Trade-off between robustness and cost for a storage loading problem: rule-based scenario generation

Abstract

Integrating uncertainties into the optimization process is crucial to obtain solutions suitable for practical needs. In particular, the considered uncertainty set has a huge impact on the quality of the computed solutions. In this paper, we consider a storage loading problem in which a set of items must be loaded into a partly filled storage area, regarding stacking constraints and taking into account stochastic data of items arriving later. We propose a robust optimization approach dealing with the stochastic uncertainty. With a focus on constructing the uncertainty set, we offer a rule-based scenario generation approach to derive such a set from the stochastic data. To evaluate the robustness of stacking solutions, we introduce the concept of a security level, which is the probability that a stacking solution is feasible when the data of the uncertain items are realized. Computational results for randomly generated problem instances are presented showing the impact of various factors on the trade-off between robustness and cost of the stacking solutions.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

References

  1. Artzner P, Delbaen F, Eber J, Heath D (1999) Coherent measures of risk. Math Finance 9(3):203–228

    Article  Google Scholar 

  2. Ben-Tal A, Nemirovski A (1998) Robust convex optimization. Math Oper Res 23(4):769–805

    Article  Google Scholar 

  3. Ben-Tal A, Nemirovski A (1999) Robust solutions of uncertain linear programs. Oper Res Lett 25(1):1–13

    Article  Google Scholar 

  4. Ben-Tal A, Nemirovski A (2000) Robust solutions of linear programming contaminated with uncertain data. Math Program 88(3):411–424

    Article  Google Scholar 

  5. Ben-Tal A, Nemirovski A (2002) Robust optimization-methodology and applications. Math Program B 92:453–480

    Article  Google Scholar 

  6. Ben-Tal A, El Ghaoui L, Nemirovski A (2009) Robust optimization. Princeton University Press, Princeton

    Book  Google Scholar 

  7. Bertsimas D, Brown DB (2009) Constructing uncertainty sets for robust linear optimization. Oper Res 57(6):1438–1495

    Article  Google Scholar 

  8. Bertsimas D, Sim M (2003) Robust discrete optimization and network flows. Math Program 98(1–3):49–71

    Article  Google Scholar 

  9. Bertsimas D, Brown DB, Caramanis C (2011) Theory and applications of robust optimization. SIAM Rev 53(3):464–501

    Article  Google Scholar 

  10. Boysen N, Emde S (2016) The parallel stack loading problem to minimize blockages. Eur J Oper Res 249(2):618–627

    Article  Google Scholar 

  11. Bruns F, Goerigk M, Knust S, Schöbel A (2014) Robust load planning of trains in intermodal transportation. OR Spectr 36(3):631–668

    Article  Google Scholar 

  12. Bruns F, Knust S, Shakhlevich NV (2016) Complexity results for storage loading problems with stacking constraints. Eur J Oper Res 249(3):1074–1081

    Article  Google Scholar 

  13. Dekker R, Voogd P, van Asperen E (2006) Advanced methods for container stacking. OR Spectr 28(4):563–586

    Article  Google Scholar 

  14. Everitt BS, Skrondal A (2010) The Cambridge dictionary of statistics, 4th edn. Cambridge University Press, Cambridge

    Book  Google Scholar 

  15. ExoAnalytics Inc. Extreme optimization, complexity made simple. http://www.extremeoptimization.com/Documentation/Statistics/Default.aspx

  16. Forbes C, Evans M, Hastings N, Peacock B (2011) Statistical distributions, 4th edn. Wiley, Hoboken

    Google Scholar 

  17. Gharehgozli AH, Yu Y, de Koster R, Udding JT (2014) A decision-tree stacking heuristic minimising the expected number of reshuffles at a container terminal. Int J Prod Res 52(9):2592–2611

    Article  Google Scholar 

  18. Goerigk M, Knust S, Le XT (2016) Robust storage loading problems with stacking and payload constraints. Eur J Oper Res 253(1):51–67

    Article  Google Scholar 

  19. Jacod J, Protter P (2004) Probability essentials. Springer, Berlin

    Book  Google Scholar 

  20. Kall P, Wallace SW (1994) Stochastic programming. Wiley, Chichester

    Google Scholar 

  21. Kang J, Ryu KR, Kim KH (2006) Deriving stacking strategies for export containers with uncertain weight information. J Intell Manuf 17(4):399–410

    Article  Google Scholar 

  22. Kim KH, Park YM, Ryu KR (2000) Deriving decision rules to locate export containers in container yards. Eur J Oper Res 124(1):89–101

    Article  Google Scholar 

  23. Koch T (2004) Rapid mathematical programming. Ph.D. thesis, Technische Universität Berlin

  24. Kouvelis P, Yu G (1997) Robust discrete optimization and its applications. Kluwer Academic Publishers, Dordrecht

    Book  Google Scholar 

  25. Le XT, Knust S (2017) MIP-based approaches for robust storage loading problems with stacking constraints. Comput Oper Res 78:138–153

    Article  Google Scholar 

  26. Lehnfeld J, Knust S (2014) Loading, unloading and premarshalling of stacks in storage areas: survey and classification. Eur J Oper Res 239(2):297–312

    Article  Google Scholar 

  27. Li M, Gabriel SA, Shim Y, Azarm S (2011) Interval uncertainty-based robust optimization for convex and non-convex quadratic programs with applications in network infrastructure planning. Netw Spatial Econ 11(1):159–191

    Article  Google Scholar 

  28. Steenken D, Winter T, Zimmermann U (2001) Stowage and transport optimization in ship planning. In: Online optimization of large scale systems. Springer, pp 731–745

  29. Trivikram D, Goerigk M (2017) An experimental comparison of uncertainty sets for robust shortest path problems. In: Proceedings of the 17th workshop on algorithmic approaches for transportation modelling, optimization, and systems (ATMOS2017)

Download references

Acknowledgements

The authors would like to thank two anonymous referees for their constructive comments. This work was partially supported by the Deutsche Forschungsgemeinschaft (DFG, Germany) under Grant Number GRK 1916/1, and the National Foundation for Science and Technology Development (NAFOSTED, Vietnam) under Grant Number 101.01-2017.315.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Sigrid Knust.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Büsing, C., Knust, S. & Le, X.T. Trade-off between robustness and cost for a storage loading problem: rule-based scenario generation. EURO J Comput Optim 6, 339–365 (2018). https://doi.org/10.1007/s13675-018-0094-x

Download citation

Keywords

  • Robust optimization
  • Stochastic uncertainty
  • Interval uncertainty
  • Storage loading
  • Stacking constraints

Mathematics Subject Classification

  • 90B06
  • 90C31