  Motivated by the increasing use of online appointment booking platforms, we
study how to offer appointment slots to customers in order to maximize the
total number of slots booked. We develop two models, non-sequential offering
and sequential offering, to capture different types of interactions between
customers and the scheduling system. In these two models, the scheduler offers
either a single set of appointment slots for the arriving customer to choose
from, or multiple sets in sequence, respectively. For the non-sequential model,
we identify a static randomized policy which is asymptotically optimal when the
system demand and capacity increase simultaneously, and we further show that
offering all available slots at all times has a constant factor of 2
performance guarantee. For the sequential model, we derive a closed-form
optimal policy for a large class of instances and develop a simple, effective
heuristic for those instances without an explicit optimal policy. By comparing
these two models, our study generates useful operational insights for improving
the current appointment booking processes. In particular, our analysis reveals
an interesting equivalence between the sequential offering model and the
non-sequential offering model with perfect customer preference information.
This equivalence allows us to apply sequential offering in a wide range of
interactive scheduling contexts. Our extensive numerical study shows that
sequential offering can significantly improve the slot fill rate (6-8% on
average and up to 18% in our testing cases) compared to non-sequential
offering.
