Please allow me to address the [set cover problem](https://en.wikipedia.org/wiki/Set_cover_problem) approach in a new answer, so  the question won't introduce more confusions. I would update this answer as I venture through this path.

And first, I have to explain myself a bit more. I've deserted maths for a decade more (not intentionally though). So my maths ability drops to every-day-maths capable. I'm picking stuffs up as I progress.


It seems I have made a wrong choice regarding $L_{j}$. As my goal is to select crew that can fulfill all jobs, $L_{j}$ won't help here. Instead list of doable jobs per worker should be the one to get.

$U \leftarrow$ all jobs

$W \leftarrow$ all workers

$L_w \leftarrow$ jobs can be done by worker $w$ (where $w \in W$ and $L_w \in U$)

Grep the [set cover](https://en.wikipedia.org/wiki/Set_cover_problem) ILP from wiki:
$$
\mathrm{minimize} \sum_{j \in L_w} x_j
$$

for all $e \in U$
$$
\sum_{j_{:e} \in L_w} x_j \ge 1
$$

for all $j \in L_w$
$$
x_j \in \{0,1\}
$$

now we have the combination of workers (first part solved), correct?

------
Below stuffs were turned out unused.

$Z \leftarrow$ slots

$n_u = |U|$

$u_j \in U$  (where $1 \le j \le n_u$)

$n_w = |W|$

$w_i \in W$ (where $1 \le i \le n_w$)

$S \leftarrow$ all skills & abilities

$s_j \leftarrow$ job $j$ skills, $s_j \in S$ (where $1 \le j \le n_u$)

$S' \in S$ (all skills used in jobs) where:
$$
S' = \bigcup_{j=1}^{n_u} s_j
$$

$s_i \leftarrow$ worker $i$ skills, $s_i \in S' \in S$ (where $1 \le i \le n_w$)

$L_j \leftarrow$ suitable workers to job $j$ (where $j \in U$)

$v_i \leftarrow$ worker $i$ score