Quantified Combinatorial Optimization

  • Thorsten EdererEmail author
  • Ulf Lorenz
  • Thomas Opfer
Conference paper
Part of the Operations Research Proceedings book series (ORP)


MIP and IP programming are state-of-the-art modeling techniques for computer-aided optimization. However, companies observe an increasing danger of disruptions that prevent them from acting as planned. One reason is input data being assumed as deterministic, but in reality, data is afflicted with uncertainties. Incorporating uncertainty in existing models, however, often pushes the complexity of problems that are in P or NP, to the complexity class PSPACE. Quantified integer linear programming (QIP) is a PSPACE-complete extension of the IP problem with variables being either existentially or universally quantified. With the help of QIPs, it is possible to model board-games like Gomoku as well as traditional combinatorial OR problems under uncertainty. In this paper, we present how to extend the model formulation of classical scheduling problems like the Job-Shop and Car-Sequencing problem by uncertain influences and give illustrating examples with solutions.


  1. 1.
    Ben-Tal, A., El Ghaoui, L., & Nemirovski, A. (2009) Robust Optimization. Princeton University.Google Scholar
  2. 2.
    Berger, A., Hoffmann, R., Lorenz, U., & Stiller, S. (2011). Online railway delay management: Hardness, simulation and computation. Simulation, 87(7), 616–629.CrossRefGoogle Scholar
  3. 3.
    Ederer, T., Lorenz, U., Martin, A., Wolf, J. (2011) Quantified Linear Programs: A Computational Study. Algorithms-ESA 2011 (pp. 203–214). Berlin: Springer.Google Scholar
  4. 4.
    Ederer, T., Lorenz, U., Opfer, T., & Wolf, J. (2011). Modelling Games with the help of Quantified Integer Linear Programs. ACG 13. Berlin: Springer.Google Scholar
  5. 5.
    Fliedner, M., & Boysen, N. (2008). Solving the car sequencing problem via branch & bound. European Journal of Operational Research, 191(3), 1023–1042.CrossRefGoogle Scholar
  6. 6.
    Grothklags S., Lorenz U., & Monien B. (2009). From state-of-the-art static fleet assignment to flexible stochastic planning of the future. Algorithmics of large and complex networks (pp. 140–165).Google Scholar
  7. 7.
    Liebchen, C., Lübbecke, M. E., Möhring, R. H., & Stiller, S. (2009). The concept of recoverable robustness, linear programming recovery, and railway applications. Robust and online large-scale optimization 1–27.Google Scholar
  8. 8.
    Subramani, K. (2004). Analyzing selected quantified integer programs. Berlin: Springer.Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Discrete OptimizationTU DarmstadtDarmstadtGermany
  2. 2.Fluid Systems TechnologyTU DarmstadtDarmstadtGermany

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