## 2 Problem Statement
### Decision Variables and Objective Function  
Maintenance, refueling, and production decisions are optimized conjointly. The variables for outage dates and refueling levels are related to T2 units, whereas the production variables concern T1 and T2 units. There is another heterogeneity of maintenance and production decisions: production decisions are considered for a discrete set of scenarios \(s\in\mathcal{S}\) to model uncertainty on power demands  
\begin{table}
\begin{tabular}{l l} \hline \hline \(\mathbf{A}_{i,k}\) & Maximal fuel level remaining in cycle \(k\) to process outage \(k+1\). \\ \(\mathbf{Bo}_{i,k}\) & Fuel level “Bore O” of cycle \(k\) \\ \(\mathbf{C}_{i,s}^{ind}\) & Proportional cost to the final remaining fuel levels for scenario \(s\). \\ \(\mathbf{C}_{i,k}^{r}\) & Proportional cost to the refueled quantity at outage \(k\). \\ \(\mathbf{C}_{i,s,t}^{f}\) & Production costs for T1 unit \(j\) at time period \(t\) and scenario \(s\). \\ \(\mathbf{D}_{s,t}\) & Power demands at time step \(t\) for the scenario \(s\). \\ \(\mathbf{Da}_{i,k}\) & Outage duration for maintenance and refueling at cycle \(k\). \\ \(\mathbf{F}_{t}=\mathbf{F}\) & Conversion factor between power and fuel in time step \(t\), constant in the data. \\ \(\mathbf{I}_{c,w}\) & Maximal offline T2 power at week \(w\) for CT21 constraint \(c\). \\ \(\mathbf{L}_{i,k,c}\) & Latency in the resource requirement in CT19 constraint \(c\) for outage \(i,k\). \\ \(\mathbf{N}_{c,w}\) & Maximal number of outages under maintenance at week \(w\) for CT20 constraint \(c\). \\ \(\pi_{s}\) & Probability of scenario \(s\). \\ \(\mathbf{Pmin}_{j,s,t}\) & Minimal power for T1 unit \(j\) at time period \(t\) and scenario \(s\). \\ \(\mathbf{Pmax}_{j,s,t}\) & Maximal power for T1 unit \(j\) at time period \(t\) and scenario \(s\). \\ \(\mathbf{Pmax}_{i,t}\) & Maximal generated power at time step \(t\). \\ \(\mathbf{Q}_{i,k}\) & Proportion of fuel that can be kept during reload in cycle \(k\) at plant \(i\) \\ \(\mathbf{R}_{i}^{10}\) & Maximal quantity of resource in CT19 constraint \(c\). \\ \(\mathbf{Rmin}_{i,k}\) & Minimal refueling at outage \(k\). \\ \(\mathbf{Rmax}_{i,k}\) & Maximal refueling at outage \(k\). \\ \(\mathbf{S}_{c}\) & Spacing/overlapping value defined by CT14-CT18 constraints. \\ \(\mathbf{Smax}_{i,k}\) & Maximal fuel level at production cycle \(k\). \\ \(t_{w}\) & Index of the first time step of week \(w\). \\ \(\mathbf{Ta}_{i,k}\) & Last possible beginning week for outage \(k\) of T2 plant \(i\). \\ \(\mathbf{To}_{i,k}\) & First, possible outage week for cycle \(k\). \\ \(\mathbf{Tu}_{i,k,c}\) & Length of the resource usage in CT19 constraint \(c\) for outage \(i,k\). \\ \(w_{t}\) & Week of the production time step \(t\). \\ \(\mathbf{Xi}_{i}\) & Initial fuel stock of T2 plant \(i\). \\ \hline \hline \end{tabular}
\end{table}
Table 2: Definitions and notations—the input parameters.  
and production capacities, whereas outage and refueling decisions are common for all the scenarios, and must guarantee the feasibility of a production planning for each scenario. The objective function minimizes the average value of the production costs considering all the scenarios. T1 Production costs are proportional to the generated power. T2 production costs are proportional to the refueling quantities over the time horizon, minus the expected stock values at the end of the period to avoid end-of-side effects.