This paper investigates an integrated production scheduling and vehicle routing problem with soft time windows and fuzzy travel times, where orders are grouped into batches for production and delivered by a limited number of multi-trip heterogeneous vehicles. A bi-objective mixed integer nonlinear programming (MINLP) model is established, which takes total cost and total early and tardy weighted penalty time as optimization objectives. First, a fuzzy superposition operation is proposed to obtain the fuzzy weighted penalty time, and it is extended to a generalized fuzzy operation law in fuzzy sets and systems. Then, we propose a knowledge-driven co-evolutionary algorithm (KDCEA) to solve this problem. The algorithm fuses a dual-subpopulation co-evolution based on different update strategies and a knowledge-driven strategy based on dynamic knowledge sets and problem-specific knowledge. Finally, the correctness of the MINLP model is verified by the CPLEX solver using the _-constraint method. A computational experiment is conducted based on different scale instances and a real-world case, and the results show the superiority of KDCEA in solving this problem.

中文翻译：

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具有软时间窗和模糊行程时间的集成生产调度与车辆路径问题的模糊叠加运算和知识驱动协同进化算法


本文研究了具有软时间窗和模糊行程时间的集成生产调度和车辆路径问题，其中订单被分组进行生产并由有限数量的多行程异构车辆交付。建立了以总成本和总早迟加权惩罚时间为优化目标的双目标混合整数非线性规划(MINLP)模型。首先，提出了一种模糊叠加运算来获得模糊加权惩罚时间，并将其推广为模糊集和系统中的广义模糊运算律。然后，我们提出了一种知识驱动的协同进化算法（KDCEA）来解决这个问题。该算法融合了基于不同更新策略的双子群体协同进化和基于动态知识集和特定问题知识的知识驱动策略。最后，CPLEX求解器使用_-constraint方法验证了MINLP模型的正确性。基于不同规模的实例和真实案例进行了计算实验，结果表明了KDCEA在解决该问题上的优越性。