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We present a two-stage methodology called Positions and Covering (P&C) to solve the twodimensional bin packing problem (2D-BPP). The objective of this classical combinatorial NP-hard problem is to pack a set of items (small rectangles) in the minimum number of bins (larger rectangles). The first stage is the key-point of the Positions and Covering, where for each item, it is generated in a pseudo-polynomial way a set of valid positions that indicate the possible ways of packing the item into the bin. In the second stage, a new set-covering formulation, strengthen with three sets of valid inequalities, is used to select the optimal non-overlapping configuration of items for each bin. Experimental results for the P&C method are presented and compared with some of the best algorithms in the literature for small and medium size instances. Furthermore, we are considering both cases of the 2D-BPP, with and without rotations of the items by 90˚. To the best of our knowledge, this is one of the first exact approaches to obtain optimal solutions for the rotation case. OPEN ACCESSCitation: Cid-Garcia NM, Rios-Solis YA (2020) Positions and covering: A two-stage methodology to obtain optimal solutions for the 2d-bin packing problem. PLoS ONE 15(4): e0229358. https://doi.

The process for agriculture planning starts by delineating the field into site-specific rectangular management zones to face within-field variability. We propose a bi-objective model that minimizes the number of these zones and maximizes their homogeneity with respect to a soil property. Then we use a method to assign the crops to the different plots to obtain the best profit at the end of the production cycle subject to water forecasts for the period, humidity sensors, and the chemical and physical properties of the zones within the plot. With this crop planning model we can identify the best management zones of the previous bi-objective model. Finally, we show a real-time irrigation method to decide the amount of water for each plot, at each irrigation turn, in order to maximize the total final yield. This is a critical decision in countries where water shortages are frequent. In this study we integrate these stages in a hierarchical process for the agriculture planning and empirically prove its efficiency.

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