Integer Programming Approaches for Appointment Scheduling with Random No-Shows and Service Durations

Abstract

We consider a single-server scheduling problem given a fixed sequence of job arrivals with random no-shows and service durations. The joint probability distribution of the uncertain parameters is assumed to be ambiguous and only the support and first moments are known. We formulate a class of distributionally robust optimization models that incorporate the worst-case expected cost and the worst-case conditional Value-at-Risk (CVaR) of appointment waiting, server idleness, and overtime as the objective or constraints. Our models flexibly adapt to different prior beliefs of no-show probabilities. We obtain exact mixed-integer nonlinear programming (MINLP) reformulations that facilitate decomposition algorithms, and derive valid inequalities to strengthen the reformulations. In particular, we derive the convex hulls for special cases of no-show beliefs, yielding polynomial-size linear programming reformulations for the least and the most conservative supports of no shows. We test various instances to demonstrate the computational efficacy of our approaches and provide insights for appointment scheduling under distributional ambiguity of multiple uncertainties.

Keywords: appointment scheduling, conditional Value-at-Risk (CVaR), distributionally robust optimization, mixed-integer nonlinear programming, convex hulls