Please allow me to address the 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 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

UPDATE

I haven't figured out other options to this problem. So I've broken up the process to offline mobile computations.

First, I've done everything on the server. Calculating all fundamental statistics, and work out a basic solution to the allocation.

Then all those results are pushed to mobile devices. When the basic solution doesn't fit the final requirement, mobile devices use the stats data from server to work out a final solution.

** Original **

Please allow me to address the 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 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