# Bounding the Likelihood of Exceeding Ward Capacity in Stochastic Surgery Scheduling
## 3 Experimental Study
### Comparison  
In Table 1, one can see the difference between optimal solutions from the WCO, RWO and actual scheduling data. First, compared to actual data, ORDS are less likely to surpass the accepted risk of regular overtime and the accepted risk of extended overtime for our models. For the regular overtime, the difference is the lowest (up to 25% lower) but the highest for the extended overtime (up to 71% lower). Comparing the ward results, one can see that the values are also lower for the models. This is apparent for both the number of beds over the given number of staffed ward beds of 6 (min, median, mean and max) and also for the maximum risk of overflow. Comparing WCO and RWO, one can see that WCO is of higher quality than RWO both in terms of overtime and ward overflow. To better understand the sources of the difference between the models, a visual representation of the actual, WCO and RWO scheduling are provided in Figures 3-5, respectively. The figures illustrate the surgeries for each day and room (indicated with 1, 3 and 6) and boxplots for the simulated ward occupancy. For the ward occupancy, the color is yellow if there is a risk of overflow, but grey if not. In practice, the scheduler will use the accumulated average surgery duration (AASD) to create ORDS by hand. Thus, the figures reflect what the scheduler can see in their planning software. Different 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. 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 tags \(u\) and \(v\) denote if the threshold for the risk of regular or extended overtime is surpassed, respectively. In the figures, three dashed lines are shown. For the ORs, two lines are shown to represent the opening hours of the ORs (the capacity parameter \(C_{d,r}\)). For the ward, a single dashed line is shown for the maximum number of staffed ward beds (\(M^{A}\)) at the specialty. \begin{table}
\begin{tabular}{l c c c c c c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**OR**} & \multicolumn{4}{c}{**Ward**} \\ \cline{2-13}  & \multicolumn{4}{c}{**Overtime**} & \multicolumn{4}{c}{**Risk of Overtime**} & \multicolumn{4}{c}{**No. Beds over**} & \multicolumn{4}{c}{**Risk of Overflow**} \\ \cline{2-13}
**Case** & **Regular** & **Extended** & **Mean** & **Median** & **Max** & **Min** & **Median** & **Mean** & **Max** & **Median** & **Mean** & **Max** \\ \hline Actual \({}^{\dagger}\) & 8 & 14 & 0.20 & 0.32 & 1.00 & 0 & 9 & 9.66 & 27 & 0.02 & 0.15 & 1.00 \\ WCO \({}^{\star}\) & 6 & 4 & 0.10 & 0.17 & 0.68 & 0 & 1 & 1.79 & 14 & 0.01 & 0.06 & 0.25 \\ RWO \({}^{\ddagger}\) & 7 & 5 & 0.08 & 0.17 & 0.69 & 0 & 1 & 1.47 & 18 & 0.01 & 0.04 & 0.26 \\ \hline \hline \end{tabular} Configurations: \({}^{\ddagger}=6\); \({}^{\star}\) (\(M^{A}\), \(\Omega\), \(|\mathcal{K}|)=(6\), 0.15, 5); \({}^{\ddagger}\) (\(M^{A}\), \(\omega)=(6\), 0.15).  
\end{table} 
Table 1: Comparison between optimal solutions of the ward combinations optimization (WCO), the robust ward optimization (RWO) and actual schedule for the planning horizon. Regular and extended _overtime_ show how often selected ORDS surpass the accepted risk for each group. _Risk of overtime_ is the probability that the selected ORDS will surpass the block capacity. _No. of beds over_ measures how many beds exceed the given number of staffed ward beds while _risk of overflow_ is the likelihood of exceeding the number of staffed ward beds. Figure 3: Visualisation of the actual surgeries for each day and OR (I, 3 and 6) and the corresponding ward occupancy (for the ward occupancy, the color is yellow if there is risk of overflow and grey if not. The dashed line represents 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.