  A very well-known machine model in scheduling allows the machines to be
unrelated, modelling jobs that might have different characteristics on each
machine. Due to its generality, many optimization problems of this form are
very difficult to tackle and typically APX-hard. However, in many applications
the number of different types of machines, such as processor cores, GPUs, etc.
is very limited. In this paper, we address this point and study the assignment
of jobs to unrelated machines in the case that each machine belongs to one of a
fixed number of types and the machines of each type are identical. We present
polynomial time approximation schemes (PTASs) for minimizing the makespan for
multidimensional jobs with a fixed number of dimensions and for minimizing the
L_p-norm. In particular, our results subsume and generalize the existing PTASs
for a constant number of unrelated machines and for an arbitrary number of
identical machines for these problems. We employ a number of techniques which
go beyond the previously known results, including a new counting argument and a
method for making the concept of sparse extreme point solutions usable for a
convex program.
