## 6 Computational Results
### Lower Bounds for Datasets B and X
#### 6.5.2 Dual Bounds Decomposing Scenario by Scenario  
Computations of dual bounds \(LB\) are tractable for each instance from datasets B and X with LP computations or truncated MILP computations for one scenario and time step aggregation. Table 10 compares the dual bounds of [16] with one resulting from \(LB(4)\), \(LB(5)\), and \(LB\) formulations with LP, LP+cuts, and MILP truncated computations. \begin{table}
\begin{tabular}{l c c c c c c c c c} \hline \hline  & **LB(4)** & **LB(4)** & **LB(4)** & **LB(5)** & **LB(5)** & **LB(5)** & **LB(5)** & **LB** & **LB** \\  & **LP** & **LP + cuts** & **MILP** & **LP** & **LP + cuts** & **MILP** & **LP + cuts** & **MILP** \\ \hline B6 & 16.58\% & 7.24\% & 7.06\% & 6.84\% & 5.67\% & 5.48\% & 5.31\% & 5.48\% & **5.07\%** \\ B7 & 15.58\% & 7.68\% & 7.54\% & 7.23\% & 6.82\% & 6.68\% & **6.34\%** & 7.99\% & 7.55\% \\ B8 & 23.60\% & 13.86\% & 13.68\% & 13.52\% & 12.75\% & 12.41\% & **12.22\%** & 16.16\% & 16.16\% \\ B9 & 21.72\% & 12.70\% & 12.55\% & 12.40\% & 11.96\% & 11.72\% & **11.51\%** & 15.81\% & 15.81\% \\ B10 & 18.03\% & 8.53\% & 8.41\% & 8.33\% & 7.52\% & 7.39\% & 7.32\% & 7.53\% & **7.01\%** \\ \hline X11 & 15.40\% & 7.95\% & 7.81\% & 7.64\% & 6.33\% & 6.17\% & 6.04\% & 5.66\% & **5.36\%** \\ X12 & 14.22\% & 7.58\% & 7.50\% & 7.42\% & 6.81\% & 6.73\% & 6.68\% & 6.88\% & **6.49\%** \\ X13 & 18.53\% & 10.74\% & 10.37\% & 10.12\% & 8.65\% & 8.31\% & 8.14\% & 7.82\% & **7.46\%** \\ X14 & 17.21\% & 10.00\% & 9.81\% & 9.62\% & 8.27\% & 8.06\% & 7.91\% & 8.10\% & **7.76\%** \\ X15 & 16.83\% & 10.21\% & 10.11\% & 10.02\% & 9.06\% & 9.16\% & 8.99\% & 9.11\% & **8.82\%** \\ \hline Total & 17.80\% & 9.65\% & 9.49\% & 9.32\% & 8.40\% & 8.21\% & 8.05\% & 9.09\% & 8.79\% \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Dual bounds for datasets B and X, computing \(LB(k_{0})\) with \(k_{0}>3\) and LB bounds with M = 1 scenario decomposition, using truncated LP, LP+cuts, and MILP computations. \begin{table}
\begin{tabular}{l c c c c c c c c c} \hline \hline  & **Primal** & **[16]** & **Gap** & **Our Duals** & **Gap** & **t(s)** & **Model** & **Cuts** & **M** \\ \hline B6 & 83,424.7 M & 69,592 M & 16.58\% & 79,179.4 M & 5.09\% & 3202 & LB(5) & yes & 10 \\ B7 & 81,099.7 M & 68,528 M & 15.58\% & 76,139.8 M & 6.20\% & 3529 & LB(5) & yes & 5 \\ B8 & 81,899.7 M & 62,594 & 23.60\% & 67,352.4 M & 17.79\% & 1521 & LB(2) & no & 1  
We note that, contrary to computations in the average scenario presented in [17; 24], the B&B convergence of \(LB\) computations was less advanced for each scenario in the computations of Table 10, and the average scenario seems to be favorable for the efficiency of the B&B search with Cplex. In addition to Tables 7 and 9, Table 10 presents the gaps obtained using formulations \(LB(k_{0})\) with \(k_{0}\) varying. Table 10 extends some empirical results mentioned for Tables 7 and 9. The LP+cuts computations of root node of the B&B tree induce a few degradation of the lower bounds compared with the full MILP computations of the same model, for a significant speed up in the computational times when computations are stopped at the root node. In a defined time limit, it is more efficient to increase the value \(k\) to compute \(LB(k)\) bounds. Computation times are not reported in Table 10, and the MILP computations reach the defined time limits, 1800 s by sub-problem. Such quality of dual bounds can thus be obtained in about 1800 s in clock time, distributing the \(S\) computations of sub-problems in a cluster of \(S\) computers with MPI distributed implementation. In Table 10, it is interesting to compare the columns related to \(LB(5)\) and \(LB\) dual bounds. For most of the instances, computing \(LB\) dual bounds induces slightly better bounds, \(LB\) outclasses \(LB(5)\) computations by more than 0.5% only for X11 and X13 instances. Contrary to Table 6, the lower bounds obtained with \(LB(5)\) dual heuristics may be better than the exact formulation.