the rest of the bags are placed such that a machine holds multiple jobs of one bag, we can repair the schedule to gain an overall feasible schedule. The constant $b'$ depends on the number of medium jobs any machine can hold in an optimal solution. We therefore set $q := \frac{1+2*\epsilon+\epsilon^2}{\epsilon^{k+1}}$ to be this number of jobs. Note that this $q$ also results from a modification that we have yet to introduce, which will increase the height of an optimal solution up to $T:= 1+2\epsilon +\epsilon^2$. We formally define this set of bags as follows:

\[def:prio\] Set $b':= (d*q+1)*q$ with $q$ being the number of medium jobs any machine can hold in an optimal schedule as introduced before the lemma and $d\in O(log_{1+\epsilon}{\frac{1}{\epsilon^{k}}})$ being the number of item sizes of large jobs. We call a bag $B_l$ *priority bag* if and only if there exists a large item size $s$ and a position $i\le b'$ such that $o_s(i) = l$. Further we define that every large bag is a priority bag as well. If a bag $B_l$ by this definition is not a priority bag, we call $B_l$ *non-priority bag*.

For the rest of the paper we set $b'$ 