  We present a new generalization of the extensible bin packing with unequal
bin sizes problem. In our generalization the cost of exceeding the bin size
depends on the index of the bin and not only on the amount in which the size of
the bin is exceeded. This generalization does not satisfy the assumptions on
the cost function that were used to present the existing polynomial time
approximation scheme (PTAS) for the extensible bin packing with unequal bin
sizes problem. In this work, we show the existence of an efficient PTAS (EPTAS)
for this new generalization and thus in particular we improve the earlier PTAS
for the extensible bin packing with unequal bin sizes problem into an EPTAS.
Our new scheme is based on using the shifting technique followed by a solution
of polynomial number of $n$-fold programming instances. In addition, we present
an asymptotic fully polynomial time approximation scheme (AFPTAS) for the
related bin packing type variant of the problem.
