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

8 Service system X i - The number of patients arriving for slot i Y i - The number of patients overflowing from slot i into slot i+1 L i - The number of services that would have been completed provided the queue does not empty min(L i,Y i−1 +X i ) - 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 c i – 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