Mixed integer linear programming models for optimal crop selection

In this paper, we propose the modeling of a real-case problem where a farmer has to optimize the use of his/her land by selecting the best mix of crops to cultivate. Complexity of the problem is due to the several factors that have to be considered simultaneously. These include the market prices variability of harvested products, the specific resource requests for each crop, the restrictions caused by limited machines availability, and the timing of operations required to complete each crop cultivation. We provide two different mathematical formulations for the analyzed problem. The first one represents a natural integer programming formulation looking for the crop-mix that maximizes the farmer's expected profit measured as the difference between revenues obtained by selling the harvested products and the production costs. Since the revenue of each crop depends on the price as quoted at the exchange market and the yield per hectare of harvested product, we define it as a random variable. Then, the second model uses the maximization of the Conditional Value-at-Risk (CVaR) as objective function and looks for the crop-mix that allows to maximize the average expected profit under a predefined quantile of worst realizations. To test and compare the proposed models with the cultivation choice made by the farmer, we use Italian historical data represented by monthly returns of different crops over a time period of 16 years. Computational results emphasize the advantage of using the CVaR model for a risk-averse farmer and provide interesting insights for farmers involved in similar problems.