Requiring exactly $n_j$ slots for job $j$ (if scheduled)

Question

Let $x_{j}(t)=1$ iff job $j$ is scheduled at time $t$. I want to say that if the job is scheduled at all, then it is scheduled at $n_j$ slots. I wrote this as:

$$x_{j}(t)\sum_{s=1}^{T}x_{j}(s)=n_jx_{j}(t).$$

Is it possible to write this as linear constraints?

If I am maximizing the number of jobs to be scheduled, can we do any better?

Answer 1

If you need consecutive slots, one way to model it is to redefine $x_j(t)$ to indicate the start at time $t$ of $n_h$ consecutive slots occupied by $j$. The "$n_j$ or none" constraint then becomes $$\sum_{t=1}^{T-n_j+1} x_j(t)\le 1\quad\forall j$$or, if you need a binary indicator $y_j$ for whether $j$ gets scheduled, $$\sum_{t=1}^{T-n_j+1}x_j(t)=y_j\quad\forall j.$$

Answer 2

Introduce a binary variable $y_j$ to indicate whether $x_j(t)>0$ for some $t$, and impose linear constraints: $$\sum_{t=1}^T x_j(t) = n_j y_j$$

Answer 3

We need an indicator of whether a job is ever scheduled at all.

$$y_j=\begin{cases} 1, & \sum\limits_{t=1}^Tx_j(t)>0 \\ 0, &\sum\limits_{t=1}^Tx_j(t)=0\end{cases}$$

$$\sum_{t=1}^T x_j(t) = n_jy_j$$

Furthermore, you want to maximize the number of jobs.

\begin{align}\max&\quad\sum_j y_j\\\text{s.t.}&\quad\sum_{t=1}^T x_j(t) = n_jy_j\\&\quad x_j(t), y_j \in \{0,1\}.\end{align}