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Annals of Operations Research

, Volume 271, Issue 2, pp 887–913 | Cite as

A perfect information lower bound for robust lot-sizing problems

  • Marcio Costa Santos
  • Michael PossEmail author
  • Dritan Nace
Original Research
  • 51 Downloads

Abstract

Robust multi-stage linear optimization is hard computationally and only small problems can be solved exactly. Hence, robust multi-stage linear problems are typically addressed heuristically through decision rules, which provide upper bounds for the optimal solution costs of the problems. We investigate in this paper lower bounds inspired by the perfect information relaxation used in stochastic programming. Specifically, we study the uncapacitated robust lot-sizing problem, showing that different versions of the problem become tractable whenever the non-anticipativity constraints are relaxed. Hence, we can solve the resulting problem efficiently, obtaining a lower bound for the optimal solution cost of the original problem. We compare numerically the solution time and the quality of the new lower bound with the dual affine decision rules that have been proposed by Kuhn et al. (Math Program 130:177–209, 2011).

Keywords

Multi-stage robust optimization Perfect information Lot-sizing problem Complexity 

Supplementary material

References

  1. Agra, A., Christiansen, M., Figueiredo, R. M. V., Hvattum, L. M., Poss, M., & Requejo, C. (2013). The robust vehicle routing problem with time windows. Computers & OR, 40(3), 856–866.  https://doi.org/10.1016/j.cor.2012.10.002.CrossRefGoogle Scholar
  2. Agra, A., Santos, M. C., Nace, D., & Poss, M. (2016). A dynamic programming approach for a class of robust optimization problems. SIAM Journal on Optimization, 26(3), 1799–1823.CrossRefGoogle Scholar
  3. Avriel, M., & Williams, A. (1970). The value of information and stochastic programming. Operations Research, 18(5), 947–954.CrossRefGoogle Scholar
  4. Ayoub, J., & Poss, M. (2016). Decomposition for adjustable robust linear optimization subject to uncertainty polytope. Computational Management Science, 13(2), 219–239.CrossRefGoogle Scholar
  5. Baumann, F., Buchheim, C., & Ilyina, A. (2014). Lagrangean decomposition for mean-variance combinatorial optimization. In Combinatorial optimization—third international symposium, ISCO 2014, Lisbon, Portugal, March 5–7, 2014. Revised Selected Papers (pp. 62–74).Google Scholar
  6. Ben-Tal, A., Golany, B., Nemirovski, A., & Vial, J.-P. (2005). Retailer-supplier flexible commitments contracts: A robust optimization approach. Manufacturing & Service Operations Management, 7(3), 248–271.CrossRefGoogle Scholar
  7. Ben-Tal, A., Goryashko, A., Guslitzer, E., & Nemirovski, A. (2004). Adjustable robust solutions of uncertain linear programs. Mathematical Programming, 99(2), 351–376.CrossRefGoogle Scholar
  8. Ben-Tal, A., & Nemirovski, A. (1998). Robust convex optimization. Mathematics of Operations Research, 23(4), 769–805.CrossRefGoogle Scholar
  9. Ben-Tal, A., & Nemirovski, A. (2000). Robust solutions of linear programming problems contaminated with uncertain data. Mathematical Programming, 88(3), 411–424.  https://doi.org/10.1007/PL00011380.CrossRefGoogle Scholar
  10. Bertsimas, D., & Dunning, I. (2014). Multistage robust mixed integer optimization with adaptive partitions. Operations Research, 64, 980–998.CrossRefGoogle Scholar
  11. Bertsimas, D., & Georghiou, A. (2015). Design of near optimal decision rules in multistage adaptive mixed-integer optimization. Operations Research, 63(3), 610–627.  https://doi.org/10.1287/opre.2015.1365.CrossRefGoogle Scholar
  12. Bertsimas, D., & Sim, M. (2003). Robust discrete optimization and network flows. Mathematical Programming, 98(1), 49–71.  https://doi.org/10.1007/s10107-003-0396-4.CrossRefGoogle Scholar
  13. Bertsimas, D., & Sim, M. (2004). The price of robustness. Operations Research, 52(1), 35–53.CrossRefGoogle Scholar
  14. Bertsimas, D., & Thiele, A. (2006). A robust optimization approach to inventory theory. Operations Research, 54(1), 150–168.CrossRefGoogle Scholar
  15. Bienstock, D., & Özbay, N. (2008). Computing robust basestock levels. Discrete Optimization, 5(2), 389–414. (in Memory of George B. Dantzig).CrossRefGoogle Scholar
  16. Billionnet, A., Costa, M., & Poirion, P. (2014). 2-Stage robust MILP with continuous recourse variables. Discrete Applied Mathematics, 170, 21–32.CrossRefGoogle Scholar
  17. Birge, J. R., & Louveaux, F. (2011). Introduction to stochastic programming. Berlin: Springer.CrossRefGoogle Scholar
  18. Bougeret, M., Pessoa, A., & Poss, M. (2016). Robust scheduling with budgeted uncertainty. Discrete Applied Mathematics Conditionally. https://hal.archives-ouvertes.fr/hal-01345283/document.
  19. Chen, X., & Zhang, Y. (2009). Uncertain linear programs: Extended affinely adjustable robust counterparts. Operations Research, 57(6), 1469–1482.CrossRefGoogle Scholar
  20. de Ruiter, F. J. C. T., Ben-Tal, A., Brekelmans, R., & den Hertog, D. (2017). Robust optimization of uncertain multistage inventory systems with inexact data in decision rules. Computational Management Science, 14(1), 45–66.  https://doi.org/10.1007/s10287-016-0253-6.CrossRefGoogle Scholar
  21. Garey, M. R., & Johnson, D. S. (2002). Computers and intractability (Vol. 29). New York: W. H. Freeman.Google Scholar
  22. Goh, J., & Sim, M. (2010). Distributionally robust optimization and its tractable approximations. Operations Research, 58(4–part–1), 902–917.CrossRefGoogle Scholar
  23. Gorissen, B. L., & den Hertog, D. (2013). Robust counterparts of inequalities containing sums of maxima of linear functions. European Journal of Operational Research, 227(1), 30–43.CrossRefGoogle Scholar
  24. Huang, C. C., & Vertinsky, W. T. Z. I. (1977). Sharp bounds on the value of perfect information. Operations Research, 25(1), 128–139.CrossRefGoogle Scholar
  25. Kouvelis, P., & Yu, G. (2013). Robust discrete optimization and its applications (Vol. 14). Berlin: Springer.Google Scholar
  26. Kuhn, D., Wiesemann, W., & Georghiou, A. (2011). Primal and dual linear decision rules in stochastic and robust optimization. Mathematical Programming, 130, 177–209.CrossRefGoogle Scholar
  27. Oostenbrink, J. B., Al, M. J., Oppe, M., & Rutten-van Mölken, M. P. (2008). Expected value of perfect information: An empirical example of reducing decision uncertainty by conducting additional research. Value in Health, 11(7), 1070–1080.CrossRefGoogle Scholar
  28. Pochet, Y., & Wolsey, L. A. (1994). Polyhedra for lot-sizing with wagner-whitin costs. Mathematical Programming, 67, 297–323.  https://doi.org/10.1007/BF01582225.CrossRefGoogle Scholar
  29. Pochet, Y., & Wolsey, L. (2013). Production planning by mixed integer programming. Berlin: Springer.Google Scholar
  30. Poss, M. (2013). Robust combinatorial optimization with variable budgeted uncertainty. 4OR, 11(1), 75–92.  https://doi.org/10.1007/s10288-012-0217-9.CrossRefGoogle Scholar
  31. Postek, K., & den Hertog, D. (2016). Multistage adjustable robust mixed-integer optimization via iterative splitting of the uncertainty set. INFORMS Journal on Computing, 28(3), 553–574.  https://doi.org/10.1287/ijoc.2016.0696.CrossRefGoogle Scholar
  32. Zeng, B., & Zhao, L. (2013). Solving two-stage robust optimization problems by a constraint-and-column generation method. Operations Research Letters, 41(5), 457–461.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversité Libre de BruxellesBrusselsBelgium
  2. 2.INOCSINRIA Lille Nord-EuropeRennesFrance
  3. 3.UMR CNRS 5506 LIRMMUniversité de MontpellierMontpellier Cedex 5France
  4. 4.UMR CNRS 7253 Heudiasyc, Université de Technologie de Compiègne, Centre de Recherches de RoyallieuSorbonne universitésCompiègneFrance

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