Journal of Optimization Theory and Applications

, Volume 171, Issue 3, pp 1033–1054 | Cite as

Robust Optimization of Schedules Affected by Uncertain Events

Article
  • 327 Downloads

Abstract

In this paper, we present a new method for finding robust solutions to mixed-integer linear programs subject to uncertain events. We present a new modeling framework for such events that result in uncertainty sets that depend parametrically on the decision taken. We also develop results that can be used to compute corresponding robust solutions. The usefulness of our proposed approach is illustrated by applying it in the context of a scheduling problem. For instance, we address uncertainty on the start times chosen for the tasks or on which unit they are to be executed. Delays and unit outages are possible causes for such events and can be very common in practice. Through our approach, we can accommodate them without altering the remainder of the schedule. We also allow for the inclusion of recourse on the continuous part of the problem, that is, we allow for the revision of some of the decisions once uncertainty is observed. This allows one to increase the performance of the robust solutions. The proposed scheme is also computationally favorable since the robust optimization problem to be solved remains a mixed-integer linear program, and the number of integer variables is not increased with respect to the nominal formulation. We finally apply the method to a concrete batch scheduling problem and discuss the effects of robustification in this case.

Keywords

Robust optimization Mixed-integer optimization Scheduling Uncertain events 

Mathematics Subject Classification

49J53 49K99 

Notes

Acknowledgments

The authors are thankful to Peyman Mohajerin Esfahani from EPFL, who helped integrating recourse in our approach, as well as to Qi Zhang from Carnegie Mellon University, for his early feedback on our manuscript. This research was supported by the Swiss National Science Foundation grant P2EZP2 159089.

References

  1. 1.
    Ben-Tal, A., Nemirovski, A.: Robust convex optimization. Math. Oper. Res. 23(4), 769–805 (1998)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Ben-Tal, A., Ghaoui, L.E., Nemirovski, A.: Robust Optimization. Princeton Series in Applied Mathematics. Princeton University Press, Princeton (2009)MATHGoogle Scholar
  3. 3.
    Bertsimas, D., Brown, D.B., Caramanis, C.: Theory and applications of robust optimization. SIAM Rev. 53, 464 (2011)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Pflug, G.C.: On-line optimization of simulated Markovian processes. Math. Oper. Res. 15(3), 381–395 (1990)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Goel, V., Grossmann, I.E.: A class of stochastic programs with decision dependent uncertainty. Math. Program. 108(2–3), 355–394 (2006)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Soyster, A.L.: Technical note—convex programming with set-inclusive constraints and applications to inexact linear programming. Oper. Res. 21(5), 1154–1157 (1973)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Thuente, D.J.: Technical note—duality theory for generalized linear programs with computational methods. Oper. Res. 28(4), 1005–1011 (1980)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Li, Z., Ding, R., Floudas, C.A.: A comparative theoretical and computational study on robust counterpart optimization: I. robust linear optimization and robust mixed integer linear optimization. Ind. Eng. Chem. Res. 50(18), 10567–10603 (2011)CrossRefGoogle Scholar
  9. 9.
    Li, Z., Tang, Q., Floudas, C.A.: A comparative theoretical and computational study on robust counterpart optimization: II. probabilistic guarantees on constraint satisfaction. Ind. Eng. Chem. Res. 51(19), 6769–6788 (2012)CrossRefGoogle Scholar
  10. 10.
    Li, Z., Floudas, C.A.: A comparative theoretical and computational study on robust counterpart optimization: III. improving the quality of robust solutions. Ind. Eng. Chem. Res. 53(33), 13112–13124 (2014)CrossRefGoogle Scholar
  11. 11.
    El Ghaoui, L., Lebret, H.: Robust solutions to least-squares problems with uncertain data. SIAM J. Matrix Anal. Appl. 18(4), 1035–1064 (1997)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Bertsimas, D., Sim, M.: The price of robustness. Oper. Res. 52(1), 35–53 (2004)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Kuhn, D., Wiesemann, W., Georghiou, A.: Primal and dual linear decision rules in stochastic and robust optimization. Math. Program. 130(1), 177–209 (2011)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Bertsimas, D., Georghiou, A.: Design of near optimal decision rules in multistage adaptive mixed-integer optimization. Oper. Res. 63(3), 610–627 (2015)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Georghiou, A., Wiesemann, W., Kuhn, D.: Generalized decision rule approximations for stochastic programming via liftings. Math. Program. 152, 1–38 (2010)MathSciNetMATHGoogle Scholar
  16. 16.
    Ben-Tal, A., Goryashko, A., Guslitzer, E., Nemirovski, A.: Adjustable robust solutions of uncertain linear programs. Math. Program. 99(2), 351–376 (2004)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Guslitser, E.: Uncertainty-immunized solutions in linear programming. Master’s thesis, Minerva Optimization Center, Technion (2002)Google Scholar
  18. 18.
    Goulart, P.J., Kerrigan, E.C., Maciejowski, J.M.: Optimization over state feedback policies for robust control with constraints. Automatica 42(4), 523–533 (2006)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Zhang, Q., Grossmann, I.E., Heuberger, C.F., Sundaramoorthy, A., Pinto, J.M.: Air separation with cryogenic energy storage: optimal scheduling considering electric energy and reserve markets. AIChE J. 61(5), 1547–1558 (2015)CrossRefGoogle Scholar
  20. 20.
    Lin, X., Janak, S.L., Floudas, C.A.: A new robust optimization approach for scheduling under uncertainty: I. bounded uncertainty. Comput. Chem. Eng. 28(6), 1069–1085 (2004)CrossRefGoogle Scholar
  21. 21.
    Janak, S.L., Lin, X., Floudas, C.A.: A new robust optimization approach for scheduling under uncertainty: II. uncertainty with known probability distribution. Comput. Chem. Eng. 31(3), 171–195 (2007)CrossRefGoogle Scholar
  22. 22.
    Vin, J.P., Ierapetritou, M.G.: Robust short-term scheduling of multiproduct batch plants under demand uncertainty. Ind. Eng. Chem. Res. 40(21), 4543–4554 (2001)CrossRefGoogle Scholar
  23. 23.
    Jia, Z., Ierapetritou, M.G.: Generate pareto optimal solutions of scheduling problems using normal boundary intersection technique. Comput. Chem. Eng. 31(4), 268–280 (2007)CrossRefGoogle Scholar
  24. 24.
    Cott, B., Macchietto, S.: Minimizing the effects of batch process variability using online schedule modification. Comput. Chem. Eng. 13(1), 105–113 (1989)CrossRefGoogle Scholar
  25. 25.
    Rodrigues, M., Gimeno, L., Passos, C., Campos, M.: Reactive scheduling approach for multipurpose chemical batch plants. Comput. Chem. Eng. 20, 1215–1220 (1996)CrossRefGoogle Scholar
  26. 26.
    Méndez, C.A., Cerdá, J.: Dynamic scheduling in multiproduct batch plants. Comput. Chem. Eng. 27(8), 1247–1259 (2003)CrossRefGoogle Scholar
  27. 27.
    Mendez, C.A., Cerdá, J.: An MILP framework for batch reactive scheduling with limited discrete resources. Comput. Chem. Eng. 28(6), 1059–1068 (2004)CrossRefGoogle Scholar
  28. 28.
    Vin, J.P., Ierapetritou, M.G.: A new approach for efficient rescheduling of multiproduct batch plants. Ind. Eng. Chem. Res. 39(11), 4228–4238 (2000)CrossRefGoogle Scholar
  29. 29.
    Ruiz, D., Cantón, J., Nougués, J.M., Espuña, A., Puigjaner, L.: On-line fault diagnosis system support for reactive scheduling in multipurpose batch chemical plants. Comput. Chem. Eng. 25(4), 829–837 (2001)CrossRefGoogle Scholar
  30. 30.
    Kanakamedala, K.B., Reklaitis, G.V., Venkatasubramanian, V.: Reactive schedule modification in multipurpose batch chemical plants. Ind. Eng. Chem. Res. 33(1), 77–90 (1994)CrossRefGoogle Scholar
  31. 31.
    Janak, S.L., Floudas, C.A., Kallrath, J., Vormbrock, N.: Production scheduling of a large-scale industrial batch plant. II. Reactive scheduling. Ind. Eng. Chem. Res. 45(25), 8253–8269 (2006)CrossRefGoogle Scholar
  32. 32.
    Tarhan, B., Grossmann, I.E., Goel, V.: Stochastic programming approach for the planning of offshore oil or gas field infrastructure under decision-dependent uncertainty. Ind. Eng. Chem. Res. 48(6), 3078–3097 (2009)CrossRefGoogle Scholar
  33. 33.
    Vujanic, R., Schmitt, M., Warrington, J., Morari, M.: Extending affine control policies to hybrid systems: robust control of a DC–DC buck converter. In: European Control Conference, pp. 1523–1528 (2013)Google Scholar
  34. 34.
    Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)CrossRefMATHGoogle Scholar
  35. 35.
    Rockafellar, R.T.: Convex analysis. In: Princeton Landmarks in Mathematics. Princeton University Press, Princeton (1997). Reprint of the 1970 original, Princeton PaperbacksGoogle Scholar
  36. 36.
    Vujanic, R., Mariéthoz, S., Goulart, P., Morari, M.: Robust integer optimization and scheduling problems for large electricity consumers. In: American Control Conference, pp. 3108–3113 (2012)Google Scholar
  37. 37.
    Kondili, E., Pantelides, C., Sargent, R.: A general algorithm for short-term scheduling of batch operations—I. MILP formulation. Comput. Chem. Eng. 17(2), 211–227 (1993)CrossRefGoogle Scholar
  38. 38.
    Biegler, L.T., Grossmann, I.E., Westerberg, A.W., Kravanja, Z.: Systematic Methods of Chemical Process Design, vol. 796. Prentice Hall PTR, Upper Saddle River (1997)Google Scholar
  39. 39.
    Diamond, S., Chu, E., Boyd, S.: CVXPY: A Python-embedded modeling language for convex optimization, version 0.2. http://cvxpy.org/ (2014)
  40. 40.
    Gurobi: Constrained optimization software. http://www.gurobi.com/ (2014)

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Automatic Control LaboratoryETH ZurichZurichSwitzerland
  2. 2.Department of Engineering ScienceUniversity of OxfordOxfordEngland, UK

Personalised recommendations