Solving 0–1 semidefinite programs for distributionally robust allocation of surgery blocks

Original Paper
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Abstract

We allocate surgery blocks to operating rooms (ORs) under random surgery durations. Given unknown distribution of the duration of each block, we investigate distributionally robust (DR) variants of two types of stochastic programming models using a moment-based ambiguous set. We minimize the total cost of opening ORs and allocating surgery blocks, while constraining OR overtime via chance constraints and via an expected penalty cost in the objective function, respectively in the two types of models. Following conic duality, we build equivalent 0–1 semidefinite programming (SDP) reformulations of the DR models and solve them using cutting-plane algorithms. For the DR chance-constrained model, we also derive a 0–1 second-order conic programming approximation to obtain less conservative solutions. We compare different models and solution methods by testing randomly generated instances. Our results show that the DR chance-constrained model better controls average and worst-case OR overtime, as compared to the stochastic programming and DR expected-penalty-based models. Our cutting-plane algorithms also outperform standard optimization solvers and efficiently solve 0–1 SDP formulations.

Keywords

Distributionally robust optimization Chance-constrained programming Stochastic programming 0–1 Semidefinite programming Cutting-plane algorithm 0–1 Second-order conic programming 

References

  1. 1.
    Berg, B.P., Denton, B.T., Erdogan, S.A., Rohleder, T., Huschka, T.R.: Optimal booking and scheduling in outpatient procedure centers. Comput. Oper. Res. 50, 24–37 (2014)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bertsimas, D., Doan, X.V., Natarajan, K., Teo, C.-P.: Models for minimax stochastic linear optimization problems with risk aversion. Math. Oper. Res. 35(3), 580–602 (2010)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Cardoen, B., Demeulemeester, E., Beliën, J.: Operating room planning and scheduling: a literature review. Eur. J. Oper. Res. 201(3), 921–932 (2010)CrossRefMATHGoogle Scholar
  4. 4.
    Chaabane, S., Meskens, N., Guinet, A., Laurent, M.: Comparison of two methods of operating theatre planning: application in belgian hospital. J. Syst. Sci. Syst. Eng. 17(2), 171–186 (2008)CrossRefGoogle Scholar
  5. 5.
    Delage, E., Ye, Y.: Distributionally robust optimization under moment uncertainty with application to data-driven problems. Oper. Res. 58(3), 595–612 (2010)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Deng, Y., Shen, S., Denton, B.T.: Chance-constrained surgery planning under conditions of limited and ambiguous data. Available at SSRN  https://doi.org/10.2139/ssrn.2432375 (2016)
  7. 7.
    Denton, B.T., Miller, A.J., Balasubramanian, H.J., Huschka, T.R.: Optimal allocation of surgery blocks to operating rooms under uncertainty. Oper. Res. 58(4), 802–816 (2010)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Erdogan, S.A., Denton, B.T.: Surgery planning and scheduling. In: Cochran, J., Cox, L., Keskinocak, P., Kharoufeh, J., Smith, J. (eds.) Wiley Encyclopedia of Operations Research and Management Science. Wiley Online Library, New York (2011)Google Scholar
  9. 9.
    Herring, W.L., Herrmann, J.W.: The single-day surgery scheduling problem: sequential decision-making and threshold-based heuristics. OR Spectr. 34(2), 429–459 (2012)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Jiang, R., Guan, Y.: Data-driven chance constrained stochastic program. Math. Program. Ser. A 158(1), 291–327 (2016)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Marques, I., Captivo, M.E., Pato, M.V.: An integer programming approach to elective surgery scheduling. OR Spectr. 34(2), 407–427 (2012)CrossRefMATHGoogle Scholar
  12. 12.
    Shapiro, A. (2000). On duality theory of conic linear problems. In: Semi-Infinite Programming, pp. 135–165. Kluwer Academic PublishersGoogle Scholar
  13. 13.
    Shapiro, A., Dentcheva, D., Ruszczyński, A.: Lectures on Stochastic Programming: Modeling and Theory, vol. 9. SIAM, Philadelphia (2009)CrossRefMATHGoogle Scholar
  14. 14.
    Shylo, O.V., Prokopyev, O.A., Schaefer, A.J.: Stochastic operating room scheduling for high-volume specialties under block booking. INFORMS J. Comput. 25(4), 682–692 (2012)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Testi, A., Tanfani, E., Torre, G.: A three-phase approach for operating theatre schedules. Health Care Manag. Sci. 10(2), 163–172 (2007)CrossRefGoogle Scholar
  16. 16.
    Todd, M.J.: Semidefinite optimization. Acta Numer. 2001(10), 515–560 (2001)MathSciNetMATHGoogle Scholar
  17. 17.
    Wagner, M.: Stochastic 0–1 linear programming under limited distributional information. Oper. Res. Lett. 36(2), 150–156 (2008)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Industrial and Operations EngineeringUniversity of MichiganAnn ArborUSA
  2. 2.Department of Industrial and Systems EngineeringSan Jose State UniversitySan JoseUSA

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