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.
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References
Artzner P, Delbaen F, Eber J, Heath D (1999) Coherent measures of risk. Math Finance 9(3):203–228
Ben-Tal A, Nemirovski A (1998) Robust convex optimization. Math Oper Res 23(4):769–805
Ben-Tal A, Nemirovski A (1999) Robust solutions of uncertain linear programs. Oper Res Lett 25(1):1–13
Ben-Tal A, Nemirovski A (2000) Robust solutions of linear programming contaminated with uncertain data. Math Program 88(3):411–424
Ben-Tal A, Nemirovski A (2002) Robust optimization-methodology and applications. Math Program B 92:453–480
Ben-Tal A, El Ghaoui L, Nemirovski A (2009) Robust optimization. Princeton University Press, Princeton
Bertsimas D, Brown DB (2009) Constructing uncertainty sets for robust linear optimization. Oper Res 57(6):1438–1495
Bertsimas D, Sim M (2003) Robust discrete optimization and network flows. Math Program 98(1–3):49–71
Bertsimas D, Brown DB, Caramanis C (2011) Theory and applications of robust optimization. SIAM Rev 53(3):464–501
Boysen N, Emde S (2016) The parallel stack loading problem to minimize blockages. Eur J Oper Res 249(2):618–627
Bruns F, Goerigk M, Knust S, Schöbel A (2014) Robust load planning of trains in intermodal transportation. OR Spectr 36(3):631–668
Bruns F, Knust S, Shakhlevich NV (2016) Complexity results for storage loading problems with stacking constraints. Eur J Oper Res 249(3):1074–1081
Dekker R, Voogd P, van Asperen E (2006) Advanced methods for container stacking. OR Spectr 28(4):563–586
Everitt BS, Skrondal A (2010) The Cambridge dictionary of statistics, 4th edn. Cambridge University Press, Cambridge
ExoAnalytics Inc. Extreme optimization, complexity made simple. http://www.extremeoptimization.com/Documentation/Statistics/Default.aspx
Forbes C, Evans M, Hastings N, Peacock B (2011) Statistical distributions, 4th edn. Wiley, Hoboken
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
Goerigk M, Knust S, Le XT (2016) Robust storage loading problems with stacking and payload constraints. Eur J Oper Res 253(1):51–67
Jacod J, Protter P (2004) Probability essentials. Springer, Berlin
Kall P, Wallace SW (1994) Stochastic programming. Wiley, Chichester
Kang J, Ryu KR, Kim KH (2006) Deriving stacking strategies for export containers with uncertain weight information. J Intell Manuf 17(4):399–410
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
Koch T (2004) Rapid mathematical programming. Ph.D. thesis, Technische Universität Berlin
Kouvelis P, Yu G (1997) Robust discrete optimization and its applications. Kluwer Academic Publishers, Dordrecht
Le XT, Knust S (2017) MIP-based approaches for robust storage loading problems with stacking constraints. Comput Oper Res 78:138–153
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
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
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
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)
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.
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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
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DOI: https://doi.org/10.1007/s13675-018-0094-x