Quadratic optimization for sustainable agriculture: A study of mixed cropping systems

Mixed cropping systems present a promising solution to conventional agriculture by combining multiple crops within a single field, thereby increasing biodiversity and soil health. However, these systems also pose new operational planning challenges regarding crop assignment and scheduling. To address this issue, we propose a novel decision problem in which crops must be assigned to different farms and fields over time, considering crop interactions, farm characteristics, growth conditions, and customer locations. We formulate this as a mixed-integer quadratically constrained optimization model and propose two column generation-based approaches that substantially outperform commercial solvers in both solution feasibility and quality for realistic-sized problems. Our computational experiments reveal that higher levels of crop compatibility promote more localized crop production with shorter transportation distances while maintaining yield benefits through beneficial crop interactions. Moreover, we observe that certain crops act primarily as ’yield catalyst’ crops, which create value by enhancing the overall productivity in the system. These findings provide practical guidelines for the implementation of mixed cropping systems, highlighting the importance of evaluating both direct revenues and companion planting benefits in agricultural planning. This research lays the groundwork for future work on sustainable agricultural innovation through mixed cropping systems. • Novel optimization model capturing complex yield benefits of companion planting in agricultural planning. • Development of adaptive column generation approaches significantly outperforming commercial solvers. • Higher levels of crop compatibility promote decentralized production networks. • System productivity is enhanced through yield catalyst crops sold on spot markets. • Possible trade-offs between operational efficiency and ecological benefits.