## 3 Related Work
### Reductions by Pre-Processing  
Facing the large size of the instances, most of the approaches competing in the EURO/ROADEF Challenge 2010 reduced the problem with pre-processing strategies. Fixing implied or necessarily optimal decisions is a valid pre-processing to compute dual bounds. Heuristic pre-processing strategies were also commonly used. Most of the previous approaches reduced instances using aggregation techniques. Firstly, the power productions are discretized with many time steps \(t\in\mathcal{T}\), and may be aggregated to their weekly average value, so that production and maintenance decisions have the same granularity. Under some hypotheses, it is proven such aggregation induces lower bounds of the original problem [17]. Secondly, many approaches used computations reduced on single scenarios, aggregating the scenarios into one average deterministic scenario as in [27], or using decomposition \begin{table}
\begin{tabular}{l} \hline \hline
**CT14**: For all constraint \(c\in\mathcal{C}_{14}\), a subset of outages, \(\mathcal{Z}_{c}\) have to be spaced by at least \(\mathbf{S}_{c}\) weeks: for \(\mathbf{S}_{c}>0\), it is a minimal spacing from the beginning of a previous outage to the beginning of a next outage, for \(\mathbf{S}_{c}\leqslant 0\), it is a maximal number of weeks where two outages can overlap. \\ \hline
**CT15**: CT15 are similar to CT14 with sets \(\mathcal{C}_{15}\), \(\mathcal{Z}_{c}\) and parameters \(\mathbf{S}_{c}\), applying only on a time interval \(\mathcal{W}_{c}\). \\ \hline
**CT16**: for each constraint \(c\in\mathcal{C}_{16}\), a subset of outages \(\mathcal{Z}_{c}\) have to be spaced by at least \(\mathbf{S}_{c}>0\) weeks from the beginning of the previous outage to the beginning of the next outage. \\ \hline
**CT17**: for each constraint \(c\in\mathcal{C}_{17}\), a subset of outages \(\mathcal{Z}_{c}\) have to be spaced by at least \(\mathbf{S}_{c}>0\) weeks from the end of the previous outage to the end of the next outage. \\ \hline
**CT18**: for each constraint \(c\in\mathcal{C}_{18}\), a subset of outages \(\mathcal{Z}_{c}\) have to be spaced by at least \(\mathbf{S}_{c}>0\) weeks from the end of the previous outage to the beginning of the next outage. \\ \hline
**CT19**: for each constraint \(c\in\mathcal{C}_{19}\), a subset of outages \((i,k)\in\mathcal{Z}_{c}\) shares a common resource (which can be a single specific tool or maintenance team) with forbid simultaneous usage of this resource. \(\mathbf{L}_{i,k,c}\) and \(\mathbf{Tu}_{i,k,c}\) indicate respectively the start and the length of the resource usage and \(\mathbf{R}_{c}^{19}\) the maximal quantity of the resource. \\ \hline
**CT20**: for each constraint \(c\in\mathcal{C}_{20}\), at most \(\mathbf{N}_{c,\mu}\) outages among a subset \(\mathcal{Z}_{c}\) overlap in weeks \(w\in\mathcal{W}_{c}\). \\ \hline
**CT21**: for each constraint \(c\in\mathcal{C}_{21}\), a time period \(\mathcal{W}_{c}\) is associated where the offline power of T2 plants \(\mathcal{I}_{c}\subset\mathcal{I}\) due to maintenance operations must be lower than \(\mathbf{I}_{c,\mu}\). \\ \hline \hline \end{tabular}
\end{table} 
Table 3: _Cont._as in [25; 26; 30; 31; 32; 33]. It is proven that lower bounds of the EURO/ROADEF Challenge 2010 can be guaranteed with \(S\) single scenario computations [17]. TW constraints CT13 can be tightened exactly, and we refer to [34] for the most detailed presentation of such pre-processing. Minimal lengths for production campaigns are implied by the initial fuel levels defined in CT8 constraints, maximal T2 productions with CT5 constraints, maximal stock before refueling with CT11 constraints. Such minimal lengths for production cycles imply that TW constraints can be tightened by induction. Tightening TW constraints, some outages can be removed when their earliest completion time exceed the time horizon. In the problem specification, there is no maximal length for a production cycle, and it is possible to have a null production phase when the fuel is consumed.