## 6 Computational Results
### Branch & Bound Solving Characteristics  
We note that Table 5 analyzes also the impact of pre-processing strategies. The specific pre-processing of Proposition 3 is not redundant with the generic MILP pre-processing of Cplex [39], and the gain in MILP reduction is significant, especially for the difficult instances B7, B8, and B9. Pre-processing of Proposition 3 has a positive impact influence in our computations of complex MILP models to reach the most advanced phases of the B&B algorithm with memory limitations. Proposition 4 reduces the number of continuous variables that will be decisive, reaching memory limits. In the following analyzes, pre-processing of Propositions 3 and 4 are always enabled.  
With Cplex 12.5, dual bounds can be computed for each instance with a single scenarios and weekly production time steps in less than one hour for the LP and LP+cuts modes, the B&B convergence characteristics are detailed using Cplex 12.5 in [17], and reactualized results with Cplex 12.8 are available in [24]. We note that 1 hour resolution time was not enough to solve the LP relaxations of such problems for instances B8 and B9 with Cplex 12.3 [23]. Computations \(LB(k)\) are quicker than \(LB\) computations, reaching more advanced phases of the B&B convergence.  
The number of continuous variables is decisive for solving capabilities. Indeed, there are the same number of binary variables with disaggregated or aggregated time steps for production variables, and the computation capabilities differ dramatically. If aggregated computations on a single scenario are tractable, disaggregated computations are tractable only for dataset A, and instances from datasets B and X induce a prohibitive memory consumption. Similarly, considering several stochastic scenarios does not change the number of binary variables, but limits are reached in terms of memory resources. This seems paradoxical, exponential complexity for MILP comes with integer variables. In this problem, increasing the number of continuous variables makes polynomial algorithms intractable. We note furthermore that this occurs also for the MILP relaxation with lighter CT6 constraints presented in [24]. Such formulation considers the same number of binary variables, but the MILP solving capability is also not efficient for the instance sizes of the Challenge. This highlights the interest of Proposition 4, in reducing the number of continuous variables to increase the MILP solving capability. With the recent versions of Cplex, dual bounds can be computed with truncated MILP solving considering up to 10 scenarios for \(LB(k)\) and \(LB\) problems, with aggregated time steps for all the instances, and with disaggregated for the instances from the dataset A.  
Lastly, we note that the variable definition has an influence in the efficiency of the branching quality. Variables \(d_{i,k,w}\leqslant d_{i,k,w+1}\) imply better branching and B&B convergence than using binaries \(x_{ijk,w}=d_{i,k,w}-d_{i,k,w-1}\) as in [26] with constraints \(\sum_{w}x_{ijk,w}\leqslant 1\). These results are coherent with [20], and the standard branching on variables implemented in Cplex induces for \(d_{i,k,w}\) variables well balanced B&B trees. However, using variables \(x_{i,k,w}\) was slightly better for the search of primal solutions. To design a primal heuristic in [26], and using few branching, variables \(x\) are relevant for the primal heuristic in [26], whereas, in our application, only the quality of dual bounds matters and choosing step variables \(d_{i,k,w}\) is more efficient.