I've read about the reduction from $2$-partition for the problem of minimizing weighted completion time with release dates but I'm not very experienced in doing reductions so I want to verify that my understanding is at least in the right direction.
First, my understanding of the problem is that we would like to complete the jobs that have high weight the earliest possible, but since there isn't any preemption and given the release dates it's hard to figure out which jobs to do first.
So in order to prove $NP$-hardness in the weak sense, I construct an instance of $2$-partition as follows,
set task weights to $1$, set $r_j = 0$ for all tasks and add an extra task with $r_j = B$, and deadline $B+1$ and a processing time of $1$, weight $0$. We add this task to split the two subsets(as required by $2$-partition). Therefore if we can solve this problem in polynomial time we can also solve $2$-partition in polynomial time.
Is my intuition correct? Also I don't expect an answer, just maybe an explanation of what I'm doing wrong