Surgery Scheduling in Flexible Operating Rooms by using a Convex Surrogate Model of Second-Stage Costs

We study the elective surgery planning problem in a hospital with operation rooms shared by elective and emergency patients. This problem can be split in two distinct phases. First, a subset of patients to be operated in the next planning period has to be selected, and the selected patients have to be assigned to a block and a tentative starting time. Then, in the online phase of the problem, a policy decides how to insert the emergency patients in the schedule and may cancel planned surgeries. The overall goal is to minimize the expectation of a cost function representing the assignment of patient to blocks, case cancellations, overtime, waiting time and idle time. We model the offline problem by a two-stage stochastic program, and show that the second-stage costs can be replaced by a convex piecewise linear surrogate model that can be computed in a preprocessing step. This results in a mixed integer program which can be solved in a short amount of time, even for very large instances of the problem. We also describe a greedy policy for the online phase of the problem, and analyze the performance of our approach by comparing it to either heuristic methods or approaches relying on sampling average approximation (SAA) on a large set of benchmarking instances. Our simulations indicate that our approach can reduce the expected costs by as much as 20% compared to heuristic methods and is able to solve problems with $1000$ patients in about one minute, while SAA-approaches fail to obtain near-optimal solutions within 30 minutes, already for $100$ patients.