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Stochastic Modeling for Clinical Scheduling

Stochastic Modeling for Clinical Scheduling. by Ji Lin Reference: Muthuraman, K., and Lawley, M. A Stochastic Overbooking Model for Outpatient Clinical Scheduling with No-shows, submitted. Outline. Introduction to Clinical Scheduling Probability model Different policies

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Stochastic Modeling for Clinical Scheduling

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  1. Stochastic Modeling for Clinical Scheduling by Ji Lin Reference: Muthuraman, K., and Lawley, M. A Stochastic Overbooking Model for Outpatient Clinical Scheduling with No-shows, submitted

  2. Outline • Introduction to Clinical Scheduling • Probability model • Different policies • Results and discussions • Recent work

  3. Traditional appointment scheduling vs. Open access scheduling • Traditional appointment scheduling • - A patient is scheduled for a future appointment time • - lead time can be very long • - In some clinics, up to 42% of scheduled patients fail to show up for pre-booked appointments • Open access scheduling • - Patients get an appointment time within a day or two of their call in. • - see doctor soon when needed • - More reliable no-show predictions

  4. Overbooking strategy • Airline industry • Fixed cost, capacity limits and fares on different class seats, • A low marginal cost of carrying additional passengers. • Either reserves or refuses a passenger. • System dynamics keeps the same for overshow situations (financial penalty)

  5. Overbooking strategy 2 • Clinical scheduling • Stochastic patient waiting time and staff overtime • The scheduler must search for an optimal appointment time • System dynamics changes (longer patient waiting times and excessive workload)

  6. Model and Assumptions • Single server • A single service period is partitioned into time slots of equal length. • Patients call-in before the first slot • Once an appointment is made, it cannot be changed. • Patients have no show probabilities and are independent from each other • All arrived patients need to be served. • Service times are exponentially distributed

  7. Call-in Procedure Choose a slot or refuse to schedule No Show Estimation Call-in

  8. Service system • Xi - The number of patients arriving for slot i • Yi - The number of patients overflowing from slot i into slot i+1 • Li - The number of services that would have been completed provided the queue does not empty • min(Li,Yi−1+Xi) - The actual number of services completed.

  9. Objective • Minimize • Patient waiting times • Stuff overtime • Maximize • Resource Utilization

  10. Weighted Profit Function • r – reward for each patient served • ci – cost for over flow from slot i to slot i+1 • Q – arrival probability matrix • R – over-flow probability matrix

  11. Attributes of this Appointment Scheduling • Static - Appointments made before the start of a session • Performance measure - Time based • Multiple block/Fixed-interval • Analytical Probability Modeling

  12. Scheduling policies • Round Robin • Myopic Optimal policy • Non Myopic Optimal policy

  13. Round Robin • assigns the ith customer to slot ((i−1) mod 8)+1.

  14. Myopic policy

  15. Simulation • Call-in process simulation

  16. Simulation(2) • Scheduled service simulation

  17. Results: The schedule and expected profit evolution

  18. Expected overflow from last slot

  19. Effect of Call-in Sequence

  20. Discussions • Myopic policy improved the max profit by approx. 30% (compare with Round Robin) • Myopic policy is not optimal, but it provides solutions within a few percent of the optimal sequential • The probability model is readily extendable easily. • Patient type need not to be finite. • Walk-in can be added into the model (only Q matrix will change) • The restriction of exponential service time can be eliminate by conditioning our expectation.

  21. Theory vs. Practice • Huge gap - Real clinic is much more complicated • More than one server • Registration, pre-exam, checkout, etc. • Physician's Restrictions • Probability model vs. simulation • The relaxed exponential service time within slots • Robustness of the policies

  22. Recent extend on optimal policy – Dynamic Programming approach

  23. Profit Function • Profit function is determined by current status and current time.

  24. Example of 2 patients and 2 call-in time periods

  25. Complexity • Optimal Policy is not stationary • For M call-in time periods and N Slots, There are final statuses • When M>>N, the Complexity is closed to (M+N)!, which is NP-hard, and not computable for large cases.

  26. Thank you!! Q&A

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