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We consider the problem of minimizing the makespan on batch processing identical machines, subject to compatibility constraints, where two jobs are compatible if they can be processed simultaneously in a same batch. These constraints are modeled by an undirected graph $G$, in which compatible jobs are represented by adjacent vertices. We show that several subproblems are polynomial. We propose some exact polynomial algorithms to solve these subproblems. To solve the general case, we propose a mixed-integer linear programming (MILP) formulation alongside with heuristic approaches. Furthermore, computational experiments are carried out to measure the performance of the proposed methods., Comment: 25 pages, 11 figures

This study aims to provide accurate information about changes in agricultural systems (AS) using phenological metrics derived from the NDVI time series. Use of such information could help land managers optimize land use choices and monitor the status of agricultural lands, under a variety of environmental and socioeconomic conditions. For this purpose, the Moderate Resolution Imaging Spectroradiometer (MODIS) NDVI data were used to derive phenological metrics over the Oum Er-Rbia basin (central Morocco). Random forest (RF), support vector machine (SVM), and K-nearest neighbor (KNN) classifiers were explored and compared on their ability to classify AS classes over the study area. Four main AS classes have been considered: (1) irrigated annual crop (IAC), (2) irrigated perennial crop (IPC), (3) rainfed area (RA), and (4) fallow (FA). By comparing the accuracy of the three classifiers, the RF method showed the best performance with an overall accuracy of 0.97 and kappa coefficient of 0.96.The RF method was then chosen to examine time variations in AS over a 16-year period (2000–2016). The AS main variations were detected and evaluated for the four AS classes. These variations have been found to be linked well with other indicators of local agricultural land management, as well as the historical agricultural drought changes over the study area. Overall, the results present a tool for decision makers to improve agricultural management and provide a different perspective in understanding the spatiotemporal dynamics of agricultural systems.