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Stochastic Modeling for Clinical Scheduling

byJi Lin

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

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Outline• Introduction to Clinical Scheduling

• Probability model

• Different policies

• Results and discussions

• Recent work

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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

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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)

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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)

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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

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Call-in Procedure

No Show Estimation

Call-inChoose a

slot or refuse to schedule

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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.

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Objective• Minimize

– Patient waiting times– Stuff overtime

• Maximize– Resource Utilization

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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

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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

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Scheduling policies• Round Robin

• Myopic Optimal policy

• Non Myopic Optimal policy

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Round Robin

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

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Myopic policy

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Simulation• Call-in process simulation

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Simulation(2)• Scheduled service simulation

17Results: The schedule and

expected profit evolution

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Expected overflow from last slot

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Effect of Call-in Sequence

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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.

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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

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Recent extend on optimal policy – Dynamic Programming approach

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Profit Function• Profit function is determined by current

status and current time.

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Example of 2 patients and 2 call-in time periods

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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.

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Thank you!!Q&A