## 2 Materials and Methods
### Equal and Near-Equal Optimal Bale Dimensions  
A complete mathematical model describing stretch film consumption for wrapping a bale with \(p_{b}\) bottom and \(p_{u}\) upper film layers using the combined 3D technique was derived in [14]; the main formula describing film usage is recalled in Appendix A with the corresponding assumptions. Symmetry of the bale is assumed. Mechanical properties of the plastic film are described by the Poisson's ratio \(v_{f}\) and unit deformation \(\varepsilon_{lf}\); thickness of the film is ignored, e.g., polyethylene film is 25 \(\upmu\)m thick [38]. The main symbols are summarized in Nomenclature, Appendix C.  
The problem of the choice of bale dimensions (diameter, height) minimizing the consumption of the film used to wrap a cylindrical bale by 3D method has been stated and solved for the first time in [20], where the necessary and sufficient condition of the existence of the unique optimal bale diameter \(D_{b}^{*}\) of the bale of pre-assumed volume \(V_{b0}\) was derived in the form of cubic equation with zero linear term coefficient [20]:  
\[\frac{2\pi p_{u}}{\Omega\Big{(}k_{f}\Big{)}}(D_{b}^{*})^{3}+\frac{\pi p_{b}}{ 1-k_{fb}}\Big{(}2\delta-b_{fr}k_{fb}\Big{)}\big{(}D_{b}^{*}\big{)}^{2}-4V_{b0 }\left(\frac{p_{b}}{1-k_{fb}}+\frac{p_{u}}{\Omega\Big{(}k_{f}\Big{)}}\right)=0, \tag{1}\]  
where \(b_{fr}\) is the width of stretched film described by Equation (A1), \(\delta\) denotes the overlap of the extreme film strips in bottom film layers at the bases of the bale, \(k_{fb}\) and \(k_{f}\) are the overlap ratios determining the width of the contact between adjacent film strips in bottom and upper film layers; function \(\Omega\Big{(}k_{f}\Big{)}\) is defined by Equation (A3). The coefficients in the first and second terms and the free term of Equation (1) depend on the pre-assumed bale volume as well as all the film and wrapping parameters. Based on the optimality condition, Equation (1), analytical and numerical analysis of the influence of film width, pre-assumed bale volume and numbers of bottom and upper film layers on optimal bale dimensions and optimal film consumption were carried out in [20], where many detailed conclusions regarding the impact of these parameters were formulated. Additionally, the influence of the bottom layers overlaps was studied in [19]. Equation (1) has one real positive root [20]. The corresponding optimal bale height \(H_{b}^{*}\) is [20]  
\[H_{b}^{*}=\frac{4V_{b0}}{\pi\big{(}D_{b}^{*}\big{)}^{2}}. \tag{2}\]  
For given \(V_{b0}\) the optimal \(D_{b}^{*}\) and \(H_{b}^{*}\) depend on film and wrapping parameters, they are linearly dependent and the difference between them is described by  
\[D_{b}^{*}-H_{b}^{*}=\frac{1}{2}H_{b}^{*}\Bigg{[}\frac{p_{b}\Omega\Big{(}k_{f} \Big{)}}{p_{u}\Big{(}1-k_{fb}\Big{)}}-1\Bigg{]}-\frac{p_{b}\Omega\Big{(}k_{f} \Big{)}}{2p_{u}\Big{(}1-k_{fb}\Big{)}}\Big{(}2\delta-b_{fr}k_{fb}\Big{)}. \tag{3}\]From a quick inspection of Equation (3) it can be seen that the relation between the optimal diameter \(D_{b}^{*}\) and height \(H_{b}^{*}\) is not evident. The conducted research [19; 20] has shown that the relations \(D_{b}^{*}>H_{b}^{*}\) and \(D_{b}^{*}<H_{b}^{*}\), as well as \(D_{b}^{*}=H_{b}^{*}\) are possible, depending on the values of the film and wrapping parameters and volume \(V_{b0}\); compare [20] (Figure 11), where the proportions diameter/height are depicted for large bales. Only the relation \(D_{b}^{*}\geq H_{b}^{*}\), desirable from the point of view of baling systems engineering, will be considered. The proportion diameter/height do not exceed 1.25 (22) (RW 1819, 150/120), or 1.2 (21) (BW 1850, 150/125), or (23) (WM 1851, 150/125); however, predominantly, it is of 1.05 order [21; 22; 23]. The goal of this paper is to study the selection of equal and nearly-equal optimal bale dimensions, therefore, the respective precise definitions are given. **Definition 1. ** (_equal optimal dimensions_).