# Outpatient Appointment Optimization: A Case Study of a Chemotherapy Service
## 6 Computational Results
### Numerical Test Results  
The relative gap for method \(m\) on an instance is calculated using the following formula:  
\[\text{GAP}_{m}=\left(\frac{\text{objective value found by m}}{\text{Optimal objective value or best LB found by MILP}}-1\right)\times 100\%\]  
Figure 8 and Table 5 show similar behaviors for the different heuristic methods with small instances (\(N_{a}<90\)). However, when the total number of treatment sessions grows (\(N_{a}\geq 90\)), CPLEX cannot find an optimal solution within the limit of 1 hour due to exponentially increased combinations (Figure 9). Additionally, from this point, the performance of solving methods can be grouped into three distinct tiers:  
* **Low performance:** including _RLIPT-ID_, _RLIPT-II_, _SPT_, _SPT_ in which the GAP vastly climbed from under 10% to over 20%, even 35% in weekend's scenario. * **Medium performance:** including _RLIPT-DD_, _RLIPT-DI_, _LIPT_, _LPT_, in which the GAP increased arguably slower than **low** group and stayed overall below 15%. * **Good performance:** including MILP and GA, which in most cases give lower GAPs than the **medium** group, generally under 10%. In general, CPLEX is effective for solving the proposed MILP. Nevertheless, when the problem grows in size and resource availability congestion appears, the GA method yields better performance. Furthermore, GA is better in terms of execution time (Figure 9 and Table 6). \begin{table}
\begin{tabular}{c c c c c c} \hline
**Instances** & \(P_{C}\) & \(P_{M}\) & **Min** & **Max** & **Mean** \\ \hline
180 daily 8 & 0.80 & 0.01 & 14.18 & 17.64 & 15.93 \\  & & 0.10 & 14.28 & 17.92 & 15.96 \\  & 1.00 & 0.01 & 14.30 & 17.37 & 15.59 \\  & & 0.10 & **14.12** & **17.24** & **15.88** \\ \hline
210 daily 9 & 0.80 & 0.01 & 4.86 & 6.97 & 5.74 \\  & & 0.10 & **4.54** & 6.92 & 5.69 \\  & 1.00 & 0.01 & 4.82 & 6.47 & 5.54 \\  & & 0.10 & 4.60 & **6.36** & **5.45** \\ \hline
90 weekly 3 & 0.80 & 0.01 & 0.37 & **0.71** & **0.50** \\  & & 0.10 & 0.39 & 0.74 & 0.54 \\  & 1.00 & 0.01 & **0.31** & 0.72 & 0.52 \\  & & 0.10 & 0.39 & 0.69 & 0.51 \\ \hline \end{tabular}
\end{table} 
Table 4: Attained GAP (%) at the 500 th generation for different difficulties and different values \((P_{C};P_{M})\). The best results are in bold. Figure 8: The relative gap of different approaches for varying instances sizes. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline
**Scenario** & \(N_{a}\) & \begin{tabular}{c} **RLIPT-** \\ **DD** \\ \end{tabular}  & \begin{tabular}{c} **RLIPT-** \\ **DI** \\ \end{tabular}  & \begin{tabular}{c} **RLIPT-** \\ **ID** \\ \end{tabular}  & \begin{tabular}{c} **RLIPT-** \\ **II** \\ \end{tabular}  & \begin{tabular}{c} **LIPT** \\ **II** \\ \end{tabular}  & \begin{tabular}{c} **LIPT** \\ **LPT** \\ \end{tabular}  & \begin{tabular}{c} **SIPT** \\ \end{tabular}  & \begin{tabular}{c} **SPT** \\ \end{tabular}  & \ \begin{tabular}{c} **GA** \\ \end{tabular}  & \begin{tabular}{c} **MILP** \\ \end{tabular}  \\ \hline daily & 15 & 0.75 & 0.96 & 1.22 & 0.95 & 0.92 & 0.75 & 1.01 &  
Finally, regarding the heuristics, we can see their GAP is decided by how they prioritized the complicated jobs regarding the heuristics. The **low** group puts the _short_ jobs at the top of their list, while the **medium** group plans the _long_ jobs first. Hence, good heuristics should focus on the number of appointments and the total processing time. On the other hand, almost no difference is observed between _RLIPT-DD_ and _RLIPT-DI_: the priority lists of both methods put the lengthier jobs' sequences higher, but the former gives priority to longer processing times. At the same time, the latter arranges for a shorter processing time. In other words, tie-breaking by processing time has little effect. **The number of treatment sessions in a sequence** is probably critical for better performance, confirming our intuition for the RLIPT priority rule.