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Tree search for the stacking problem

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Abstract

The stacking problem is a hard combinatorial optimization problem with high practical interest in, for example, steel storage or container port operations. In this problem, a set of items is stored in a warehouse for a period of time, and a crane is used to place them in a limited number of stacks. Since the entrance and exit of items occurs in an arbitrary order, items may have to be relocated in order to reach and deliver other items below them. The objective of the problem is to find a feasible sequence of movements that delivers all items, while minimizing the total number of movements.

We study the scalability of an exact approach to this problem, and propose two heuristic methods to solve it approximately. The two heuristic approaches are a multiple simulation algorithm using semi-greedy construction heuristics, and a stochastic best-first tree search algorithm. The two methods are compared in a set of challenging instances, revealing a superior performance of the tree search approach in most cases.

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References

  • Avriel, M., Penn, M., Shpirer, N., & Witteboon, S. (1998). Stowage planning for container ships to reduce the number of shifts. Annals of Operations Research, 76, 55–71.

    Article  Google Scholar 

  • Avriel, M., Penn, M., & Shpirer, N. (2000). Container ship stowage problem: complexity and connection to the coloring of circle graphs. Discrete Applied Mathematics, 103(1–3), 271–279.

    Article  Google Scholar 

  • Caserta, M., Voß, S., & Sniedovich, M. (2011). Applying the corridor method to a blocks relocation problem. OR Spectrum, 33, 915–929.

    Article  Google Scholar 

  • Dekker, R., Voogd, P., & Asperen, E. (2007). Advanced methods for container stacking. In K. H. Kim & H.-O. Günther (Eds.), Container terminals and cargo systems (pp. 131–154). Berlin: Springer.

    Chapter  Google Scholar 

  • Gavril, F. (1973). Algorithms for a maximum clique and a maximum independent set of a circle graph. Networks, 3, 261–273.

    Article  Google Scholar 

  • Golumbic, M. C. (2004). Annals of discrete mathematics: Algorithmic graph theory and perfect graphs. Amsterdam: Elsevier.

    Google Scholar 

  • Hartmann, S. (2004). A general framework for scheduling equipment and manpower at container terminals. OR Spectrum, 26, 51–74.

    Article  Google Scholar 

  • Kim, K. H., & Hong, G.-P. (2006). A heuristic rule for relocating blocks. Computers and Operations Research, 33, 940–954.

    Article  Google Scholar 

  • Law, A. M. (2006). Simulation modeling and analysis (4th ed.). New York: McGraw-Hill.

    Google Scholar 

  • Pedroso, J. P., & Kubo, M. (2010). Heuristics and exact methods for number partitioning. European Journal of Operational Research, 202, 73–81.

    Article  Google Scholar 

  • Pochet, Y., & Wolsey, L. A. (2006). Production planning by mixed integer programming. Berlin: Springer.

    Google Scholar 

  • Rei, R. J., Kubo, M., & Pedroso, J. P. (2008). Simulation-based optimization for steel stacking. In H. A. Le Thi, P. Bouvry, & T. Pham Dinh (Eds.), Communications in computer and information science: Vol. 14. Modelling, computation and optimization in information systems and management sciences (pp. 254–263). Berlin: Springer.

    Chapter  Google Scholar 

  • Rei, R. J., & Pedroso, J. P. (2009). Heuristic search for the stacking problem. In A. Viana et al. (Eds.), EUMe2009, The European chapter on metaheuristics’ workshop on debating the future: new areas of application and innovative approaches, Porto, Portugal (pp. 109–114).

    Google Scholar 

  • Rei, R. J., & Pedroso, J. P. (2012). Stacking problem instances and instance generator. Internet repository, version 1.0. http://www.dcc.fc.up.pt/~jpp/code/stacking.

  • Ruml, W. (2001). Stochastic tree search: where to put the randomness? In H. H. Hoos & T. G. Stützle (Eds.), Proceedings of the IJCAI-01 workshop on stochastic search (pp. 43–47).

    Google Scholar 

  • Ruml, W. (2002). Adaptive tree search. PhD thesis. Cambridge: Harvard University.

  • Unger, W. (1988). On the k-colouring of circle-graphs. In R. Cori & M. Wirsing (Eds.), Lecture notes in computer science: Vol. 294. STACS 88 (pp. 61–72). Berlin: Springer.

    Chapter  Google Scholar 

  • Unger, W. (1992). The complexity of colouring circle graphs. In A. Finkel & M. Jantzen (Eds.), Lecture notes in computer science: Vol. 577. STACS 92 (pp. 389–400). Berlin: Springer.

    Chapter  Google Scholar 

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Acknowledgements

We would like to thank Prof. Mikio Kubo, from the Tokyo University of Marine Science and Technology, Japan, for his important contributions. We would also like to thank three anonymous reviewers for their constructive comments on previous versions of this paper. The work was partially funded by PhD grant SFRH/BD/66075/2009 from the Portuguese Foundation for Science and Technology (FCT).

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Authors and Affiliations

  1. INESC Porto and Faculdade de Ciências, Universidade do Porto, Rua do Campo Alegre, 4169-007, Porto, Portugal

    Rui Rei & João Pedro Pedroso

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  1. Rui Rei
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  2. João Pedro Pedroso
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Correspondence to João Pedro Pedroso.

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Rei, R., Pedroso, J.P. Tree search for the stacking problem. Ann Oper Res 203, 371–388 (2013). https://doi.org/10.1007/s10479-012-1186-2

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