## 3 Results
### Application to UPHES Management
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \(n_{batch}=1\) & **min** & **mean** & **max** & **sd** \\ \hline TuRBO & \(-2469\) & \(-898\) & \(\mathbf{368}\) & \(917\) \\ KB-\(q\)-EGO & \(-1582\) & \(-765\) & \(-27\) & \(\mathbf{416}\) \\ mic-\(q\)-EGO & \(-\mathbf{1511}\) & \(\mathbf{-609}\) & \(-39\) & \(487\) \\ MC-based \(q\)-EGO & \(-2123\) & \(-690\) & \(108\) & \(778\) \\ BSP-EGO & \(-2248\) & \(-1126\) & \(139\) & \(883\) \\ \hline \(n_{batch}=2\) & **min** & **mean** & **max** & **sd** \\ \hline TuRBO & \(-2106\) & \(19\) & \(1338\) & \(1058\) \\ KB-\(q\)-EGO & \(-1474\) & \(167\) & \(1147\) & \(773\) \\ mic-\(q\)-EGO & \(-\mathbf{429}\) & \(\mathbf{253}\) & \(\mathbf{1369}\) & \(473\) \\ MC-based \(q\)-EGO & \(-517\) & \(79\) & \(768\) & \(\mathbf{404}\) \\ BSP-EGO & \(-708\) & \(-101\) & \(723\) & \(503\) \\ \hline \(n_{batch}=4\) & **min** & **mean** & **max** & **sd** \\ \hline TuRBO & \(-1766\) & \(78\) & \(819\) & \(837\) \\ KB-\(q\)-EGO & \(-344\) & \(168\) & \(784\) & \(\mathbf{287}\) \\ mic-\(q\)-EGO & \(-\mathbf{96}\) & \(\mathbf{559}\) & \(\mathbf{1097}\) & \(405\) \\ MC-based \(q\)-EGO & \(-548\) & \(65\) & \(824\) & \(436\) \\ BSP-EGO & \(-389\) & \(90\) & \(746\) & \(412\) \\ \hline \(n_{batch}=8\) & **min** & **mean** & **max** & **sd** \\ \hline TuRBO & \(-972\) & \(86\) & \(\mathbf{1259}\) & \(713\) \\ KB-\(q\)-EGO & \(-703\) & \(-168\) & \(547\) & \(423\) \\ mic-\(q\)-EGO & \(-568\) & \(\mathbf{416}\) & \(1184\) & \(473\) \\ MC-based \(q\)-EGO & \(-\mathbf{362}\) & \(163\) & \(862\) & \(\mathbf{381}\) \\ BSP-EGO & \(-1102\) & \(78\) & \(504\) & \(463\) \\ \hline \(n_{batch}=16\) & **min** & **mean** & **max** & **sd** \\ \hline TuRBO & \(-827\) & \(\mathbf{51}\) & \(\mathbf{1162}\) & \(\mathbf{692}\) \\ KB-\(q\)-EGO & \(-2048\) & \(-963\) & \(658\) & \(752\) \\ mic-\(q\)-EGO & \(-2383\) & \(-706\) & \(684\) & \(1049\) \\ MC-based \(q\)-EGO & \(-2475\) & \(-1225\) & \(-14\) & \(948\) \\ BSP-EGO & \(-3509\) & \(-1183\) & \(408\) & \(1275\) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Minimum, maximum, average profit values (EUR) as well as standard deviation of the UPHES management problem obtained with 10 runs of each method according to batch size. Figure 9: Study on the scalability of each approach. Solid lines indicate the average over \(10\) runs, and dashed lines of same colors indicate the standard deviation. (**a**) Number of simulations as a function of the batch size. (**b**) Number of cycles as a function of the batch size. Increasing the batch size to 4 makes the conceptual differences between algorithms appear. Indeed, we can see that sequential AP heuristics of \(q\)-EGO like algorithms are quite time-consuming compared to APs of TuRBO and MC-based \(q\)-EGO. Even if BSP-EGO carries out \(2\times n_{batch}\) sub-APs, its parallel execution makes it as fast as MC-based \(q\)-EGO. Although the number of cycles is slightly smaller than for lower batch sizes, it is the best compromise regarding final expected profits of Table 7.  
Reaching \(n_{batch}=8\) allows to perform more simulations for any method, but at the cost of the number of cycles. The productivity per computing core is clearly diminished and as suggested by the final profits, it is not profitable in this context except maybe for TuRBO. Finally, consistently with benchmark section observations, the significant decrease in the number of cycles involves a drop in the number of simulations which is almost comparable to \(n_{batch}=4\). Nevertheless, as suggested in the BOTorch documentation [12], the gap between sequential heuristics and MC-based AF becomes smaller for large batch sizes because of the complexity of multi-points AF optimization. In addition, it has to be noted that relying on different single point AF, as in mic-\(q\)-EGO is efficient to reduce the acquisition time and to improve the quality of the batch of candidates for large batch sizes.