# Bounding the Likelihood of Exceeding Ward Capacity in Stochastic Surgery Scheduling
## 1 Introduction  
Today, hospital managers seek ways to maximize the use of existing resources in response to aging populations and increasing cost of care. In this respect, operating rooms (ORs) have received considerable attention. They are both an expensive resource and a significant source of income [1, 2]. However, patient flow from ORs is constrained by downstream resources, such as intensive care units (ICU) [3], the post-anesthesia care unit (PACU) [4] and wards [5, 6]. To achieve increased OR utilization, surgeries need to be scheduled in such a way as to maximize throughput and to avoid last-minute cancellations due to overtime or downstream bottlenecks. This is a challenging task due to the stochastic nature of patient arrivals [7], surgery times [8, 9], length of stay [3, 10], and the competing objectives of different stakeholders in the surgery process [11, 12] that create artificial variability for elective surgeries [13]. In practice, surgery scheduling is commonly divided into three distinct decision levels: strategic, tactical, and operational [14, 15]. The first decision level is the strategic level. At this level, decisions are made about the overall surgical capacity, including ORs and equipment. These decisions are long-term, i.e., they affect the scheduling for at least a year [14]. The second decision level is the tactical level, where a cyclic master surgical schedule (MSS) is decided. The MSS specifies the allocation of available OR time to surgical specialties and/or operators for each day and room [15]. This strategy is referred to as block scheduling. An open operating room is here referred to as a block, and the opening hours are referred to as block length. Moreover, these schedules may also refer to finding an ideal mix of surgical procedures [16, 17, 18, 19, 20]. Tactical decisions are medium-term, i.e., they are usually made months in advance. Finally, at the operational level, patients waiting for elective surgeries are scheduled to blocks and times based on the MSS [21]. These decisions are short-term, i.e., they are usually made at least one week in advance. Commonly, a distinction is made between assigning patients to a block and sequencingpatients within a block [22; 23], even if both may be done at the same time [24]. Addressing the stochastic and integrative nature of the surgery process is important for implementing surgery scheduling systems in practice. In this paper, we focus on the assignment of elective surgeries at the operational level, taking into account uncertainties in both surgery times and length of stay (LOS) in the ward. Sequencing within blocks is not considered. Recent literature reviews [15; 21; 25; 26; 27] give a general overview of the vast literature on surgery scheduling. Studies addressing uncertainties in both surgery times and LOS either use stochastic programming [3; 10; 28] or robust optimization [2; 29]. In stochastic programming (SP), scenarios are used to represent uncertainty for surgery times and LOS in the downstream resources [3]. In this case, expected values such as overtime and ward numbers are minimized, guaranteeing good performance on average. The constraints are soft in this case. In robust optimization (RO), constraints are dealt with in a hard manner, leading to conservative solutions. Robust optimization (RO) uses uncertainty sets to guarantee feasibility towards the worst-case outcome based on the selected level of robustness [2; 29]. For example, uncertainty sets are created to represent the uncertainty for surgery times and LOS. RO is also an alternative to SP when distributions of stochastic factors are hard to identify [2]. Alternatively, distributional RO is an intermediate approach between SP and the RO, where the worst-case distribution is chosen [30]. The focus of this work is to minimize the likelihood of exceeding the limited number of staffed ward beds for a high-patient-throughput surgery schedule while minimizing the number of ORs resulting in overtime in the planning horizon.