Load Balancing Problem: The input consists of $n$ jobs $J_1, J_2, \dots J_n$ and an integer $m$ denoting the number of machines. The size of $J_i$ is a non-negative number $s_i$. The goal is to assign the jobs to machines while minimizing the makespan (minimize the largest load of any machine).
Below is the primal linear program that I written where $l$ is the makespan load and $x_{ij}$ is an indicator variable that equals $1$ when job $J_i$ is assigned to machine $j$ and $0$ otherwise.
$min: l$
$\sum_{j=1}^{m}x_{ij} = 1 $ $\forall 1\leq i\leq n$
$l \leq \sum_{i=1}^{n}s_ix_{ij}$ $\forall 1\leq j \leq m$
$x_{ij} \geq 0$
How would you write the primal for this dual? I am confused on how to convert this into a dual because of the summations and the irregular format. I have tried writing dual linear programs for similar programs (max-flow to min-cut LP), but struggle to write the dual each time.
I am studying for an exam and this was a practice problem provided in the review. Any tips on how to approach creating duals for these more complicated linear programs would be very appreciated. Thank you!