  The $1|B,r_j|\sum w_jU_j$ scheduling problem takes as input a batch setup
time $\Delta$ and a set of $n$ jobs, each having a processing time, a release
date, a weight, and a due date; the task is to find a sequence of batches that
minimizes the weighted number of tardy jobs. This problem was introduced by
Hochbaum and Landy in 1994; as a wide generalization of {\sc Knapsack}, it is
$\mathsf{NP}$-hard.
  In this work we provide a multivariate complexity analysis of the
$1|B,r_j|\sum w_jU_j$ problem with respect to several natural parameters. That
is, we establish a thorough classification into fixed-parameter tractable and
$\mathsf{W}[1]$-hard problems, for parameter combinations of (i) $\#p$ =
distinct number of processing times, (ii) $\#w$ = number of distinct weights,
(iii) $\#d$ = number of distinct due dates, (iv) $\#r$ = number of distinct
release dates, and (v) $b$ = batch sizes. Thereby, we significantly extend the
work of Hermelin et al. (2018) who analyzed the parameterized complexity of the
non-batch variant of this problem without release dates.
  As one of our key results, we prove that $1|B,r_j|\sum w_jU_j$ is
$\mathsf{W}[1]$-hard parameterized by the number of distinct processing times
and distinct due dates. To the best of our knowledge, these are the first
parameterized intractability results for scheduling problems with few distinct
processing times. Further, we show that $1|B,r_j|\sum w_jU_j$ is
fixed-parameter tractable with respect to parameter $\#p+\#d+\#r$ and with
respect to parameter $\#w+\#d$ if there is just a single release date. Both
results hold even if the number of jobs per batch is limited by some integer
$b$.
