Journal of Combinatorial Optimization

, Volume 37, Issue 1, pp 123–149 | Cite as

Online scheduling for outpatient services with heterogeneous patients and physicians

  • Huiqiao Su
  • Guohua WanEmail author
  • Shan Wang


In outpatient services, it is critical to schedule patients for physicians to reduce both patients waiting and physicians overtime working. In this paper, we regard the problem as an online scheduling problem and based on analysis of a real data set from a big hospital in China, we develop a dynamic programming model to solve the problem. We propose a Policy Iteration Algorithm to find the optimal solution in the steady state, and obtain the structural properties of the policy. We conduct numerical experiments to compare the performance of the policy with that of the two policies used in practice by simulating various scenarios. The numerical results show that the policy has the best performance across all scenarios, especially when the system is heavily loaded. We also discuss the managerial implications of the study for practitioners. The model and solution method can be easily extended to multi-server case and can be applied to the general service scheduling problems with heterogeneous customers and service providers.


Health care operations Online scheduling Scheduling policy Markov decision process 



The authors are grateful to Shanghai General Hospital for providing data and help with this research. The authors are listed alphabetically and they contribute equally to this work.


  1. Ahn JH, Hornberger JC (1996) Involving patients in the cadaveric kidney transplant allocation process: a decision-theoretic perspective. Manag Sci 42(5):629–641CrossRefzbMATHGoogle Scholar
  2. Begen MA, Queyranne M (2011) Appointment scheduling with discrete random durations. Math Operat Res 36(2):240–257MathSciNetCrossRefzbMATHGoogle Scholar
  3. Bennett CC, Hauser K (2013) Artificial intelligence framework for simulating clinical decision-making: a markov decision process approach. Artif Intell Med 57(1):9–19CrossRefGoogle Scholar
  4. Breg BP, Denton BP (2017) Fast approximation methods for online scheduling of outpatient procedure centers. INFORMS J Comput 29(4):631–644Google Scholar
  5. Cao XR (2007) Stochastic learning and optimization: a sensitivity-based approach. Springer, BerlinCrossRefzbMATHGoogle Scholar
  6. Cayirli T, Veral E (2003) Outpatient scheduling in health care: a review of literature. Prod Oper Manag 12(4):519–549CrossRefGoogle Scholar
  7. Cayirli T, Veral E, Rosen H (2006) Designing appointment scheduling systems for ambulatory care services. Health Care Manag Sci 9(1):47–58CrossRefGoogle Scholar
  8. Cayirli T, Yang KK, Quek SA (2012) A universal appointment rule in the presence of no-shows and walk-ins. Prod Oper Manag 21(4):682–697CrossRefGoogle Scholar
  9. Chen G, Shen ZJM (2007) Probabilistic asymptotic analysis of stochastic online scheduling problems. IIE Trans 39(5):525–538CrossRefGoogle Scholar
  10. Chou MC, Liu H, Queyranne M, Simchi-Levi D (2006) On the asymptotic optimality of a simple on-line algorithm for the stochastic single-machine weighted completion time problem and its extensions. Oper Res 54(3):464–474MathSciNetCrossRefzbMATHGoogle Scholar
  11. Fetter RB, Thompson JD (1966) Patients’ waiting time and doctors’ idle time in the outpatient setting. Health Serv Res 1(1):66Google Scholar
  12. Ge X (2016) Personal communication, October 14, 2016, Shanghai General Hospital, ShanghaiGoogle Scholar
  13. Green LV, Savin S, Wang B (2006) Managing patient service in a diagnostic medical facility. Oper Res 54(1):11–25CrossRefzbMATHGoogle Scholar
  14. Gupta D, Denton B (2008) Appointment scheduling in health care: challenges and opportunities. IIE Trans 40(9):800–819CrossRefGoogle Scholar
  15. Hassin R, Mendel S (2008) Scheduling arrivals to queues: a single-server model with no-shows. Manag Sci 54(3):565–572CrossRefzbMATHGoogle Scholar
  16. Hu C, Lovejoy WS, Shafer SL (1996) Comparison of some suboptimal control policies in medical drug therapy. Oper Res 44(5):696–709CrossRefzbMATHGoogle Scholar
  17. Kaandorp GC, Koole G (2007) Optimal outpatient appointment scheduling. Health Care Manag Sci 10(3):217–229CrossRefGoogle Scholar
  18. Kim SH, Whitt W (2014) Are call center and hospital arrivals well modeled by nonhomogeneous Poisson processes? Manuf Serv Oper Manag 16(3):464–480CrossRefGoogle Scholar
  19. Knowledge@Wharton (2013) ticking time bombs: Chinas health care system faces issues of access, quality and cost.
  20. Kolesar P (1970) A Markovian model for hospital admission scheduling. Manag Sci 16(6):B-384CrossRefGoogle Scholar
  21. Kong Q, Lee CY, Teo CP, Zheng Z (2013) Scheduling arrivals to a stochastic service delivery system using copositive cones. Oper Res 61(3):711–726MathSciNetCrossRefzbMATHGoogle Scholar
  22. Kopach R, DeLaurentis PC, Lawley M, Muthuraman K, Ozsen L, Rardin R, Wan H, Intrevado P, Qu X, Willis D (2007) Effects of clinical characteristics on successful open access scheduling. Health Care Manag Sci 10(2):111–124CrossRefGoogle Scholar
  23. LaGanga LR, Lawrence SR (2012) Appointment overbooking in health care clinics to improve patient service and clinic performance. Prod Oper Manag 21(5):874–888CrossRefGoogle Scholar
  24. Li J, Dong M, Ren Y, Yin K (2015) How patient compliance impacts the recommendations for colorectal cancer screening. J Comb Optim 30(4):920–937MathSciNetCrossRefzbMATHGoogle Scholar
  25. Magni P, Quaglini S, Marchetti M, Barosi G (2000) Deciding when to intervene: a markov decision process approach. Int J Med Inform 60(3):237–253CrossRefGoogle Scholar
  26. Megow N, Vredeveld T (2006) Approximation in preemptive stochastic online scheduling. Springer, BerlinCrossRefzbMATHGoogle Scholar
  27. Megow N, Uetz M, Vredeveld T (2006) Models and algorithms for stochastic online scheduling. Math Oper Res 31(3):513–525MathSciNetCrossRefzbMATHGoogle Scholar
  28. Patrick J (2012) A markov decision model for determining optimal outpatient scheduling. Health Care Manag Sci 15(2):91–102MathSciNetCrossRefGoogle Scholar
  29. Puterman ML (1990) Markov decision processes. Handb Oper Res Manag Sci 2:331–434MathSciNetzbMATHGoogle Scholar
  30. Qu X, Peng Y, Shi J, LaGanga L (2015) An MDP model for walk-in patient admission management in primary care clinics. Int J Prod Econ 168:303–320CrossRefGoogle Scholar
  31. Rising EJ, Baron R, Averill B (1973) A systems analysis of a university-health-service outpatient clinic. Oper Res 21(5):1030–1047CrossRefGoogle Scholar
  32. Schaefer AJ, Bailey MD, Shechter SM, Roberts MS (2005) Modeling medical treatment using Markov decision processes. In: Operations research and health care. Springer, pp 593–612Google Scholar
  33. Schulz AS (2008) Stochastic online scheduling revisited. In: Combinatorial Optimization and Applications. Springer, pp 448–457Google Scholar
  34. Sevcik KC (1974) Scheduling for minimum total loss using service time distributions. J ACM (JACM) 21(1):66–75MathSciNetCrossRefzbMATHGoogle Scholar
  35. Sloan TW (2007) Safety-cost trade-offs in medical device reuse: a Markov decision process model. Health Care Manag Sci 10(1):81–93CrossRefGoogle Scholar
  36. Swartzman G (1970) The patient arrival process in hospitals: statistical analysis. Health Serv Res 5(4):320–9Google Scholar
  37. Tan Z, Zhang A (2013) Online and semi-online scheduling. Springer, New YorkCrossRefzbMATHGoogle Scholar
  38. Wang S, Liu N, Wan G (2016) Managing appointment-based services in the presence of walk-in customers. In: Working paper. Columbia University, New YorkGoogle Scholar
  39. Weiss G (1995) On almost optimal priority rules for preemptive scheduling of stochastic jobs on parallel machines. Adv Appl Probab 27:821–839MathSciNetCrossRefzbMATHGoogle Scholar
  40. Wiesche L, Schacht M, Werners B (2016) Strategies for interday appointment scheduling in primary care. Health Care Manag Sci.
  41. Zacharias C, Pinedo M (2014) Appointment scheduling with no-shows and overbooking. Prod Oper Manag 23(5):788–801CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Antai College of Economics and ManagementShanghai Jiao Tong UniversityShanghaiChina

Personalised recommendations