# Bounding the Likelihood of Exceeding Ward Capacity in Stochastic Surgery Scheduling
## 3 Experimental Study
### Parameter Analysis  
Parameter analyses are performed to explore the trade-offs between computational requirements and quality of solutions of the WCO and RWO for a given range of model parameters. The results are presented in Tables 2 and 3. In Table 2, one can see that changing the parameters \((M^{A},\Omega,|\mathcal{K}|)\) has little effect on the number of times the optimal solution surpasses the accepted risk of overtime, both regular and extended. As we have already noted in Figure 4, most of the ORDSs that surpass the threshold on the accepted risk of entering overtime are composed of few surgeries and are unavoidable for any solutions that require all surgeries to be scheduled. Changing the parameters affects the median and the maximum number of beds going over the given bed limit and the maximum risk of overflow. Decreasing the bounds of the risk of overflow (values of \(\Omega\)), for the same resolution in the discretization of the LOS distribution \(|\mathcal{K}|\) and the number of staffed beds (\(M^{A}\)), lowers the median and the maximum number of beds over the given limits for each solution. For example, with \(M^{A}=6,\ |\mathcal{K}|=5\), and \(\Omega=1.00\) (no bounds on the risk of ward overflow), the median number of beds over the given bed limit has a maximum of 22. A similar effect can be seen for the mean, median, and maximum risk of ward overflow, with the largest drop being in the value of the maximum risk. Decreasing the discretization error for the same values of \(\Omega\) and \(M^{A}\) lowers the maximum risk of overflow and also median/maximum number of beds over. For \(M^{A}=6\) and \(\Omega=0.15\), the maximum risk of ward overflow decreases by 35% when \(|\mathcal{K}|\) increases from 4 to 6. Similarly, the maximum number of beds over decreases. It was impossible for some settings to find a feasible solution. This was evident when a low value for \(\Omega\) was imposed. \begin{table}
\begin{tabular}{l c c c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**OR**} & \multicolumn{3}{c}{**Ward**} & \multicolumn{3}{c}{**MIP**} \\ \cline{2-10}
**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{2-10}
**(\(M^{A}\), \(\Omega\), \(|\mathcal{K}|\))** & **Regular** & **Extended** & **Min** & **Median** & **Mean** & **Max** & **Median** & **Mean** & **Max** & **(s)** \\ \hline (5, 1.00, 5) & 6 & 4 & 3 & 16 & 16.18 & 44 & 0.16 & 0.32 & 0.98 & 95 \\ (5, 0.75, 5) & 6 & 4 & 1 & 10 & 11.13 & 46 & 0.15 & 0.28 & 0.89 & 133 \\ (5, 0.50, 5)  
Altering the values of the different parameters impacts the time it takes to solve the WCO. First, the maximum staff ward beds (\(M^{A}\)) and the bounds on the risk of overflow (\(\Omega\)) influences the computational time. For \(M^{A}=6\), feasible solutions are found for all values of \(\Omega\) for \(|\mathcal{K}|=5\), but computation time increases from 121 s to 3365 s. A similar effect can be identified for other settings. Second, computational time will also be dependent on the discretization accuracy of the LOS distributions. As \(|\mathcal{K}|\) is increased from four to six for \(M^{A}=6\) and \(\Omega=0.15\), the computation time increases from 362 s. to 6505 s (17 times). For some cases, when increasing the values of \(\Omega\), it was impossible to find feasible solutions within the limit on computational time. In Table 3, one can identify similar effects by changing the values of \(\omega\) when using the RWO. When \(\omega\) decreases, one can see that the number of beds over decreases (both median and max). The same applies to the risk of ward overflow. For most settings, the regular and extended overtime remains the same. However, for the configuration of \((6,\ 0.15)\), regular and extended overtime values increase. The results are similar for the WCO and the RWO for the number of beds over (median and max), but they are generally slightly lower for the WCO.