Eight relations of The Scheduling Zoo do not compare two specifications field by field, and are not threshold steps: the more special problem has to be encoded in the more general one by changing the instance.

Just in time inside on time. On a single machine, a job with processing time $p_j$ and due date $d_j$ completes exactly at its due date if and only if it is on time and starts no earlier than $d_j - p_j$. Hence $1 \mid\mid \sum Z_j$ is a special case of $1 \mid r_j \mid \sum U_j$ with the release dates $r_j = d_j - p_j$, and likewise with weights.

A single machine inside a two-machine flow shop. A job of the single machine becomes a job with an operation of length $0$ on the first machine followed by its operation on the second machine. Hence the decision version of $1 \mid\mid \sum Z_j$ reduces to that of $F2 \mid\mid \sum Z_j$, and likewise with weights.

Flow shops and parallel machines inside the two-stage flexible flow shop. A two-machine flow shop is a two-stage flexible flow shop with one machine in the second stage, and a problem on identical parallel machines is one in which every first operation has length $0$. Hence the decision versions of $F2 \mid\mid \sum Z_j$ and of $P \mid\mid \sum Z_j$ reduce to that of $FF(1,m) \mid\mid \sum Z_j$, and likewise with weights.

The relations across machine models are stated for decision versions, because a problem on parallel machines has instances without machines and hence without schedules, which a shop problem cannot represent by an instance but only by an unreachable threshold.

Each reduction across machine models is stated with its effect on the numbers of jobs and machines: the number of jobs is unchanged; the flow shop has two machines, and the flexible flow shop of a problem on $m$ parallel machines has $m + 1$, that of a flow shop two.

Theorem. These eight relations hold, with these numbers of jobs and machines.