Skip to main content
SpringerLink
Log in

Robust Optimization of Schedules Affected by Uncertain Events

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

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

    Article  MathSciNet  MATH  Google Scholar 

  2. Ben-Tal, A., Ghaoui, L.E., Nemirovski, A.: Robust Optimization. Princeton Series in Applied Mathematics. Princeton University Press, Princeton (2009)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Pflug, G.C.: On-line optimization of simulated Markovian processes. Math. Oper. Res. 15(3), 381–395 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  5. Goel, V., Grossmann, I.E.: A class of stochastic programs with decision dependent uncertainty. Math. Program. 108(2–3), 355–394 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  7. Thuente, D.J.: Technical note—duality theory for generalized linear programs with computational methods. Oper. Res. 28(4), 1005–1011 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bertsimas, D., Sim, M.: The price of robustness. Oper. Res. 52(1), 35–53 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  14. Bertsimas, D., Georghiou, A.: Design of near optimal decision rules in multistage adaptive mixed-integer optimization. Oper. Res. 63(3), 610–627 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  15. Georghiou, A., Wiesemann, W., Kuhn, D.: Generalized decision rule approximations for stochastic programming via liftings. Math. Program. 152, 1–38 (2010)

    MathSciNet  MATH  Google Scholar 

  16. Ben-Tal, A., Goryashko, A., Guslitzer, E., Nemirovski, A.: Adjustable robust solutions of uncertain linear programs. Math. Program. 99(2), 351–376 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  17. Guslitser, E.: Uncertainty-immunized solutions in linear programming. Master’s thesis, Minerva Optimization Center, Technion (2002)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  24. Cott, B., Macchietto, S.: Minimizing the effects of batch process variability using online schedule modification. Comput. Chem. Eng. 13(1), 105–113 (1989)

    Article  Google Scholar 

  25. Rodrigues, M., Gimeno, L., Passos, C., Campos, M.: Reactive scheduling approach for multipurpose chemical batch plants. Comput. Chem. Eng. 20, 1215–1220 (1996)

    Article  Google Scholar 

  26. Méndez, C.A., Cerdá, J.: Dynamic scheduling in multiproduct batch plants. Comput. Chem. Eng. 27(8), 1247–1259 (2003)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

  34. Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)

    Book  MATH  Google Scholar 

  35. Rockafellar, R.T.: Convex analysis. In: Princeton Landmarks in Mathematics. Princeton University Press, Princeton (1997). Reprint of the 1970 original, Princeton Paperbacks

  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)

  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)

    Article  Google Scholar 

  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. Diamond, S., Chu, E., Boyd, S.: CVXPY: A Python-embedded modeling language for convex optimization, version 0.2. http://cvxpy.org/ (2014)

  40. Gurobi: Constrained optimization software. http://www.gurobi.com/ (2014)

Download references

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.

Author information

Authors and Affiliations

  1. Automatic Control Laboratory, ETH Zurich, Zurich, Switzerland

    Robin Vujanic & Manfred Morari

  2. Department of Engineering Science, University of Oxford, Oxford, England, UK

    Paul Goulart

Authors
  1. Robin Vujanic
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Paul Goulart
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Manfred Morari
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Robin Vujanic.

Additional information

Communicated by Christodoulos Floudas.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Vujanic, R., Goulart, P. & Morari, M. Robust Optimization of Schedules Affected by Uncertain Events. J Optim Theory Appl 171, 1033–1054 (2016). https://doi.org/10.1007/s10957-016-0920-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-016-0920-3

Keywords

Mathematics Subject Classification

Search

Navigation