## 2 Materials and Methods
### Methodology  
In this paper, a model-based analytical approach was applied, which addressed the goals (derivation of the conditions for equal and near-equal optimal bale dimensions and solving the problems of wrapping parameter design guaranteeing such equilibria) by using mathematical tools. The solutions of the optimality conditions for the bale diameter were obtained by applying numerical tools. Firstly, the necessary and sufficient conditions under which for the given bale volume the optimal bale diameter is near-equal to its optimal height with order \(\varepsilon_{0}\) were derived using the optimality conditions expressed by Equations (1) and (2); Proposition 1 abstracts these results (Section 2.3.1). The conditions for optimal bale dimensions equilibria are given by algebraic equations. One of them uniquely relates bale volume to order \(\varepsilon_{0}\) and film and wrapping parameters, i.e., this condition is satisfied for a given volume. The complementary condition determined by two simple equations relating only to \(\varepsilon_{0}\) and film and wrapping parameter applies regardless of the bale volume. Based on these conditions, using differential calculus, the relationships between the volume of the bale with near-equal optimal dimensions and the order \(\varepsilon_{0}\), width of the film and overlaps were analyzed (Section 2.3.2). Next, laying \(\varepsilon_{0}=0\) the conditions for exact equilibrium of the optimal bale dimensions was obtained directly from the conditions of their near-equilibrium (Section 2.3.3). Knowing the conditions of equilibrium and near-equilibrium of optimal bale dimensions, the tasks of designing the wrapping process in such a way that these conditions would be satisfied were formulated and solved. Due to the dichotomous nature of the equilibrium conditions, design tasks for a given bale volume and those in which the bale volume is arbitrary were considered separately; these are covered in Section 3.1 and Section 3.2, respectively. Consequently, four design problems were solved; however, the solutions for equal optimal bale dimensions resulted directly from those for near-equilibrium by substituting \(\varepsilon_{0}=0\) (Sections 3.3 and 3.4). It was assumed that the film parameters (width, mechanical parameters \(v_{f}\), \(\varepsilon_{lf}\)) are known, i.e., a practically available plastic film can be used, e.g., a commercial PE film used traditionally due to its mechanical properties and low costs [26; 27; 39]. Also the global number of film layers \(p_{l}\) wrapped on the bale's lateral surface was taken as a given. Many studies have investigated the number of desired film layers for baled silage preservation, for example [7; 29; 40; 41]. Mostly, four, six or eight layers of film are applied [6; 24; 35]; however, ten, twelve, and even sixteen film layers in which the silages are wrapped are also considered [7; 22; 23; 29]. Therefore no specific assumptions were taken concerning the number of global film layers; the numerical studies were conducted for \(p_{l}\) from four to sixteen. The standard overlaps 50%, 67% or 75% resulting in uniform film coverage [17] were assumed for upper film layers. Thus, the bottom layers overlaps \(\delta\), \(k_{fb}\) and film layers decomposition \(p_{l}=p_{b}+p_{u}\) had to be selected. The upper and lower limits were taken for the overlaps, the selection of which should take into account the knowledge and experience of baling as well as the possibilities of the wrappers available. Consequently, the considered design problems consisted in solving a set of algebraic equations (resulting from the conditions of the equilibria of optimal bale dimensions) and inequalities (the constraints of variables) with respect to three decision variables. Two variables--the overlaps \(\delta\), \(k_{fb}\)--were continuous, while the third--the number of bottom film layers \(p_{b}\)--was integer. These mixed (hybrid) sets of equations and inequalities were solved analytically, separately for fixed and free bale volume.