# Bounding the Likelihood of Exceeding Ward Capacity in Stochastic Surgery Scheduling
## 3 Experimental Study
### Comparison
The symbol \(+\) denotes if a patient arrived during the execution of the schedule, \(*\) denotes one-week priority and ICU denotes if ICU is required. The dashed lines represent the distinct opening hours of the ORs (\(C_{d,r}\)). The tags \(u\) and \(v\) denote if the threshold for the risk of regular or extended overtime is surpassed, respectively. Inspecting Figure 3, one may note the imbalance in the utilization of the ORs. Observing the AASD for each day and room, the ORs are, in many instances, utilized close to full capacity, while in others, they are under-utilized. For example, on days 3, 9, 15, 17, 18, 22, and 24, the AASD of at least one room is more than \(7,5\) h. However, on days 4, 5, 8, 10, 11, 16, 19, 23 and 25, at least one room has a low OR utilization (low OR hours). As the AASD is closer to full capacity, there is an increased risk of overtime, as can be seen from the tags denoting the surpassing of the accepted risk of regular (\(u\)) and extended (\(v\)) overtime above each OR in the figure. On some days, operating close to, or over, full capacity cannot be avoided. One can see this occurs on days 1, 17, 18, 22, and 23, when the average duration of a single surgery is close to the available opening time. The imbalances detected in the utilization of the ORs are also apparent in our simulation of ward occupancy (see Figure 3). The utilization of the ward is lowest during the weekends but increases at the beginning of each week and commonly reaches a maximum on Thursdays before decreasing again. In the first week, there is a high risk of overflow already on Wednesday (day 3) to Saturday (day 6) and again on day 18. The risk of overflow is relatively low and the ward occupancy is balanced in the second week (days 8-14). However, imbalances and a higher risk of overflow reappears in the last two weeks (days 15-28), even if they are not as severe as in the first week. In Figures 4 and 5, the optimal solutions are visualized for the WCO and RWO schedules in the same way as was done in Figure 3 for the actual schedule. Of the 103 surgeries included in both schedules, 28 are scheduled by the model on the same day as they were actually performed on for the WCO; this number is 23 for the RWO. This suggests that the optimized schedules and the actual one are different. Analyzing the figures, one can identify that, for the optimized schedules, the utilization of both the wards and the ORs is more evened out for the entire planning horizon. For example, the daily ward admissions are relatively balanced, with most days in the range of 1-2 admissions per day. Additionally, the risk of overflow is lower. For the ORs, one can observe that overtime has been concentrated to fewer days, leading to improved and more evened out utilization. Moreover, single surgeries that have, on their own, a high risk of entering overtime (those spanning the whole day) are never combined with other surgeries in the optimal solution (see e.g., Figure 4 on day 10 in room 1), whereas this occurs in the actual data (see e.g., Figure 3 on day 18 in room 1). Comparing the schedules of the WCO to the RWO, one can see that they differ in terms of overtime and ward utilization. More overtime is apparent in the RWO in the second week, but less is apparent in the third week. In terms of the wards, both solutions contain three days with a risk of overflow but on different days. Since RWO is more conservative as it depends on the worst-case outcome for the ward LOS, there is generally lower ward utilization. Figure 4: Visualisation of the optimal solution for each day and OR (1, 3 and 6) for the WCO and the corresponding ward occupancy (for the ward occupancy, the color is yellow if there is risk of overflow, but is grey if not. The dashed line represent the maximum number of staffed ward beds (\(M^{A}\))). Colors reflect the operators and the text is the code of the surgical procedure (if the surgical procedure code is in upper case, ward admission is required.