In recent years, there has been an irresistible need for the agri-food industry to prioritize the development of efficient supply chain systems considering circular economy concepts. Agri-food products are critical to sustaining human life, as they provide essential nutrients required for maintaining bodily functions. Producing the appropriate quantity of agri-food products to match the potential demand is crucial for efficient supply to industries and markets. In addition, the principles of the circular economy can be employed to effectively utilize or reintroduce agri-food waste back into the network. This work formulates a novel mixed-integer linear programming model to propose a closed-loop agri-food supply chain network design for the soybean industry under sustainability and circular economy terms. The multi-objective network struggles to manage the total net profit while monitoring CO₂ emissions and the satisfaction of customers within the network. Given the NP-hard nature of the network, the solution approach embraces a set of conventional and hybrid metaheuristics to surmount its complexity effectively. To this end, four multi-objective optimizers and three hybrid algorithms are utilized. Hence, the model's compatibility with real conditions is investigated using fifteen practical tests. The results of interval plots and the hybrid multi-criteria decision-making techniques emphasize that optimizers can solve all sizes of problems. However, MOGWO is better suited for addressing small-size problems, whereas MOHHSA is highly effective for tackling problems of medium and large sizes.

近年来，农产品行业迫切需要优先发展考虑循环经济理念的高效供应链系统。农产品对于维持人类生命至关重要，因为它们提供维持身体机能所需的必需营养素。生产适量的农产品以满足潜在需求对于行业和市场的有效供应至关重要。此外，可以利用循环经济的原理来有效利用农业食品废物或将其重新引入网络。这项工作制定了一种新颖的混合整数线性规划模型，为可持续发展和循环经济条件下的大豆产业提出了闭环农业食品供应链网络设计。₂网络内的排放量和客户满意度。考虑到网络的NP困难性质，解决方案方法采用一组传统和混合元启发法来有效克服其复杂性。为此，使用了四个多目标优化器和三种混合算法。因此，通过十五次实际测试来研究模型与实际条件的兼容性。区间图和混合多标准决策技术的结果强调优化器可以解决各种规模的问题。然而，MOGWO 更适合解决小规模问题，而 MOHHSA 对于解决中型和大型问题非常有效。