An integrated distributionally robust model for two-echelon patient appointment scheduling

We develop a distributionally robust optimization (DRO) model for the outpatient appointment scheduling problem of a set of patients served in two serial stages, consultation and examination. The arrival sequence of patients is known, and the problem of scheduling is to assign appointment time for each patient to minimize total cost with random service time for two serial stages. A max–min problem is formulated for the two-stage appointment scheduling as a whole, in which the waiting time exhibits a high degree of coupling due to the continuous two-stage process. To address this, we devise a two-stage network maximum flow model that provides an equivalent linear expression for the waiting time. For the inner maximum problem, we employ a conic programming approach for equivalent representation, incorporate the scheduling decision of the outer minimum problem, and convert the model to its equivalent copositive programming by taking the conic duality. We conduct numerical experiments and sensitivity analysis using real and simulated data, and the results verify the effectiveness of our proposed DRO model.