We introduce a scheduling problem derived from radiotherapy scheduling.
It consists in scheduling a set of tasks $\mathcal{J}$ ($j \in \mathcal{J}$) with release dates $r_j$ and due dates $d_j$. Tasks are processed in parallel on multiple heterogeneous machines $m \in \mathcal{M}$ with compatibilities and different processing times.
Those machines also have availabilities constraints such as opening windows and unavailable time slots.\\
The problem is also prone to uncertainty because tasks have an \textit{appearance date} which indicates when the task is visible by the information system and an \textit{acceptance date} which tells when the task is available for treatment. At each decision time point, the set of tasks is partitioned into two subsets:
\begin{itemize}
\item the set of \textit{certain} tasks, for those the appearance date and the acceptance dates are past. In this case, the release date $r_j$ is known exactly. We denote this set by $\mathcal{J}^C$.
\item the set of \textit{uncertain} tasks, in this case only the appearance date is past and the acceptance date is still unknown, which has an impact on the release date, $r_j \in \llbracket r_j^{min}, r_j^{max} \rrbracket$ . We denote this set by $\mathcal{J}^I$.
\end{itemize}
The objective criterion is to minimize the sum of completion times $\sum_{j \in \mathcal{J}} C_j$.