Deciding whether every job of an instance of $R || ∑_j Z_j$ can be just in time is NP-hard. Every language in NP reduces to interval scheduling with eligible machine sets, by the second theorem of $lax-470956$, and that problem reduces to this one: every job is given the due date $d_j + 1$ and the processing time it has, and on a machine it may not use, the processing time $d_j + 1$, so that it covers the time point $1$, which one extra job per machine, due at $1$ and of length $1$ on every machine, occupies. Those extra jobs overlap one another and so sit on distinct machines, hence on all of them, and a job that sits on a machine it may not use would overlap the extra job of that machine. The map is computed by a word RAM program, which is polynomial in the length of the code, since the extra jobs are only added when there is a job, and then there are as many machines as the matrix of the input has columns.