EURO Journal on Computational Optimization

, Volume 6, Issue 4, pp 339–365 | Cite as

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

  • Christina Büsing
  • Sigrid KnustEmail author
  • Xuan Thanh Le
Original Paper


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.


Robust optimization Stochastic uncertainty Interval uncertainty Storage loading Stacking constraints 

Mathematics Subject Classification

90B06 90C31 



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.


  1. Artzner P, Delbaen F, Eber J, Heath D (1999) Coherent measures of risk. Math Finance 9(3):203–228CrossRefGoogle Scholar
  2. Ben-Tal A, Nemirovski A (1998) Robust convex optimization. Math Oper Res 23(4):769–805CrossRefGoogle Scholar
  3. Ben-Tal A, Nemirovski A (1999) Robust solutions of uncertain linear programs. Oper Res Lett 25(1):1–13CrossRefGoogle Scholar
  4. Ben-Tal A, Nemirovski A (2000) Robust solutions of linear programming contaminated with uncertain data. Math Program 88(3):411–424CrossRefGoogle Scholar
  5. Ben-Tal A, Nemirovski A (2002) Robust optimization-methodology and applications. Math Program B 92:453–480CrossRefGoogle Scholar
  6. Ben-Tal A, El Ghaoui L, Nemirovski A (2009) Robust optimization. Princeton University Press, PrincetonCrossRefGoogle Scholar
  7. Bertsimas D, Brown DB (2009) Constructing uncertainty sets for robust linear optimization. Oper Res 57(6):1438–1495CrossRefGoogle Scholar
  8. Bertsimas D, Sim M (2003) Robust discrete optimization and network flows. Math Program 98(1–3):49–71CrossRefGoogle Scholar
  9. Bertsimas D, Brown DB, Caramanis C (2011) Theory and applications of robust optimization. SIAM Rev 53(3):464–501CrossRefGoogle Scholar
  10. Boysen N, Emde S (2016) The parallel stack loading problem to minimize blockages. Eur J Oper Res 249(2):618–627CrossRefGoogle Scholar
  11. Bruns F, Goerigk M, Knust S, Schöbel A (2014) Robust load planning of trains in intermodal transportation. OR Spectr 36(3):631–668CrossRefGoogle Scholar
  12. Bruns F, Knust S, Shakhlevich NV (2016) Complexity results for storage loading problems with stacking constraints. Eur J Oper Res 249(3):1074–1081CrossRefGoogle Scholar
  13. Dekker R, Voogd P, van Asperen E (2006) Advanced methods for container stacking. OR Spectr 28(4):563–586CrossRefGoogle Scholar
  14. Everitt BS, Skrondal A (2010) The Cambridge dictionary of statistics, 4th edn. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  15. ExoAnalytics Inc. Extreme optimization, complexity made simple.
  16. Forbes C, Evans M, Hastings N, Peacock B (2011) Statistical distributions, 4th edn. Wiley, HobokenGoogle 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):2592–2611CrossRefGoogle Scholar
  18. Goerigk M, Knust S, Le XT (2016) Robust storage loading problems with stacking and payload constraints. Eur J Oper Res 253(1):51–67CrossRefGoogle Scholar
  19. Jacod J, Protter P (2004) Probability essentials. Springer, BerlinCrossRefGoogle Scholar
  20. Kall P, Wallace SW (1994) Stochastic programming. Wiley, ChichesterGoogle Scholar
  21. Kang J, Ryu KR, Kim KH (2006) Deriving stacking strategies for export containers with uncertain weight information. J Intell Manuf 17(4):399–410CrossRefGoogle 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):89–101CrossRefGoogle Scholar
  23. Koch T (2004) Rapid mathematical programming. Ph.D. thesis, Technische Universität BerlinGoogle Scholar
  24. Kouvelis P, Yu G (1997) Robust discrete optimization and its applications. Kluwer Academic Publishers, DordrechtCrossRefGoogle Scholar
  25. Le XT, Knust S (2017) MIP-based approaches for robust storage loading problems with stacking constraints. Comput Oper Res 78:138–153CrossRefGoogle 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):297–312CrossRefGoogle 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):159–191CrossRefGoogle 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 731–745Google Scholar
  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)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and EURO - The Association of European Operational Research Societies 2018

Authors and Affiliations

  • Christina Büsing
    • 1
  • Sigrid Knust
    • 2
    Email author
  • Xuan Thanh Le
    • 3
  1. 1.Lehrstuhl II für MathematikRWTH Aachen UniversityAachenGermany
  2. 2.Institute of Computer ScienceUniversity of OsnabrückOsnabrückGermany
  3. 3.Institute of MathematicsVietnam Academy of Science and TechnologyHanoiVietnam

Personalised recommendations