This paper presents the problem of batching and scheduling jobs belonging to incompatible job families on unrelated-parallel machines. More specifically, we investigate cost-efficient approaches for solving batching and scheduling problems concerning the desired lower bounds on batch sizes (\({LB}_{b}\)), which indirectly has a considerable impact on the production cost. Batch scheduling is a more realistic extension of the traditional group scheduling approach, in which the jobs belonging to a job family can be processed as multiple batches. The objective is to minimize the total weighted job completion time and tardiness subject to a machine- and sequence-dependent setup time, dynamic machine availability and job release times, customer segments and job priority, and different machine capability and eligibility criteria for processing. Solving this type of batch scheduling problem is a big challenge due to the high computational complexity incurred by both the sequencing assignment and batching composition. A machine learning random forest classification algorithm is used for the \({LB}_{b}\) determination. Then, an efficient mixed-integer linear programming model (MILP) is developed based on the flow conservation constraints of jobs on machines to reduce the computational complexity. By mapping the MILP model onto a network formulation, an equivalent integer set partitioning type formulation is developed for a branch-and-price optimization algorithm. Computational experiments carried out over different sets of instances, indicate the efficiency and effectiveness of the optimization algorithm, compared to the linear relaxation and relaxed MILP models. Regarding the only available benchmark in the literature, the optimization algorithm yields optimal solutions with affordable computational time.

本文提出了在不相关的并行机器上对属于不兼容作业族的作业进行批处理和调度的问题。更具体地说，我们研究了解决批处理和调度问题的具有成本效益的方法，这些问题涉及所需的批处理大小下限 ( \({LB}_{b}\))，这间接地对生产成本产生了相当大的影响。批处理调度是传统组调度方法的一种更现实的扩展，其中属于一个作业族的作业可以作为多个批处理进行处理。目标是根据机器和顺序相关的设置时间、动态机器可用性和作业发布时间、客户群和作业优先级以及不同的机器能力和加工资格标准，最大限度地减少总加权作业完成时间和迟到。由于排序分配和批处理组合都会产生高计算复杂性，因此解决这种类型的批处理调度问题是一个很大的挑战。机器学习随机森林分类算法用于\({LB}_{b}\)决心。然后，基于机器上作业的流守恒约束，开发了一种高效的混合整数线性规划模型（MILP），以降低计算复杂度。通过将 MILP 模型映射到网络公式，为分支和价格优化算法开发了等效的整数集划分类型公式。与线性松弛和松弛 MILP 模型相比，在不同实例集上进行的计算实验表明了优化算法的效率和有效性。关于文献中唯一可用的基准，优化算法产生具有可承受计算时间的最优解。