## 6 Computational Results
### Lower Bounds for Dataset A  
For none of the instance of dataset A, the hypothesis (40) holds to guarantee dual bounds with time step aggregation and Proposition 2. However, computations with disaggregated time steps are tractable with few stochastic scenarios, and dual bounds are computable using Proposition 5. Table 6 compares the dual bounds for the EURO/ROADEF Challenge 2010, computing dual bounds of \(LB\) with LP and LP+cut relaxations, and MILP computations truncated in 30 min, to the bounds obtained with surrogate relaxations \(LB(k)\) with \(k=4\) and \(k=5\). In Table 6, it is preferable to compute LB bounds instead of \(LB(k)\) bounds for the dataset A. Indeed, dual bounds are computable with LB bounds, avoiding the additional gaps to the primal solutions with approximations \(LB(k)\). For instances A1, A2 and A3, LB bounds are computable to MILP optimality with up to 10 scenarios. It induces lower bounds of a very good quality for these instances. For instances A4 and A5, the computation times do not allow for converging to optimality with MILP computations of LB in the defined time limits. Furthermore, the memory space is limiting, and computations with 3600 s per scenario are too much memory-consuming. We note that, for the A5 instance, it is preferable to optimize conjointly by groups of 3 instead of 5. This is due to the stopped MILP computation, and it is preferable to reach more advanced phases of the B&B algorithm with three scenarios rather than using more scenarios and stopping earlier than the B&B algorithm. Otherwise, increasing the number of scenarios increases the quality of the dual bounds in Table 6, and some relations were proven thanks to lemma 2 with optimal computations. This holds also empirically for the stopped computations on instance A4. Computation times are not reported in Table 6. Using Machine A, 6395 s (i.e., more than 1 h 45 min) are measured to solve to optimality \(LB\) with the 10 scenarios of instance A1, but the time printed by Cplex is only 1014 s (less than 20 min). With three scenarios, the total time measuring the iterated MILP computations using Machine A, compared to the sum of the Cplex solving time are:  
* for instance A1: we measure 1191 s (nearly 20 min) for the total MILP computations, and summing Cplex time we find nearly 135 s (2 min 15 s). * for instance A2: we measure 10,385 s (nearly three hours) for the total MILP computations, and summing Cplex time we find 2100 s (i.e., 35 min). * for instance A3: we measure 11,835 (more than three hours) for the total MILP computations, and summing Cplex time we find 2418 s (i.e., nearly 40 min). The differences are not due to Algorithm 1, very quick and not counted in these reported time measures. The differences correspond to the loading time of MILP matrix, before the MILP pre-processing. We note that using Machine B instead of Machine A for the results in Table 6 solves this problem. With Machine B, most of the computation time is devoted to the B&B solving, and loading the MILP models was around 10% of the total computational time. To explain the difficulty met with Machine A, huge matrices are stored by Cplex, before removing many variables and constraints with MILP pre-processing. Using OPL and having to define variables with rectangular multi-indexes arrays,