Don’t Rock the Boat: Algorithms for Balanced Dynamic Loading and Unloading

  • Sándor P. Fekete
  • Sven von Höveling
  • Joseph S. B. Mitchell
  • Christian Rieck
  • Christian Scheffer
  • Arne Schmidt
  • James R. Zuber
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10807)


We consider dynamic loading and unloading problems for heavy geometric objects. The challenge is to maintain balanced configurations at all times: minimize the maximal motion of the overall center of gravity. While this problem has been studied from an algorithmic point of view, previous work only focuses on balancing the final center of gravity; we give a variety of results for computing balanced loading and unloading schemes that minimize the maximal motion of the center of gravity during the entire process.

In particular, we consider the one-dimensional case and distinguish between loading and unloading. In the unloading variant, the positions of the intervals are given, and we search for an optimal unloading order of the intervals. We prove that the unloading variant is NP-complete and give a 2.7-approximation algorithm. In the loading variant, we have to compute both the positions of the intervals and their loading order. We give optimal approaches for several variants that model different loading scenarios that may arise, e.g., in the loading of a transport ship with containers.



We would like to thank anonymous reviewers for providing helpful comments and suggestions improving the presentation of this paper. J. Mitchell is partially supported by the National Science Foundation (CCF-1526406).


  1. 1.
    Amiouny, S.V., Bartholdi, J.J., Vate, J.H.V., Zhang, J.: Balanced loading. Oper. Res. 40(2), 238–246 (1992)CrossRefzbMATHGoogle Scholar
  2. 2.
    Bischoff, E.E., Marriott, M.D.: A comparative evaluation of heuristics for container loading. Eur. J. Oper. Res. 44(2), 267–276 (1990)CrossRefzbMATHGoogle Scholar
  3. 3.
    Bischoff, E.E., Ratcliff, M.: Issues in the development of approaches to container loading. Omega 23(4), 377–390 (1995)CrossRefGoogle Scholar
  4. 4.
    Christensen, S.G., Rousøe, D.M.: Container loading with multi-drop constraints. Int. Trans. Oper. Res. 16(6), 727–743 (2009)CrossRefzbMATHGoogle Scholar
  5. 5.
    Davies, A.P., Bischoff, E.E.: Weight distribution considerations in container loading. Eur. J. Oper. Res. 114(3), 509–527 (1999)CrossRefzbMATHGoogle Scholar
  6. 6.
    Fasano, G.: A MIP approach for some practical packing problems: balancing constraints and tetris-like items. 4OR 2(2), 161–174 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Fekete, S.P., von Höveling, S., Mitchell, J.S.B., Rieck, C., Scheffer, C., Schmidt, A., Zuber, J.R.: Don’t rock the boat: algorithms for balanced dynamic loading and unloading. CoRR, abs/1712.06498 (2017).
  8. 8.
    Gehring, H., Bortfeldt, A.: A genetic algorithm for solving the container loading problem. Int. Trans. Oper. Res. 4(5–6), 401–418 (1997)CrossRefzbMATHGoogle Scholar
  9. 9.
    Gehring, H., Menschner, K., Meyer, M.: A computer-based heuristic for packing pooled shipment containers. Eur. J. Oper. Res. 44(2), 277–288 (1990)CrossRefzbMATHGoogle Scholar
  10. 10.
    Gilmore, P., Gomory, R.E.: Multistage cutting stock problems of two and more dimensions. Oper. Res. 13(1), 94–120 (1965)CrossRefzbMATHGoogle Scholar
  11. 11.
    Limbourg, S., Schyns, M., Laporte, G.: Automatic aircraft cargo load planning. JORS 63(9), 1271–1283 (2012)CrossRefGoogle Scholar
  12. 12.
    Lurkin, V., Schyns, M.: The airline container loading problem with pickup and delivery. Eur. J. Oper. Res. 244(3), 955–965 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Mathur, K.: An integer-programming-based heuristic for the balanced loading problem. Oper. Res. Lett. 22(1), 19–25 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Mongeau, M., Bes, C.: Optimization of aircraft container loading. IEEE Trans. Aerosp. Electron. Syst. 39(1), 140–150 (2003)CrossRefGoogle Scholar
  15. 15.
    Øvstebø, B.O., Hvattum, L.M., Fagerholt, K.: Optimization of stowage plans for roro ships. Comput. Oper. Res. 38(10), 1425–1434 (2011)CrossRefzbMATHGoogle Scholar
  16. 16.
    Øvstebø, B.O., Hvattum, L.M., Fagerholt, K.: Routing and scheduling of roro ships with stowage constraints. Transp. Res. Part C: Emerg. Technol. 19(6), 1225–1242 (2011)CrossRefzbMATHGoogle Scholar
  17. 17.
    Paterson, M., Peres, Y., Thorup, M., Winkler, P., Zwick, U.: Maximum overhang. Am. Math. Mon. 116(9), 763–787 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Paterson, M., Zwick, U.: Overhang. Am. Math. Mon. 116(1), 19–44 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Pollaris, H., Braekers, K., Caris, A., Janssens, G.K., Limbourg, S.: Vehicle routing problems with loading constraints: state-of-the-art and future directions. OR Spectr. 37(2), 297–330 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Sevastianov, S.: On some geometric methods in scheduling theory: a survey. Discret. Appl. Math. 55(1), 59–82 (1994)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Sevastianov, S.: Nonstrict vector summationin multi-operation scheduling. Ann. Oper. Res. 83, 179–212 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Souffriau, W., Demeester, P., Berghe, G.V., De Causmaecker, P.: The aircraft weight and balance problem. Proc. ORBEL 22, 44–45 (1992)Google Scholar
  23. 23.
    Vancroonenburg, W., Verstichel, J., Tavernier, K., Berghe, G.V.: Automatic air cargo selection and weight balancing: a mixed integer programming approach. Transp. Res. Part E: Logist. Transp. Rev. 65, 70–83 (2014)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Computer ScienceTU BraunschweigBraunschweigGermany
  2. 2.Department of Applied Mathematics and StatisticsStony Brook UniversityStony BrookUSA

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