This paper studies a scheduling problem in which serial and parallel batching decisions must be simultaneously taken. There is a single machine with a fixed capacity that can process multiple jobs at the same time (i.e., parallel batching). Throughout the planning horizon, jobs are released at different instants and they belong to different families. Each job requires one unit of the machine capacity and only jobs of the same family can be processed at the same time. Once a group of jobs is decided to be processed, a preparation time occurs and all jobs are considered completed when the last job of the group finishes its processing (i.e., serial batching). We aim to minimize total flow time (i.e., sum of job completion times, $\sum C_j$) and total weighted tardiness ($\sum w_j{T}_j$) objectives. First, we formulate this problem as an integer linear programming model. Then, we develop a column generation algorithm to solve the linear relaxation which is integrated in a branch and bound tree in search for the integer solution. We use a highly efficient heuristic branching technique for $\sum w_j{T}_j$ objective and prove its optimality for $\sum C_j$ objective. Numerical test results show that the algorithm can handle large instances in reasonable computational times.

本文研究了一个必须同时进行串行和并行批处理决策的调度问题。有一台容量固定的机器，可以同时处理多个作业（即并行批处理）。在整个规划范围内，工作在不同的时刻发布，并且属于不同的家庭。每个作业需要一个单位的机器容量，并且只能同时处理同一系列的作业。一旦决定处理一组作业，就会出现准备时间，并且当该组的最后一个作业完成其处理（即串行批处理）时，所有作业都被视为已完成。我们的目标是最小化总流程时间（即工作完成时间的总和，$\sum C_j$) 和总加权迟到 ($\sum {瓦}_j{吨}_j$) 目标。首先，我们将此问题表述为整数线性规划模型。然后，我们开发了一种列生成算法来解决线性松弛问题，该算法集成在分支定界树中以寻找整数解。我们使用高效的启发式分支技术$\sum {瓦}_j{吨}_j$ 客观并证明其最优性 $\sum C_j$客观的。数值测试结果表明，该算法可以在合理的计算时间内处理大实例。