## 3. Results and Discussion
### Volume-Fixed Optimal Equal Bale Dimensions Design  
**Problem 4. **_Given: film parameters \(b_{f}\), \(v_{f}\), \(\epsilon_{lf}\), overlap ratio \(k_{f}\), Equation (20), number of global film layers \(p_{l}\) and hole volume \(V_{l0}\). Find film layers decomposition \((p_{b},p_{u})\) and overlaps \(\delta\), \(k_{fb}\) satisfying constraints from Equations (22) and (23) such that the optimal bale dimensions \(D_{b}^{*}\), \(H_{b}^{*}\) resulting in the volume \(V_{l0}\) are equal and film usage is minimal._  
Lying \(\epsilon_{0}=0\) in inequalities expressed by Equations (29) and (30) and in Equations (34) and (35) based on Proposition 5 the following solution results. **Proposition 7. **_The solution to Problem 4 exists if and only if there exist \(k_{fb}\) and an integer \(p_{b}\) such that inequalities_  
\[\stackrel{{\equiv}}{{k}}_{fb}(p_{b})=\max\Bigl{\{}k_{fb}(p_{b} ),k_{fb,min}\Bigr{\}}\leq k_{fb}\leq\min\left\{\widetilde{k}_{fb}(p_{b}), \frac{2\delta_{max}+\left(\frac{p_{l}}{p_{b}}-2\right)\sqrt[3]{\frac{4V_{l0}}{ \pi}}}{\left(\frac{p_{l}}{p_{b}}-1\right)\sqrt[3]{\frac{4V_{l0}}{\pi}}+b_{fr}}, k_{fb,max}\right\} \tag{41}\]  
_and inequality_  
\[k_{fb}\
eq\frac{p_{l}-2p_{b}}{p_{l}-p_{b}} \tag{42}\]  
_are satisfied together with the applicability condition, Equation (21), where_  
\[\widetilde{k}_{fb}(p_{b})=\frac{2\delta_{min}+\left(\frac{p_{l}}{p_{b}}-2 \right)\sqrt[3]{\frac{4V_{l0}}{\pi}}}{\left(\frac{p_{l}}{p_{b}}-1\right) \sqrt[3]{\frac{4V_{l0}}{\pi}}+b_{fr}}, \tag{43}\]  
_and \(\widetilde{k}_{fb}(p_{b})\) is defined by the next equation with \(\stackrel{{\equiv}}{{k}}_{fb}(p_{b})\) introduced in Equation (41)_  
\[\widetilde{k}_{fb}(p_{b})=1-\frac{2}{\frac{b_{fr}}{\sqrt[3]{\frac{4V_{l0}}{ \pi}}}\left[\frac{2}{\frac{1-\widetilde{k}_{fb}(p_{b})}{\pi}-\left(\frac{p_{l }}{p_{b}}-1\right)}\right]+\left(\frac{p_{l}}{p_{b}}-1\right)}. \tag{44}\]  
_For any such \(p_{b}\) and \(k_{fb}\) the overlap \(\delta\) is given by_  
\[\delta=\frac{\sqrt[3]{\frac{4V_{l0}}{\pi}}}{2}\left[1-\left(\frac{p_{l}}{p_{b} }-1\right)\left(1-k_{fb}\right)\right]+\frac{b_{fr}k_{fb}}{2}, \tag{45}\] \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(\delta_{min}\) [\(\mathbf{m}\)] & \(\mathbf{b}_{f}\) [\(\mathbf{m}\)] & \(\mathbf{p}_{l}\) & \(\mathbf{p}_{b}\) & \(k_{fb}\) [\(-\)] & \(\delta\) [\(\mathbf{m}\)] & \(\mathbf{FC}\) [\(\mathbf{m}^{-1}\)] \\ \hline \multirow{4}{*}{0.1} & \multirow{4}{*}{0.75} & 8 & 3 & 0.400 & 0.114 & 41.047 \\  & & 14 & 5 & 0.444 & 0.127 & 72.524 \\  & & 16 & 6 & 0.400 & 0.114 & 82.095 \\ \cline{2-6}  & & 8 & 3 & 0.400 & 0.137 & 39.449 \\  & & 10 & 4 & 0.333 & 0.114 & 49.478 \\  & & 14 & 5 & 0.444 & 0.152 & 76.222 \\  & & 16 & 6 & 0.400 & 0.137 & 78.898 \\ \cline{2-6}  & & 14 & 5 & 0.444 & 0.152 & 76.222 \\ \hline \multicolumn{2}{c}{0.15} & \multicolumn{1}{c}{} & 14 & 5 & 0.444 & 0.152 & 76.222 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The numbers of global \(p_{l}\) and bottom \(p_{b}\) film layers and bottom layers overlaps \(k_{fb}\), \(\delta\) for which Problem 3 of volume-free equal optimal bale dimensions design has a solution for bale from Example 3; film width \(b_{f}=0.75,\ 0.9\) m, the bounds from Equations (22) and (23): \(k_{fb,min}=0.2,\ k_{fb,max}=0.45\), \(\delta_{min}=0.1,\ 0.15\) m, \(\delta_{max}=0.3\) m, \(4\leq p_{l}\leq 16\). Film usage \(FC\), Equation (A2) for the assumed bale volume \(V_{l0}=1.357\) m\({}^{3}\).