Online Supplement to: Adaptive Appointment Systems with Patient Preferences

Let vectors α(t) = (α1(t), · · · , α l m(t)) and β (t) = (β 1(t), · · · , β l b(t)) denote estimated panel l physician and time-block acceptance probabilities after the tth update. We assume that each patient’s PCP is always acceptable to him/her and set αl(t) = 1 for each l and t. This is not a requirement of our model but rather a reasonable assumption in the application domain. It does simplify some computations because the clinic only needs to estimate each panel’s acceptance probabilities for the (m − 1) non-PCP. However, our approach can be reworked if it were important to consider a situation in which a patient’s PCP would not be acceptable to him/her, but a non-PCP would be for the same time block. In practice, patients are free to change their designated PCP as often as they wish. It is therefore unlikely that a patient will prefer not to visit his/her PCP at an acceptable time. The estimated acceptance probabilities after the tth update, αi(t) (resp. β l j(t)), can be obtained as the relative frequency that panel l’s patients include physician i (time-block j) in the acceptable set. That is, αi(t) = [α l i(0)θ + ∑t z=1 N l i,z]/(θ + t), where θ ≥ 0 is the prior count or the weight given to a subjective estimate of acceptance probability prior to information updating, and N l i,z = 1 if the zth panel l arrival includes physician i in the acceptable set and 0 otherwise. Similarly, β j(t) = [β l j(0)θ + ∑t z=1 H l j,z ]/(θ + t), where H j,z = 1 if time block j is included in the zth panel l arrival’s acceptable set, and 0 otherwise. A higher value of θ represents a higher level of confidence in the initial estimates. There can be a variety of ways to obtain initial estimates. For example, αi(0) can be obtained by calculating αi(0) = ∑ k 6=l n l k,0/n l 0 for i 6= l, where n l k,0 and n l 0 represent respectively the total number of appointments with physician k and with all physicians booked by panel l patients. That is, when a patient booked with any one of the non PCP, we may assume that each non PCP was equally acceptable to the patient. This is because we only observe actual bookings and not acceptable sets in the historical data. Similarly, β j(0) = h l j,0/n l 0 for time block j, where h l j,0 represents the total number of block j appointments by all patients in panel l and n0 is the total number of bookings from panel l patients in the historical records. The updating procedure does not depend on which acceptable combination is actually booked by the clinic. It utilizes knowledge of the composition of acceptable sets from the web-based appointment request system. To better understand the accuracy of the estimating procedure, consider a primary-care physician who on average receives 25 booking requests each day. In this instance, 500 updates are reached in about 20 days of operation. After these many updates, the standard error of the estimated acceptance probability is reduced to less than 0.02 for each (i, j) combination. Twenty days is a relatively short time in our application domain during which aggregate patient choices are unlikely to change much. Therefore, our approach converges rapidly to true probabilities and remains accurate so long as these probabilities do not