It’s a complex math problem that cannot be solved with pen and paper
Improving chair utilization for an infusion treatment center is complex largely because of the wide range of treatment durations — some treatments require less than 45 minutes while others require more than 8 hours. If a large majority of treatments required approximately the same length of treatment, the problem would be a lot simpler.
The key is to mine the pattern of prior appointments in order to develop a realistic estimate of the volume and mix of appointment types for each day of the week. Armed with that information, it is possible to create an optimal pattern of “slots” — which reflect the number of “appointment starts” of each duration (i.e., 1 hour duration, 3 hour duration, etc.) at each appointment time (i.e., 9:10am, 9:20 am, etc.). The slots have to take as many of the real operational constraints into account. These include the hours of operation, the number of chairs, various “rules” that depend on clinic schedules as well as patient-centric policies (e.g., treatments longer than 4 hours should be assigned to a bed and not a chair).
The Holy Grail is to move the chair utilization graph from the typical “triangle that peaks somewhere between 11am and 2pm each day to a “trapezoid that ramps up smoothly between 7am and 9am and then stays flat from 9am until 4pm and then ramps down smoothly from 4pm until 7pm”. Mathematical optimization makes this possible.