# Department Seminar by Jeroen De Mast, University of Amsterdam

Monday, April 24, 2017 — 4:00 PM EDT

## Mathematical optimization and a multiple-case study

Primary and secondary care are offered through outpatient clinics, where patients visit the hospital for a consultation or treatment. Some clinics work on a walk-in basis, but most work with scheduled appointments. Variation in the duration of consultations, random no-shows and walk-ins, and other sources of variability and uncertainty result in waiting time for patients and idle time for the clinician. The challenge, then, is to determine appointment schedules that achieve the best-possible synchronization of the patient’s arrival with the availability of the clinician despite this variability in the process.

1. Mathematical optimization

There is a large body of research in OR that designs procedures for computing schedules that strike a good balance between expected waiting time for patients and idle time for the clinician. Determining expectations of the waiting and idle time is, in general, analytically intractable, and the challenge is to find effective approximation and simplification techniques. We developed an approach where we approximate the service times by phase-type distributions. These approximate realistic distributions well, and in addition, their properties allow a recursive algorithm to determine the expected waiting and idle times analytically. Standard optimization techniques can be used to compute near-optimal schedules from there.

Next, we show how the approach can be extended in an elegantly simple manner to incorporate no-shows and walk-ins. The procedure outperforms simpler approximations almost uniformly in its precision, and it outperforms approaches based on simulation vastly in computation time and ease of use.

2. Multiple-case study

The way in which the problem of appointment scheduling is usually framed in mathematical optimization approaches has very meager substantiation, neither from theory in operations management, nor from field studies analyzing the objectives and challenges in real healthcare operations. We conducted a multiple-case study, conducted in 10 clinics, aimed at two research questions:

•    What is the structure of the problem of appointment scheduling in outpatient clinics?
•    What are current practices and underlying assumptions and conceptualizations in the field?

Based on the results, we propose that there are better ways to address the problem than scheduling, and these reflect modern approaches for production management in industry. Rather than absorbing the effects of variability in buffers of waiting and idle time, it is better to pursue that variability is reduced, and to absorb the remaining variability by flexibility of clinicians and other resources. Moreover, given the complexity and dynamic behavior of service processes in outpatient clinics, we recommend optimization approaches based on feedback adjustment, instead of the currently studied approaches based on optimal design.

Location
M3 - Mathematics 3
Room 3127
200 University Avenue West
Waterloo, ON N2L 3G1

### April 2018

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