Daily scheduling of surgical operations is a complicated and recurrent problem in the literature on health care optimization. In this study, we present an often overlooked approach to this problem that incorporates a rolling and overlapping planning horizon. The basis of our modeling approach is a Markov decision process, where patients are scheduled to a date and room on a daily basis. Acknowledging that both state and action space are only partially observable, we employ our model using a simulation-based method, where actions are derived from a heuristic search procedure. We test the potential of using this modeling approach on the resulting hospital costs and number of patients that are outsourced to avoid violating constraints on capacity. Using data from a Danish hospital, we find a distinct improvement in performance when compared with a policy that resembles a manual planner. Further analysis shows that substantial improvements can be attained by employing other simple policies.