## 1 Introduction  
Currently, the demand to limit consumption of the film used to wrap bales of agriculture materials has been receiving increasing attention to reduce both costs and damage to the environment caused by plastic waste [1, 2, 3, 4]. For this purpose, both experimental studies comparing film usage for different wrapping conditions and methods [5, 6, 7, 8, 9] and model-based analytical approaches useful for estimation of the film usage [10, 11, 12, 13, 14] and its minimization [15, 16, 17, 18] were used. In the last few years it has been shown that by appropriate optimal selection of the film width [15, 16], overlaps between adjacent film strips [17], and film width and overlaps, together, [18] it is possible to reduce film consumption by up to 20%. The model-based optimization research concerned mainly conventional [15, 17, 18] and IntelliWrap [16] wrapping methods. The monograph [19] comprehends the issues of modelling of the film consumption for wrapping round bales and related optimization problems solved in the last few years. Also, an appropriate choice of bale size dimensions (diameter, height) may guarantee decreasing film consumption [11, 19]. In [11], where the dependence of the film consumption on the bale diameter for the conventional wrapping technique was investigated, the analytical analysis showed that the larger the bale diameter is, the lower is the film consumption per unit of bale volume, which led to the conclusion that the use of the bale with the largest permissible diameter ensures the smallest film consumption. These studies were based on a rough model which describes film usage as a continuous function of bale dimensions, film width and a number of wrapped film strips; however, the model did not take into account mechanical properties of the stretch film and the direct relation between the number of wrapped film strips and bale and film parameters. The concept of optimal (from the point of view of film consumption) selection of bale dimensions for the assumed bale volume was first introduced in [15] for the conventional wrapping method. A more accurate model was used that describes the consumption of the film as a function of the bale and film dimensions, mechanical parameters of the film (Poisson ratio, unit deformation), overlap ratio and the number of bale rotations [12]. The film consumption per unit of the bale volume is used as a measure of film usage. Since it is very difficult to find the optimal bale dimensions minimizing the original exact film usage index due to the discontinuity of this index, near-optimal parameters, being as important as optimal parameters for engineering applications, are sought. It was shown in [15] that in the case of using the conventional wrapping method, the optimal bale height is twice its optimal diameter. This optimality rule, which holds also for the IntelliWrap method [19], has only a theoretical character and is a consequence, among others, of multiple overlapping segments on the bale cylinder top and bottom, where there are 2-16 times more film layers than on the bale's lateral surface [10; 19]. However, in the case of the combined 3D wrapping method [6; 14], the optimal bale dimensions turn out to be useful and applicable from the engineering practice point of view. In the 3D wrapping method, which offers the potential to minimize film usage [6; 14; 20] as well as to enhance the quality of silage [6], biaxial rotation of the film applicators results in two types of film layers wrapped perpendicularly: the bottom layers are wrapped around the bale's lateral surface and the upper layers are wrapped along the bale's longitudinal axis. For a detailed description of the wrapping process, see [6] and the producer's documentations [21; 22; 23]. The problem of the choice of the best bale dimensions to guarantee the minimal film consumption was solved in [20], where the optimality conditions were established in the form of algebraic cubic equations, which can easily be solved using both analytical and numerical methods.