, Martinez A, Tamarit J (2013) A branch & bound algorithm for cutting and packing irregularly shaped pieces. Int J Prod Econ 145(2):463–477

    Google Scholar 

  11. Araujo O, Armentano V (2007) A multi-start random constructive heuristic for the container loading problem. Pesquisa Operacional 27(2):311–331

    Google Scholar 

  12. Arenales M, Morabito R (1995) An AND/OR-graph approach to the solution of two-dimensional non-guillotine cutting problems. Eur J Oper Res 84:599–617

    Google Scholar 

  13. Art RC (1966) An approch to the two dimensional, irregular cutting stock problem. IBM Cambridge Scientific Center Report, pp 1–35

    Google Scholar 

  14. Baker B, Coffman E, Rivest R (1980) Orthogonal packings in two dimensions. SIAM J Comput 9(4):846–855

    Article  MathSciNet  MATH  Google Scholar 

  15. Below G, Scheithauer G, Mukhacheva E (2008) One-dimensional heuristics adapted for two-dimensional rectangular strip packing. J Oper Res Soc 59:823–832

    Article  MATH  Google Scholar 

  16. Bennell JA, Dowsland KA (2001) Hybridising tabu search with optimisation techniques for irregular stock cutting. Manag Sci 47(8):1160–1172

    Article  Google Scholar 

  17. Bennell JA, Oliveira JF (2008) The geometry of nesting problems: a tutorial. Eur J Oper Res 184(2):397–415

    Article  MathSciNet  MATH  Google Scholar 

  18. Bennell JA, Song X (2010) A beam search implementation for the irregular shape packing problem. J Heuristics 16(2):167–188

    Article  MATH  Google Scholar 

  19. Bennell J, Lee L, Potts C (2013) A genetic algorithm for two-dimensional bin packing with due dates. Int J Prod Econ 145:547–560

    Article  Google Scholar 

  20. Bortfeldt A, Gehring H (2001) A hybrid algorithm for the container loading problem. Eur J Oper Res 131:143–161

    Article  MATH  Google Scholar 

  21. Burke E, Kendall G, Whitwell G (2004) A new placement heuristic for the orthogonal stock-cutting problem. Oper Res 52(4):655–671

    Article  MATH  Google Scholar 

  22. Burke E, Kendall G, Whitwell G (2009) A simulated annealing enhancement of the best-fit heuristic for the orthogonal stock-cutting problem. INFORMS J Comput 21(3):505–516

    Article  MATH  Google Scholar 

  23. Burke E, Hyde M, Kendall G (2011) A squeaky wheel optimization methodology for two-dimensional strip packing. Comput Oper Res 38:1035–1044

    Article  MATH  Google Scholar 

  24. Ceschia S, Schaerf A (2013) Local search for a multi-drop multi-container loading problem. J Heuristics 19:275–294

    Article  Google Scholar 

  25. Chazelle B (1983) The bottom-left bin-packing heuristic: an efficient implementation. IEEE Trans Comput 32(8):697–707

    Article  MATH  Google Scholar 

  26. Chen W, Zhai P, Zhu H, Zhang Y (2014) Hybrid algorithm for the two-dimensional rectangular layer-packing problem. J Oper Res Soc 65:1068–1077

    Article  Google Scholar 

  27. Cherri L, Toledo F, Carravilla MA (2016) A model-based heuristic for the irregular strip packing problem. Pesquisa Operacional 36(3):447–468

    Article  Google Scholar 

  28. Christensen S, Rousoe D (2009) Container loading with multi-drop constraints. Int Trans Oper Res 16:727–743

    Article  MATH  Google Scholar 

  29. da Silveira J, Miyazawa F, Xavier E (2013) Heuristics for the strip packing problem with unloading constraints. Comput Oper Res 40:991–1003

    Article  MathSciNet  MATH  Google Scholar 

  30. Dowsland KA, Dowsland WB, Bennell JA (1998) Jostling for position: local improvement for irregular cutting patterns. J Oper Res Soc 49:647–658

    Article  MATH  Google Scholar 

  31. Dowsland KA, Vaid S, Dowsland WB (2002) An algorithm for polygon placement using a bottom-left strategy. Eur J Oper Res 141(2):371–381

    Article  MathSciNet  MATH  Google Scholar 

  32. Dyckhoff H (1981) A new linear programming approach to the cutting stock problem. Oper Res 29:1092–1104

    Article  MathSciNet  MATH  Google Scholar 

  33. Egeblad J, Nielsen BK, Odgaard A (2007) Fast neighborhood search for two- and three-dimensional nesting problems. Eur J Oper Res 183(3):1249–1266

    Article  MathSciNet  MATH  Google Scholar 

  34. Eley M (2002) Solving container loading problems by block arrangement. Eur J Oper Res 141:393–409

    Article  MathSciNet  MATH  Google Scholar 

  35. Elkeran A (2013) A new approach for sheet nesting problem using guided cuckoo search and pairwise clustering. Eur J Oper Res 231(3):757–769

    Article  MathSciNet  MATH  Google Scholar 

  36. Fanslau T, Bortfeldt A (2010) A tree search algorithm for solving the container loading problem. INFORMS J Comput 22(2):222–235

    Article  MathSciNet  MATH  Google Scholar 

  37. Faroe O, Pisinger D, Zachariasen M (2003) Guided local search for the three-dimensional bin-packing problem. INFORMS J Comput 15(3):267–283

    Article  MathSciNet  MATH  Google Scholar 

  38. Fischetti M, Luzzi I (2009) Mixed-integer programming models for nesting problems. J Heuristics 15(3):201–226

    Article  MATH  Google Scholar 

  39. Gilmore P, Gomory R (1961) A linear programming approach to the cutting-stock problem. Oper Res 9:849–859

    Article  MathSciNet  MATH  Google Scholar 

  40. Gilmore P, Gomory R (1963) A linear programming approach to the cutting-stock problem – part II. Oper Res 11:863–888

    Article  MATH  Google Scholar 

  41. Gilmore P, Gomory R (1965) Multistage cutting stock problems of two and more dimensions. Oper Res 13:94–120

    Article  MATH  Google Scholar 

  42. Gomes AM, Oliveira JF (2002) A 2-exchange heuristic for nesting problems. Eur J Oper Res 141(2):359–370

    Article  MathSciNet  MATH  Google Scholar 

  43. Gomes AM, Oliveira JF (2006) Solving irregular strip packing problems by hybridising simulated annealing and linear programming. Eur J Oper Res 171(3):811–829

    Article  MATH  Google Scholar 

  44. Gonçalves J, Resende M (2013) A biased random key genetic algorithm for 2D and 3D bin packing problems. Int J Prod Econ 145:500–510

    Article  Google Scholar 

  45. Hifi M, M’Hallah R (2009) A literature review on circle and sphere packing problems: models and methodologies. Adv Oper Res 2009(150624):1–22

    MATH  Google Scholar 

  46. Hifi M, Michrafy M, Sbihi A (2004) Heuristic algorithms for the multiple-choice multidimensional knapsack problem. J Oper Res Soc 55:1323–1332

    Article  MATH  Google Scholar 

  47. Hopper E, Turton B (2001) An empirical investigation of meta-heuristic and heuristic algorithms for a 2D packing problem. Eur J Oper Res 128:34–57

    Article  MATH  Google Scholar 

  48. Imamichi T, Yagiura M, Nagamochi H (2009) An iterated local search algorithm based on nonlinear programming for the irregular strip packing problem. Discret Optim 6(4):345–361

    Article  MathSciNet  MATH  Google Scholar 

  49. Iori M, Martello S, Monaci M (2003) Metaheuristic algorithms for the strip packing problem. In: Pardalos PM, Korotkikh V (eds) Optimization in industry: new frontiers. Kluwer Academic Publishers, Dordrecht, pp 159–179

    Chapter  Google Scholar 

  50. Jakobs S (1996) On genetic algorithms for the packing of polygons. Eur J Oper Res 88: 165–181

    Article  MATH  Google Scholar 

  51. Lai K, Chan J (1997) Developing a simulated annealing algorithm for the cutting stock problem. Comput Ind Eng 32:115–127

    Article  Google Scholar 

  52. Lesh N, Mitzenmacher M (2006) Bubblesearch: a simple heuristic for improving priority-based greedy algorithms. Inf Process Lett 97(4):161–169

    Article  MathSciNet  MATH  Google Scholar 

  53. Lesh N, Marks J, McMahon A, Mitzenmacher M (2003) New heuristic and interactive approaches to 2D rectangular strip packing. Technical report TR2003-18. Mitsubishi Electric Research Laboratories, Cambridge

    Google Scholar 

  54. Leung T, Chan C, Troutt M (2003) Application of a mixed simulated annealing-genetic algorithm heuristic for the two-dimensional orthogonal packing problem. Eur J Oper Res 145:530–542

    Article  MathSciNet  MATH  Google Scholar 

  55. Leung S, Zhang D, Sim K (2011) A hybrid simulated annealing metaheuristic algorithm for the two-dimensional knapsack packing problem. Eur J Oper Res 215:57–69

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  57. Leung SC, Lin Y, Zhang D (2012) Extended local search algorithm based on nonlinear programming for two-dimensional irregular strip packing problem. Comput Oper Res 39(3):678–686

    Article  MathSciNet  MATH  Google Scholar 

  58. Liu D, Teng H (1999) An improved BL-algorithm for genetic algorithm of the orthogonal packing of rectangles. Eur J Oper Res 112:413–420

    Article  MATH  Google Scholar 

  59. Liu D, Tan K, Huang S, Goh C, Ho W (2008) On solving multiobjective bin packing problems using evolutionary particle swarm optimization. Eur J Oper Res 190:357–382

    Article  MathSciNet  MATH  Google Scholar 

  60. Lodi A, Martello S, Vigo D (2004) TSpack: a unified tabu search code for multi-dimensional bin packing problems. Ann Oper Res 131:203–213

    Article  MathSciNet  MATH  Google Scholar 

  61. Martinez Sykora A (2013) Nesting problems: exact and heuristic algorithms. Phd thesis, Univesity of Valencia

    Google Scholar 

  62. Martinez-Sykora A, Alvarez-Valdes R, Bennell J, Tamarit JM (2015) Constructive procedures to solve 2-dimensional bin packing problems with irregular pieces and guillotine cuts. Omega 52:15–32

    Article  Google Scholar 

  63. Milenkovic VJ, Daniels K (1999) Translational polygon containment and minimal enclosure using mathematical programming. Int Trans Oper Res 6:525–554

    Article  MathSciNet  Google Scholar 

  64. Moura A, Oliveira JF (2008) An integrated approach to the vehicle routing and container loading problems. OR Spect 31(4):775–800

    Article  MathSciNet  MATH  Google Scholar 

  65. Mukhacheva E, Belov G, Kartak V, Mukhacheva A (2000) Linear one-dimensional cutting-packing problems: numerical experiments with sequential value correction method (SVC) and a modified branch-and-bound method (MBB). Pesquisa Operacional 20:153–168

    Article  MATH  Google Scholar 

  66. Oliveira JF, Ferreira JS (1993) Algorithms for nesting problems. In: Vidal RVV (ed) Applied simulated annealing. Lecture notes in economics and mathematical systems. Springer, Berlin, pp 256–279

    Google Scholar 

  67. Oliveira JF, Gomes AM, Ferreira JS (2000) TOPOS – a new constructive algorithm for nesting problems. OR Spektr 22(2):263–284

    Article  MathSciNet  MATH  Google Scholar 

  68. Parreño F, Alvarez-Valdes R, Oliveira J, Tamarit J (2010) Neighbourhood structures for the container loading problem: a VNS implementation. J Heuristics 16(1):1–22

    Article  MATH  Google Scholar 

  69. Pisinger D (2002) Heuristics for the container loading problem. Eur J Oper Res 141:382–392

    Article  MathSciNet  MATH  Google Scholar 

  70. Pureza V, Morabito R (2006) Some experiments with a simple tabu search algorithm for the manufacturers pallet loading problem. Comput Oper Res 33:804–819

    Article  MATH  Google Scholar 

  71. Ren J, Tian Y, Sawaragi T (2011) A tree search method for the container loading problem with shipment priority. Eur J Oper Res 214:526–535

    Article  MATH  Google Scholar 

  72. Riehme J, Scheithauer G, Terno J (1996) The solution of two-stage guillotine cutting stock problems having extremely varying order demands. Eur J Oper Res 91:543–552

    Article  MATH  Google Scholar 

  73. Sato AK, Martins TC, Tsuzuki MSG (2012) An algorithm for the strip packing problem using collision free region and exact fitting placement. Comput Aided Des 44(8):766–777

    Article  Google Scholar 

  74. Segenreich SA, Braga LMPF (1986) Optimal nesting of general plane figures: a Monte Carlo heuristical approach. Comput Graph 10(3):229–237

    Article  Google Scholar 

  75. Song X, Bennell Ja (2013) Column generation and sequential heuristic procedure for solving an irregular shape cutting stock problem. J Oper Res Soc 65(7):1037–1052

    Article  Google Scholar 

  76. Stadtler H (1990) A one-dimensional cutting stock problem in the aluminium industry and its solution. Eur J Oper Res 44:209–223

    Article  MATH  Google Scholar 

  77. Stoyan YG, Novozhilova MV, Kartashov AV (1996) Mathematical model and method of searching for a local extremum for the non-convex oriented polygons allocation problem. Eur J Oper Res 92(1):193–210

    Article  MATH  Google Scholar 

  78. Takahara S, Kusumoto Y, Miyamoto S (2003) Solution for textile nesting problems using adaptive meta-heuristics and grouping. Soft Comput 7:154–159

    Article  MATH  Google Scholar 

  79. Toledo FMB, Carravilla MA, Ribeiro C, Oliveira JF, Gomes AM (2013) The dotted-board model: a new MIP model for nesting irregular shapes. Int J Prod Econ 145:478–487

    Article  Google Scholar 

  80. Umetani S, Yagiura M, Imahori S, Imamichi T, Nonobe K, Ibaraki T (2009) Solving the irregular strip packing problem via guided local search for overlap minimization. Int Trans Oper Res 16(6):661–683

    Article  MathSciNet  MATH  Google Scholar 

  81. Valerio de Carvalho J (2002) LP models for bin packing and cutting stock problems. Eur J Oper Res 141:253–273

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  83. Wei L, Lim A (2015) A bidirectional building approach for the 2D constrained guillotine knapsack packing problem. Eur J Oper Res 242:63–71

    Article  MathSciNet  MATH  Google Scholar 

  84. Wei L, Oon W, Zhu W, Lim A (2011) A skyline heuristic for the 2D rectangular packing and strip packing problems. Eur J Oper Res 215:337–346

    MathSciNet  MATH  Google Scholar 

  85. Wu Y, Huang W, Lau S, Wong C, Young G (2002) An effective quasi-human based heuristic for solving the rectangle packing problem. Eur J Oper Res 141:341–358

    Article  MathSciNet  MATH  Google Scholar 

  86. Zhu W, Oon W, Lim A, Weng Y (2012) The six elements to block-building approaches for the single container loading problem. Appl Intell 37:431–445

    Article  Google Scholar 

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Author information

Authors and Affiliations

  1. Universitat de València, Burjassot, Spain

    Ramón Alvarez-Valdes

  2. INESC TEC and Faculty of Engineering, University of Porto, Porto, Portugal

    Maria Antónia Carravilla & José Fernando Oliveira

Authors
  1. Ramón Alvarez-Valdes
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  2. Maria Antónia Carravilla
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  3. José Fernando Oliveira
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Corresponding author

Correspondence to Ramón Alvarez-Valdes .

Editor information

Editors and Affiliations

  1. Department of Statistics and Operations, University of Valencia, Valencia, Spain

    Rafael Martí

  2. University of Florida Industrial and Systems Engineering, Gainesville, Florida, USA

    Pardalos Panos

  3. Modeling and Optimization Group, Amazon (United States) , Seattle, Washington, USA

    Mauricio G. C. Resende

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Alvarez-Valdes, R., Carravilla, M.A., Oliveira, J.F. (2018). Cutting and Packing. In: Martí, R., Panos, P., Resende, M. (eds) Handbook of Heuristics. Springer, Cham. https://doi.org/10.1007/978-3-319-07153-4_43-1

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  • DOI: https://doi.org/10.1007/978-3-319-07153-4_43-1

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  • Online ISBN: 978-3-319-07153-4

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