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A fast heuristic for a three-dimensional non-convex domain loading problem

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Abstract

In this paper we tackle a three-dimensional non-convex domain loading problem. We have to efficiently load identical small boxes into a highly irregular non-convex domain. The boxes to be loaded have a particular shape. If d is the length of the smallest edge of the box, its dimensions are d × nd × md, n ≤ m, with n and m integer values. This loading problem arises from an industrial design problem where it is necessary to obtain good solutions with very low computation time. We propose a fast heuristic based on an approximate representation of the non-convex domain in terms of cubes of dimension d and on the decomposition of the whole problem in several two-dimensional subproblems related to ‘planes’ of height d. The proposed heuristic shows good performances in terms of quality of solution and computation times. The results on several real test cases, coming from the industrial application, are shown.

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References

  • Althaus E, Baumann T, Schömer E, Werth K (2007) Trunk packing revisited. Lect Notes Comput Sci 4525: 420–432

    Article  Google Scholar 

  • Bischoff EE, Ratcliff MSW (1995) Issues in the development of approaches to container loading. Omega Int J Manag Sci 23(4): 377–390

    Article  Google Scholar 

  • Bischoff EE (2006) Three-dimensional packing of items with limited load bearing strength. Eur J Oper Res 168(3): 952–966

    Article  Google Scholar 

  • Bortfeldt A, Gehring H (1998) A tabu search algorithm for weakly heterogeneous container loading problems. OR Spectr 20: 237–250

    Article  Google Scholar 

  • Bortfeldt A, Gehring H (2001) A hybrid genetic algorithm for the container loading problem. Eur J Oper Res 131(1): 143–161

    Article  Google Scholar 

  • Bortfeldt A, Gehring H, Mack D (2003) A parallel tabu search algorithm for solving the container loading problem. Parallel Comput 29(4): 641–662

    Article  Google Scholar 

  • Chen CS, Lee SM, Shen QS (1995) An analytical model for the container loading problem. Eur J Oper Res 80(1): 68–76

    Article  Google Scholar 

  • Davies AP, Bischoff EE (1999) Weight distribution considerations in container loading. Eur J Oper Res 114(3): 509–527

    Article  Google Scholar 

  • Dychoff H (1990) A typology of cutting and packing problems. Eur J Oper Res 44(2): 145–159

    Article  Google Scholar 

  • Dyckhoff H, Scheithauer G, Terno J (1997) Cutting and packing (C&P). In: Dell’Amico M, Maffioli F, Martello S (eds) Annotated bibliographies in combinatorial optimization. Wiley, Chichester

    Google Scholar 

  • Egeblad J, Pisinger D (2009) Heuristic approaches for the two and three-dimensional knapsack packing problem. Comput Oper Res 36: 1026–1049

    Article  Google Scholar 

  • Eisenbrand F, Funke S, Reichel J, Schömer E (2003) Packing a trunk. In: Di Battista G, Zwick U (eds) ESA 2003, vol 2832. LNCS. Springer, Heidelberg, pp 618–629

    Google Scholar 

  • Eisenbrand F, Funke S, Karrenbauer A, Reichel J, Schömer E (2005) Packing a trunk: now with a twist! In: SPM 05: proceedings of the 2005 ACM symposium on solid and physical modeling. ACM Press, New York, pp 197–206

  • Eley M (2002) Solving container loading problems by block arrangement. Eur J Oper Res 141(2): 393–409

    Article  Google Scholar 

  • Fasano G (1999) Cargo analytical integration in space engineering: a three dimensional packing model. In: Ciriani T, Gliozzi S, Johnson EL (eds) Operations research in industry. Macmillan, New York, pp 232–246

    Google Scholar 

  • Fasano G (2003) MIP models for solving three-dimensional packing problems arising in space engineering. In: Ciriani T, Fasano G, Gliozzi S, Johnson EL (eds) Operations research in space and air. Kluwer, Boston, pp 43–56

    Google Scholar 

  • Fasano G (2004) A MIP approach for some practical packing problems: balancing constraints and tetris like items. 4OR 2(2): 161–174

    Article  Google Scholar 

  • Fasano G (2008) MIP-based heuristic for non-standard 3D-packing problems. 4OR 6(3): 291–310

    Article  Google Scholar 

  • Fekete S, Schepers J (2004) A combinatorial characterization of higher-dimensional orthogonal packing. Math Oper Res 29(2): 353–368

    Article  Google Scholar 

  • Fekete S, Schepers J (2007) An exact algorithm for higher-dimensional orthogonal packing. Oper Res (in press)

  • Gehring H, Bortfeldt A (1997) A genetic algorithm for solving the container loading problem. Int Trans Oper Res 4: 401–418

    Article  Google Scholar 

  • Gehring H, Bortfeldt A (2002) A parallel genetic algorithm for solving the container loading problem. Int Trans Oper Res 9(4): 497–511

    Article  Google Scholar 

  • George JA, Robinson DF (1980) A heuristic for packing boxes into a container. Comput Oper Res 7(3): 147–156

    Article  Google Scholar 

  • George JA (1992) A method for solving container packing for a single size of box. J Oper Res Soc 43(4): 307–312

    Google Scholar 

  • Huang W, He K (2009) A new heuristic algorithm for cuboids packing with no orientation constraints. Comput Oper Res 36(2): 425–432

    Article  Google Scholar 

  • Martello S, Pisinger D, Vigo D, den Boef E, Korst J (2007) Algorithms for general and robot-packable variants of the three-dimensional bin packing problem. ACM Trans Math Softw 33(1): article 7

    Google Scholar 

  • Moura A, Oliveira JF (2005) A GRASP approach to the container-loading problem. IEEE Intell Syst 20: 50–57

    Article  Google Scholar 

  • Padberg M (2000) Packing small boxes into a big box. Math Methods Oper Res 52: 1–21

    Article  Google Scholar 

  • Parreño F, Alvarez-Valdes R, Oliveira JF, Tamarit JM (2008a) A maximal-space algorithm for the container loading problem. INFORMS J Comput 20(3): 412–422

    Article  Google Scholar 

  • Parreño F, Alvarez-Valdes R, Oliveira JF, Tamarit JM (2008b) Neighborhood structures for the container loading problem: a VNS implementation. J Heuristics. doi:10.1007/s10732-008-9081-3

  • Pisinger D (2002) Heuristics for the container loading problem. Eur J Oper Res 141(2): 382–392

    Article  Google Scholar 

  • Scheithauer G, Stoyan YG, Romanova T (2005) Mathematical modeling of interactions of primary 3D objects. Cybern Syst Anal 41(3): 332–342

    Article  Google Scholar 

  • Stoyan YG (1980) On one generalization of the dense allocation function. In: Reports Ukrainian SSR Academy of Science Series A, vol 8. pp 70–74

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

    Article  Google Scholar 

  • Stoyan Y, Scheithauer G, Gil N, Romanova T (2004) Φ-functions for complex 2D-objects. 4OR 2: 69–84

    Google Scholar 

  • Stoyan Y, Terno J, Scheithauer G, Gil N, Romanova T (2002) Φ-functions for primary 2D-objects. Studia Inform Univ 2(1): 1–32

    Google Scholar 

  • Terno J, Scheithauer G, Sommerweiss U, Riehme J (2000) An efficient approach for the multi-pallet loading problem. Eur J Oper Res 123(2): 372–381

    Article  Google Scholar 

  • Wang Z, Kevin WL, Zhang X (2008) A heuristic for the container loading problem: a tertiary-tree-based dynamic space decomposition approach. Eur J Oper Res 191(1): 86–99

    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 

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Authors and Affiliations

  1. Dipartimento di Ingegneria, Universitá del Sannio, Corso Garibaldi 104, 82100, Benevento, Italy

    Maurizio Boccia

  2. ELASIS S.C.p.A., FIAT Group, Via Ex Aeroporto, 80038, Pomigliano d’Arco (Napoli), Italy

    Serena di Muro & Francesco Mosca

  3. Dipartimento di Informatica e Sistemistica, Universitá degli Studi di Napoli “Federico II”, Via Claudio 21, 80125, Napoli, Italy

    Antonio Sforza & Claudio Sterle

Authors
  1. Maurizio Boccia
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  2. Serena di Muro
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  3. Francesco Mosca
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  4. Antonio Sforza
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  5. Claudio Sterle
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Correspondence to Antonio Sforza.

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Boccia, M., di Muro, S., Mosca, F. et al. A fast heuristic for a three-dimensional non-convex domain loading problem. 4OR-Q J Oper Res 9, 83–101 (2011). https://doi.org/10.1007/s10288-010-0133-9

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  • DOI: https://doi.org/10.1007/s10288-010-0133-9

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