Optimal and Robustly Optimal Consumption of Stretch Film Used for Wrapping Cylindrical Baled Silage

Abstract

A conventional method for wrapping round bales of agricultural materials by wrappers with a rotating table or with rotating arms is considered. In contemporary agriculture, the demand for minimal consumption of the film used to wrap bales is very high, in order to apply this method with lower cost and less damage to the environment. A combined model-based problem of such a design, focusing on the width of stretch film and the overlap between adjacent film strips that minimizes film consumption, was mathematically formulated and solved. It was proven that the complete set of optimal film widths is defined by a simple algebraic equation described in terms of film, bale, and wrapping parameters. The optimal overlap ratios were found to be irreducible fractions in which the dividend is the divisor minus one; however, only the first three factions, 1 2 , 2 3 ,   and   3 4 , are practically significant. Next, the robustness to disturbances in the functioning of an actual bale wrapper, which leads to overlap ratio uncertainty, is examined. It was shown that, unfortunately, the optimal film widths applied together with the optimal overlaps do not provide any robustness to overlap variations. To overcome this inconvenience, the problems of a choice of the best commercially available film width guaranteeing minimal film consumption or maximal tolerance on the overlap uncertainty were formulated and solved. A new algorithm for a robust design of wrapping parameters was developed, motivated, and numerically verified to achieve a trade-off between satisfactory robustness and low film usage. For the resulting wrapping parameters, near-optimal film usage was achieved; the relative errors of the minimal film consumption approximation did not exceed 4%. It was proven that for the overlap, slightly more than 50%, i.e., 51% or 52%, provides both optimality and robustness of the overlap over disturbances, which are ensured regardless of the number of film layers. Moreover, it was found that for these overlaps and for the commercially available film widths selected according to the algorithm, the film consumption was more than twice as small than the film usage for exactly 50% overlap, if the actual overlap was smaller than pre-assumed. Similarly, an overlap of slightly more than the commonly used 67% will result in about 30% to 40% reduction in film usage in the presence of unfavorable disturbances, depending on the number of film layers and wrapping parameters.