## 3 Problem Statement
### Model  
The subset of tasks (or jobs) \(j\in\mathcal{C}\) refers to the group of _constellation tasks_, i.e., the ones assigned to all the satellites which any can execute. The tasks that do not belong to subset \(\mathcal{C}\) are called _individual tasks_ ( \(\mathcal{I}\)) and are assigned to a single satellite. An optimal allocation of the available energy resources must distribute them between both described task groups--constellation and individual. However, since the resources are limited, a group of tasks can not be favored--with more allocated resources--without deteriorating the other group. Therefore, there are multiple optimal solutions to a problem in this form. By splitting the cost function \(F\) defined in Equation (2) into two terms--\(A\) and \(B\), respectively, corresponding to constellation tasks and individual tasks--a certain degree of flexibility is achieved. This allows the application of methodologies such as the \(\epsilon\)-constrained method presented in [20], which is capable of identifying the Pareto-frontier of the problem. Maximizing \(F\) means optimizing energy usage by scheduling tasks according to their priorities \(u_{s,j,t}\), therefore, improving the mission value and the quality of service (QoS). \[F:\ \max_{x_{s,j,t}}\ \underbrace{\sum_{(s,j,t)\in\mathcal{C}}u_{s,j,t}x_{s,j,t}} _{\text{A}}+\underbrace{\sum_{(s,j,t)\in\mathcal{I}}u_{s,j,t}x_{s,j,t}}_{ \text{B}}+\underbrace{\sum_{(s,s^{\prime},j,t)\in\mathcal{R}\subset\mathcal{C }}\beta_{s,s^{\prime},j,t}}_{\text{C}} \tag{2}\]  
Notice that while the \(C\) term is presented at the cost function, it will not be considered in the \(\epsilon\)-constrained methodology, being in charge of directing the redundancy of the tasks, which will be discussed later. \begin{table}
\begin{tabular}{l l} \hline \hline
**Notation** & **Description** \\ \hline \(M\) & A large number which would not be part of any optimal solution, used in big-M constraints. \(M\in\mathbb{N}\). \\ \(V_{b}\) & Nominal battery voltage. \(V_{b}\in\mathbb{R}^{+}\). \\ \(SoC_{s,0}\) & Initial state of charge (SoC) of the battery of subsystem \(s\). \(SoC_{s,0}\in\mathbb{R}^{+},\leq 1\). \\ \(\epsilon_{c}\) & Battery charging efficiency. \(\epsilon_{c}\in\mathbb{R}^{+},\leq 1\). \\ \(\epsilon_{d}\) & Battery discharging efficiency. \(\epsilon_{d}\in\mathbb{R}^{+},\leq 1\). \\ \(\mathbb{Q}\) & Battery nominal capacity (Ampere-hour). \(Q\in\mathbb{R}^{+}\). \\ \(\rho\) & Minimum accepted battery SoC. \(\rho\in\mathbb{R}^{+}\), \(<1\). \\ \(r_{s,t}\) & Power input from solar panels at subsystem \(s\) and instant \(t\). \(r_{s,t}\in\mathbb{R}^{+}\). \\ \(q_{s,j}\) & Power required to execute job \(j\) in subsystem \(s\). \(q_{s,j}\in\mathbb{R}^{+}\). \\ \(u_{s,j,t}\) & Dynamic priority of job \(j\), at instant \(t\) in subsystem \(s\). \(u_{s,j,t}\in\mathbb{W}\). \\ \(\sum_{s^{\prime},max}^{min}\) & Minimum/maximum time required to execute job \(j\) in subsystem \(s\). \(t_{s,j}^{min}/max\in\mathbb{N}\). \\ \(y_{s,j}^{min}/max\) & Minimum/maximum desired number of executions (startups) of job \(j\) in subsystem \(s\). \(y_{s,j}^{min}/max\in\mathbb{W}\). \\ \(g_{j}^{min}/max\) & Minimum/maximum desired number of executions (startups) of job \(j\), considering the whole constellation (all subsystems). \(g_{j}^{min}/max\in\mathbb{W}\). \\ \(p_{s,j}^{min}/max\) & Minimum/maximum period between executions of job \(j\) in subsystem \(s\). \(p_{s,j}^{min}/max\in\mathbb{N}\). \\ \(w_{s,j}^{min}/max\) & Beginning(min)/ending(max) of execution window for job \(j\) in subsystem \(s\). \(w_{s,j}^{min}/max\in\mathbb{W}\). \\ \(\psi_{j}^{min}/max\) & Minimum/maximum number of subsystems allowed to execute job \(j\) at the same instant (simultaneously). \\ \(\psi_{j}^{min}/max\in\mathbb{W}\). \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Model inputs.