Skip to main content
SpringerLink
Log in

Tetris-like Items

  • Chapter
  • First Online:
Solving Non-standard Packing Problems by Global Optimization and Heuristics

Part of the book series: SpringerBriefs in Optimization ((BRIEFSOPTI))

Abstract

The packing of tetris-like items, i.e. clusters of mutually orthogonal rectangular parallelepipeds, inside a given domain, is discussed here. Orthogonal rotations are admitted and additional conditions can be present. An MIP formulation is described at quite a detailed level.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • Addis, B., Locatelli, M., Schoen, F.: Efficiently packing unequal disks in a circle: a computational approach which exploits the continuous and combinatorial structure of the problem. Oper. Res. Lett. 36(1), 37–42 (2008a)

    Article  MATH  MathSciNet  Google Scholar 

  • Addis, B., Locatelli, M., Schoen, F.: Disk packing in a square: a new global optimization approach. INFORMS J. Comput. 20(4), 516–524 (2008b)

    Article  MATH  MathSciNet  Google Scholar 

  • Allen, S.D., Burke, E.K., Mareček, J.: A space-indexed formulation of packing boxes into a larger box. Oper. Res. Lett. 40(1), 20–24 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  • Beasley, J.E.: An exact two-dimensional non-guillotine cutting tree search procedure. Oper. Res. 33(1), 49–64 (1985)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  • Bortfeldt, A., Wäscher, G.: Container loading problems—a state-of-the-art review. FEMM working papers 120007, Otto-von-Guericke University Magdeburg, Faculty of Economics and Management, Magdeburg (2012)

    Google Scholar 

  • Castillo, I., Kampas, F.J., Pintér, J.D.: Solving circle packing problems by global optimization: numerical results and industrial applications. Eur. J. Oper. Res. 191(3), 786–802 (2008)

    Article  MATH  Google Scholar 

  • Chen, C.S., Lee, S.M., Shen, Q.S.: An analytical model for the container loading problem. Eur. J. Oper. Res. 80, 68–76 (1995)

    Article  MATH  Google Scholar 

  • Christensen, S.G., Rousøe, D.M.: Container loading with multi-drop constraints. Int. Trans. Oper. Res. 16(6), 727–743 (2009)

    Article  MATH  Google Scholar 

  • Egeblad, J.: Placement of two- and three-dimensional irregular shapes for inertia moment and balance. In: Morabito, R., Arenales, M.N., Yanasse, H.H. (eds.) Int. Trans. Oper. Res. (Special Issue on Cutting, Packing and Related Problems) 16(6), 789–807 (2009)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Fasano, G.: Satellite Optimal Layout. Application of Mathematical and Optimization Techniques. IBM Europe Institute, Garmisch-Partenkirchen, Germany (1989)

    Google Scholar 

  • Fasano, G.: Cargo analytical integration in space engineering: a three-dimensional packing model. In: Ciriani, T., Gliozzi, S., Johnson, E.L. (eds.) Operations Research in Industry, pp. 232–246. Macmillan, London (1999)

    Google Scholar 

  • Fasano, G.: MIP models for solving 3-dimensional packing problems arising in space engineering. In: Ciriani, T., Fasano, G., Gliozzi, S., Tadei, R. (eds.) Operations Research in Space and Air, pp. 43–56. Kluwer Academic Publisher, Boston (2003)

    Chapter  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Gensane, T.: Dense packing of equal spheres in a cube. Electron. J. Combinator. 11(1), 1–17 (2004)

    MathSciNet  Google Scholar 

  • Hadjiconstantinou, E., Christofides, N.: An exact algorithm for general, orthogonal, two-dimensional knapsack problems. Eur. J. Oper. Res. 83(1), 39–56 (1995)

    Article  MATH  Google Scholar 

  • Junqueira, L., Morabito, R., Yamashita, D.S.: Mip-based approaches for the container loading problem with multi-drop constraints. Ann. Oper. Res. 199(1), 51–75 (2011)

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Junqueira, L., Morabito, R., Yamashita, D.S., Yanasse, H.H.: Optimization models for the three-dimensional container loading problem with practical constraints. In: Fasano, G., Pintér, J.D. (eds.) Modeling and Optimization in Space Engineering, pp. 271–294. Springer Science + Business Media, New York (2013)

    Google Scholar 

  • Kahng, A.B., Wang, Q.: Implementation and extensibility of an analytic placer. IEEE Trans. Comput. Aided Des. Integrated Circ. Syst. 24(5), 734–747 (2005)

    Article  Google Scholar 

  • Kampas, F.J., Pintér, J.D.: Packing Equal-Size Circles in a Triangle. http://library.wolfram.com/infocenter/TechNotes/6202 (2005). Accessed 30 Aug 2013

  • Kim, J.G., Kim, Y.D.: A linear programming-based algorithm for floorplanning in VLSI design. IEEE Trans. Comput. Aided Des. Integrated Circ. Syst. 22(5), 584–592 (2003)

    Article  Google Scholar 

  • Kleinhans, J.M., Sigl, G., Johannes, F.M., Antreich, K.J.: GORDIAN: VLSI placement by quadratic programming and slicing optimization. IEEE Trans. Comput. Aided Des. Integrated Circ. Syst. 10(3), 356–365 (1991)

    Article  Google Scholar 

  • Lai, K.K., Xue, J., Xu, B.: Container packing in a multi-customer delivering operation. Comput. Ind. Eng. 35(1–2), 323–326 (1998)

    Article  Google Scholar 

  • Limbourg, S., Schyns, M., Laporte, G.: Automatic aircraft cargo load planning. J. Oper. Res. Soc. 63, 1271–1283 (2012)

    Article  Google Scholar 

  • Locatelli, M., Raber, U.: Packing equal circles into a square: a deterministic global optimization approach. Discrete Appl. Math. 122, 139–166 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  • Minoux, M., Vajda, S.: Mathematical Programming: Theory and Algorithms. Wiley, London (1986)

    MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  • Moura, A., Oliveira, J.F.: A GRASP approach to the container-loading problem. IEEE Intell. Syst. 4(20), 50–57 (2005)

    Article  Google Scholar 

  • Padberg, M.W.: Packing small boxes into a big box. Office of Naval Research, N00014-327, New York University (1999)

    Google Scholar 

  • Pan, Y., Shi, L.: On the equivalence of the max-min transportation lower bound and the time-indexed lower bound for single-machine scheduling problems. Math. Programm Ser. A 110(3), 543–559 (2007)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Pisinger, D., Sigurd, M.: The two-dimensional bin packing problem with variable bin sizes and costs. Discrete Optim. 2(2), 154–167 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  • Ratcliff, M.S.W., Bischoff, E.E.: Allowing for weight considerations in container loading. OR Spektrum 20(1), 65–71 (1998)

    Article  MATH  Google Scholar 

  • Silva, J.L.C., Soma, N.Y., Maculan, N.: A greedy search for the three-dimensional bin packing problem: the packing static stability case. Int. Trans. Oper. Res. 10(2), 141–153 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  • Stoyan, Y.G., Yaskov, G.: Packing identical spheres into a rectangular parallelepiped. In: Bortfeldt, A., Homberger, J., Kopfer, H., Pankratz, G., Strangmeier, R. (eds.) Intelligent Decision Support. Current Challenges and Approaches, pp. 47–67. GWV Fachverlage GMbH, Wiesbaden, Germany (2008)

    Chapter  Google Scholar 

  • Stoyan, Y.G., Yaskov, G., Scheithauer, G.: Packing of various radii solid spheres into a parallelepiped. Cent. Eur. J. Oper. Res. 11(4), 389–407 (2003)

    MATH  MathSciNet  Google Scholar 

  • Sutou, A., Dai, Y.: Global optimization approach to unequal global optimization approach to unequal sphere packing problems in 3D. J. Optim. Theor. Appl. 114(3), 671–694 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  • Vielma, J.P., Nemhauser, G.L.: Modeling disjunctive constraints with a logarithmic number of binary variables and constraints. Math. Program. 128(12), 49–72 (2009)

    MathSciNet  Google Scholar 

  • Williams, H.P.: Model Building in Mathematical Programming. Wiley, London (1993)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Engineering and Advanced Studies, Thales Alenia Space, Turin, Italy

    Giorgio Fasano

Authors
  1. Giorgio Fasano
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Giorgio Fasano

About this chapter

Cite this chapter

Fasano, G. (2014). Tetris-like Items. In: Solving Non-standard Packing Problems by Global Optimization and Heuristics. SpringerBriefs in Optimization. Springer, Cham. https://doi.org/10.1007/978-3-319-05005-8_2

Download citation

Publish with us

Policies and ethics

Search

Navigation