Just-in-time scheduling in a two-stage flexible flow shop is strongly NP-hard, already when every weight is one.

The reduction is from Hitting Set. For $2 \le k \le n$, the shop built from a family $F_1, \dots, F_m$ over $\{1, \dots, n\}$ admits $R \cdot m \cdot (2k-1)$ just-in-time jobs exactly when the family has a hitting set of size $k$.

One direction schedules: a hitting set assigns to each of its $k$ elements one machine, which runs in every epoch either the selection job of the element that hits that epoch's set or the two dummies of the element it is responsible for, and the preprocessing of those dummies fits into the epoch exactly. The other direction extracts: the preprocessing budget of an epoch admits at most $2(k-1)$ dummies, so a solution of the target size holds exactly one selection job and $2(k-1)$ dummies in every epoch, the element a machine is responsible for never decreases from one segment to the next, and with more than $k(n-1)$ gaps between segments some gap sees no change at all — the $k$ elements of that segment hit every set.

The numbers of the constructed shop are polynomial in $n$, $m$ and $k$, and hence in the length of the Hitting Set instance, so the hardness is strong: no algorithm polynomial in the magnitudes of the due dates can exist unless $\mathrm{P} = \mathrm{NP}$.