## 6 Computational Results
### Lower Bounds for Dataset A
\begin{table}
\begin{tabular}{l c c c c c c c c c} \hline \hline  & & **LB(4)** & **LB(5)** & **LB** & **LB** & **LB** & **LB** & **LB** & **LB** \\  & **[16].1** & **[16].2** & **MILP** & **MILP** & **LP** + **cuts** & **MILP** & **MILP** & **MILP** & **MILP** \\  & & & **M = 1** & **M = 1** & **M = 1** & **M = 1** & **M = 2** & **M = 3** & **M = 5** \\ \hline A1 & 5.09\% & 2.31\% & **0.54**\% & **0.08**\% & 0.29\% & 0.09\% & **0.03**\% & **0.02**\% & **0.02**\% & **0.01**\% \\ A2 & 10.83\% & 4.09\% & **1.42**\% & **0.33**\% & 0.64\% & 0.35\% & **0.24**\% & **0.22**\% & **0.21**\% & **0.21**\% \\ A3 & 10.72\% & 3.77\% & **1.85**\% & **0.71**\% & 1.06\% & 0.48\% & **0.36**\% & **0.35**\% & **0.34**\% & **0.33**\% \\ A4 & 26.07\% & 8.22\% & **3.36**\% & **2.38**\% & 2.82\% & 1.93\% & 1.33\% & 1.27\% & 1.27\% & 1.22\% \\ A5 & 24.22\% & 9.70\% & **4.25**\% & **3.24**\% & 3.46\% & 2.80\% & 2.38\% & 2.16\% & 2.01\% & 2.14\% \\ \hline Total & 14.20\% & 5.24\% & **2.11**\% & **1.19**\% & 1.49\% & 1.00\% & 0.77\% & 0.71\% & 0.68\% & 0.69\% \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Dual bounds for the dataset A, comparison of former dual bounds of the literature with the dual bounds \(LB(k_{0})\) with \(k_{0}>3\) and \(LB\) without aggregation of production time steps and scenario decomposition. M indicates the maximal number of scenarios in sub-problems. MILP computation of dual bounds are stopped after \(2400\times M\) seconds using Machine B. Bold values denote stopped computations with optimal dual bounds in the 0.001% tolerance for MILP computations. **LP** and **LP + cuts** are also the terminating values, no additional gap is implied by the time limit. all the variables \(d_{i,k,w}\) and \(p_{i,k,s,t}^{T2}\) are initialized firstly before pre-processing operations detect that constraints (3), (4) and (7) allow for deleting most of the variables \(d_{i,k,w}\) and \(p_{i,k,s,t}^{T2}\). Production variables \(p_{i,k,s,t}^{T2}\) have the most impact on these difficulties. One way to reduce the MILP loading time would be an implementation using Cplex API to define only the required variables \(d_{i,k,w}\) and \(p_{i,k,s,t}^{T2}\) taking into account constraints (3), (4) and (7), so that Cplex will load a smaller matrix before the MILP solving. In Table 6, the improvements with partial scenario decomposition are significant, but not so important to close the gap to the optimal dual bounds. For instance A1, we have the optimal values of best and the worst lower bounds of LB with scenario decomposition from Proposition 5: any scenario decomposition implies a gap to the BKS between 0.04%, the worst bound with \(M=1\), and 0.01%, the best bound with \(M=10\). The little impact of scenario decomposition is explained with Figure 1, showing the demand curves for each scenario in instance A5. The stochastic scenarios are distributed with a tube structure related to the meteorological seasonality, which is highly correlated to the power demands. It explains that few improvements are observed, the scenarios are relatively close, and induce optimal maintenance dates in similar zones of the time horizon. One may find, for any subset of scenarios, a good compromise in the dates of maintenance operations for each considered scenario. It induces also that choosing badly partitions of scenarios in Proposition 5 may induce very few differences with the scenario decomposition with \(M=1\). The significant gap improvements show the interest of a careful partitioning procedure with Algorithm 1.