# An efficient computational method for large scale surgery scheduling problems with chance constraints

- 164 Downloads

## Abstract

We propose an efficient solution method based on a decomposition of set-partitioning formulation of an integrated surgery planning and scheduling problem with chance constraints. The studied problem is characterized by a set of identical operating rooms (ORs), a set of surgeries with uncertain durations, a set of surgeons, and surgery dependent turnover times. The decision variables include the number of ORs to open, assignments of surgeries and surgeons to ORs in admissible periods, and the sequence of surgeries to be performed in a period. The objective is to minimize the cost of opening ORs and the penalties associated with turnover times.In the proposed formulation, the column generation subproblem is decomposed over ORs and time periods. The structure of the subproblem is further exploited and transformed to a shortest path problem. A search algorithm has been devised to efficiently solve the resulting subproblem, subject to some optimality and feasibility conditions. The proposed computational method outperforms the standard chance constrained model and reduces the solution time significantly. Furthermore, extensive simulation experiments have been carried out to compare the performance of three variants of the underlying formulations and evaluate the sensitivity of the decisions to the probability of exceeding a session length.

## Keywords

Surgery planning and scheduling Set-partitioning formulation Integer chance constrained programming Integer programming## References

- 1.Ahmed, S., Papageorgiou, D.J.: Probabilistic set covering with correlations. Oper. Res.
**61**, 438–452 (2013)MathSciNetCrossRefMATHGoogle Scholar - 2.Baldacci, R., Mingozzi, A.: A unified exact method for solving different classes of vehicle routing problems. Math. Program.
**120**(2), 347–380 (2009)MathSciNetCrossRefMATHGoogle Scholar - 3.Barnhart, C., Johnson, E., Nemhauser, G., Savelsbergh, M., Vance, P.: Branch-and-price: column generation for solving huge integer programs. Oper. Res.
**46**, 316–329 (1998)MathSciNetCrossRefMATHGoogle Scholar - 4.Batun, S., Denton, B.T., Huschka, T.R., Schaefer, A.J.: Operating room pooling and parallel surgery processing under uncertainty. INFORMS J. Comput.
**23**, 220–237 (2011)MathSciNetCrossRefMATHGoogle Scholar - 5.Cardoen, B., Demeulemeester, E., Beliën, J.: Operating room planning and scheduling: a literature review. Eur. J. Oper. Res.
**201**, 921–932 (2010)CrossRefMATHGoogle Scholar - 6.Charnes, A., Cooper, W.: Chance-constrained programming. Manag. Sci.
**6**, 73–79 (1959)MathSciNetCrossRefMATHGoogle Scholar - 7.Deng, Y., Shen, S., Denton, B.: Chance-constrained surgery planning under uncertain or ambiguous surgery durations (2015, unpublished). https://papers.ssrn.com/sol3/papers.cfm?abstractid=2432375
- 8.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**, 802–816 (2010)MathSciNetCrossRefMATHGoogle Scholar - 9.Erdogan, S.A., Denton, B.T.: Wiley Encyclopedia of Operations Research and Management Science. Wiley, New York (2011). ch. Surgery planning and schedulingGoogle Scholar
- 10.Etzioni, D., Liu, J., Maggard, M., Ko, C.: The aging population and its impact on the surgery workforce. Ann. Surg.
**238**, 170–177 (2003)Google Scholar - 11.Fukasawa, R., Longo, H., Lysgaard, J., Aragão, M.P.D., Reis, M., Uchoa, E., Werneck, R.F.: Robust branch-and-cut-and-price for the capacitated vehicle routing problem. Math. Program.
**106**, 491–511 (2006)MathSciNetCrossRefMATHGoogle Scholar - 12.Gul, S., Denton, B.T., Fowler, J.W.: A progressive hedging approach for surgery planning under uncertainty. INFORMS J. Comput.
**27**, 755–772 (2015)MathSciNetCrossRefMATHGoogle Scholar - 13.Liu, X., Kucukyavuz, S., Luedtke, J.: Decomposition algorithms for two-stage chance-constrained programs. Math. Progam. Ser. B
**157**, 219–243 (2014)MathSciNetCrossRefMATHGoogle Scholar - 14.Luedtke, J.: A branch-and-cut decomposition algorithm for solving chance-constrained mathematical programs with finite support. Math. Program.
**146**, 1–26 (2013)MathSciNetGoogle Scholar - 15.Lulli, G., Sen, S.: A branch-and-price algorithm for multistage stochastic integer programming with application to stochastic batch-sizing problems. Manag. Sci.
**50**, 786–796 (2004)CrossRefMATHGoogle Scholar - 16.Meng, F., Qi, J., Zhang, M., Ang, J., Chu, S., Sim, M.: A robust optimization model for managing elective admission in a public hospital. Oper. Res.
**63**, 1452–1467 (2015)MathSciNetCrossRefMATHGoogle Scholar - 17.Min, D., Yih, Y.: Scheduling elective surgery under uncertainty and downstream capacity constraints. Eur. J. Oper. Res.
**206**, 642–652 (2010)MathSciNetCrossRefMATHGoogle Scholar - 18.Nemirovski, A., Shapiro, A.: Convex approximations of chance constrained programs. SIAM J. Optim.
**17**, 969–996 (2006)MathSciNetCrossRefMATHGoogle Scholar - 19.Neyshabouri, S., Berg, B.: Adaptive elective surgery planning under duration and length-of-stay uncertainty: a robust optimization approach. (2015)Google Scholar
- 20.Noorizadegan, M.: On vehicle routing with uncertain demands. Ph.D. thesis, Warwick Business School (2013)Google Scholar
- 21.Pessoa, A., de Aragao, M.P., Uchoa, E.: A robust branch-cut-and-price algorithm for the heterogeneous fleet vehicle routing problem. Lect. Notes Comput. Sci.
**4525**, 150–160 (2007)CrossRefMATHGoogle Scholar - 22.Pulido, R., Aguirre, A.M., Ortega-Mier, M., García-Sánchez, A., Méndez, C.A.: Managing daily surgery schedules in a teaching hospital: a mixed-integer optimization approach. BMC Health Serv. Res.
**14**, 1–13 (2014)CrossRefGoogle Scholar - 23.Saxena, A., Goyal, V., Lejeune, M.A.: Mip reformulations of the probabilistic set covering problem. Math. Program.
**121**, 1–31 (2010)MathSciNetCrossRefMATHGoogle Scholar - 24.Sherali, H.D., Zhu, X.: Two-stage stochastic mixed-integer programs: algorithms and insights. In: Gao, D.Y., Sherali, H.D. (eds.) Advances in Applied Mathematics and Global Optimization, pp. 405–435. Springer Science+Business Media, Boston, MA (2009)Google Scholar
- 25.Shylo, O.V., Prokopyev, O.A., Schaefer, A.J.: Stochastic operating room scheduling for high-volume specialties under block booking. INFORMS J. Comput.
**25**, 682–692 (2013)MathSciNetCrossRefGoogle Scholar - 26.Song, Y., Luedtke, J.R., Küçükyavuz, S.: Chance-constrained binary packing problems. INFORMS J. Comput.
**26**, 735–747 (2014)MathSciNetCrossRefMATHGoogle Scholar - 27.Wang, Z., Crowcroft, J.: Quality-of-service routing for supporting multimedia applications. IEEE Sel. Areas Commun.
**14**, 1228–1234 (1996)CrossRefGoogle Scholar - 28.Wang, Y., Tang, J., Fung, R.Y.K.: A column-generation-based heuristic algorithm for solving operating theater planning problem under stochastic demand and surgery cancellation risk. Int. J. Prod. Econ.
**158**, 28–36 (2014)CrossRefGoogle Scholar - 29.Zhang, B., Murali, P., Dessouky, M.M., Belson, D.: A mixed integer programming approach for allocating operating room capacity. J. Oper. Res. Soc.
**60**, 663–673 (2009)CrossRefMATHGoogle Scholar - 30.Zhang, Z., Xie, X.: Simulation-based optimization for surgery appointment scheduling of multiple operating rooms. IIE Trans.
**47**, 998–1012 (2015)CrossRefGoogle Scholar