A Mixed Integer Programming Reformulation of the Mixed Fruit-Vegetable Crop Allocation Problem.

Mixed fruit-vegetable cropping systems are a promising way of ensuring environmentally sustainable agricultural production systems in response to the challenge of being able to fulfill local market requirements. Indeed, they combine productions and they make a better use of biodiversity. These agroforestry systems are based on a complex set of interactions modifying the utilization of light, water and nutrients. Thus, designing such a system must optimize the use of these resources: by maximizing positive interactions (facilitation) and minimizing negative ones (competition). To attain these objectives, the system's design has to include the spatial and temporal dimensions, taking into account the evolution of above-and belowground interactions over a time horizon. For that, we define the Mixed Fruit-Vegetable Crop Allocation Problem (MFVCAP) using a discrete representation of the land and the interactions between vegetable crops and fruit trees. First, we give a direct formulation as a binary quadratic program (BQP). Then we reformulate the problem using a Benders decomposition approach. The master problem has 0/1 binary variables and deals with tree positioning. The subproblem deals with crop quantities. The BQP objective function becomes linear in the continuous subproblem by exploiting the fact that it depends only on the quantity of crops assigned to land units having shade, root, or nothing. This problem decomposition allows us to reformulate the MFVCAP into a Mixed Integer linear Program (MIP). The detailed spatial-temporal crop allocation plan is easy to obtain after solving the MIP. Experimental results show the efficiency of our approach compared to a direct solving of the original BQP formulation.