With all the previous considerations in mind, we now define the C2 scheduling problem.
With all the previous considerations in mind, we now define the C2 scheduling problem.
\todo{Look in Myszkowski for how to phrase it.}
In addition to what was said before, we make the following assumptions.
No preemption:
Tasks can not be interrupted once they are started.
Task durations are integer:
Durations are unitless numbers which can stand for days, hours, or minutes.
Fractions are not necessary.
Capability requirements are known in advance and do not change while tasks are being executed.
Let $C$ be a set of capabilities.
Let $T$ and $R$ be sets of task and resource identifiers, a unique one for every task and every resource.
\paragraph{Tasks}
A task is a tuple $\tuple{t, d, \vec{c}, \vec{p}}$ where $t$ is a unique task identifier, $d \in\N$ is the task duration, $\vec{c}$ is a list of capability requirements, and $\vec{p}$ is a list of predecessor task identifiers.
\paragraph{Resources}
A resource is a tuple $\tuple{r, \vec{c}}$, where $r$ is a unique resource identifier, and $\vec{c}$ is the list of capabilities of the resource.
\paragraph{Resource affinity constraints}
A resource affinity constraint is an equality of the following form.
\begin{IEEEeqnarray*}{c}
\tuple{t,i} = \tuple{u,j} = \ldots
\end{IEEEeqnarray*}
The meaning of such an equality is that the resource assigned to capability requirement $c_i$ of task $t$ must be the same as the one assigned to capability requirement $c_j$ of task $u$, and so on.
We require that all entries in a resource affinity constraint refer to the same capability.
\paragraph{Assignments}
An assignment is a set of tuples $\tuple{t, i, r}$ that assigns a resource $r$ to every capability requirement $c_i$ of every task $t$.
\paragraph{Schedules}
A schedule is an assignment together with a set of tuples $\tuple{t, s}$ that assigns a start time $s \in\N$ to every task $t$.
The end time of a task is its start time plus its duration.
\paragraph{Validity}
We say that an assignment is \emph{valid} iff
\begin{itemize}[noitemsep]
\item all resources assigned to the same task are different
\item the assignment respects the resource affinity constraints.
\end{itemize}
We say that a schedule is \emph{valid} iff
\begin{itemize}[noitemsep]
\item its assignment is valid
\item it respects the task ordering, which means every task's start time $s_t$ is after the end time of all of its predecessors $\vec{p}_t$
\item no resource is assigned to two tasks that are overlapping in time.
