## 3. Results and Discussion
### Volume-Fixed Near-Equal Optimal Bale Dimensions Design  
Based on Proposition 1, case (i), the following problem of bale dimensions optimal design was formulated. **Problem 2**. : _Given: film parameters \(b_{f}\), \(v_{f}\), \(\varepsilon_{lf}\), upper layers overlap ratio \(k_{f}\), Equation (20), number of global film layers \(p_{l}\), bale volume \(V_{\mathbb{R}0}\), and order of near-equilibrium \(\varepsilon_{0}\). Find film layers decomposition \((p_{b},p_{u})\) and the bottom layers overlaps \(\delta\), \(k_{fb}\) satisfying constraints expressed by Equations (22) and (23) such that the optimal bale dimensions \(D_{b}^{*}\), \(H_{b}^{*}\) yielding bale volume \(V_{\mathbb{R}0}\) are near-equal with order \(\varepsilon_{0}\) and results in the minimal film usage._  
Based on Proposition 1 and the properties of film consumption index \(FC\), Equation (A2), with non-continuous ceiling function the solution of the above problem is derived in Appendix B. The next proposition abstracts this solution. **Proposition 5**. : _The solution to Problem 2 exists if and only if there exists an integer \(p_{b}\) and \(k_{fb}\) such that inequalities:_  
\[\breve{k}_{fb}(p_{b})\leq k_{fb}\  
\[k_{fb}\
eq\frac{(p_{l}-p_{b})(1+2\epsilon_{0})-p_{b}}{(p_{l}-p_{b})(1+2\epsilon_{0 })}=k_{fb,n\epsilon}(p_{b}) \tag{30}\]  
_are satisfied together with the applicability condition, Equation (21), where_  
\[\overset{\equiv}{k}_{fb}(p_{b})=\frac{2\delta_{max}\big{(}\sqrt[3]{1+ \epsilon_{0}}\big{)}^{2}-\sqrt[3]{\frac{4V_{00}}{\pi}}+(1+2\epsilon_{0})\big{(} \frac{p_{l}}{p_{b}}-1\big{)}\sqrt[3]{\frac{4V_{00}}{\pi}}}{(1+2\epsilon_{0}) \big{(}\frac{p_{l}}{p_{b}}-1\big{)}\sqrt[3]{\frac{4V_{00}}{\pi}}+b_{fr}\big{(} \sqrt[3]{1+\epsilon_{0}}\big{)}^{2}}, \tag{31}\]  
\[k_{fb}(p_{b})=\max\left\{\frac{2\delta_{min}\big{(}\sqrt[3]{1+\epsilon_{0}} \big{)}^{2}-\sqrt[3]{\frac{4V_{00}}{\pi}}+(1+2\epsilon_{0})\big{(}\frac{p_{l} }{p_{b}}-1\big{)}\sqrt[3]{\frac{4V_{00}}{\pi}}}{(1+2\epsilon_{0})\big{(}\frac {p_{l}}{p_{b}}-1\big{)}\sqrt[3]{\frac{4V_{00}}{\pi}}+b_{fr}\big{(}\sqrt[3]{1+ \epsilon_{0}}\big{)}^{2}},k_{fb,min}\right\}, \tag{32}\]  
_and \(k_{fb}(p_{b})\) is given by_  
\[\hat{k}_{fb}(p_{b})=1-\frac{\frac{b_{fr}\big{(}\sqrt[3]{1+\epsilon_{0}}\big{)} ^{2}}{\sqrt[3]{\frac{4V_{00}}{\pi}}}}{\sqrt[3]{\frac{4V_{00}}{\pi}}}\left| \frac{2}{\frac{1-\epsilon_{0}}{\pi}\big{(}\frac{1}{1-\epsilon_{0}(p_{b})}- \big{(}\frac{p_{l}}{p_{b}}-1\big{)}(1+2\epsilon_{0})\big{)}}\right|+\Big{(} \frac{p_{l}}{p_{b}}-1\big{)}(1+2\epsilon_{0})}. \tag{33}\]  
_For any such \(p_{b}\) and \(k_{fb}\) the overlap \(\delta\) is given by equation_  
\[\delta=\frac{\sqrt[3]{\frac{4V_{00}}{\pi}}}{2\big{(}\sqrt[3]{1+\epsilon_{0}} \big{)}^{2}}\bigg{[}1-\bigg{(}\frac{p_{l}}{p_{b}}-1\bigg{)}\big{(}1-k_{fb} \big{)}(1+2\epsilon_{0})\bigg{]}+\frac{b_{fr}k_{fb}}{2}, \tag{34}\]  
_the optimal hole dimensions are described by Equations (5) and (6) and film usage_  
\[FC(p_{b})=\frac{4b_{f}\sqrt[3]{1+\epsilon_{0}}}{\Big{(}\epsilon_{lf}+1\Big{)} \bigg{[}\sqrt[3]{\frac{4V_{00}}{\pi}}\bigg{]}^{2}}\Bigg{\{}p_{b}\left[\frac{ \sqrt[3]{\frac{4V_{00}}{\pi}}\bigg{[}\frac{2}{1-k_{fb}(p_{b})}-\big{(}\frac{p _{l}}{p_{b}}-1\big{)}(1+2\epsilon_{0})\bigg{]}}{b_{fr}\big{(}\sqrt[3]{1+ \epsilon_{0}}\big{)}^{2}}\right]+\frac{2(2+\epsilon_{0})}{\pi(1+\epsilon_{0}) }\bigg{[}\frac{\pi\sqrt[3]{\frac{4V_{00}}{\pi}}(1+\epsilon_{0})(p_{l}-p_{b}) }{2b_{fr}\big{(}\sqrt[3]{1+\epsilon_{0}}\big{)}^{2}}\bigg{]}\Bigg{\}}, \tag{35}\]  
_where \(\lceil x\rceil\) denotes ceiling function [45]. Integer \(p_{b}^{*}\) solving Problem 2 is such that_  
\[FC(p_{b}^{*})=\min_{p_{b}}FC(p_{b}). \tag{36}\]  
It may be easily shown that if the following inequality  
\[b_{fr}\left[1-\frac{1}{\Big{(}\frac{p_{l}}{p_{b}}-1\Big{)}(1+2\epsilon_{0})} \right]<2\delta_{min} \tag{37}\]  
holds, then  
\[k_{fb,n\epsilon}(p_{b})<\overline{k}_{fb}(p_{b}) \tag{38}\]  
and condition from Equation (30) can be neglected. In all the tested examples the above inequality has been satisfied, compare Table 3 below.