Two problems built on a problem $s$ of the family with operations, as used for interval scheduling with eligible machine sets (lax-767944).

A job is kept by a schedule if it is on time, for an objective that counts late jobs, or just in time, for one that counts jobs not just in time; for the other objectives no job is kept.

At most $k$ scheduled jobs. An instance is an instance $x$ of $s$ together with a number $k$. A schedule is a schedule of $x$ together with a set $S$ of at most $k$ jobs, the jobs it schedules; it is feasible if the schedule is feasible for $s$. Its cost is the weight of the jobs that are not in $S$ or not kept, so that a cost of at most $D$ says that some schedule of at most $k$ jobs keeps weight at least $W - D$, with $W$ the total weight. This is the weighted question parameterized by the number $k$ of scheduled jobs: a schedule of at most $k$ jobs and weight at least a target.

Prescribed machine loads. An instance is an instance $x$ of $s$ together with a number $q_i$ for every machine $i$. A schedule is feasible if it is feasible for $s$ and every machine $i$ runs exactly $q_i$ jobs, a job running on a machine if a piece of one of its operations occupies it. The cost is the cost in $s$; with the threshold $0$ and an objective that counts jobs not just in time, the question is whether all jobs can be scheduled with these loads (Das, Opler and Valla).