# Bounding the Likelihood of Exceeding Ward Capacity in Stochastic Surgery Scheduling
## 4 Conclusions  
Compared to actual scheduling data, the results suggest that utilization of both the wards and the operating rooms (ORs) is more evened out using the ward combination optimization (WCO) for the same level of throughput. Since ORs and wards are operating  
\begin{table}
\begin{tabular}{l c c c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**OR**} & \multicolumn{6}{c}{**Ward**} & \multicolumn{6}{c}{**MIP**} \\ \cline{2-11}
**Configuration** & \multicolumn{3}{c}{**Overtime 1**} & \multicolumn{3}{c}{**No. Beds over 2**} & \multicolumn{3}{c}{**Risk of Overflow 3**} & \multicolumn{3}{c}{**CPU**} \\ \cline{2-11}
**(\(M^{A}\), \(\omega\))** & **Regular** & **Extended** & **Min** & **Median** & **Mean** & **Max** & **Median** & **Mean** & **Max** & **(s)** \\ \hline (5, 1.00) & 6 & 4 & 2 & 19 & 19.65 & 47 & 0.17 & 0.28 & 1.00 & 18 \\ (5, 0.75) & 6 & 4 & 0 & 10 & 10.76 & 31 & 0.20 & 0.27 & 0.86 & 27 \\ (5, 0.50) & 6 & 4 & 0  
close to full capacity, overtime cannot be avoided. However, using the WCO, the risk of overtime is concentrated to fewer days, leading to fewer disruptions of the schedule and improved utilization over the entire planning horizon. The robust ward optimization (RWO) is more conservative, as it depends on the worst-case outcome for the ward LOS. As a result, lower quality solutions in terms of OR overtime and ward utilization are produced. Similar to [28], we observe a trade-off between the robustness of the solution and computational tractability. The likelihood of exceeding the available staffed ward beds is affected by the approximation of the empirical distribution of LOS in the ward and the threshold set for accepted risk of overflow. Reducing the discretization error will increase robustness, but at the cost of increased computational time. Similarly, increasing the threshold for accepted risk also increases computational time, but this time due to the reduced number of feasible ward combinations. Higher quality solutions are achieved using the WCO and RWO compared to actual scheduling data. First, the risk of overtime is lower compared to actual data. Second, the overall ward numbers are substantially lower, suggesting that using WCO or RWO can hedge against the risk of exceeding the number of staffed ward beds and thus reduce the risk of last-minute cancellations. Nevertheless, as we noticed from the scheduler's notes, unforeseen last-minute cancellations disrupt the schedule and cause imbalances in the utilization of ORs and the ward. Thus, a direction for future research is to anticipate the uncertainty of arrivals of elective patients during long-term operational planning horizons. Another possible future research is extending the RWO further by discritizing the worst-case LOS for a single patient into at least three intervals. This would reduce the possibility of exceeding the staffed ward beds when there is a large number of patients in ward. Conceptualization, A.O.S., T.P.R. and R.J.S. ; Data curation, A.O.S. and T.P.R. ; Formal analysis, A.O.S., T.P.R. and R.J.S. ; Funding acquisition, T.P.R. and R.J.S. ; Investigation, A.O.S. ; Methodology, A.O.S., T.P.R. and R.J.S. ; Project administration, R.J.S. ; Resources, A.O.S. and T.P.R. ; Software, A.O.S. and T.P.R. ; Supervision, T.P.R. and R.J.S. ; Validation, A.O.S. and T.P.R. ; Visualization, A.O.S. ; Writing--original draft, A.O.S. ; Writing--review & editing, T.P.R. and R.J.S. All authors have read and agreed to the published version of the manuscript. This project was supported by the Icelandic Technology Development Fund grant number 175373-0611. Not applicable. The authors would like to acknowledge the staff and the managers at Landspitali for giving insights and support to this project. We also want acknowledge the reviewers for their comments which helped improving the paper. The authors declare no conflict of interest.