## 3 Results
### Application to UPHES Management  
Results displayed in Figures 3-7 present the average profit value of the UPHES management problem according to the number of simulations performed by the algorithms. Dashed lines are added to indicate the standard deviation around the average objective value. Optimization runs are repeated 10 times with 10 different initial sets, and each algorithm is run once with each initial set. Therefore, within each figure, every curve has the same starting point. Since the limiting factor is time, each execution of the optimization algorithms does not perform the same number of cycles (and simulations). Consequently, the curves only display the results for which all data are available, and the rest are truncated. Thereby, the final expected profit is not always visible on the graphs. Nevertheless, the final results at the end of the time budget are visible in Table 7. The table also presents the minimum, maximum and standard deviation of each set of optimizations, for all batch sizes and all approaches. First, looking at the average profit curves for batch sizes 1 and 2 in Figures 3 and 4, we observe very few differences in the execution of the algorithms. The average values are similar, with maybe a thin advantage for the mic-\(q\)-EGO algorithm thanks to its smaller variance. Table 7 confirms the previous observation that mic-\(q\)-EGO performs slightly better in terms of average profit, and has the highest minimal expected profit among the 10 executions compared to other approaches. Then in this context, mic-\(q\)-EGO seems to be the safest choice. However, according to the \(p\)-value of the pairwise Student's \(t\)-test summarized in Figure 8, no significant difference is observed (\(p\)-value \(>\) 0.1). Figure 4: Evolution of the best-known objective value according to the number of executed cycles with \(n_{batch}=2\). Solid lines indicate the average over 10 runs, and dashed lines of same colors indicate the standard deviation. Figure 3: Evolution of the best-known objective value according to the number of executed cycles with \(n_{batch}=1\). Solid lines indicate the average over 10 runs, and dashed lines of same colors indicate the standard deviation. Figure 5: Evolution of the best-known objective value according to the number of executed cycles with \(n_{batch}=4\). Solid lines indicate the average over 10 runs, and dashed lines of same colors indicate the standard deviation. Figure 6: Evolution of the best-known objective value according to the number of executed cycles with \(n_{batch}=8\). Solid lines indicate the average over 10 runs, and dashed lines of same colors indicate the standard deviation. Figure 8: Heatmap of the \(p\)-values of pairwise Student’s \(t\)-tests. Figure 7: Evolution of the best known objective value according to the number of executed cycles with \(n_{batch}=16\). Solid lines indicate the average over 10 runs, and dashed lines of same colors indicate the standard deviation. Regarding results for \(n_{batch}=4\) in Figure 5, from around 150 iterations on, the mic-\(q\)-EGO appears to outperform the other approaches. With 150 simulations, the latter reaches a final expected profit equivalent to its competitors with twice as many simulations. Others are visually equivalent, with only a noticeable difference regarding the standard deviation. Indeed, as for the previously mentioned curves (\(n_{batch}=1,\,2\)), TuRBO has the highest variance. Except for mic-\(q\)-EGO, which has the highest minimum, average and maximum profit for \(n_{batch}=4\), all the approaches have similar behavior and final outcomes. The advantage of mic-\(q\)-EGO is confirmed by the \(t\)-tests displayed in Figure 8 where a significant advantage for mic-\(q\)-EGO is visible. The \(p\)-value of comparison with KB-\(q\)-EGO, BSP-EGO and MC-based \(q\)-EGO is around 0.02 which provides strong evidence in favor of mic-\(q\)-EGO, while the comparison against TuRBO gives a \(p\)-value of 0.12 (less evidence, but strong assumption).