Data-driven Risk-sensitive Appointment Scheduling : A Wasserstein Distributionally Robust Optimization Approach

We consider an optimal appointment scheduling problem for a single-server healthcare delivery system with random durations, focusing on the tradeoff between overtime work and patient delays which are measured under conditional value-at-risk (CVaR). To address the distributional ambiguity, we propose a data-driven distributionally robust optimization (DRO) approach with the ambiguity set constructed from a type-1 Wasserstein ball centered at an empirical service time distribution. We first derive the equivalent representation of the Wasserstein worst-case CVaR measure and then show that the Wasserstein worst-case CVaR measure is equivalent to its sample average approximation counterpart adjusted by a regularization term, which allows us to reformulate the Wasserstein DRO model as a tractable constrained optimization problem. We then show that the Lagrangian of the regularization reformulation possesses L#-convexity, a desired combinatorial property which guarantees a local optimum to be a global optimum and allows us to prove the strong duality through convex extension and develop an efficient algorithm. The regularization reformulation also implies that the L#-convexity can be preserved under Wasserstein DRO. We further extend our analysis to joint sequencing and scheduling problems. Our numerical study shows that Wasserstein DRO indeed provides robust out-of-sample performance and how the Wasserstein ball radius is selected empirically