  We study a single-server appointment scheduling problem with a fixed sequence
of appointments, for which we must determine the arrival time for each
appointment. We specifically examine two stochastic models. In the first model,
we assume that all appointees show up at the scheduled arrival times yet their
service durations are random. In the second model, we assume that appointees
have random no-show behaviors and their service durations are random given that
they show up at the appointments. In both models, we assume that the
probability distribution of the uncertain parameters is unknown but can be
partially observed via a set of historical data, which we view as independent
samples drawn from the unknown distribution. In view of the distributional
ambiguity, we propose a data-driven distributionally robust optimization (DRO)
approach to determine an appointment schedule such that the worst-case (i.e.,
maximum) expectation of the system total cost is minimized. A key feature of
this approach is that the optimal value and the set of optimal schedules thus
obtained provably converge to those of the true model, i.e., the stochastic
appointment scheduling model with regard to the true probability distribution
of the uncertain parameters. While our DRO models are computationally
intractable in general, we reformulate them to copositive programs, which are
amenable to tractable semidefinite programming problems with high-quality
approximations. Furthermore, under some mild conditions, we recast these models
as polynomial-sized linear programs. Through an extensive numerical study, we
demonstrate that our approach yields better out-of-sample performance than two
state-of-the-art methods.
