The allocation of feedstuff to intensively managed dairy cows to achieve different objectives is challenging due to the inherent complexity of the system and the combinatorial problem that has to be solved. Pareto-based multi-objective optimization approaches using evolutionary algorithms can help to address these challenges and show the trade-offs and synergies among various objectives. Here we present a framework for multi-objective optimization with the Differential Evolution (DE) algorithm applied to dairy feeding systems with grazing and concentrate supply to generate an approximation of the Pareto front. The available feed resources are located in different feeding areas, and the number of animals and groups of animals with similar feeding requirements are distributed across these areas for feeding purposes. To evaluate the DE algorithm, we performed two in-silico experiments to: (1) compare the solutions quality of single-objective DE with exact Linear Programming (LP) solutions, and (2) assess the influence of different stocking rates (number of cows/ha) on milk production, feed allocation and economic performance indicators. The DE solutions that minimize the feeding costs for different stocking rates (1.1–2.6 cows/ha) closely approached the solutions derived with LP, confirming the quality of the heuristic algorithm. The multi-objective model scenarios demonstrated that increasing stocking density would enhance milk production and gross margin per unit of area at largely unchanged productivity per animal by shifting the feed ration from roughage to a large proportion of supplementary concentrate feed. At low stocking rates solutions with high productivity and gross margin and a large proportion of roughage in the ration and limited supplementary feeding were identified. We conclude that the multi-objective optimization with a Pareto-based DE algorithm is highly effective to explore the interrelations among conflicting objectives and to find suitable solutions.

中文翻译：

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奶牛饲养管理的多目标优化模型

由于系统固有的复杂性和必须解决的组合问题，将饲料分配给集约化管理的奶牛以实现不同的目标具有挑战性。使用进化算法的基于帕累托的多目标优化方法可以帮助解决这些挑战并展示各种目标之间的权衡和协同作用。在这里，我们提出了一个多目标优化框架，将差分进化 (DE) 算法应用于具有放牧和精料供应的乳品喂养系统，以生成帕累托前沿的近似值。可用的饲料资源分布在不同的饲养区，具有相似饲养需求的动物数量和动物群分布在这些区域以供饲养。为了评估 DE 算法，我们进行了两个计算机实验：(1) 将单目标 DE 的解决方案质量与精确线性规划 (LP) 解决方案进行比较，以及 (2) 评估不同的放养率（奶牛数量/公顷）对牛奶的影响产量、饲料分配和经济绩效指标。DE 解决方案最大限度地降低了不同放养率（1.1-2.6 头奶牛/公顷）的饲养成本，与 LP 得出的​​解决方案非常接近，证实了启发式算法的质量。多目标模型情景表明，通过将饲料日粮从粗饲料转变为大量补充精料，增加饲养密度将提高单位面积的牛奶产量和毛利率，而每头动物的生产力基本保持不变。在低放养率下，确定了具有高生产率和毛利率的解决方案，以及日粮中粗饲料比例大和补充饲料有限的解决方案。我们得出结论，使用基于帕累托的 DE 算法的多目标优化对于探索冲突目标之间的相互关系并找到合适的解决方案非常有效。