The paper aims to present a multi-product, multi-stage, multi-period, and multi-objective, probabilistic mixed-integer linear programming model for a sustainable forward and reverse logistics network problem. It looks at original and return products to determine both flows in the supply chain—forward and reverse—simultaneously. Besides, to establish centres of forward and reverse logistics activities and make a decision for transportation strategy in a more close-to-real manner, the demand is considered uncertain. We attempt to represent all major dimensions in the objective functions: First objective function is minimizing the processing, transportation, fixed establishing cost and costs of CO2 emission as environmental impacts. Furthermore, the processing time of reverse logistics activities is developed as the second objective function. Finally, in the third objective function, it is tried to maximize social responsibility. Indeed, a complete sustainable approach is developed in this paper. In addition, this model provides novel environmental constraint and social matters in the objective functions as its innovation and contribution. Another contribution of this paper is using probabilistic programming to manage uncertain parameters. Moreover, a non-dominated sorting genetic algorithm (NSGA-II) is configured to achieve Pareto front solutions. The performance of the NSGA-II is compared with a multi-objective particle swarm optimization (MOPSO) by proposing 10 appropriate test problems according to five comparison metrics using analysis of variance (ANOVA) to validate the modeling approach. Overall, according to the results of ANOVA and the comparison metrics, the performance of NSGA-II algorithm is more satisfying compared with that of MOPSO algorithm.

中文翻译：

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需求不确定下的正向和逆向物流系统可持续多目标优化模型

本文旨在为可持续的正向和逆向物流网络问题提出一个多产品、多阶段、多周期、多目标、概率混合整数线性规划模型。它查看原始产品和退货产品，以同时确定供应链中的正向和反向流动。此外，建立正向和逆向物流活动中心并以更接近真实的方式做出运输战略决策，需求被认为是不确定的。我们试图在目标函数中表示所有主要维度：第一个目标函数是最小化作为环境影响的 CO2 排放的加工、运输、固定建设成本和成本。此外，逆向物流活动的处理时间被开发为第二个目标函数。最后，在第三个目标函数中，力求社会责任最大化。事实上，本文提出了一种完整的可持续方法。此外，该模型在目标函数中提供了新的环境约束和社会问题作为其创新和贡献。本文的另一个贡献是使用概率规划来管理不确定参数。此外，配置了非支配排序遗传算法（NSGA-II）来实现帕累托前沿解决方案。通过使用方差分析 (ANOVA) 根据五个比较指标提出 10 个适当的测试问题，将 NSGA-II 的性能与多目标粒子群优化 (MOPSO) 进行比较，以验证建模方法。总体而言，根据方差分析的结果和比较指标，