Feed formulation using multi-objective Bayesian optimization

Animal diet design has been addressed mainly by optimizing analytical functions that describe digestible energy and essential nutrients, along with a set of restrictions regarding minimum nutritional content in the feed formulation. This approach results in limitations since theoretical models are not flexible enough to incorporate variables related to environmental or zootechnical conditions that affect production efficiency or to include multiple objectives regarding current challenges associated with the adaptability to new environmental contexts and the reduction of ecological footprint. Unlike analytical methods, heuristic approaches can deal with variables from multiple sources using surrogate data-driven models of the objectives functions but commonly require thousands of evaluations of the target function, which is unfeasible in the context of animal diet formulation. This work proposes the use of Bayesian Optimization as an alternative solution to address the animal diet design problem since it is intended to optimize costly-to-evaluate target functions and is able to deal with noisy sampling, which is helpful in handling the intrinsic variability in the nutrient content of raw materials. A multi-objective swine diet design problem is used to evaluate the suitability of Bayesian optimization to optimize three target functions: digestible energy, lysine, and cost, and the solutions are compared with a fractional stochastic programming approach. The analytical formulation of the problem is not considered by the Bayesian optimization approach, but target functions are modeled through surrogate Bayesian models, where only input and output responses are used to drive the optimization process. Results show that a multi-objective Bayesian optimization process is able to find better solutions than previously proposed methods, improving in 10.71%, 14.77%, and 3.79% the three objectives defined. Experiments using batches of query samples per iteration show that the optimization process can also be accelerated by sampling the objective functions simultaneously.