In this paper we study a problem with periodic time slots in which a
processing time and periodically repeating due dates are given for
each job, where the period is the same for all jobs. The objective
is to minimize the number of periodic time slots in which all jobs
can be scheduled just-in-time. We show that when set-up times
between jobs are considered, the problem becomes unary NP-hard
even in the single machine case. When set-up
times are not considered, we study several subproblems which arise
by putting additional restriction on processing times. We show that
under a rather strict (although natural) restriction the problem is
solvable in polynomial time for an arbitrary number of identical
parallel machines, and we present a simple polynomial time algorithm
solving the problem. We also show, that when the restriction on
processing times is lifted, the problem without set-up times becomes
binary NP-hard already for two machines and unary NP-hard when the
number of machines is a part of the input.