of all small item sizes and $S_{ml}$ to be the set of all medium and large item sizes. With $B_x^s := \bigcup\limits_{l\le b:l\not\in A} B_l^s$ , we denote the set of all jobs of non-priority bags of a certain size $s$. With $height(p)$ we will denote the height of the pattern $p$ which will be the sum off all included jobs in the pattern. We also remember that we set $T=1+2\epsilon+\epsilon^2$ to be the optimal height of our given instance. Furthermore we define a characteristic function $\chi_p$ for patterns $p\in \mathscr{P}$ to indicate how many jobs of a bag appear in a pattern as follows: Given a potential entry $B_l^s$ of $p$ with $s\in S_{ml}$ and $l \in A \cup \{x\}$ we define $$\chi_p(B_l^s) :=  
\begin{cases}
       z &\quad\text{if } B_l^s \in p \text{ and } B_l^s \text{ appears } z \text{ times in  } p\\
       0 &\quad\text{otherwise.} \\ 
\end{cases}$$

We extend this definition for full bags, let $B_l$ be any bag and set $$\chi_p(B_l) :=  
\begin{cases}
       1 &\quad\text{if } \exists s\in S_{ml}:\chi_p(B_l^s) \neq 0 \\
       0 &\quad\text{otherwise.} \\ 
\end{cases}$$ Note that in this definition $\chi_p(B_l)=0$ in case of $B_l$ being a non-priority bag, as the $ B_x$ occurrences count for 